Get premium membership and access revision papers, questions with answers as well as video lessons.

Core Technical 1 Financial Mathematics [Including Examiners Report(2005-2011)] Question Paper

Core Technical 1 Financial Mathematics [Including Examiners Report(2005-2011)]

Course:

Exam Year:

Faculty of Actuaries Institute of Actuaries
EXAMINATION
6 April 2005 (am)
Subject CT1 Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 11 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the
Formulae and Tables and your own electronic calculator.
Faculty of Actuaries
CT1 A2005 Institute of Actuaries
CT1 A2005 2
1 A bond is priced at £95 per £100 nominal, has a coupon rate of 5% per annum
payable half-yearly, and has an outstanding term of five years.
An investor holds a short position in a forward contract on £1 million nominal of this
bond, with a delivery price of £98 per £100 nominal and maturity in exactly one year,
immediately following the coupon payment then due.
The continuously compounded risk-free rates of interest for terms of six months and
one year are 4.6% per annum and 5.2% per annum, respectively.
Calculate the value of this forward contract to the investor assuming no arbitrage. [5]
2 An investment fund had a market value of £2.2 million on 31 December 2001 and
£4.2 million on 31 December 2004. It had received a net cashflow of £1.44 million
on 31 December 2003.
The money weighted rate of return and the time weighted rate of return for the period
from 31 December 2001 to 31 December 2004 are equal (to two decimal places).
Calculate the market value of the fund immediately before the net cashflow on
31 December 2003. [7]
3 A computer manufacturer is to develop a new chip to be produced from 1 January
2008 until 31 December 2020. Development begins on 1 January 2006. The cost of
development comprises £9 million payable on 1 January 2006 and £12 million
payable continuously during 2007.
From 1 January 2008 the chip will be ready for production and it is assumed that
income will be received half yearly in arrear at a rate of £5 million per annum.
(i) Calculate the discounted payback period at an effective rate of interest of 9%
per annum. [6]
(ii) Without doing any further calculations, explain whether the discounted
payback period would be greater than, less than or equal to that given in part
(i) if the effective interest rate were substantially greater than 9% per annum.
[2]
[Total 8]
CT1 A2005 3 PLEASE TURN OVER
4 The force of interest, (t) , is a function of time and at any time t (measured in years)
is given by
0.07 0.005 for 8
( )
0.06 for 8
t t
t
t
(i) Calculate the accumulation at time t = 10 of £500 invested at time t = 0. [3]
(ii) Calculate the present value at time t = 0 of a continuous payment stream at the
rate of £200e0.1t paid from t = 10 to t = 18. [5]
[Total 8]
5 A university student receives a 3-year sponsorship grant. The payments under the
grant are as follows:
Year 1 £5,000 per annum paid continuously.
Year 2 £5,000 per annum paid monthly in advance.
Year 3 £5,000 per annum paid half yearly in advance.
Calculate the total present value of these payments at the beginning of the first year
using a rate of interest of 8% per annum convertible quarterly. [8]
6 At time t = 0 an investor purchased an annuity-certain which paid her £10,000 per
annum annually in arrear for three years. The purchase price paid by the investor was
£25,000.
The value of the retail price index at various times was as shown in the table below:
Time t (years): t = 0 t = 1 t = 2 t = 3
Retail price index: 170.7 183.3 191.0 200.9
(i) Calculate, to the nearest 0.1%, the following effective rates of return per
annum achieved by the investor from her investment in the annuity:
(a) the real rate of return; and
(b) the money rate of return
[7]
(ii) By considering the average rate of inflation over the three-year period, explain
the relationship between your answers in (a) and (b) of (i). [2]
[Total 9]
CT1 A2005 4
7 A loan of nominal amount £100,000 is to be issued bearing coupons payable quarterly
in arrear at a rate of 5% per annum. Capital is to be redeemed at 103 on a single
coupon date between 15 and 20 years after the date of issue, inclusive. The date of
redemption is at the option of the borrower.
An investor who is liable to income tax at 20% and capital gains tax of 25% wishes to
purchase the entire loan at the date of issue. Calculate the price which the investor
should pay to ensure a net effective yield of at least 4% per annum. [9]
8 A small insurance fund has liabilities of £4 million due in 19 years time and £6
million in 21 years time. The manager of the fund has sold the assets previously held
and is creating a new portfolio by investing in the zero-coupon bond market. The
manager is able to buy zero-coupon bonds for whatever term he requires and has
adequate monies at his disposal.
(i) Explain whether it is possible for the manager to immunise the fund against
small changes in the rate of interest by purchasing a single zero-coupon bond.
[2]
(ii) In fact, the manager purchases two zero-coupon bonds, one paying £3.43
million in 15 years time and the other paying £7.12 million in 25 years time.
The current interest rate is 7% per annum effective.
Investigate whether the insurance fund satisfies the necessary conditions to be
immunised against small changes in the rate of interest.
[8]
[Total 10]
9 The one-year forward rate of interest at time t = 1 year is 5% per annum effective.
The gross redemption yield of a two-year fixed interest stock issued at time t = 0
which pays coupons of 3% per annum annually in arrear and is redeemed at 102 is
5.5% per annum effective.
The issue price at time t = 0 of a three-year fixed interest stock bearing coupons of
10% per annum payable annually in arrear and redeemed at par is £108.9 per £100
nominal.
(i) Calculate the one-year spot rate per annum effective at time t = 0. [4]
(ii) Calculate the one-year forward rate per annum effective at time t = 2 years.
[3]
(iii) Calculate the two-year par yield at time t = 0. [3]
[Total 10]
CT1 A2005 5 PLEASE TURN OVER
10 (i) In any year, the interest rate per annum effective on monies invested with a
given bank has mean value j and standard deviation s and is independent of the
interest rates in all previous years.
Let Sn be the accumulated amount after n years of a single investment of 1 at
time t = 0.
(a) Show that [ ] = (1 )n
E Sn j .
(b) Show that Var [ ] = (1 2 2 2 )n (1 )2n
Sn j j s j .
[5]
(ii) The interest rate per annum effective in (i), in any year, is equally likely to be
i1 or i2 (i1 i2 ) . No other values are possible.
(a) Derive expressions for j and s2 in terms of i1 and i2.
(b) The accumulated value at time t = 25 years of £1 million invested with
the bank at time t = 0 has expected value £5.5 million and standard
deviation £0.5 million.
Calculate the values of i1 and i2.
[8]
[Total 13]
CT1 A2005 6
11 (i) A loan is repayable over 20 years by level instalments of £1,000 per annum
made annually in arrear. Interest is charged at the rate of 5% per annum
effective for the first 10 years, increasing to 7% per annum effective for the
remaining term.
Show that the amount of the original loan is £12,033.56. (Minor discrepancies
due to rounding will not be penalised). [2]
(ii) The following are the details from the loan schedule for year x, i.e. the year
running from exact duration x 1 years to exact duration x years.
Instalment paid at the end of the year
Loan outstanding at the
beginning of the year Interest Capital
Year x £8,790.48 £439.52 £560.48
Determine the value of x. [4]
(iii) At the beginning of year 11, it is agreed that the increase in the rate of interest
will not take place, so that the rate remains at 5% per annum effective for the
remainder of the loan. The annual instalment will continue to be payable at
the same level so that there may be a reduced term and a reduced final
instalment.
(a) Calculate by how many years, if any, the repayment schedule is
shortened.
(b) Calculate the amount of the reduced final instalment.
(c) Calculate the reduction in the total interest paid during the existence of
the loan as a result of the interest rate not increasing.
[7]
[Total 13]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
EXAMINATION
April 2005
Subject CT1 Financial Mathematics
Core Technical
EXAMINERS REPORT
Introduction
The attached subject report has been written by the Principal Examiner with
the aim of helping candidates. The questions and comments are based around
Core Reading as the interpretation of the syllabus to which the examiners are
working. They have however given credit for any alternative approach or
interpretation which they consider to be reasonable.
M Flaherty
Chairman of the Board of Examiners
15 June 2005
Faculty of Actuaries
Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 2
1 r T t f S I Ke
where:
t is the present time
T is the time of maturity of the forward contract
r is the continuously compounded risk-free rate of interest for the interval from t
to T
S is the spot price of the security at time t
I is the present value, at the risk-free interest rate, of the income generated by the
security during the interval from t to T
K is the delivery price of the forward contract
f is the value of a long position in the forward contract
Here, working with £100 nominal,
S = 95, K = 98, T t =1, r = 0.052
I 2.5 e 0.046 0.5 e 0.052 1 4.81648
f 95 4.81648 98e 0.052 2.85071
The value of the investor s short position in a forward contract on £1 million is
therefore
1,000,000
10,000 2.85071
100
f
= £28,507
2 MWRR: 2.2 3 1 i 1.44 1 i 4.2
Estimate i 6%, LHS 4.1466
i 7%, LHS 4.2359
4.2 4.1466
0.06 0.01
4.2359 4.1466
i
= 6.60% p.a. to two decimal places
Let F = Fund value before net cashflow on 31 December 2003
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 3
Then,
TWRR = 6.60% p.a. means that
3 4.2
1.066
2.2 1.44
F
F
0.63452
1.44
F
F
0.63452 F 0.63452 x 1.44 F
F =£2.5m
3 (i) Work in millions:
1 PV of liabilities 9 12v a at 9%
9 12 .
i
v v
9 12 0.917432 1.044354
= 19.54811
The assets up to k 2 years from 1 January 2006 have:
2 2 2
2
5 5 k k
i
PV v a v a
i
5 0.84168 1.022015 k a
4.301048 k a
With k 6, PV 4.301048 4.4859
= 19.2941
The next payment of 2.5 million at k = 6.5 is made at time
8.5 and has present value = 2.5 v8.5 1.2018
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 4
This would make PV of assets (20.5m) > PV of liabilities (19.5m)
Discounted payback period = 8.5 years.
(ii) The income of the development is received later than the costs are incurred.
Hence an increase in the rate of interest will reduce the present value of the
income more than the present value of the outgo. Hence the DPP will increase.
4 (i) Accumulation =
10
500 0
s ds
e
=
8 10
0 8
0.07 0.005 0.06
500
s ds ds
e
=
8
2 10
8
0
0.005
0.07 0.06
2
500
S S S
e
= 500e0.40 0.12
= 841.01
(ii) 0
18 0.1
10
200 .
t t s ds PV e e dt
8
0 8
18 0.1 0.07 0.005 ) 0.06
10
200 .
t
s ds ds
e t e
18 0.1 0.40 0.48 0.06
10
200e t . e . e t dt
0.08 18 0.04
10
200e e t dt
0.08 18 0.04
10
200
0.04
e t
e
5000 e0.08 e0.72 e0.40 3047.33
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 5
5 Present Value = 12 2 2
1 1 1
5000 a v.a v .a at i%
where 4 1 i 1.02 i 8.24322% p.a. effective
1
0.0824322 1
. .
1.0824322 1.0824322
i
a v
Ln
0.9614201
and
1
12 12
1 12
1
1.0824322 .
v
a
i
where
1.0824322 =
12
12
12 1 0.0794725
12
i
i
12
1
a 0.9645970
and
1
2 2
1 2
1
1.0824322 .
v
a
i
where 1.0824322
2
2
2 1 0.0808000
2
i
i
2
1
a = 0.9805844
So PV 5000 0.9614201 v 0.9645970 v2 0.9805844 13, 447.39
Examiners Comment: There are other valid methods for obtaining the required
answer which also received full credit.
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 6
6 (i) (a) Work in t = 0 monetary values
25000 = 10000 170.7 2 170.7 3 170.7
183.3 191.0 200.9
v v v
where
1
1
v
i
with i = real rate of return
Try 4% RHS = 24770.94
3% RHS = 25241.25
25241.25 25000
0.03 0.01
25241.25 24770.94
i
= 0.0351 i.e. 3.5%
(b) 25000 = 10000 3 a at i%p.a.
3
a 2.5
From tables,
3
a 2.5313 at 9%
= 2.4869 at 10%
2.5313 2.5
0.09 0.01
2.5313 2.4869
i
= 0.097
i.e. 9.7% p.a.
(ii) We should find that
1
1
1
i
e
i
where e = average annual rate of inflation over the period.
Hence 1
1 1.097
1.06
1 1.035
i
i
which implies 6% p.a. inflation over the period
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 7
The actual average inflation rate was:
3 200.9
1 5.6%
170.7
e e p.a.
The inflation rate would not be expected to be exactly 6% p.a. since the Retail
Price Index is not increasing by a constant amount each year.
7
4
4
4 1 1.04 0.039414
4
i
i
1
0.05
1 0.80 0.038835
1.03
g t
4
i 1 t1 g
Capital gain on contract
Assume redeemed as late as possible (ie: after 20 years) to obtain minimum yield.
Price of stock, P:
P 4
20
100000 0.05 0.80 a
103000 0.25 103000 P v20at 4%
4 20
20
20
4000 77250
1 0.25
a v
P
v
4000 1.014877 13.5903 77250 0.45639
1 0.25 0.45639
= 102,072.25
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 8
8 (i) No, because the spread (convexity) of the liabilities would always be greater
than the spread (convexity) of the assets 3rd Redington condition would
never be satisfied.
(ii) Conditions required: (a) VA VL
(b) ' '.
VA VL
(c) " "
VA VL
where differentiation can be in respect of delta or i. In this solution, it is in
respect of delta.
(a) VA 3.43v15 7.12v25 @7%
= 2.5550
4 19 6 21 VL v v
= 2.5551
VA VL (ignoring rounding)
(b) V 'A 3.43 15v15 7.12 25v25
= 51.444
' 4 19 19 6 21 21 V L v v
= 51.445
' '
V A V L (ignoring rounding)
(c) V"A 3.43 152v15 7.12 252v25
= 1099.627
" 4 192 19 6 212 21 V L v v
1038.322
" " V A V L
all 3 conditions are satisfied.
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 9
Examiners Comment: There are other valid methods for obtaining the
required answer which also received full credit.
9 (i) From two year stock information:
Price = 2
3a2 102v at 5.5%
= 3 1.84632 + 102 0.89845
= 97.1811
Therefore, from one-year forward rate information,
1 1 1,1
3 3 102
97.1811
1 i 1 i 1 f
where 1 i =one-year spot rate
f1,1= one-year forward rate from t = 1
1 1
3 105
97.1811
1 i 1 i 1.05
1
103
97.1811
1 i
i1 5.9877%p.a.
(ii) From three-year stock information:
108.9 =
1 1 2,1
10 10 110
1 i 1 ii 1.05 1 i 1.05 1 f
where f2,1=one-year forward rate from t = 2
Hence
2,1
10 10 110
108.9
1.059877 1.059877 1.05 1.059877 1.05 1 f
2,1
110
108.9 9.4351 8.9858
1.11287 1 f
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 10
f2,1 9.245%p.a.
(iii) Let y2%p.a. be the two-year par yield
100 = 2
1 1 1,1 1 1,1
1 1 100
1 1 1 1
y
i i i f i f
100 = 2
1 1 100
1.059877 1.059877 1.05 1.059877 1.05
y
100 y2 1.84208 89.8577
y2 5.506%p.a.
10 (i) (a) Let it be the (random) rate of interest in year t . Let Sn be the
accumulation of a single investment of 1 unit after n years:
E Sn E 1 i1 1 i2 1 in
E Sn E 1 i1 E 1 i2 E 1 in as it are independent
E it j
1 n
E Sn j
(b)
2 2
E Sn E 1 i1 1 i2 1 in
2 2 2
E 1 i1 E 1 i2 E 1 in (using independence)
2 2 2
E 1 2i1 i1 E 1 2i2 i2 E 1 2in in
1 2 2 2
n
j s j
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 11
as 2 2 2 2
E ii V it E it s j
Var 1 2 2 2 1 2
n n
Sn j s j j
(ii) (a) 1 2
1
Interest
2
E j i i
2 2 2 Var Interest s E Interest E Interest
2
2 2
1 2 1 2
1 1
2 2
i i i i
= 2 2
1 2 1 2
1 1
.
4 2
i i i i
2
1 2
1
2
i i
(b) 25
E S25 1 j 5.5
j 0.0705686
Var
2 2 25 50 2
S25 1 2 j j s 1 j 0.5
2 2 25 1 2 0.0705686 0.0705686 s 50 1.0705686 0.25
s2 0.000377389
Hence, 2 2
1 2
1
0.000377389
4
s i i
i1 i2 0.0388530 (taking positive root since i1 i2 )
i1 i2 2 0.07056862 = 0.1411372
2i1 0.0388530 0.1411372
i1 0.089995 8.9995%p.a.
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 12
and 2 i 0.051142 5.1142%p.a.
11 (i) Loan 5% 10 7%
10 5% 10 1000 a v a
= 1000 7.7217 0.61391 7.0236
= 12033.56
(ii) Note
439.52
0.05 10
8790.48
x
5% 11 7%
11 5% 10 8790.48 1000 x
x a v a
11
11
1
8.79048 7.0236
0.05
x
x
v
v
8.79048 20 20 7.0236 v11 x
11 11.20952
0.86384
12.9764
v x at 5%
x 8
(iii) Let Y = reduced final payment
n = new total term of loan
Loan outstanding after 10 years = 7%
10 1000 a = £7,023.60
After change is made:
7023.60 = 10
11 1000 n at 5%
n a Yv
try n = 20 (i.e., keep to original term)
RHS = 1000 7.1078 Y 0.61391
Y 137.15
doesn t work
try n = 19
Subject CT1 (Financial Mathematics Core Technical) April 2005 Examiners Report
Page 13
RHS = 1000 6.4632 Y 0.64461
Y 869.36
Hence:
(a) Term shortened by 1 year
(b) Final instalment = £869.36
(c) Under original terms, total interest paid is:
20 1000 12033.56 7966.44
Under changed terms, total interest paid is:
18 1000 869.36 12033.56 6835.80
difference = £1,130.64
END OF EXAMINERS REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
7 September 2005 (am)
Subject CT1 Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 11 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the
Formulae and Tables and your own electronic calculator.
Faculty of Actuaries
CT1 S2005 Institute of Actuaries
CT1 S2005 2
1 Describe how cashflows are exchanged in an interest rate swap . [2]
2 An investor has earned a money rate of return from a portfolio of bonds in a particular
country of 1% per annum effective over a period of ten years. The country has
experienced deflation (negative inflation) of 2% per annum effective during the
period.
Calculate the real rate of return per annum over the ten years. [2]
3 Calculate the time in days for £1,500 to accumulate to £1,550 at:
(a) a simple rate of interest of 5% per annum
(b) a force of interest of 5% per annum
[4]
4 The force of interest (t) at time t is a + bt2 where a and b are constants. An amount of
£200 invested at time t = 0 accumulates to £210 at time t = 5 and £230 at time t = 10.
Determine a and b. [5]
5 (i) Calculate the present value of £100 over ten years at the following rates of
interest/discount:
(a) a rate of interest of 5% per annum convertible monthly
(b) a rate of discount of 5% per annum convertible monthly
(c) a force of interest of 5% per annum
[4]
(ii) A 91-day treasury bill is bought for $98.91 and is redeemed at $100.
Calculate the annual effective rate of interest obtained from the bill. [3]
[Total 7]
6 (i) State the features of a eurobond. [3]
(ii) An investor purchases a eurobond on the date of issue at a price of £97 per
£100 nominal. Coupons are paid annually in arrear. The bond will be
redeemed at par twenty years from the issue date. The rate of return from the
bond is 5% per annum effective.
(a) Calculate the annual rate of coupon paid by the bond.
(b) Calculate the duration of the bond.
[6]
[Total 9]
CT1 S2005 3 PLEASE TURN OVER
7 A bank makes a loan to be repaid in instalments annually in arrear. The first
instalment is 50, the second 48 and so on with the payments reducing by 2 per annum
until the end of the 15th year after which there are no further payments. The rate of
interest charged by the lender is 6% per annum effective.
(i) Calculate the amount of the loan. [6]
(ii) Calculate the interest and capital components of the second payment. [3]
(iii) Calculate the amount of capital repaid in the instalment at the end of the
fourteenth year. [3]
[Total 12]
8 An insurance company has just written contracts that require it to make payments to
policyholders of £1,000,000 in five years time. The total premiums paid by
policyholders amounted to £850,000. The insurance company is to invest half the
premium income in fixed interest securities that provide a return of 3% per annum
effective. The other half of the premium income is to be invested in assets that have
an uncertain return. The return from these assets in year t, it, has a mean value of
3.5% per annum effective and a standard deviation of 3% per annum effective. (1 + it)
is independently and lognormally distributed.
(i) Deriving all necessary formulae, calculate the mean and standard deviation of
the accumulation of the premiums over the five-year period. [9]
(ii) A director of the company suggests that investing all the premiums in the
assets with an uncertain return would be preferable because the expected
accumulation of the premiums would be greater than the payments due to the
policyholders.
Explain why this still may be a more risky investment policy. [2]
[Total 11]
CT1 S2005 4
9 (i) Explain what is meant by the expectations theory for the shape of the yield
curve. [2]
(ii) Short-term, one-year annual effective interest rates are currently 8%; they are
expected to be 7% in one years time, 6% in two years time and 5% in three
years time.
(a) Calculate the gross redemption yields (spot rates of interest) from
1-year, 2-year, 3-year and 4-year zero coupon bonds assuming the
expectations theory explanation of the yield curve holds.
(b) The price of a coupon paying bond is calculated by discounting
individual payments from the bond at the zero-coupon bond yields
in (a).
Calculate the gross redemption yield of a bond that is redeemed at par
in exactly four years and pays a coupon of 5 per annum annually in
arrear.
(c) A two-year forward contract has just been issued on a share with a
price of 400p. A dividend of 4p is expected in exactly one year.
Calculate the forward price using the above spot rates of interest,
assuming no arbitrage. [12]
[Total 14]
10 An investor purchased a bond with exactly 15 years to redemption. The bond,
redeemable at par, has a gross redemption yield of 5% per annum effective. It pays
coupons of 4% per annum, half yearly in arrear. The investor pays tax at 25% on the
coupons only.
(i) Calculate the price paid for the bond. [3]
(ii) After exactly eight years, immediately after the payment of the coupon then
due, this investor sells the bond to another investor who pays income tax at a
rate of 25% and capital gains tax at a rate of 40%. The bond is purchased by
the second investor to provide a net return of 6% per annum effective.
(a) Calculate the price paid by the second investor.
(b) Calculate, to one decimal place, the annual effective rate of return
earned by the first investor during the period for which the bond was
held. [10]
[Total 13]
CT1 S2005 5
11 (i) Explain what is meant by the following terms:
(a) equation of value
(b) discounted payback period from an investment project
[4]
(ii) An insurance company is considering setting up a branch in a country in
which it has previously not operated. The company is aware that access to
capital may become difficult in twelve years time. It therefore has two
decision criteria. The cashflows from the project must provide an internal rate
of return greater than 9% per annum effective and the discounted payback
period at a rate of interest of 7% per annum effective must be less than twelve
years.
The following cashflows are generated in the development and operation of
the branch.
Cash Outflows
Between the present time and the opening of the branch in three years time the
insurance company will spend £1.5m per annum on research, development and
the marketing of products. This outlay is assumed to be a constant continuous
payment stream. The rent on the branch building will be £0.3m per annum
paid quarterly in advance for twelve years starting in three years time. Staff
costs are assumed to be £1m in the first year, £1.05m in the second year, rising
by 5% per annum each year thereafter. Staff costs are assumed to be incurred
at the beginning of each year starting in three years time and assumed to be
incurred for 12 years.
Cash Inflows
The company expects the sale of products to produce a net income at a rate of
£1m per annum for the first three years after the branch opens rising to £1.9m
per annum in the next three years and to £2.5m for the following six years.
This net income is assumed to be received continuously throughout each year.
The company expects to be able to sell the branch operation 15 years from the
present time for £8m.
Determine which, if any, of the decision criteria the project fulfils.
[17]
[Total 21]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
EXAMINATION
September 2005
Subject CT1 Financial Mathematics
Core Technical
EXAMINERS REPORT
Faculty of Actuaries
Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 2
As is in some recent diets, the questions requiring descriptions of concepts, definitions or
verbal reasoning (such as Q1, Q8(ii) and Q9(i)) tended not to be well answered with
candidates producing vague statements which did not demonstrate that they understood the
relevant points. It is important that candidates understand the subject well enough to express
important topics and issues in their own words as well as in mathematical language. In show
that questions or questions where students are asked to derive formulae (such as Q8 part (i))
candidates are required to show detailed steps in deriving the results required in order to
obtain full marks.
Please note that differing answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates were not penalised for this. However, candidates were penalised
where excessive rounding had been used or where insufficient working had been shown.
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 3
1 One party agrees to pay to the other a regular series of fixed amounts for a certain
term. In exchange the second party agrees to pay a series of variable amounts based
on the level of a short term interest rate.
2 If f = the rate of inflation; j = the real rate of return and i = the money rate of return,
then j = (i f)/(1 + f). In this case, f = 2%, i= 1% and therefore j = 3.061%.
3 (a) Let the answer be t days
1,500(1 + 0.05 t/365) = 1,550
t = 243.333 days
(b) Let the answer be t days
1,500e0.05(t/365) = 1,550
0.05 (t/365) = ln (1,550/1500)
t = 239.366 days
4
5 2 3 5 1
3 0
0
210 200exp a bt dt 200exp at bt 200 5a 41.667b
10 10 2 1 3
3 0
0
230 200exp a bt dt 200exp at bt 200 10a 333.333b
ln(1.05) 5a 41.667b
ln(1.15) 10a 333.333b
The second expression less twice the first expression gives:
ln(1.15) 2ln(1.05) 250b b 0.0001687
ln(1.15) 333.333 0.0001687
0.0083520
10
a
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 4
5 (i) (a) 100 (1 + 0.05/12) 12 10 = £60.716
(b) 100 (1 0.05/12)12 10 = £60.590
(c) 100 e 10 = £60.6531
(ii) 98.91 = 100(1 + i) 91/365
ln(1 + i) = ( 65/91) ln(98.91/100) = 0.04396
therefore i = 0.04494
6 (i)
Used for medium or long-term borrowing
Unsecured
Regular annual coupon payments
Generally repayable at par
Generally issued by large companies and on behalf of governments
Yields depend on risk and marketability
Generally innovative market designed to attract different types of investor
Issued internationally (normally by a syndicate of banks)
Can be issued in any currency (not necessarily the domestic currency of
the borrower)
(ii) (a) 97 = 20 ga + 100v20 at 5% per annum effective
20 a = 12.4622; v20 = 0.37689 therefore 97 = 12.4622g + 100
0.37689
g = (97 37.689)/12.4622 = 4.75927
(b) Duration = Ct tvt/ Ctvt where Ct is the amount of the cash flow at
time t
(Ia)20 = tvt Therefore duration of the eurobond is:
(4.75927 20 (Ia) + 100 20v20)/(4.75927 20 a + 100v20)
20 (Ia) = 110.9506 all other values have been used in (a) above
therefore duration is:
(4.75927 110.9506 + 100 20 0.37689)/(4.75927
12.4622 + 100 0.37689) =1281.8239/97 = 13.2147
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 5
7 (i) Value of loan = 50v + 48v2 + 46v3 + 44v4 + + 22v15
= 52(v +v2 +v3 + + v14 + v15) 2(v + 2v2 +4v3 + + 28v14+30 v15)
= 52 a15 - 2 (Ia)15
15 (Ia) = 67.2668
15 a = 9.7122
Therefore amount of the loan is 52 9.7122 - 2 67.2668 = 370.501
Candidates who derived an appropriate formula for a decreasing annuity directly or
who calculated the value of the loan by summing the individual terms received full
credit.
(ii) Interest component in first year is 0.06 370.504 = 22.23024; therefore
capital component is 50 22.23024 = 27.76976.
Capital remaining after first instalment is 370.504 27.76976 = 342.73424.
Interest paid in second instalment is 0.06 342.73424 = 20.56405
Capital in second instalment is 48 20.56405 = 27.43595.
(iii) At the end of the thirteenth year, the capital outstanding is:
24v + 22v2 = 24 0.94340 + 22 0.89000 = 42.2216
The interest due in the fourteenth instalment 0.06 42.2216 = 2.53330
The capital payment is therefore 24 2.53330 = 21.46670
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 6
8 (i) Let it be the (random) rate of interest in year t . Let S5be the accumulation of
a single investment of 1 unit after 5 years:
5
5
1
5
1
1
1
t
t
t
t
E S E i
E i
as it are independent
5
E S5 E 1 it
E 1 it 1 E it = 1.035
5
E S5 1.035 1.187686
5 5
2 2 2
5
1 1
1 t 1 t
t t
E S E i E i (using independence)
2 5 5 5 2 2
2 5
1 1 2 1 2
1 2
t t t t t
t t t
E i E i i E i E i
E i Var i E i
2 2
5 5 5
2 5 10 1 2 t t t 1 t
Var S E S E S
E i Var i E i E i
2
0.035
0.03
t
t
E i
Var i
2 2 5 10
5 1 2 0.035 0.03 0.035 1.035
1.416534 1.410598
0.0059356
Var S
Mean value of the accumulation of premiums is:
5
425000 5 425000(1.03) 425000 1.187686 425000 1.15927
997458
E S
Standard deviation is 425000SD S5 425000 0.0059356 32743.21
Candidates who obtained slightly different answers by first deriving the parameters of
the lognormal distribution received full credit.
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 7
(ii) Investing all premiums in the risky assets is likely to be more risky because,
although there may be a higher probability of the assets accumulating to more
than £1 million, the standard deviation would be twice as high so the
probability of a large loss would be greater.
9 (i) Bond yields are determined by investors expectations of future short-term
interest rates, so that returns from longer-term bonds reflect the returns from
making an equivalent series of short-term investments
(ii) (a) Let it be the spot yield over t years:
One year: yield is 8% therefore i1 = 0.08
two years: (1 + i2)2 = 1.08 1.07 therefore i2 = 0.074988
three years: (1 + i3)3 = 1.08 1.07 1.06 therefore i3 = 0.06997
four years: (1 + i4)4 = 1.08 1.07 1.06 1.05 therefore i4 = 0.06494
(b) Price of the bond is 5[(1.08) 1 + (1.074988) 2 + (1.06997) 3]
+ 105 (1.06494) 4 = 13.03822 + 81.6373 = 94.67552
Find gross redemption yield from
94.67552 = 5 4 a + 100v4
try 7%; 4 a = 3.3872; v4 = 0.76290
gives RHS = 93.226
GRY must be lower, try 6%; 4 a = 3.4651; v4 = 0.79209
gives RHS = 96.5345
interpolate between 6% and 7%.
i = 0.07 0.01 (94.67552 93.226)/(96.5345 93.226)
i = 0.07 0.0043812 = 0.06562
(c) Present value of the dividend is 4v calculated at 8% per annum
effective = 3.70370.
Therefore forward price is
F = (400 3.70370) 1.08 1.07 = 457.9600
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 8
10 (i) Price paid by first investor is P1
5%
(2) 15
1 15
(2)
15
15
1
4 100
1.012348
0.48102
10.3797
4 1.012348 10.3797 100 0.48102
42.0315 48.1020 90.1335
P a v
i
i
v
a
P
(ii) (a)
2
2
2 1 1.06 0.059126
2
i
i
g 1 t1 0.04 0.75 0.03
2
i 1 t1 g
Capital gain on contract
Price paid by second investor is P2
(2) 7 7
2 7 6% 6% 2 6%
7 (2) 7
2 6% 7 6% 6%
(2)
7
7
2
0.75 4 100 0.4 100
1 0.4 0.75 4 0.6 100
1.014782
0.66506
5.5824
0.75 4 1.014782 5.5824 60 0.66506
1 0.4 0.66506
77.5207
P a v P v
P v a v
i
i
v
a
P
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 9
(b) Rate of return earned by the first investor is the solution to:
(2) 8
8
(2)
8
8
(2)
8
8
90.1335 0.75 4 77.5207
2%
1.004975
0.85349
7.3255
88.2490
1.5%
1.003736
0.88771
7.4859
91.3575
90.1335 88.2490
0.02 0.005 1.697% 1.7%
91.3575 88.2490
a v
i
i
i
v
a
RHS
i
i
i
v
a
RHS
i
11 (i) (a) An equation of value expresses the equality of the present value of
positive and negative (or incoming and outgoing) cash flows that are
connected with an investment project, investment transaction etc.
(b) The discounted payback period from an investment project is the first
time at which the net present value of the cash flows from the project is
positive.
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 10
(ii) Consider first the NPV at 9% per annum effective. Working in £million.
Present value of cash outflows:
(4) 3 3 4 2 5 11 14
3 9% 12 9% 9% 9% 9% 9% 9%
12 12
1.5 0.3 1.05 1.05 1.05
1.5 1.044354 2.5313 0.3 1.055644 7.1607 0.77218
1 1.05
0.77218 5.71647 7.60679 13.32326
1 1.05
a a v v v v v
v
v
Present value of cash inflows:
15
6 9% 3 9% 9 9% 6 9% 15 9% 9 9% 9%
15
15 9 6 3
1.9 2.5 8
2.5 0.6 0.9 8
1.044354 2.5 8.0607 0.6 5.9952 0.9 4.4859 2.5313 8 0.27454
12.6253
a a a a a a v
a a a a v
Hence NPV of project @ 9% = 12.6253 13.3233 = £0.698 million
so the IRR is less than 9% p.a. effective
To find whether the discounted payback period is less than 12 years at 7% per
annum effective, we need to find the NPV @ 7% of first twelve years
cashflows
Present value of cash outflows:
(4) 3 3 4 2 5 8 11
3 7% 9 7% 7% 7% 7% 7% 7%
9 9
1.5 0.3 1.05 1.05 1.05
1.5 1.034605 2.6243 0.3 1.043380 6.5152 0.81630
1 1.05
0.81630 5.73739 6.82096 12.55835
1 1.05
a a v v v v v
v
v
Subject CT1 (Financial Mathematics Core Technical) September 2005 Examiners Report
Page 11
Present value of cash inflows:
6 7% 3 7% 9 7% 6 7% 12 7% 9 7%
12 9 6 3
1.9 2.5
2.5 0.6 0.9
1.034605 2.5 7.9427 0.6 6.5152 0.9 4.7665 2.6243
9.3461
a a a a a a
a a a a
NPV is negative so the discounted payback period is more than 12 years.
Project fulfils neither the discounted payback period criterion nor the internal
rate of return criterion.
END OF EXAMINERS REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
4 April 2006 (am)
Subject CT1 Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 12 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the
Formulae and Tables and your own electronic calculator.
Faculty of Actuaries
CT1 A2006 Institute of Actuaries
CT1 A2006 2
1 An investment is discounted for 28 days at a simple rate of discount of 4.5% per
annum. Calculate the annual effective rate of interest. [3]
2 An annuity certain with payments of £150 at the end of each quarter is to be replaced
by an annuity with the same term and present value, but with payments at the
beginning of each month instead.
Calculate the revised payments, assuming an annual force of interest of 10%. [3]
3 At time t = 0 the n-year spot rate of interest is equal to (2.25 + 0.25n)% per annum
effective (1 n 5).
(a) Calculate the 2-year forward rate of interest from time t = 3 expressed as an
annual effective rate of interest.
(b) Calculate the 4-year par yield.
(c) Without performing any further calculations, explain how you would expect
the gross redemption yield of a 4-year bond paying annual coupons of 3.5% to
compare with the par yield calculated in (b).
[7]
4 An investor, who is liable to income tax at 20% but is not liable to capital gains tax,
wishes to earn a net effective rate of return of 5% per annum. A bond bearing
coupons payable half-yearly in arrear at a rate 6.25% per annum is available. The
bond will be redeemed at par on a coupon date between 10 and 15 years after the date
of issue, inclusive. The date of redemption is at the option of the borrower.
Calculate the maximum price that the investor is willing to pay for the bond. [5]
5 A share currently trades at £10 and will pay a dividend of 50p in one month s time. A
six-month forward contract is available on the share for £9.70. Show that an investor
can make a risk-free profit if the risk-free force of interest is 3% per annum. [4]
6 An actuarial student has created an interest rate model under which the annual
effective rate of interest is assumed to be fixed over the whole of the next ten years.
The annual effective rate is assumed to be 2%, 4% and 7% with probabilities 0.25,
0.55 and 0.2 respectively.
(a) Calculate the expected accumulated value of an annuity of £800 per annum
payable annually in advance over the next ten years.
(b) Calculate the probability that the accumulated value will be greater than
£10,000.
[4]
CT1 A2006 3 PLEASE TURN OVER
7 A company has entered into an interest rate swap. Under the terms of the swap the
company makes fixed annual payments equal to 6% of the principal of the swap. In
return, the company receives annual interest payments on the principal based on the
prevailing variable short-term interest rate which currently stands at 5.5% per annum.
(a) Describe briefly the risks faced by a counterparty to an interest rate swap.
(b) Explain which of the risks described in (a) are faced by the company. [4]
8 An ordinary share pays annual dividends. A dividend of 25p per share has just been
paid. Dividends are expected to grow by 2% next year and by 4% the following year.
Thereafter, dividends are expected to grow at 6% per annum compound in perpetuity.
(i) State the main characteristics of ordinary shares. [4]
(ii) Calculate the present value of the dividend stream described above at a rate of
interest of 9% per annum effective from a holding of 100 ordinary shares. [4]
(iii) An investor buys 100 shares in (ii) for £8.20 each. He holds them for two
years and receives the dividends payable. He then sells them for £9
immediately after the second dividend is paid.
Calculate the investor s real rate of return if the inflation index increases by
3% during the first year and by 3.5% during the second year assuming
dividends grow as expected. [4]
[Total 12]
9 The force of interest (t) is a function of time and at any time t, measured in years, is
given by the formula:
2
0.04
( ) 0.008
0.005 0.0003
t t
t t
0 5
5 10
10
t
t
t
(i) Calculate the present value of a unit sum of money due at time t = 12. [5]
(ii) Calculate the effective annual rate of interest over the 12 years. [2]
(iii) Calculate the present value at time t = 0 of a continuous payment stream
that is paid at the rate of e 0.05t per unit time between time t = 2 and time
t = 5. [3]
[Total 10]
CT1 A2006 4
10 A piece of land is available for sale for £5,000,000. A property developer, who can
lend and borrow money at a rate of 15% per annum, believes that she can build
housing on the land and sell it for a profit. The total cost of development would be
£7,000,000 which would be incurred continuously over the first two years after
purchase of the land. The development would then be complete.
The developer has three possible project strategies. She believes that she can sell the
completed housing:
in three years time for £16,500,000
in four years time for £18,000,000
in five years time for £20,500,000
The developer also believes that she can obtain a rental income from the housing
between the time that the development is completed and the time of sale. The rental
income is payable quarterly in advance and is expected to be £500,000 in the first year
of payment. Thereafter, the rental income is expected to increase by £50,000 per
annum at the beginning of each year that the income is paid.
(i) Determine the optimum strategy if this is based upon using net present value
as the decision criterion. [9]
(ii) Determine which strategy would be optimal if the discounted payback period
were to be used as the decision criterion. [2]
(iii) If the housing is sold in six years time, the developer believes that she can
obtain an internal rate of return on the project of 17.5% per annum. Calculate
the sale price that the developer believes that she can receive. [6]
(iv) Suggest reasons why the developer may not achieve an internal rate of return
of 17.5% per annum even if she sells the housing for the sale price calculated
in (iii). [2]
[Total 19]
CT1 A2006 5 PLEASE TURN OVER
11 An actuarial student has taken out two loans.
Loan A: a five-year car loan for £10,000 repayable by equal monthly instalments of
capital and interest in arrear with a flat rate of interest of 10.715% per
annum.
Loan B: a five-year bank loan of £15,000 repayable by equal monthly instalments of
capital and interest in arrear with an effective annual interest rate of 12% for
the first two years and 10% thereafter.
The student has a monthly disposable income of £600 to pay the loan interest after all
other living expenses have been paid.
Freeloans is a company which offer loans at a constant effective interest rate for all
terms between three years and ten years. After two years, the student is approached
by a representative of Freeloans who offers the student a 10-year loan on the capital
outstanding which is repayable by equal monthly instalments of capital and interest in
arrear. This new loan is used to pay off the original loans and will have repayments
equal to half the original repayments.
(i) Calculate the final disposable income (surplus or deficit) each month after the
loan payments have been made. [5]
(ii) Calculate the capital repaid in the first month of the third year assuming that
the student carries on with the original arrangements. [5]
(iii) Estimate the capital repaid in the first month of the third year assuming that
the student has taken out the new loan. [5]
(iv) Suggest, with reasons, a more appropriate strategy for the student. [2]
[Total 17]
CT1 A2006 6
12 A pension fund has liabilities of £3 million due in 3 years time, £5 million due in 5
years time, £9 million due in 9 years time, and £11 million due in 11 years time.
The fund holds two investments, X and Y. Investment X provides income of £1
million payable at the end of each year for the next five years with no capital
repayment. Investment Y is a zero coupon bond which pays a lump sum of £R at the
end of n years (where n is not necessarily an integer). The interest rate is 8% per
annum effective.
(i) Investigate whether values of £R and n can be found which ensure that the
fund is immunised against small changes in the interest rate.
You are given that
5
2
1
t 40.275
t
t v at 8%. [8]
(ii) (a) The interest rate immediately changes to 3% per annum effective.
Calculate the revised present values of the assets and liabilities of the
fund.
(b) Explain your answer to (ii)(a). [4]
[Total 12]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
EXAMINATION
April 2006
Subject CT1 Financial Mathematics
Core Technical
EXAMINERS REPORT
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
M Flaherty
Chairman of the Board of Examiners
June 2006
Comments
Individual comments are shown after each question.
General comments
As is in some recent diets, the questions requiring verbal reasoning (such as Q3(c), Q7(b),
Q10(iv) and Q11(iv)) tended not to be well answered with candidates producing vague
statements which did not demonstrate that they understood the relevant points.
Please note that different answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this.
However, candidates may be penalised where excessive rounding has been used or where
insufficient working is shown.
Faculty of Actuaries
Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 2
1 Annual rate of interest is i where
1
28 28/365
1 1
365
d
i
This gives
365/ 28 28 0.045
1 1 4.611%
365
i
Comments on question 1: This was generally well answered.
2 We require X where:
(4) (12)
(4) (12)
(12) (4) 600 12 50 n 50
n n
n
a d
a Xa X
a i
12
4
(12) 1/12
(4) 1/ 4
12 1 1 12 1 0.099584
4 1 1 4 1 0.101260
d d e
i i e
Hence X 49.1724 or £49.17
Comments on question 2: Candidates were not penalised for assuming that the annuities
were for a specific term even though this was not needed for the calculations.
3 (a)
5 5
2 5
3,2 3 3 3,2
3
1 1.035
1 4.255%
1 1.03
y
f f
y
(b) Par yield is yc4 where
1 2 3 4 4
2 3 4 4
yc4 vy vy vy vy vy 1
Thus 1 2 3 4 4
yc4 1.025 1.0275 1.03 1.0325 1.0325 1
4
0.12009
3.230%
3.71785
yc
(c) The par yield is equal to the gross redemption yield for a par yield bond.
Coupons for the 3.5% bond are higher than for the par yield bond. Thus a
lower proportion of the total proceeds are included within the redemption
payment which is when spot yields/discount rates are highest. The present
value of the proceeds of the 3.5% bond will be higher and so the gross
redemption yield will be lower than that of the par yield bond and thus less
than the par yield.
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 3
Comments on question 3: Part (a) was answered well but some candidates struggled with
the calculation of the par yield in part (b). In part (c) the marks were awarded for a clear
explanation. Many candidates, who just stated their conclusion, were unable to explain their
reasoning clearly and so failed to score full marks on this part.
4 i 2 0.049390
g 1 t1 0.0625 0.80 0.05
2
i 1 t1 g
Capital loss on contract
Assume redeemed as early as possible (i.e.: after 10 years) to obtain minimum
yield.
Price of stock per £100 nominal, P:
P 2
10
100 0.0625 0.80 a 100v10at 5%
2 10
10
P 5 a 100v
5 1.012348 7.7217 100 0.61391
39.0852 61.3910 £100.4762
Comments on question 4: Well answered although some candidates who recognised that the
investor faced a capital loss did not recognise that this meant that the minimum yield would
be obtained if the bond was redeemed at the earliest possible date.
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 4
5 An investor can borrow £10 at the risk-free rate, buy one share for £10, enter into the
forward contract to sell the share in six months time.
The initial cashflow is zero.
After one month the 50p dividend from the share is invested at the risk-free rate. After
six months the share can be sold for £9.70, the dividend proceeds are worth
5
0.03 12 0.5e and the borrowing is repaid at 10 e0.015 . This gives a net cashflow of 9.7
+
5
0.03 12 0.5e 10 e0.015 = 0.0552
The investor has made a deal with zero initial cost, no risk of future loss and a riskfree
future profit.
Comments on question 5: The majority of candidates were able to calculate the nonarbitrage
forward price by use of the appropriate formula. However, marks were lost for not
clearly explaining how a risk-free profit could thus be made.
6 (a) Expected accumulated value
10 0.02 10 0.04 10 0.07
11 0.02 11 0.04 11 0.07
800 0.25 0.55 0.2
800 0.25 1 0.55 1 0.2 1
800 0.25 11.1687 0.55 12.4864 0.2 14.7836
0.25 8934.96 0.55 9989.12 0.2 11826.88
£10,093.13
s s s
s s s
(b) Accumulation is only over £10,000 if the interest rate is 7% p.a. which has
probability 0.2
Comments on question 6: The most poorly answered question on the paper. This model of
interest rates had not been examined recently and the majority of candidates assumed instead
that the interest rate changed each year (in line with previous examination questions on this
topic).
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 5
7 (a) The counterparty faces market risk which is the risk that market conditions
will change so that the present value of the net outgo under the agreement
increases.
The counterparty also faces credit risk which is the risk that the other
counterparty will default on its payments.
(b) The company still faces the market risk since the interest rates could fall
further which will make the value of the swap even more negative to the
company.
The company does not currently face a credit risk since the value of the swap
is positive to the other counterparty.
Comments on question 7: Part (a) was answered well but many candidates failed to
recognise in (b) that the company would not currently face credit risk in this example.
8 (i) Main characteristics of ordinary shares:
Issued by commercial undertakings and other bodies.
Entitle holders to receive all net profits of the company in the form of
dividends after interest on loans and other fixed interest stocks has been
paid.
Higher expected returns than for most other asset classes
but risk of capital losses
and returns can be variable.
Lowest ranking form of finance.
Low initial running yield but dividends should increase with inflation.
Marketability varies according to size of company.
Voting rights in proportion to number of shares held.
(ii) Present value of future dividends
100 0.25 1.02v 1.02 1.04v2 1.02 1.04 1.06v3 1.02 1.04 1.062v4
2 2 2
2
25 1.02 25 1.02 1.04 1 1.06 1.06
1.09
25 1.02 25 1.02 1.04
0.03
23.3945 811.0092 834.4037 £834.40
v v v v
v v
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 6
(iii) Real rate of return is i such that:
2
2
2
100 100 100
820 100 0.25 1.02 100 0.25 1.02 1.04
103 103 103.5
100 100
900
103 103.5
24.7573 869.1150
v v
v
v v
24.7573 24.75732 4 869.1150 820
0.95719
2 869.1150
v
(taking positive root)
Hence i = 4.47%
Comments on question 8: Despite being a bookwork question, part (i) was answered patchily
with few students getting all of the required points. Part (ii) was answered well. In part (iii),
it was expected that students would solve the quadratic equation. However, full credit was
given to students who used interpolation methods.
9 (i)
5 5
0 0
0.04 0.04 0.2 (0,5) 1.22140
dt t A e e e
10 2 10
5 5
0.008 0.004 0.3 (5,10) 1.34986
tdt t
A e e e
12 2 3 12 2
10 10
0.005 0.0003 0.0025 0.0001 0.1828 (10,12) 1.20057
t t dt t t
A e e e
Required present value
1 1 1
A 0,5 A 5,10 A 10,12 1.22140 1.34986 1.20057 1.97941
= 0.50520
(ii) Equivalent effective annual rate is i where 12 1 i 1.97941 i 5.855%
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 7
(iii) Present Value at time t = 0
0
5 5
0.05 0.04 0.05 0.04
2 2
5 0.09 5 0.18 0.45
0.09
2 2
2.1960
0.09 0.09
t
t ds t t
t
t
e e dt e e dt
e e e
e dt
Comments on question 9: Well answered.
10 (i) Net present value of costs
2 2 5,000,000 3,500,000 5,000,000 3,500,000
5,000,000 3,500,000 1.073254 1.6257 11,106,762
i
a a
Net present value of benefits
2 4 2 (4)
2 2
450,000 50,000 n
n n n
v a v Ia S v
2 2
(4) 2 (4) 2 450,000 50,000 n
n n n
i i
v a v Ia S v
d d
where n is the year of sale and Sn are the sale proceeds if the sale is made in
year n.
If n = 3 the NPV of benefits
450,000 0.75614 1.092113 0.86957
50,000 0.75614 1.092113 0.86957
16,500,000 0.65752
323,137 35,904 10,849,080 11, 208,121
Hence net present value of the project is 11,208,121 11,106,762 = 101,359
Note that if n = 4 the extra benefits in year 4 consist of an extra £1.5 million
on the sale proceeds and an extra £650,000 rental income. This is clearly less
than the amount that could have been obtained if the sale had been made at the
end of year 3 and the proceeds invested at 15% per annum. Hence selling in
year 4 is not an optimum strategy.
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 8
If n = 5 the NPV of benefits
450,000 0.75614 1.092113 2.2832
50,000 0.75614 1.092113 4.3544
20,500,000 0.49718
848, 450 179,791 10,192,190 11, 220,431
Hence net present value of the project is 11,220,431 11,106,762 = 113,669
Hence the optimum strategy if net present value is used as the criterion is to
sell the housing after 5 years.
(ii) If the discounted payback period is used as the criterion, the optimum strategy
is that which minimises the first time when the net present value is positive.
By inspection, this is when the housing is sold after 3 years.
(iii) We require
2 4 2 (4)
2 2 2
5,000,000 3,500,000 450,000 50,000 n
n n n
i
a v a v Ia S v at 17.5%
LHS
2
0.175
0.175
1 1 0.72431
5,000,000 3,500,000 5,000,000 3,500,000
0.16127
v
10,983,227
RHS
4 4
2 0.175 2 4 0.175 6
0.175 4 0.175 4 6 0.175
1 4
450,000 50,000
v a v
v v S v
d d
1
4 4
d0.175 4 1 v 0.15806
4
4
1
3.1918
v
a
d
Therefore we have on the RHS
6
6
3.1918 2.0985
450,000 0.72431 3.0076 50,000 0.72431 0.37999
0.15806
980, 296 250,502 0.37999
S
S
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 9
For equality 6
10,983,227 1,230,798
£25,665,000
0.37999
S
(iv) Reasons investor may not achieve the internal rate of return:
Allowance for expenses when buying/selling which may be significant.
There may be periods when the property is unoccupied and no rental
income is received.
Rental income may be reduced by maintenance expenses.
Tax on rental income and/or sale proceeds
Comments on question 10: A significant number of candidates assumed that the development
costs amounted to £7 million per annum and subsequently found that no strategy would lead
to a profit. Otherwise the calculations were performed well. In part (iv), credit was given for
other valid answers. Despite this, few students scored full marks on this part.
11 (i) Let X A, XB be the monthly repayments under Loans A and B respectively.
For loan A:
Flat rate of interest = 10.715%
60 60 10000
£255.96
5 50000
A A A
A
A
X L X
X
L
For loan B:
(12) 2 (12)
2 12% 12% 310%
2
12 2 12% 12 3
12% 10%
15000 12
1,250
1,250
1.053875 1.6901 0.79719 1.045045 2.4869
B B
B
L X a v a
X
i i
a v a
i i
XB £324.43
Hence student s overall surplus = 600 X A XB = £19.61
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 10
(ii) Effective rate of interest under loan A is i% where
12 12
5 5
12 255.96a 10000 a 3.2557
Try i = 20%: 12
5
a = 3.2557
So capital outstanding after 24 months is 12 255.96 12
3
a at 20%
12 255.96 1.088651 2.1065 7043.74
Capital outstanding under B is 12 324.43 12
3
a at 10%
12 324.43 1.045045 2.4869 10118.02
So interest paid in month 25 under loans A and B
20% 10%
12 12
7043.74 10118.02 107.84 80.68 £188.52
12 12
i i
and capital repaid
255.96 107.84 324.43 80.68 148.12 243.75 £391.87
(iii) Under the new loan the capital outstanding is the same as under the original
arrangement = 17161.76.
The monthly repayment
255.96 324.43
£290.20
2
The effective rate of interest on the new loan A is i where
12 12
10 10
12 290.20a 17161.76 a 4.9281
Try i = 20%: 12
10
a 4.5642
Try i = 15%: 12
10
a 5.3551
By interpolation
5.3551 4.9281
15% 20% 15% 17.7%
5.3551 4.5642
i
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 11
Hence interest paid in month 25
17.7%
12
17161.76 234.66
12
i
and capital repaid is £290.20 £234.66 = £55.54
(iv) The new strategy reduces the monthly payments but repays the capital more
slowly. The student could consider the following options:
Keeping loan B and taking out a smaller new loan to repay loan A
(which has the highest effective interest rate).
Taking out the new loan for a shorter term to repay the capital more
quickly.
Comments on question 11: In part (i) some candidates struggled to deal with the flat rate of
Loan A whilst others failed to deal with the change in interest rate of Loan B. Part (ii) was
answered well. In part (iii), different answers for the effective rate of interest (and hence the
interest paid) for the new loan could be obtained according to the actual interpolation used
and full credit was given for a range of answers. If calculated exactly, the effective rate of
interest is actually 17.5%. In part (iv), credit was again given for any valid strategy suitably
explained.
12 (i) We will consider three conditions necessary for immunisation
(1) A L V V (all expressions in terms of £m)
5
n
A V a Rv at 8%
3.9927 Rvn
3 3 5 5 9 9 11 11 L V v v v v at 8%
15.0044
Rvn 11.0117
(2) ' '
A L V V where ' A & ' L
A L
V V
V V
'
5
11.3651
n
A
n
V Ia nRv
nRv
' 9 3 25 5 81 9 121 11
116.5741
L V v v v v
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 12
nRvn 105.2090
9.5543
105.2090
9.5543
11.0117
11.0117 1.08 £22.9720
n
R m
Alternatively:
' '
A L V V where ' A & ' L
A L
V V
V V
i i
' 1
5
1
1
11.3651
10.5233
n
A
n
n
V v Ia nRv
v nRv
nRv
' 9 4 25 6 81 10 121 12
107.9389
L V v v v v
nRvn 1 97.4156
9.5543
97.4156
9.5543
11.0117
11.0117 1.08 £22.9720
n
v
R m
(3) '' ''
A L V V (where
2 2
'' ''
2 2 A & L
A L
V V
V V )
5
'' 2 2
1
40.275 9.5543 2 22.9720 9.5543
1045.483
t n
A
t
V t v n Rv
v
'' 27 3 125 5 729 9 1331 11
1042.031
L V v v v v
Subject CT1 (Financial Mathematics Core Technical) April 2006 Examiners Report
Page 13
Alternatively (differentiating with respect to i):
5
'' 2 2
1
5
2 2 2 2
5
1
11.5543
1 1
1
0.85734 40.275 0.85734 11.3651 9.5543 10.5543 22.9720
34.53 9.74 952.00 996.27
t n
A
t
t n
t
V t t v n n Rv
v t v v Ia n n Rv
v
'' 3 3 4 5 5 5 6 7 9 9 10 11 11 11 12 13
993.32
L V v v v v
Thus n 9.5543, R £22.9720m will satisfy all three conditions and so will
achieve immunisation.
(ii) (a) Value of assets at 3% 9.5543
5 a Rvn 4.5797 22.9720v £21.900m
Value of liabilities at 3% = 3v3 5v5 9v9 11v11 £21.903m
Hence fund has a deficit of approximately £3,000.
(b) Immunisation will only enable to be a fund to be protected against a
small change in interest rates. It will not be necessarily protected
against sudden large changes as in this case.
Comments on question 12: Part (i) was answered surprisingly poorly, given that it required
the same techniques as those required in previous examination questions on the same topic.
Full credit was given to students who observed directly that the spread of the assets around
the mean term was greater than the spread of the liabilities. Few students answered part (ii)
fully and the examiners felt that students should have recognised that immunisation would
not protect the fund against such a large change in interest rates even if they had not
answered part (i) correctly.
END OF EXAMINERS REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
12 September 2006 (am)
Subject CT1 Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 12 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the
Formulae and Tables and your own electronic calculator.
Faculty of Actuaries
CT1 S2006 Institute of Actuaries
CT1 S2006 2
1 (a) Distinguish between a future and an option.
(b) Explain why convertibles have option-like characteristics.
[3]
2 An individual makes an investment of £4m per annum in the first year, £6m per
annum in the second year and £8m per annum in the third year. The investments are
made continuously throughout each year. Calculate the accumulated value of the
investments at the end of the third year at a rate of interest of 4% per annum effective.
[3]
3 An individual has invested a sum of £10m. Exactly one year later, the investment is
worth £11.1m. An index of prices has a value of 112 at the beginning of the
investment and 120 at the end of the investment. The investor pays tax at 40% on all
money returns from investment. Calculate:
(a) The money rate of return per annum before tax.
(b) The rate of inflation.
(c) The real rate of return per annum after tax.
[4]
4 An investor is able to purchase or sell two specially designed risk-free securities, A
and B. Short sales of both securities are possible. Security A has a market price of
20p. In the event that a particular stock market index goes up over the next year, it
will pay 25p and, in the event that the stock market index goes down, it will pay 15p.
Security B has a market price of 15p. In the event that the stock market index goes up
over the next year, it will pay 20p and, in the event that the stock market index goes
down, it will pay 12p.
(i) Explain what is meant by the assumption of no arbitrage used in the pricing
of derivative contracts. [2]
(ii) Find the market price of B, such that there are no arbitrage opportunities and
assuming the price of A remains fixed. Explain your reasoning. [2]
[Total 4]
5 (i) Calculate the time in days for £3,600 to accumulate to £4,000 at:
(a) a simple rate of interest of 6% per annum
(b) a compound rate of interest of 6% per annum convertible quarterly
(c) a compound rate of interest of 6% per annum convertible monthly
[4]
(ii) Explain why the amount takes longest to accumulate in (i)(a) [1]
[Total 5]
CT1 S2006 3 PLEASE TURN OVER
6 The rate of interest is a random variable that is distributed with mean 0.07 and
variance 0.016 in each of the next 10 years. The value taken by the rate of interest in
any one year is independent of its value in any other year. Deriving all necessary
formulae calculate:
(i) The expected accumulation at the end of ten years, if one unit is invested at the
beginning of ten years. [3]
(ii) The variance of the accumulation at the end of ten years, if one unit is invested
at the beginning of ten years. [5]
(iii) Explain how your answers in (i) and (ii) would differ if 1,000 units had been
invested. [1]
[Total 9]
7 A life insurance fund had assets totalling £600m on 1 January 2003. It received net
income of £40m on 1 January 2004 and £100m on 1 July 2004. The value of the fund
was:
£450m on 31 December 2003;
£500m on 30 June 2004;
£800m on 31 December 2004.
(i) Calculate, for the period 1 January 2003 to 31 December 2004, to three
decimal places:
(a) The time weighted rate of return per annum.
(b) The linked internal rate of return, using sub intervals of a calendar
year.
[8]
(ii) Explain why the linked internal rate of return is higher than the time weighted
rate of return. [2]
[Total 10]
8 The force of interest (t) at time t is at bt2where a and b are constants. An amount
of £100 invested at time t = 0 accumulates to £150 at time t = 5 and £230 at time
t = 10.
(i) Calculate the values of a and b. [5]
(ii) Calculate the constant force of interest that would give rise to the same
accumulation from time t = 0 to time t = 10. [2]
(iii) At the force of interest calculated in (ii), calculate the present value of a
continuous payment stream of 20e0.05t paid between from time t = 0 to time
t = 10. [4]
[Total 11]
CT1 S2006 4
9 An individual took out a loan of £100,000 to purchase a house on 1 January 1980.
The loan is due to be repaid on 1 January 2010 but the borrower can repay the loan
early if he wishes. The borrower pays interest on the loan at a rate of 6% per annum
convertible monthly, paid in arrears. The loan instalments only cover the interest on
the loan. At the same time, the borrower took out a thirty-year investment policy,
which was expected to repay the loan, and into which monthly premiums were paid,
in advance, at a rate of £1,060 per annum. The individual was told that premiums in
the investment policy were expected to earn a rate of return of 7% per annum
effective. After twenty years, the individual was informed that the premiums had
only earned a rate of return of 4% per annum effective and that they would continue
to do so for the final ten years of the policy. The borrower agrees to increase his
monthly payments into the investment policy to £5,000 per annum for the final ten
years.
(a) Calculate the amount to which the investment policy was expected to
accumulate at the time it was taken out.
(b) Calculate the amount by which the investment policy would have fallen short
of repaying the loan had extra premiums not been paid for the final ten years.
(c) Calculate the amount of money the individual will have, after using the
proceeds of the investment policy to repay the loan, after allowing for the
increase in premiums.
(d) Suggest another course of action the borrower could have taken which would
have been of higher value to him, explaining why this higher value arises.
(e) Calculate the level annual instalment that the investor would have had to pay
from outset if he had repaid the loan in equal instalments of interest and
capital.
[11]
10 A financial regulator has brought in a new set of regulations and wishes to assess the
cost of them. It intends to conduct an analysis of the costs and benefits of the new
regulations in their first twenty years.
The costs are estimated to be as follows:
The cost to companies who will need to devise new policy terms and computer
systems is expected to be incurred at a rate of £50m in the first year increasing by
3% per annum over the twenty year period.
The cost to financial advisers who will have to set up new computer systems and
spend more time filling in paperwork is expected to be incurred at a rate of £60m
in the first year, £19m in the second year, £18m in the third year, reducing by £1m
every year until the last year, when the cost incurred will be at a rate of £1m.
The cost to consumers who will have to spend more time filling in paperwork and
talking to their financial advisers is expected to be incurred at a rate of £10m in
the first year, increasing by 3% per annum over the twenty year period.
CT1 S2006 5 PLEASE TURN OVER
The benefits are estimated as follows:
The benefit to consumers who are less likely to buy inappropriate policies is
estimated to be received at a rate of £30m in the first year, £33m in the second
year, £36m in the third year and so on, rising by £3m per year until the end of
twenty years.
The benefit to companies who will spend less time dealing with complaints from
customers is estimated to be received at a rate of £12m per annum for twenty
years.
Calculate the net present value of the benefit or cost of the regulations in their first
twenty years at a rate of interest of 4% per annum effective. Assume that all costs and
benefits occur continuously throughout the year.
[12]
11 (i) Describe the characteristics of an index-linked government bond. [3]
(ii) On 1 July 2002, the government of a country issued an index-linked bond of
term seven years. Coupons are paid half-yearly in arrears on 1 January and 1
July each year. The annual nominal coupon is 2%. Interest and capital
payments are indexed by reference to the value of an inflation index with a
time lag of eight months.
You are given the following values of the inflation index.
Date Inflation index
November 2001 110.0
May 2002 112.3
November 2002 113.2
May 2003 113.8
The inflation index is assumed to increase continuously at the rate of 2½% per
annum effective from its value in May 2003.
An investor, paying tax at the rate of 20% on coupons only, purchased the
stock on 1 July 2003, just after a coupon payment had been made.
Calculate the price to this investor such that a real net yield of 3% per annum
convertible half yearly is obtained and assuming that the investor holds the
bond to maturity. [10]
[Total 13]
CT1 S2006 6
12 A pension fund has the following liabilities: annuity payments of £160,000 per annum
to be paid annually in arrears for the next 15 years and a lump sum of £200,000 to be
paid in ten years. It wishes to invest in two fixed-interest securities in order to
immunise its liabilities. Security A has a coupon rate of 8% per annum and a term to
redemption of eight years. Security B has a coupon rate of 3% per annum and a term
to redemption of 25 years. Both securities are redeemable at par and pay coupons
annually in arrear.
(i) Calculate the present value of the liabilities at a rate of interest of 7% per
annum effective. [2]
(ii) Calculate the discounted mean term of the liabilities at a rate of interest of 7%
per annum effective. [4]
(iii) Calculate the nominal amount of each security that should be purchased so
that both the present value and discounted mean terms of assets and liabilities
are equal. [7]
(iv) Without further calculation, comment on whether, if the conditions in (iii) are
fulfilled, the pension fund is likely to be immunised against small, uniform
changes in the rate of interest. [2]
[Total 15]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
EXAMINATION
September 2006
Subject CT1 — Financial Mathematics
Core Technical
EXAMINERS’ REPORT
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
M A Stocker
Chairman of the Board of Examiners
November 2006
© Faculty of Actuaries
© Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’Report
Page 2
Comments
As in many recent diets, the questions requiring verbal reasoning (e.g. Question 4(i)) tended
not to be well answered with candidates producing vague statements which did not
demonstrate that they understood the relevant points
Please note that differing answers may be obtained from those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this.
However, candidates may be penalised where excessive rounding has been used or where
insufficient working is shown.
Comments on solutions presented to individual questions for this September 2006 paper are
given below.
Question 1
Generally well answered. To gain full marks candidates were required to specify the
difference between futures and options rather than just defining each contract separately.
Question 2
Well answered. This was a question where some candidates were penalised if answers had
been rounded excessively.
Question 3
Generally well answered. Another possible solution is to use 1 1 0.6 1 0.6 0.11
1 1.07143
j i
f
+ + ×
+ = =
+
which leads to the same answer.
Question 4
For full marks in part (i), an answer should have included a description of the ‘risk-free’
concept (rather than just saying arbitrage profits are impossible). Many students had
difficulty with part (ii).
Question 5
Full marks were given if either 365 or 365.25 days were used in the calculation. Most
students scored well on this question.
Question 6
This question was well answered. For full marks, candidates were required to show detailed
steps in deriving the result required including a definition of the initial terms used and a
correct explanation of the relevance of the independence assumption.
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’ Report
Page 3
Question 7
This question was poorly answered to the surprise of the examiners. Many candidates
struggled to deal with the linked internal rate of return.
Question 8
Well answered.
Question 9
This question appeared to reward candidates who had a good understanding of the topic.
Whilst the best candidates usually scored close to full marks on this question, weaker or lessprepared
candidates often scored very badly.
Whilst the question did state that payments were made monthly, the examiners recognised
that there was some potential for misinterpretation as to the frequency of the loan repayments
in part (e) and took this into account. Thus students who used the formula Xa30 =100,000
with (12) i = 6% & i =6.168% to get an answer of £7,396 in this part were awarded full marks.
Question 10
Generally well answered.
Question 11
This was the worst answered question on the paper by some margin with very few candidates
scoring close to full marks. This may be because this type of question has not appeared in
recent diets. Candidates needed to show that they could derive logically the amounts that will
be paid, the real values of those amounts and their present values in real terms. Appropriate
formulae then needed to be developed.
Question 12
Many candidates answered this question well although a minority scored very badly (possibly
due to time pressure).
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’Report
Page 4
1 (i) A future is a contract binding buyer and seller to deliver or take delivery of an
asset at a given price at a given time in the future. An option is a contract that
gives the buyer the option to deliver or take delivery of the asset at the given
price. The seller of the option must deliver/take delivery if the buyer of the
option wishes to exercise the option.
(ii) Convertibles have option-like characteristics because they give the holder the
option to purchase equity in a company on pre-arranged terms.
2 The accumulated value is
4s3 + 2s2 + 2s1
( ) 3 2 1 = i 4s + 2s + 2s
d
0.04 (4 3.1216 2 2.0400 2)
0.039221
18.9352
= × + × +
=
3 (a) The money rate of return is i where (1+i) = 11.1/10
i = 0.11 or 11%
(b) The rate of inflation is f where (1+f) = 120/112
f = 0.07143 or 7.143%
(c) The net real rate of return per annum is j
where 0.6 0.6 0.11 0.07143
1 1.07143
j i f
f
- × -
= =
+
= -0.005068 or -0.5068%
4 (i) The no arbitrage assumption means that it is assumed that an investor is unable
to make a risk-free trading profit.
(ii) In all states of the world, security B pays 80% of A. Therefore its price must
be 80% of A’s price, or the investor could obtain a better payoff by only
purchasing one security and make risk-free profits by selling one security short
and buying the other. The price of B must therefore be 16p.
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’ Report
Page 5
5 (i) (a) Let the answer be t days
3,600(1 + 0.06 × t/365) = 4,000
t = 675.9 days
(b) Let the answer be t days
3,600( )4
0.06 365
4 1
t
+ = 4,000
(4t/365) ln(1.015) = ln (4,000/3,600)
t = 645.7 days
(c) Let the answer be t days
3,600( )12
0.06 365
12 1
t
+ = 4,000
(12t/365) ln(1.005) = ln (4,000/3,600)
t = 642.5 days
(ii) (i)(a) takes longest because, under conditions of simple interest, interest does
not earn interest.
6 (i) Let it be the (random) rate of interest in year t . Let S10 be the accumulation of
the unit investment after 10 years:
E (S10 ) = E ??(1+ i1 )(1+ i2 )…(1+ i10 )??
E (S10 ) = E[1+ i1]E[1+ i2 ]…E[1+ i10 ] as {it}are independent
E[it ] = j
( ) ( )10 10
? E S10 = 1+ j =1.07 =1.96715
(ii) ( 2 ) ( )( ) ( ) 2
E S10 E 1 i1 1 i2 1 i10 = ?? + + + ? ? ??? ? ??
…
( )2 ( )2 ( )2
= E 1+ i1 E 1+ i2 …E 1+ i10 (using independence)
( 2 ) ( 2 ) ( 2 )
= E 1+ 2i1 + i1 E 1+ 2i2 + i2 …E 1+ 2i10 + i10
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’Report
Page 6
= ?E (1+ 2it + it2 )?10
? ? ( )2 2 10 = 1+ 2 j + s + j
as 2 [ ] [ ]2 2 2
E ??ii ?? =V it + E it = s + j
[ ] ( ) ( ) 2 2 10 20 ? Var Sn = 1+ 2 j + s + j - 1+ j
( ) ( ) 2 10 20 = 1+ 2×0.07 + 0.016 + 0.07 - 1.07 = 0.5761
(iii) If 1,000 units had been invested, the expected accumulation would have been
1,000 times bigger. The variance would have been 1,000,000 times bigger.
7 (i) (a) (1 )2 450 500 800 1.015%
600 450 40 500 100
+ i = ?i =
+ +
(b) First sub-interval is first year. Money weighted rate of return is 1 i
where ( 1 ) 1
1 450 25%
600
+ i = ?i = -
Second sub-interval is second year. Money weighted rate of return is i2
where ( ) ( )1
2
490 1+ i2 +100 1+ i2 = 800
Then ( )1
2
2
2
100 100 4 490 ( 800) 100 1256.1847 1
2 490 980
i
- ± - × × - - ±
+ = =
×
= 1.17978 (taking positive root)
(1+ i2 ) =1.39188?i2 = 39.188%
Linked internal rate of return is i
where (1+ i)2 = 0.75×1.39188?i = 2.1719%
(ii) The linked IRR is higher because it relies on two money weighted rates of
return. With the calculation of the second money weighted rate of return, there
is more money in the fund when the fund is performing well (in the second
half of the year).
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’ Report
Page 7
8 (i) ( ) [ ]
5 2 1 2 1 3 5
2 3 0
0
150 100exp at bt dt 100exp at bt 100exp 12.5a 41.667b
?? ?? = ? + ? = ? + ? = + ? ? ? ? ? ?
?
( ) [ ]
10 10 2 1 2 1 3
2 3 0
0
230 100exp at bt dt 100exp at bt 100exp 50a 333.333b
?? ?? = ? + ? = ? + ? = + ? ? ? ? ? ?
?
ln(1.5) =12.5a + 41.667b
ln(2.3) = 50a + 333.333b
The second expression less four times the first expression gives:
ln(2.3) - 4ln(1.5) =166.667b?b = -0.0047337
ln(2.3) 333.333 0.0047337 0.0482162
50
a
- ×-
= =
(ii) 100e10d = 230?10d = ln 2.3?d = 0.08329
(iii) Present Value
10
0.05 0.08329
0
= ? 20e te- tdt
10
0.03329
0
= ? 20e- tdt
0.03329 10
0
20
0.03329
? e- t ?
= ? ?
??- ??
= 20×8.5058 =170.116
9 (a) Premiums were expected to accumulate to
(12)
30 1,060s???? at 7% (12) 30 1,060 i s 1,060 1.037525 94.4608 £103,885.77
d
= = × × =
(b) Premiums would have accumulated to
(12)
30 1,060s???? at 4% (12) 30 1,060 i s 1,060 1.021537 56.0849 £60,730.37
d
= = × × =
The shortfall is 100,000 – 60,730.37 = £39,269.63
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’Report
Page 8
(c) Accumulation will be
(12) ( )10 (12)
20 4% 10 4% 1,060??s?? 1.04 + 5,000s????
( ) ( ) ( )
10
12 20 12 10 1,060 1.04 5,000
1,060 1.021537 29.7781 1.48024 5,000 1.021537 12.0061
£109,053.12
i s i s
d d
= +
= × × × + × ×
=
Therefore the excess is £9,053.12
(d) The investor has earned a return of 4 % by investing extra premiums in the
investment policy. The investor could have obtained a lower present value of
total payments on the loan by paying off part of the loan instead. This is
because the interest being paid on the loan was greater than the interest he was
earning on his premiums.
(e) If he had repaid the loan by a level annuity, the annual instalment would have
been X where
360 100,000
12
X a = at 0.5% (or (12)
30 Xa =100,000 with (12) i = 6% & i = 6.168%)
360
12 100,000 1,200,000 £7,194.61
166.7916
X
a
×
= = =
10 Present value of companies’ and consumers’ costs is (in £ million)
i (50 +10)(v +1.03v2 +1.032v3 + +1.0319v20 )
d
…
( ( ) ( ) )
( ( ) )
2 19
20
60 1 1.03 1.03 1.03
1 1.03 1 1.80611 0.45639 60 1.019869 60 0.96154
1 1.03 1 1.03 0.96154
1.019869 60 0.96154 18.27680 1075.383
i v v v v
i v v
v
= + + + +
d
- ? - × ? = = × × ×? ? d - ? - × ?
= × × × =
…
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’ Report
Page 9
Present value of costs to financial advisors (in £ million)
i (60v +19v2 +18v3 + + v20 )
d
…
( )
( ) ( )
2 3 20
20 20 20 20
40 20 19 18
40 21 40 21
iv i v v v v
iv i a Ia i v a Ia
= + + + + +
d d
= + - = + -
d d d
…
1.019869 (40 0.96154 21 13.5903 125.1550)
1.019869 198.7029 202.651
= × × + × -
= × =
Total PV of all costs = £1278.034 million
Present value of benefits (in £ million)
( 2 3 20 )
20 i 30v + 33v + 36v + + 87v + i 12a
d d
…
( )
( ( ) )
( )
2 3 20
20 20
20 20
27 3 6 9 60 12
3 39
1.019869 3 125.1550 39 13.5903
1.019869 905.4867
i a v v v v a
i Ia a
+ + + + + +
d
= +
d
= × + ×
= ×
…
= 923.478
Net present value of costs = PV(costs) – PV(benefits)
= 1278.034 – 923.478 = £354.556 million
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’Report
Page 10
11 (i)
• Payments guaranteed by government.
• Can be various different indexation provisions but, in general, protection is
given against a fall in the purchasing power of money.
• Fairly liquid (i.e. large issue size and ability to deal in large quantities)
compared with corporate issues, but not compared with conventional
issues.
• Normally coupon and capital payments both indexed to increases in a
given price index with a lag.
• Low volatility of return and low expected real return.
• More or less guaranteed real return if held to maturity (can vary due to
indexation lag).
• Nominal return is not guaranteed.
(ii) The first coupon the investor will receive will be on 31st December 2003. The
net coupon per £100 nominal will be:
0.8×1× (Index May 2003/Index November 2001) = 0.8 1 113.8
110
× ×
In real present value terms, this is
( )0.5
0.8113.8
110 1
v
+ r
where r = 2.5% per annum and v is calculated at 1.5% (per half year)
The second coupon on 30th June 2004 per £100 nominal will be
0.8 1 113.8 (1 )0.5
110
× × + r
In real present value terms, this is ( ) ( )
2
0.8 1 0.5 113.8
110 1
r v
r
+
+
The third coupon on 31st December 2004 per £100 nominal will be
0.8 1 113.8 (1 )
110
× × + r
In real present value terms, this is ( )
( )
3
1.5
0.8 1 113.8
110 1
r v
r
+
+
Continuing in this way, the last coupon payment on 30 June 2009 per £100
nominal will be 0.8 1 113.8 (1 )5.5
110
× × + r
In real present value terms, this is ( )
( )
12
5.5
6
0.8 1 113.8
110 1
r v
r
+
+
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’ Report
Page 11
By similar reasoning, the real present value of the redemption payment is
( )
( )
12
5.5
6
100 1 113.8
110 1
r v
r
+
+
The present value of the succession of coupon payments and the capital
payment can be written as:
( ) ( ( ) )
( )
( )
2 12 12
0.5
12
12 1.5% 1.5%
1 113.8 0.8 100
1 110
1 113.8 0.8 100
1.0124224 110
1.02185 0.8 10.9075 100 0.83639
94.3833
P vv v v
r
a v
= + + + +
+
= +
= × × + ×
=
…
12 (i) Present value of liabilities is 10
15 160,000a + 200,000v at 7%
160,000 9.1079 200,000 0.50835
£1,558,934
= × + ×
=
(ii) Discounted mean term (DMT) of liabilities is
( 2 15 ) 10
10
15
1 160,000 2 160,000 15 160,000 200,000 10
160,000 200,000
v v v v
a v
× × + × × + + × × + × ×
=
+
…
( ) 10
15
10
15
160,000 200,000 10
160,000 200,000
Ia v
a v
× + × ×
=
+
160,000 61.5540 200,000 10 0.50835
1,558,934
× + × ×
=
10,865,340
1,558,934
= = 6.9697 years (½ mark deducted for no units)
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’Report
Page 12
(iii) Let the nominal amounts in each security equal A and B respectively.
If the present values of assets and liabilities are to be equal then:
( 8 ) ( 25 )
8 25 A 0.08a + v + B 0.03a + v =1,558,934 (1)
If the DMTs of the assets and liabilities are equal, then:
( ( ) 8 ) ( ( ) 25 )
8 25 0.08 8 0.03 25
6.9697
1,558,934
A Ia + v + B Ia + v
=
or ( ( ) 8 ) ( ( ) 25 )
8 25 A 0.08 Ia + 8v + B 0.03 Ia + 25v =10,865,340 (2)
From (1)
(0.08 5.9713 0.58201) (0.03 11.6536 0.18425) 1,558,934
1.059714 0.533858 1,558,934
A B
A B
× + + × + =
? + =
From (2)
(0.08 24.7602 8 0.58201) (0.03 112.3301 25 0.18425) 10,865,340
6.636896 7.976153 10,865,340
A B
A B
× + × + × + × =
? + =
Therefore
6.636896 1,558,934 0.533858 7.976153 10,865,340
1.059714
7.976153 6.636896 0.533858 10,865,340 6.636896 1,558,934
1.059714 1.059714
1,101,872.85 £237,850
4.632647
B B
B
B
? - ?? ? + =
? ?
? × ? × ? ? - ? = -
? ?
? = =
1,558,934 0.533858 £1,351,266
1.059714
A B
? - ? = ? ? =
? ?
Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’ Report
Page 13
(iv) It appears that the asset payments are more spread out than the liability
payments. The third condition for immunisation is that that convexity of the
assets is greater than that of the liabilities, or that the asset times are more
spread around the discounted mean term than the liability times. From
observation is appears likely that this condition is met.
END OF EXAMINERS’ REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
12 April 2007 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 11 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the
Formulae and Tables and your own electronic calculator.
© Faculty of Actuaries
CT1 A2007 © Institute of Actuaries
CT1 A2007—2
1 An investor pays £400 every half-year in advance into a 25-year savings plan.
Calculate the accumulated fund at the end of the term if the interest rate is 6% per
annum convertible monthly for the first 15 years and 6% per annum convertible halfyearly
for the final 10 years. [5]
2 The force of interest d(t ) is a function of time and at any time, measured in years, is
given by the formula:
( )
( )
( )
0.04 0.01 0 4
0.12 0.01 4 8
0.06 8
t t t
t t t
t t
d = + = =
d = - < =
d = <
Calculate the present value at time t = 0 of a payment stream, paid continuously from
time t = 9 to t = 12, under which the rate of payment at time t is 50e0.01t .
[6]
3 An ordinary share pays annual dividends. The next dividend is due in exactly eight
months’ time. This dividend is expected to be £1.10 per share. Dividends are expected
to grow at a rate of 5% per annum compound from this level and are expected to
continue in perpetuity. Inflation is expected to be 3% per annum. The price of the
share is £21.50.
Calculate the expected effective annual real rate of return for an investor who
purchases the share. [7]
4 An investor entered into a long forward contract for a security five years ago and the
contract is due to mature in seven years’ time. The price of the security was £95 five
years ago and is now £145. The risk-free rate of interest can be assumed to be 3% per
annum throughout the 12-year period.
Assuming no arbitrage, calculate the value of the contract now if:
(i) The security will pay dividends of £5 in two years’ time and £6 in four years’
time. [3]
(ii) The security has paid and will continue to pay annually in arrear a dividend of
2% per annum of the market price of the security at the time of payment. [3]
[Total 6]
CT1 A2007—3 PLEASE TURN OVER
5 In a particular bond market, n-year spot rates per annum can be approximated by the
function 0.08 - 0.04e-0.1n .
Calculate:
(i) The price per unit nominal of a zero coupon bond with term nine years. [2]
(ii) The four-year forward rate at time 7 years. [3]
(iii) The three-year par yield. [3]
[Total 8]
6 A fund had a value of £21,000 on 1 July 2003. A net cash flow of £5,000 was
received on 1 July 2004 and a further net cash flow of £8,000 was received on 1 July
2005. Immediately before receipt of the first net cash flow, the fund had a value of
£24,000, and immediately before receipt of the second net cash flow the fund had a
value of £32,000. The value of the fund on 1 July 2006 was £38,000.
(i) Calculate the annual effective money weighted rate of return earned on the
fund over the period 1 July 2003 to 1 July 2006. [3]
(ii) Calculate the annual effective time weighted rate of return earned on the fund
over the period 1 July 2003 to 1 July 2006. [3]
(iii) Explain why the values in (i) and (ii) differ. [2]
[Total 8]
7 An insurance company has liabilities of £87,500 due in 8 years’ time and £157,500
due in 19 years’ time. Its assets consist of two zero coupon bonds, one paying
£66,850 in four years’ time and the other paying £X in n years’ time. The current
interest rate is 7% per annum effective.
(i) Calculate the discounted mean term and convexity of the liabilities. [5]
(ii) Determine whether values of £X and n can be found which ensure that the
company is immunised against small changes in the interest rate. [5]
[Total 10]
CT1 A2007—4
8 A company has borrowed £800,000 from a bank. The loan is to be repaid by level
instalments, payable annually in arrear for 10 years from the date the loan is made.
The annual repayments are calculated at an effective rate of interest of 8% per annum.
(i) Calculate the amount of the level annual payment and the total amount of
interest which will be paid over the 10 year term. [3]
(ii) At the beginning of the eighth year, immediately after the seventh payment
has been made, the company asks for the term of the loan to be extended by
two years. The bank agrees to do this on condition that the rate of interest is
increased to an effective rate of 12% per annum for the remainder of the term
and that payments are made quarterly in arrear.
(a) Calculate the amount of the new quarterly payment.
(b) Calculate the capital and interest components of the first quarterly
instalment of the revised loan repayments.
[6]
[Total 9]
9 A property developer is constructing a block of offices. It is anticipated that the
offices will take six months to build. The developer incurs costs of £40 million at the
beginning of the project followed by £3 million at the end of each month for the
following six months during the building period. It is expected that rental income
from the offices will be £1 million per month, which will be received at the start of
each month beginning with the seventh month. Maintenance and management costs
paid by the developer are expected to be £2 million per annum payable monthly in
arrear with the first payment at the end of the seventh month. The block of offices is
expected to be sold 25 years after the start of the project for £60 million.
(i) Calculate the discounted payback period using an effective rate of interest of
10% per annum. [7]
(ii) Without doing any further calculations, explain whether your answer to (i)
would change if the effective rate of interest were less than 10% per annum.
[3]
[Total 10]
CT1 A2007—5
10 A loan is issued bearing interest at a rate of 9% per annum and payable half-yearly in
arrear. The loan is to be redeemed at £110 per £100 nominal in 13 years’ time.
(i) The loan is issued at a price such that an investor, subject to income tax at
25%, and capital gains tax at 30%, would obtain a net redemption yield of 6%
per annum effective. Calculate the issue price per £100 nominal of the stock.
[5]
(ii) Two years after the date of issue, immediately after a coupon payment has
been made, the investor decides to sell the stock and finds a potential buyer,
who is subject to income tax at 10% and capital gains tax at 35%. The
potential buyer is prepared to buy the stock provided she will obtain a net
redemption yield of at least 8% per annum effective.
(a) Calculate the maximum price (per £100 nominal) which the original
investor can expect to obtain from the potential buyer.
(b) Calculate the net effective annual redemption yield (to the nearest 1%
per annum effective) that will be obtained by the original investor if
the loan is sold to the buyer at the price determined in (ii) (a).
[10]
[Total 15]
11 £80,000 is invested in a bank account which pays interest at the end of each year.
Interest is always reinvested in the account. The rate of interest is determined at the
beginning of each year and remains unchanged until the beginning of the next year.
The rate of interest applicable in any one year is independent of the rate applicable in
any other year.
During the first year, the annual effective rate of interest will be one of 4%, 6% or 8%
with equal probability.
During the second year, the annual effective rate of interest will be either 7% with
probability 0.75 or 5% with probability 0.25.
During the third year, the annual effective rate of interest will be either 6% with
probability 0.7 or 4% with probability 0.3.
(i) Derive the expected accumulated amount in the bank account at the end of
three years. [5]
(ii) Derive the variance of the accumulated amount in the bank account at the end
of three years. [8]
(iii) Calculate the probability that the accumulated amount in the bank account is
more than £97,000 at the end of three years. [3]
[Total 16]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
EXAMINATION
April 2007
Subject CT1 — Financial Mathematics
Core Technical
EXAMINERS’ REPORT
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
M A Stocker
Chairman of the Board of Examiners
June 2007
© Faculty of Actuaries
© Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 2
Comments
Please note that different answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this.
However, candidates may be penalised where excessive rounding has been used or where
insufficient working is shown.
Q1.
Whilst most candidates made a good attempt at this question on basic compound interest
accumulation, comparatively few students completed the question without error.
Q2.
Well answered.
Q3.
Most students answered this question well although candidates were expected to note that the
sum of the geometric progression would only converge if the rate of return was below the
dividend growth rate. Depending on the interpolation used, the final answer can justifiably
vary from that given.
Q4.
This proved to be the most difficult question on the paper. Other related methods to
determine the answers were available e.g. calculating the forward price of each contract and
working out the present value of the difference in these prices.
Q5.
Well answered.
Q6.
The calculations in parts (i) and (ii) were generally well done. Again, depending on the
interpolation used, the final answer can justifiably vary from that given although the
examiners penalised the use of too wide a range of interpolation.
The explanation in part (iii) was very poorly handled. In such cases, the examiners are not
simply looking for a statement lifted directly from the Core Reading. Instead, candidates are
expected to apply the relevant theory to the actual situation described in the question.
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 3
Q7.
Generally well answered.
Q8.
This was the best answered question on the paper.
Q9.
Many candidates struggled with this question, firstly in determining when the various
costs/payments would be made and then in manipulating the resulting equation(s). A common
error was not to recognise that the DPP should be expressed as a whole number of months
since payments at the relevant time were being made at monthly intervals. In part (ii) little
credit was given for a correct conclusion without any accompanying explanation.
Q10.
This question seemed to provide a significant differentiation between candidates with many
scoring well and a sizeable minority scoring very badly. This seemed surprising given that
this topic is regularly examined. A common omission on part (ii)(b) was not to state whether
a capital gain had been made.
Q11.
The workings for parts (i) and (ii) were often too brief (the questions said ‘Derive…’). Note
that the final answer in part (ii) can justifiably vary significantly according to the rounding
used in intermediate calculations. Part (iii) was poorly done with many candidates assuming
a lognormal distribution for this discrete example.
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 4
1 Fund after 25 years =
( ) *% 20 3%
30 20 400S???? i × 1.03 + 400S????
where 1+ i* = (1.005)6
?i* = 3.03775% per ½-year
( )30
30
1.0303775 1
@3.03775% 1.0303775
0.0303775
s
? - ?
= × ? ?
?? ??
????
= 49.3215
( )20
20
1.03 1
@3% 1.03
0.03
S
? - ?
= × ? ?
?? ??
???? = 27.6765
Hence fund =
400× 49.3215×(1.03)20 + 400× 27.6765
= 35632.06 + 11070.60
= £46,702.66
2 (i)
( ) 0 12 0.01
9
50 .
t t t dt PV e e dt
-? d = ?
where
( ) ( ) ( ) 4 8
0 0 4 8
0.04 0.01 0.12 0.01 0.06
t t
? d t dt = ? + t dt + ? - t dt + ? dt
[ ] 2 4 2 8
0 4 8
= ??0.04t + 0.005t ?? + ??0.12t - 0.005t ?? + 0.06t t
= [0.24]+[0.64 - 0.40]+[0.06t - 0.48]
= 0.06t
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 5
Hence
12 0.01 0.06
9
PV = ? 50e t . e- t dt
12 0.05
9
50
t
e dt
-
= ?
12
0.05
9
50
0.05
= ? - e- t ? ?? ??
= –548.812 + 637.628
= 88.816
3 Let i = money rate of return
i' = real rate of return
?1+ i = (1+ i')(1.03) here
( )4 ( ( ) )
21.50 = 1+ i 12 · 1.10v +1.05×1.10v2 + 1.05 2 ×1.10v3 +????
( ) ( )
( )
4
12
1.05
1
1.05
1
1
1 1.101
i
i
i v
8
+
+
? - ? = + × ? ? ?? - ??
? ?
( )8 ( )
12 1.05
1
1.10 1 1
1 i 1 +i
= ×
+ -
assuming i > 0.05
( )8
12 1.05
1
19.5455 1 1
1 i 1 +i
= ×
+ -
Try i = 10% RHS = 20.6456
11% RHS = 17.2566
0.10 20.6456 19.5455 0.01 0.10325
20.6456 17.2566
i -
? = + × =
-
comes from 1 1.10325 7.1% p.a.
1.03
?i' + i' = ?i' =
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 6
4 (i) The current value of the forward price of the old contract is:
95 (1.03)5 5(1.03) 2 6(1.03) 4 - - × - -
whereas the current value of the forward price of a new contract is:
145 5(1.03) 2 6(1.03) 4 - - - -
Hence, current value of old forward contract is:
145 - 95(1.03)5 = £34.87
(ii) The current value of the forward price of the old contract is:
95(1.02) 12 (1.03)5 86.8376 - =
whereas the current value of the forward price of a new contract is:
145(1.02) 7 126.2312 - =
? current value of old forward contract is:
126.23 - 86.84 = £39.39
5 (i) Let Yk = spot rate for k year term
Pk = Price per unit nominal for k year term
Y9 = 0.063737
9
9
9
1 0.57344
1
P
Y
? ?
= ? ? = ? + ?
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 7
(ii) 0.1(7)
Y7 0.08 0.04 e 0.060137 = - - =
0.1(11)
Y11 0.08 0.04e 0.066685 = - - =
( ) ( )
( )
( )
( )
11 11
4 11
7,4 7 7
7
1 1.066685
1
1 1.060137
Y
f
Y
+
+ = =
+
= 1.35165
? 4-year forward rate is 7.824% at time 7.
(iii) Y1 = 0.04381, Y2 = 0.04725, Y3 = 0.05037
( )( ) 3 1 2 3 3
1 1 2 3 3 = Yc vY + vY + vY + vY
3 Yc = 0.05016 i.e. 5.016% p.a.
6 (i) Work in £000’s
MWRR is i such that:
21(1+ i)3 + 5(1+ i)2 + 8(1+ i) = 38
Try i = 5%, LHS = 38.223
i = 4%, LHS = 37.350
By interpolation i = 4.74% p.a.
(ii) TWRR is i such that:
(1 )3 24 32 38 6.21% p.a.
21 29 40
+ i = × × ?i =
(iii) MWRR is lower than TWRR because of the large cash flow on 1/7/05; the
overall return in the final year is much lower than in the first 2 years, and the
payment at 1/7/05 gives this final year more weight in the MWRR, but does
not affect the TWRR.
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 8
7 Let PVL be PV of liabilities, DMTL be DMT of liabilities, CL be convexity of
liabilities.
(i) PVL = 87,500v8 +157,500v19 at 7%
= 94,475.86
87,500 8 8 157,500 19 19
L 94,475.86
DMT v v × + ×
? = at 7%
1,234,857.56
94, 475.86
=
= 13.070615 years
87,500 8 9 10 157,500 19 20 21
L 94,475.86
C v v × × + × ×
= at 7%
17,657,158.78
94, 475.86
=
= 186.895985
(ii) Firstly, PVs should be equal:
?66,850v4 + Xvn = 94, 475.86 at 7%
? Xvn = 43, 476.31507
Secondly, DMTs should be equal
?66,850× 4v4 + Xnvn =1, 234,857.56
? Xnvn =1,030,859.38
?n = 23.710827 years
? X = 43, 476.31507×1.07n
= 216,255.12
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 9
Lastly, verify 3rd condition
(66,850 4 5 6 216,255.12 ( 1) (n 2) )/ 94,475.86
CA v n n v = × × + + +
= 23,140,343.20/94,475.86
= 244.93393
> CL
Hence, immunisation is achieved.
8 (i) 8%
10 800,000 = P a = P×6.7101
? P =119,223.26
Total amount of interest = 10 × 119,223.26 – 800,000
= £392,232.60
(ii) (a) Capital o/s at start of 8th year
= 119,223.26 8%
3 a =119, 223.26*2.5771 = 307, 250.26
Let new payment be P' per annum, then
(4)
5
12%
P'a = P'*1.043938*3.6048 = 307, 250.26
? P' = 81,646.28
?q'ly payment = 20,411.57
(b) Capital o/s after 7 years = 307,250.26
? Interest in 1st q'ly payment ( )1
= 30,7250.26*?? 1.12 4 -1?? = 8,829.56
? ?
? capital component = 20,411.57 -8,829.56 = 11,582.01
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 10
9 (i) The discounted payback period is the first point at which the present value of
the income exceeds the present value of the outgoings. The present value of
all payments and income up to time t is given by (working in £m)
( ) ( ) ( )
( ) ( ) ( )
1 1
2 2
1 1 1 1
2 2 2 12
1 1
2 2
1 1 1
2 2 2
12 12 12
12 12 1 12
12
40 36 2 12
40 36 2 12
t t
t t
PV a v a v a
a va v a
- - +
- -
= - - - +
= - - - + ? + ? ? ?
? ?
????
= ( )
( )
1 1 0.5
2 2
1 12
2
40 36 12 10 1 vt
i
a v v - - + + - -
( )
( )
1
2
1
2
12
12
a 1 v at 10%
i
-
= 1 0.9534626 0.48634
0.0956897
-
= =
? 0.56758 = 1 - vt-0.5
? vt-0.5 = 0.43242
? t = ( )
( )
log 0.43242
log 0.90909 + 0.5
? t = 9.296
Hence, the discounted pay back period is 9 years and 4 months.
(ii) If the effective rate of interest were less than 10% p.a. then the present values
of the income and outgo would both increase. However, the bigger impact
would be on the present value of the income since the bulk of the outgo occurs
in the early years when discounting has less effect. Hence, the DPP would
decrease.
10 (i) i(2) = 0.059126
( 1 )
1 0.09 0.75=0.06136
1.10
g -t = ×
(2) ( )
?i < 1- t1 g
? No capital gain
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 11
Price of £100 nominal stock
(2) 13
13 = 0.75×9 a +110v at 6%
= 0.75 × 9 × 1.014782 × 8.8527 + 110 × 0.46884
= 60.639 + 51.572
= £112.21
(ii) (a) i(2) = 0.078461
( 1 )
1 0.09 0.90= 0.073636
1.10
g - t = ×
(2) ( )
?i > 1- t1 g
? Capital gain
Price, P = 0.90 × 9 × (2) ( ( ) ) 11
11 a + 110 - 110 - P ×0.35 v at 8%
0.90 9 1.019615 7.1390 0.65 110 0.42888
1 0.35 0.42888
P × × × + × ×
? =
- ×
89.62508 105.455
0.849892
= =
(b) No capital gain made
112.21 = 0.75 × 9 × (2) 2
2 a +105.455v
Try i = 3%, RHS = 112.41
i = 4%, RHS = 110.36
? yield = 3% p.a. to nearest 1%
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 12
11 (i) Let 3 Accumulated f S = und after 3 years of investment of 1 at time 0
it = Interest rate for year t
Then, fund after 3 years
= 80,000 S3 = 80000(1+ i1 )(1+ i2 )(1+ i3 )
( ) 1 ( )
1 3 E i = 0.04+0.06+0.08 =0.06
E (i2 ) = 0.75×0.07+0.25×0.05=0.065
E (i3 ) = 0.7×0.06 + 0.3×0.04 = 0.054
Then:
E[80000S3 ] = 80,000 E[S3 ]
= E ??80,000(1+ i1 )(1+ i2 )(1+ i3 )??
= 80,000 E (1+ i1 ). E (1+ i2 ). E (1+ i3 )
since it ' s are independent
= 80,000 × 1.06 × 1.065 × 1.054 = £95,188.85
(ii) Var[ ] 2 [ ]
80000S3 = 80,000 ×Var S3
where [ ] 2 ( [ ])2
Var S3 = E ??S3 ?? - E S3
2 ( )2 ( )2 ( )2
E S3 E 1 i1 1 i2 1 i3 ? ? = ? + + + ? ? ? ?? ??
( )2 ( )2 ( )2
E 1 i1 . E 1 i2 . E 1 i3 = ? + ? ? + ? ? + ? ?? ?? ?? ?? ?? ??
using independence
( [ ] 2 ) ( [ ] 2 )
= 1+ 2E i1 + E ??i1 ?? . 1+ 2E i2 + E ??i2 ?? ( [ ] ) 2
· 1+ 2 E i3 + E ??i3 ??
Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report
Page 13
Now,
( 2 ) 1 ( 2 2 2 )
1 3 E i = 0.04 + 0.06 + 0.08 = 0.0038667
( 2 ) 2 2
E i2 = 0.75×0.07 + 0.25×0.05 = 0.0043
( 2 ) 2 2
E i3 = 0.7×0.06 + 0.3×0.04 = 0.0030
Hence, 2
E ?S3 ?
? ?
= (1+ 2×0.06 + 0.0038667) ×(1+ 2×0.065 + 0.0043) ×(1+ 2×0.054 + 0.003)
=1.41631
Hence Var[80,000S3 ]
2 [ ]
= 80,000 Var S3
= 80,0002 (1.41631-(1.18986)2 )
= 3,476,355
(iii) Note: 80,000 × 1.08 × 1.07 × 1.06 = 97,995 > 97,000
But, if in any year, the highest interest rate for the year is not achieved then the
fund after 3 years falls below £97,000.
Hence, answer is probability that highest interest rate is achieved in each year
1 0.75 0.7 0.175
3
= × × =
END OF EXAMINERS’ REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
25 September 2007 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 11 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the
Formulae and Tables and your own electronic calculator.
© Faculty of Actuaries
CT1 S2007 © Institute of Actuaries
CT1 S2007—2
1 A 90-day government bill is purchased for £96 at the time of issue and is sold after 45
days to another investor for £97.90. The second investor holds the bill until maturity
and receives £100.
Determine which investor receives the higher rate of return. [2]
2 An investor purchases a share for 769p at the beginning of the year. Halfway through
the year he receives a dividend, net of tax, of 4p and immediately sells the share for
800p. Capital gains tax of 30% is paid on the difference between the sale and the
purchase price.
Calculate the net annual effective rate of return the investor obtains on the investment.
[4]
3 An insurance company offers a customer two payment options in respect of an
invoice for £456. The first option involves 24 payments of £20 paid at the beginning
of each month starting immediately. The second option involves 24 payments of
£20.50 paid at the end of each month starting immediately. The customer is willing to
accept a monthly payment schedule if the annual effective interest rate per annum he
pays is less than 5%.
Determine which, if any, of the payment options the customer will accept. [4]
4 State the characteristics of an equity investment. [4]
5 A one-year forward contract is issued on 1 April 2007 on a share with a price of 900p
at that date. Dividends of 50p per share are expected on 30 September 2007 and 31
March 2008. The 6-month and 12-month spot, risk-free rates of interest are 5% and
6% per annum effective respectively on 1 April 2007.
Calculate the forward price at issue, stating any assumptions. [4]
6 The annual effective forward rate applicable over the period t to t + r is defined as
ft,r where t and r are measured in years. f0,1 = 4%, f1,1 = 4.25% f2,1 = 4.5%,
f2,2 = 5%. Calculate the following:
(i) f3,1 [1]
(ii) All possible zero coupon (spot) yields that the above information allows you
to calculate. [4]
(iii) The gross redemption yield of a four-year bond, redeemable at par, with a 3%
coupon payable annually in arrears. [6]
(iv) Explain why the gross redemption yield from the four-year bond is lower than
the one-year forward rate up to time 4, f3,1 [2]
[Total 13]
CT1 S2007—3 PLEASE TURN OVER
7 The force of interest,d(t) , is a function of time and at any time t (measured in years)
is given by
0.04 0.01 for 0 10
( )
0.05 for 10
t t
t
t
? + = =
d = ? > ?
(i) Derive, and simplify as far as possible, expressions for v(t)where v(t) is the
present value of a unit sum of money due at time t. [5]
(ii) (a) Calculate the present value of £1,000 due at the end of 15 years.
(b) Calculate the annual effective rate of discount implied by the
transaction in (a). [4]
(iii) A continuous payment stream is received at a rate of 20e-0.01t units per
annum between t = 10 and t = 15. Calculate the present value of the payment
stream.
[4]
[Total 13]
8 A pension fund makes the following investments (£m):
1 January 2004 1 July 2004 1 January 2005 1 January 2006
12.5 6.6 7.0 8.0
The rates of return earned on money invested in the fund were as follows:
1 January 2004
to 30 June 2004
1 July 2004 to
31 December 2004
1 January 2005 to
31 December 2005
1 January 2006 to
31 December 2006
5% 6% 6.5% 3%
You may assume that 1 January to 30 June and 1 July to 31 December are precise half
year periods.
(i) Calculate the linked internal rate of return per annum over the three years from
1 January 2004 to 31 December 2006, using semi-annual sub-intervals. [3]
(ii) Calculate the time weighted rate of return per annum over the three years from
1 January 2004 to 31 December 2006. [3]
(iii) Calculate the money weighted rate of return per annum over the three years
from 1 January 2004 to 31 December 2006. [4]
(iv) Explain the relationship between your answers to (i), (ii) and (iii) above. [2]
[Total 12]
CT1 S2007—4
9 The expected effective annual rate of return from a bank’s investment portfolio is 6%
and the standard deviation of annual effective returns is 8%. The annual effective
returns are independent and (1+it ) is lognormally distributed, where it is the return in
year t.
Deriving any necessary formulae:
(i) calculate the expected value of an investment of £2 million after ten years. [6]
(ii) calculate the probability that the accumulation of the investment will be less
than 80% of the expected value. [3]
[Total 9]
10 A government is holding an inquiry into the provision of loans by banks to consumers
at high rates of interest. The loans are typically of short duration and to high risk
consumers. Repayments are collected in person by representatives of the bank making
the loan. Campaigners on behalf of the consumers and campaigners on behalf of the
banks granting the loans are disputing one particular type of loan. The initial loans are
for £2,000. Repayments are made at an annual rate of £2,400 payable monthly in
advance for two years.
The consumers’ association case
The consumers’ association asserts that, on this particular type of loan, consumers
who make all their repayments pay interest at an annual effective rate of over 200%.
The banks’ case
The banks state that, on the same loans, 40% of the consumers default on all their
remaining payments after exactly 12 payments have been made. Furthermore half of
the consumers who have not defaulted after 12 payments default on all their
remaining payments after exactly 18 payments have been made. The banks also argue
that it costs 30% of each monthly repayment to collect the payment. These costs are
still incurred even if the payment is not made by the consumer. Furthermore, with
inflation of 2.5% per annum, the banks therefore assert that the real rate of interest
that the lender obtains on the loan is less than 1.463% per annum effective.
(i) (a) Calculate the flat rate of interest paid by the consumer on the loan
described above.
(b) State why the flat rate of interest is not a good measure of the cost of
borrowing to the consumer. [4]
(ii) Determine, for each of the cases above, whether the assertion is correct. [10]
[Total 14]
CT1 S2007—5
11 A pension fund has liabilities to pay pensions each year for the next 60 years. The
pensions paid will be £100m at the end of the first year, £105m at the end of the
second year, £110.25m at the end of the third year and so on, increasing by 5% each
year. The fund holds government bonds to meet its pension liabilities. The bonds
mature in 20 years time and pay an annual coupon of 4% in arrears.
(i) Calculate the present value of the pension fund’s liabilities at a rate of interest
of 3% per annum effective. [4]
(ii) Calculate the nominal amount of the bond that the fund needs to hold so that
the present value of the assets is equal to the present value of the liabilities. [3]
(iii) Calculate the duration of the liabilities. [6]
(iv) Calculate the duration of the assets. [4]
(v) Using your calculations in (iii) and (iv), estimate by how much more the value
of the liabilities would increase than the value of the assets if there were a
reduction in the rate of interest to 1.5% per annum effective. [4]
[Total 21]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
EXAMINATION
September 2007
Subject CT1 — Financial Mathematics
Core Technical
MARKING SCHEDULE
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
M A Stocker
Chairman of the Board of Examiners
December 2007
© Faculty of Actuaries
© Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 2
Comments
Please note that different answers may be obtained from those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this.
However, candidates may be penalised where excessive rounding has been used or where
insufficient working is shown.
It should be noted that the rubric of the examination paper does ask for candidates to show
their calculations where this is appropriate. Candidates often failed to show sufficient clarity
and detail in their working and lost marks as a result.
Q1.
Well answered.
Q2.
Well answered.
Q3.
Whilst this question was generally answered well, some candidates lost marks by not stating
the conclusions that arose from their calculations i.e. that neither deal was acceptable.
Q4.
This question was very poorly answered which was disappointing given that this was a
bookwork question.
Q5.
Reasonably well answered but some candidates failed to obtain full marks by not stating the
required assumption.
Q6.
Parts (i) and (ii) were well answered but part (iii) was a good differentiator with weaker
candidates failing to recognise the correct method for calculating the gross redemption yield.
As with many previous diets, many candidates in part (iv) had great difficulty in giving a
clear explanation of their calculations.
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 3
Q7.
Generally well answered. Some candidates lost marks by not giving an explicit formula for
v(t) when t = 10.
Q8.
This question was very poorly answered to the surprise of the examiners who felt that the
question should have been relatively straightforward.
Q9.
Part (i) can be done much more simply than by using the method given in this report but the
calculations given would still need to be done for part (ii).
Q10.
This question was the worst answered on the paper. Part (ii) did successfully differentiate
between candidates with weaker candidates appearing to struggle to apply the theory to a
real-life situation.
Q11.
The first three parts were generally answered well by the candidates who attempted the
question. Many struggled to complete part (iv) although it is possible that this was due to
time pressure. When calculating DMTs, candidates were expected to give the answer in terms
of the correct units.
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 4
1 The first investor receives the higher rate of return if:
97.9 100
96 97.9
>
This inequality does not hold, therefore the second investor receives the higher rate of
return.
2 Start by working in half years. The half yearly effective return is i such that:
769 = 4v + 800v – 0.3(800 – 769)v
769 = (804 - 240 + 230.7)v
v = 769 0.967661
794.7
= therefore i = 3.3420%
Annual effective rate is (1.033422 – 1) = 6.7957%
3 The annual rate of payment for the first deal is 240.
This deal is acceptable if:
240 (12)
2 a???? < 456 at a rate of interest of 5%
240 (12)
2 a???? = 240×1.8594× 1.026881 = 458.252
Therefore first deal is not acceptable
The annual rate of payment on the second deal is 246.
This deal is acceptable if:
246 (12)
2 a = 246×1.8594×1.022715 = 467.803
Therefore second deal is also not acceptable
4 Main characteristics of equity investments:
• Issued by commercial undertakings and other bodies.
• Entitle holders to receive all net profits of the company in the form of
dividends after interest on loans and other fixed interest stocks has been paid.
• Higher expected returns than for most other asset classes …
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 5
• …but risk of capital losses
• … and returns can be variable.
• Lowest ranking form of finance.
• Low initial running yield…
• … but dividends should increase with inflation.
• Marketability varies according to size of company.
• Voting rights in proportion to number of shares held.
5 Assuming no arbitrage:
Present value of dividends is (in£):
0.5v1/2 (at 5%) + 0.5v (at 6%) = 0.5(0.97590+0.94340) = 0.95965
Hence forward price is: F = (9-0.95965)× 1.06 = £8.5228
6 (i) f3,1 is such that 1.045× f3,1 = 1.052. Therefore f3,1= 5.5024%
(ii) One-year spot rate is same as one-year forward rate = 4%
Two-year spot rate is i2 such that (1+i2 )2 = 1.04×1.0425.
Therefore i2 = 4.1249%
Three-year spot rate is i3 such that (1+i3 )3 = 1.04×1.0425×1.045.
Therefore i3 = 4.2498%
Four year spot rate is such that (1+i4 )4 = 1.04×1.0425×1.045×1.055024
Therefore i4 = 4.5615%
(iii) Present value of the payments from the bond is:
P = 3(1.04-1 + 1.041249-2 + 1.042498-3 + 1.045615-4)
+ 100×1.045615-4
Therefore P = 3(0.96154 + 0.92234 + 0.88262 + 0.83659)
+ 100× 0.83659 = 94.468
Equation of value to find the gross redemption yield from the bond is such
that:
94.468 = 3 4 a + 100v4
Try i = 4.5%
v4 = 0.83856, 4 a = 3.58753, RHS = 94.619
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 6
Try i = 5%
v4 = 0.82270, 4 a = 3.5460, RHS = 92.908
Interpolation:
Yield = 0.045 + 0.005× (94.619 – 94.468) /(94.619 – 92.908)
= 4.544%
(iv) The yield from the bond is lower than the one-year forward rate up to time 4
because the bond can be seen to be a series of zero coupon bonds (1 year, 2
years etc.) each with lower yields than the forward rate. The gross redemption
yield from the bond is, in effect, an average of spot rates that are themselves a
weighted average of earlier forward rates.
7 (i) For t = 10
( )
2 2
0 0
0.04 0.01 0.04 0.005 0.04 0.005
t t sds s s t t v t e e e
- + -? + ? = ? = ? ? = - -
For t > 10
( ) ( ) [ ] ( ) ( ) 10 10 10 0.05 0.9 0.05 0.9 0.05 10 0.4 0.05
t ds s t v t v e e e e e t e t = -? = - - = - - - = - +
(ii) (a) Present value 1000e (0.4 0.05 15) 1000e 1.15 316.637 = - + × = - =
(b) 1000(1- d)15 = 316.637?d = 7.380%
(iii) Present value
15 (0.4 0.05 ) 0.01
10
= ? e- + t 20e- tdt
15 0.4 0.06
10
= 20? e- e- tdt
( )
0.06 15
0.4 0.4
10
20 20 6.77616 + 9.14686 31.783
0.06
e t e e
-
- ? ? -
= ? ? = - =
?? - ??
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 7
8 (i) Linked internal rate of return is found by linking the money weighted rate of
return from the sub-periods.
(LIRR)3 = 1.05×1.06×1.065× 1.03
Therefore LIRR = 0.06879 or 6.879%
(ii) The TWRR requires the value of the fund every time a payment is made.
Size of the fund after six months is: 12.5× (1.05) = 13.125
Size of the fund after one year is: (13.125 + 6.6) × 1.06 = 20.909
Size of the fund after two years is: (20.909 + 7) × 1.065 = 29.723
Size of the fund after three years is: (29.723 + 8) × 1.03 = 38.855
The TWRR is i where i is the solution to:
(1+i)3 = (13.125/12.5) ×[20.909/(13.125+6.6)] × [29.723/(20.909+7)]
×[38.855/(29.723+8)]
or just use the rates of return given to give:
(1+i)3 = 1.05×1.06×1.065× 1.03
giving i = 6.879%
(iii) For MWRR, we need to know the size of the fund at the end of the period. We
can use the values above to give:
MWRR is solution to: 12.5(1+i)3 + 6.6(1+i)2.5 + 7(1+i)2 + 8(1+i) = 38.855
Solve by iteration and interpolation, starting with i = 7%.
i = 7% gives LHS = 39.704
i = 6% gives LHS = 38.868
i = 5.5% gives LHS = 38.454
Interpolate between 5.5% and 6%.
i = 0.055 + 0.005× (38.855-38.454)/(38.868-38.454) = 5.98%
(iv) (i) and (ii) are the same because there are no cash flows within sub-periods to
“distort” the LIRR away from the TWRR. The MWRR is lower because the
fund has a smaller amount of money in it at the beginning when rates of return
are higher.
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 8
9 (i) (1+ it ) ~ Lognormal (µ,s2 )
ln (1+ it ) ~ N (µ,s2 )
ln (1+ it )10 = ln (1+ it ) + ln (1+ it ) +…+ ln (1+ it ) ~ N (10µ,10s2 )
since it ' s are independent
(1+ it )10 ~ Lognormal (10µ,10s2 )
[½] for correct use of independence assumption
( )
( ) ( ) ( )
( )
2
2 2 2
2
2 2
2
1 exp 1.06
2
1 exp 2 exp 1 0.08
0.08 exp 1 0.0056798
1.06
t
t
E i
Var i
? s ?
+ = ??µ + ?? =
? ?
+ = µ + s ? s - ? =
? ?
= ? s - ??s =
? ?
exp 0.0056798 1.06 ln1.06 0.0056798 0.055429
2 2
?µ + ? = ?µ = - = ? ?
? ?
10µ = 0.55429 ,10s2 = 0.056798
Let S10 be the accumulation of one unit after 10 years:
( 10 )
exp 0.55429 0.056798 1.790848
2
E S = ?? + ?? =
? ?
Expected value of investment = 2,000,000E (S10 ) = £3.5817m
(ii) We require P[S10 < 0.8×1.790848 =1.4327]
P[ln S10 < ln1.4327] where ln S10~N(0.55429,0.056798)
(0,1) ln1.4327 0.55429
0.056798
P N ? - ?
? ? < ?
? ?
? P ??N (0,1) < -0.8171?? = 0.207 ˜ 21%
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 9
10 (i) (a) The flat rate of interest is: (2× 2,400 – 2,000)/(2×2,000) = 70%
(b) The flat rate of interest is not a good measure of the cost of borrowing
because it takes no account of the timing of payments and the timing of
repayment of capital.
(ii) If the consumers’ association is correct, then the present value of the
repayments is greater than the loan at 200%
i.e. (12) 2 2,000 2, 400 i a
d
<
i =2; 2 a = 0.44444; d(12) = 1.04982 gives RHS = 2,032
The consumers’ association is correct.
If the banks are correct, then the present value of the payments received by the
bank, after expenses, is less than the amount of the loan at a nominal (before
inflation) rate of interest of (1.01463× 1.025 -1) per annum effective = 0.04.
i.e. (12) 2 (12) 1.5 (12) 1 (12) 2 2,000 720 i a 720 i a 960 i a 0.3 2, 400 i a
d d d d
> + + - ×
(12)
i
d
= 1.021529; 2 a = 1.8861; 1 a = 0.9615;
1.5
1.5
1 1.04 1.4283
0.04
a
- -
= =
So RHS = 720×1.021529×1.8861+ 720×1.021529× 1.4283 +
960×1.021529× 0.9615 – 0.3×2,400×1.021529×1.8861
= 1,387.23+ 1,050.52 + 942.91 – 1,387.23 = 1,993.43
Therefore, the banks are also correct.
11 (i) Present value of the fund’s liabilities (in £m) is:
( )
( ( ) ( ) )
2 23 5960
2 59
100 1.05 1.05 1.05
100 1 1.05 1.05 1.05
v v v v
v v v v
+ + + +
= + + + +
…
…
( ) ( )
( )
60 1.05 60
1.03
1.05
1.03
1 1.05 1-
100 100 0.97087
1 1.05 1-
97.087 111.7795 £10,852
v
v
v
m
? ? ? ? - ? ? = ? ? = × ? - ? ? ? ? ? ? ? ? ?
= × =
(ii) Let the nominal holding of bonds = N in £m
The present value of the bonds must equal £10,852m
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 10
Therefore 20
0.04Na20 + Nv = 10,852 at 3%
20 a = 14.8775, v20 = 0.55368
So 10,852 = 0.04N×14.8775 + N×0.55368
N = 10,852 /(0.04×14.8775 + 0.55368) = £9,446.54m
(iii) The numerator for the duration of the liabilities can be expressed as follows:
100v (1× 1 + 1.05v× 2 + 1.052v2 ×3+…+1.0559v59 ×60)
= 1.03
1.05
100 v (1.05v× 1 + 1.052v2 × 2 + 1.053v3 ×3+…+1.0560v60 ×60)
The part inside the brackets can be regarded as ( )60 Ia evaluated at a rate of
interest i such that v = 1.05/1.03; the discount factor outside the brackets
should be evaluated at 3%
1.03
1.05
100 v = 100
1.05
= 95.2381
For the ( )60 Ia function, v = 1.019417; i = -0.019048; ( ) 60 1+ i a = 111.7727
( )
60
60
111.7727 60 1.019417 = 4118.567
0.019048
Ia - ×
=
-
Therefore numerator for duration is: 95.2381×4118.567 = 392,244
Therefore the duration is: 392,244/10,852 = 36.1 years.
(iv) The duration of the assets can be expressed as the sum of payments times time
of receipt times present value factors divided by total present value.
The equation for the numerator is
0.04× 9,446.54( )20 Ia + 9,446.54×20×v20 at 3%
( )20 Ia = 141.6761, v20 = 0.55368
Numerator is: 158,141
Therefore the duration is: 158,141/10,852 = 14.6 years.
(v) Duration of the liabilities is 36.1 years. Therefore volatility of the liabilities is:
36.1/1.03 = 35. If there were a reduction in interest rates to 1.5%, the liabilities
would increase in value by approximately 35× 1.5 = 52.5%
Subject CT1 (Financial Mathematics Core Technical) — September 2007 — Examiners’ Report
Page 11
Duration of the assets is 14.6 years. Therefore volatility of the assets is:
14.6/1.03 = 14.2. If there were a reduction in interest rates to 1.5%, the assets
would increase in value by approximately 14.2× 1.5 = 21.3%.
The liabilities would increase in value by an additional 31.2% of their original
value i.e. by £3,386 more than the value of the assets.
END OF EXAMINERS’ REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
15 April 2008 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 10 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
© Faculty of Actuaries
CT1 A2008 © Institute of Actuaries
CT1 A2008—2
1 An eleven month forward contract is issued on 1 March 2008 on a stock with a price
of £10 per share at that date. Dividends of 50 pence per share are expected to be paid
on 1 April and 1 October 2008.
Calculate the forward price at issue, assuming a risk-free rate of interest of 5% per
annum effective and no arbitrage. [4]
2 Describe the characteristics of the following investments:
(a) Eurobonds
(b) Certificates of deposit [4]
3 A mortgage company offers the following two deals to customers for twenty-five year
mortgages.
Product A
A mortgage of £100,000 is offered with level repayments of £7,095.25 made annually
in arrear. There are no arrangement or exit fees.
Product B
A mortgage of £100,000 is offered whereby a monthly payment in advance is
calculated such that the customer pays an effective rate of return of 4% per annum
ignoring arrangement and exit fees. In addition the customer also has to pay an
arrangement fee of £6,000 at the beginning of the mortgage and an exit fee of £5,000
at the end of the twenty-five year term of the mortgage.
Compare the annual effective rates of return paid by customers on the two products.
[8]
4 A loan of nominal amount £100,000 is to be issued bearing coupons payable quarterly
in arrear at a rate of 7% per annum. Capital is to be redeemed at 108% on a coupon
date between 15 and 20 years after the date of issue, inclusive. The date of redemption
is at the option of the borrower.
An investor who is liable to income tax at 25% and capital gains tax at 35% wishes to
purchase the entire loan at the date of issue.
Calculate the price which the investor should pay to ensure a net effective yield of at
least 5% per annum. [8]
CT1 A2008—3 PLEASE TURN OVER
5 The n –year spot rate of interest, in , is given by:
in = a - bn
for n =1,2 and 3, and where a and b are constants.
The one-year forward rates applicable at time 0 and at time 1 are 6.1% per annum
effective and 6.5% per annum effective respectively. The 4–year par yield is 7% per
annum.
Stating any assumptions:
(i) calculate the values of a and b. [4]
(ii) calculate the price per £1 nominal at time 0 of a bond which pays annual
coupons of 5% in arrear and is redeemed at 103% after 4 years. [5]
[Total 9]
6 (i) An investor is considering the purchase of an annuity, payable annually in
arrear for 20 years. The first payment is £500. Using a rate of interest of 8%
per annum effective, calculate the duration of the annuity when:
(a) the payments remain level over the term.
(b) the payments increase at a rate of 8% per annum compound. [6]
(ii) Explain why the answer in (i)(b) is higher than the answer in (i)(a). [2]
[Total 8]
7 The shares of a company currently trade at £2.60 each, and the company has just paid
a dividend of 12p per share. An investor assumes that dividends will be paid annually
in perpetuity and will grow in line with a constant rate of inflation. The investor
estimates the assumed inflation rate from equating the price of the share with the
present value of all estimated future gross dividend payments using an effective
interest rate of 6% per annum.
(i) Calculate the investor’s estimation of the effective inflation rate per
annum based on the above assumptions. [4]
(ii) Suppose that the actual inflation rate turns out to be 3% per annum effective
over the following twelve years, but that all the investor’s other assumptions
are correct.
Calculate the investor’s real rate of return per annum from purchase to sale, if
she sold the shares after twelve years for £5 each immediately after a dividend
has been paid. You may assume that the investor pays no tax. [6]
[Total 10]
CT1 A2008—4
8 An investor is considering investing in a capital project.
The project requires an outlay of £500,000 at outset and further payments at the end
of each of the first 5 years, the first payment being £100,000 and each successive
payment increasing by £10,000.
The project is expected to provide a continuous income at a rate of £80,000 in the first
year, £83,200 in the second year and so on, with income increasing each year by 4%
per annum compound. The income is received for 25 years.
It is assumed that, at the end of 15 years, a further investment of £300,000 will be
required and that the project can be sold to another investor for £700,000 at the end of
25 years.
(i) Calculate the net present value of the project at a rate of interest of 11% per
annum effective. [9]
(ii) Without doing any further calculations, explain how the net present value
would alter if the interest rate had been greater than 11% per annum effective.
[3]
[Total 12]
9 The force of interest, d(t ) , is a function of time and at any time t, measured in years,
is given by the formula:
( )
0.06 0 4
0.10 0.01 4 7
0.01 0.04 7
t
t t t
t t
= = ??
d = - < = ??
? - <
(i) Calculate the value at time t = 5 of £1,000 due for payment at time t = 10. [5]
(ii) Calculate the constant rate of interest per annum convertible monthly which
leads to the same result as in (i) being obtained. [2]
(iii) Calculate the accumulated amount at time t = 12 of a payment stream, paid
continuously from time t = 0 to t = 4, under which the rate of payment at time t
is ?(t ) =100e0.02t . [6]
[Total 13]
CT1 A2008—5
10 An insurance company holds a large amount of capital and wishes to distribute some
of it to policyholders by way of two possible options.
Option A
£100 for each policyholder will be put into a fund from which the expected annual
effective rate of return from the investments will be 5.5% and the standard deviation
of annual returns 7%. The annual effective rates of return will be independent and
(1+it ) is lognormally distributed, where it is the rate of return in year t. The
policyholder will receive the accumulated investment at the end of ten years.
Option B
£100 will be invested for each policyholder for five years at a rate of return of 6% per
annum effective. After five years, the accumulated sum will be invested for a further
five years at the prevailing five-year spot rate. This spot rate will be 1% per annum
effective with probability 0.2, 3% per annum effective with probability 0.3, 6% per
annum effective with probability 0.2, and 8% per annum effective with probability
0.3. The policyholder will receive the accumulated investment at the end of ten years.
Deriving any necessary formulae:
(i) Calculate the expected value and the standard deviation of the sum the
policyholders will receive at the end of the ten years for each of options A and
B. [17]
(ii) Determine the probability that the sum the policyholders will receive at the
end of ten years will be less than £115 for each of options A and B. [5]
(iii) Comment on the relative risk of the two options from the policyholders’
perspective. [2]
[Total 24]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
Subject CT1 — Financial Mathematics
Core Technical
EXAMINERS’ REPORT
April 2008
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
M A Stocker
Chairman of the Board of Examiners
June 2008
© Faculty of Actuaries
© Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 2
Comments
Comments on solutions presented to individual questions for this April 2008 paper are given
below.
Please note that different answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this. However, candidates may be penalised
where excessive rounding has been used or where insufficient working is shown.
Question 1 Well answered.
Question 2 As has often been the case when words rather than numbers have been
required, this bookwork question was answered poorly.
Question 3 Generally well answered, although some students treated the fees on Product
B paid by the customer as a cost to the mortgage company.
Question 4 Well answered although many candidates’ working was unclear when
performing the CGT test.
Question 5 Part (i) was answered well but in part (ii) many candidates failed to recognise
the need to calculate the 4-year spot rate before calculating the bond price.
Question 6 Part (i) of this question did appear to differentiate between stronger
candidates who often scored very well and weaker candidates who often failed
to score at all. As with many previous diets, many candidates in part (ii) had
difficulty in giving a clear explanation of their results.
Question 7 This question was answered relatively poorly with, particularly in part (ii),
candidates often appearing confused between real and money rates of interest.
Question 8 Most candidates managed to make a reasonable attempt at this question
although marks were often lost in part (i) through a combination of
calculation errors and insufficient working being shown. Candidates generally
made a better attempt at the explanation required in part (ii) when compared
to similar questions both on this paper and in previous diets.
Question 9 Well answered.
Question 10 Part (i) (for Option A) can be done much more simply than by using the
method given in this report but the calculations given would still need to be
done for part (ii). It was disappointing to see many candidates incorrectly
calculate the mean accumulated value for Option B by using the mean rate of
interest. Few candidates brought together the answers from (i) and (ii) to fully
answer part (iii).
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 3
1 The present value of the dividends, I, is:
1 7 ( )
I = 0.5v 12 + 0.5v 12 = 0.5 0.99594 + 0.97194 = 0.98394 calculated at i = 5%
Hence forward price is (again calculated at i = 5%):
( )( )11
10 0.98394 1 12 9.42845
£9.43
F = - + i =
=
2 (a) Eurobonds
?? form of unsecured medium or long-term borrowing
?? issued in a currency other than the issuer's home currency outside the
issuer's home country
?? pay regular interest payments and a final capital repayment at par.
?? issued by large companies, governments and supra-national organisations.
?? yields depend upon the issuer and issue size but will typically be slightly
lower than for the conventional unsecured loan stocks of the same issuer.
?? issuers have been free to add novel features to their issues in order to
make them appeal to different investors.
?? usually issued in bearer form
(b) Certificates of Deposit
?? a certificate stating that some money has been deposited
?? issued by banks and building societies
?? terms to maturity are usually in the range 28 days to 6 months.
?? interest is payable on maturity
?? security and marketability will depend on the issuing bank
?? active secondary market
3 For Product A, the annual rate of return satisfies the equation:
25
25
7,095.25 100,000
14.0939
a
a
=
? =
This equates to the value of 25 a at 5%. Hence the annual effective rate of return is
5%.
For Product B, the annual rate of payment is X such that:
(12)
25 Xa???? =100,000 at 4%
( )
( )
12
25 12 25 1.021537 15.6221 = 15.95855
= 100,000 6, 266.23
15.95855
a i a
d
X
= = ×
? =
????
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 4
The equation of value to calculate the rate of return from Product B is:
( )
25
12 25 6,000 + 5,000v 6,266.23 i a 100,000
d
+ =
Clearly the rate of return must be greater than 4%. Try 5%.
LHS = 6,000 + 5,000×0.29530 + 6,266.2335×1.026881×14.0939 = 98,166
At 5% the present value of the payments is less than the amount of the loan at 5% so
the rate of return must be less than 5%. Try 4%:
LHS = 6,000 + 5,000×0.37512 + 100,000 = 107,876
Interpolate between 4% and 5% to get the effective rate of return, i:
0.04 0.01 107,876 100,000 4.81%
107,876 98,166
i ? - ?
= + ? ? ˜ ? - ?
(actual answer is 4.80%)
Therefore Product B charges a lower effective annual return than Product A.
4
( )
( )
4 4
1 1.05 4 0.049089
4
i i
? ?
?? + ?? = ? =
? ?
( ) 1
1 0.07 0.75 0.04861
1.08
g -t = × =
(4) ( )
1 ?i > 1- t g
? Capital gain on contract and we assume loan is redeemed as late as possible (i.e.
after 20 years) to obtain minimum yield.
Let Price of stock = P
(4)
20 P = 0.07×100,000×0.75× a
+(108,000 - 0.35(108,000 - P))v20at 5%
(4) 20
20
20
5250 70, 200
1 0.35
a v
P
v
+
? =
-
5250 1.018559 12.4622 70,200 0.37689
1 0.35 0.37689
× × + ×
=
- ×
= 107,245.38
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 5
5 Assuming no arbitrage.
(i) ( )2 ( )( )
1 0 2 1 1 i = f and 1+ i = 1+ i 1+ f .
Hence a – b = 0.061
?a = b + 0.061
( )2 1 2 1.061 1.065
1 2 1.061 1.065
a b
a b
+ - = ×
? + - = ×
0.002
0.059
b
a
? =-
? =
(ii) Firstly, find the 4-year spot rate. Consider £1 nominal:
1 = 0.07 ( ) 1 2 3 4 4
2 3 4 4
i i i i i v + v + v + v + v
= 0.07 ( ) 4
1.061 1 1.063 2 1.065 3 1.07 4 i - + - + - + ×v
( )4
4 ? 1+ i = 1.31429212
4 ?i = 7.0713% p.a
Let bond price per £1 nominal be P. Then
( ) 1 2 3 4 4
0.05 2 3 4 1.03 4 i i i i i P = v + v + v + v + v
= ( ) 0.05 1.061 1 1.063 2 1.065 3 1.08 1.070713 4 - + - + - + × -
= 0.9545
i.e. 95.45 pence per £1 nominal
6 (i) (a) The duration is:
( )
( )
2 3 20
2 3 20
20
20
500 2 3 20
at 8%
500( )
78.9079 8.037 years
9.8181
v v v v
v v v v
Ia
a
+ + + +
+ + + +
= = =
…
…
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 6
(b) The duration is:
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
2 2 3 19 20
2 2 3 19 20
1
2
500 1.08 2 1.08 3 1.08 20
at 8%
500 1.08 1.08 1.08
1 2 3 20 20 21
10.5 years
20 20
v v v v
v v v v
v
v
? + × + × + + × ?
? ?
? + + + + ?
? ?
+ + + + ×
= = =
…
…
…
(ii) The duration in (i)(b) is higher because the payments increase over time so
that the weighting of the payments is further towards the end of the series.
7 (i) 260 =12(v (1+ e) + v2 (1+ e)2 + v3 (1+ e)3 +......)
where v = 1
1.06
and e denotes inflations rate.
Then,
260 = 12 % where 1 1
1 1
a at j e
8 j i
+
=
+ +
i.e. 0.06
1
j e
e
-
=
+
260 12
0.046153846
0.01324 i.e 1.324% pa
j
j
e
? =
? =
? =
(ii) 260 =12(1.03v +1.032 v2 +.....+1.0312 v12 ) +500v12
12
12 % %
12 500
j i
= a + v where 0.03
1.03
j i -
=
Try i =10%, RHS = 255.67
i = 9%, RHS = 279.35
Hence, 0.09 279.35 260 0.01
279.35 255.67
i -
= + ×
-
= 0.098
Let i' = real return
Then (1+ i')(1+ e) =1+ i
1 1.0982
1.03
? + i' = ?i' = 6.62% pa
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 7
8 (i) Working in £000s
Outlay
Pv = 500 + 90a5 +10(Ia)5 @11%
5
5
1 3.695897
0.11
a v -
= =
( )
5 5
5
5
5 1.11 3.695897 5
0.11 0.11
a v v Ia
- × -
= =
????
= 10.319900
? PV = 500 + 90×3.695897 +10×10.3199
= 935.8297
Income
( ( )2 2 ( )24 24 )
1 1 1 1 PV = 80 a +1.04v a + 1.04 v a +?? ??+ 1.04 v a
( )
( )
25
1
1 1.04
80
1 1.04
v
a
v
? - ?
= ×? ?
?? - ??
where 1
. 0.11 . 1 0.949589
ln1.11 1.11
a i v
d
= = =
? PV = 80×0.949589×12.74554 = 968.2421
PV of cost of further investment
= 300v15 = 62.7013
PV of sale = 700v25 = 51.5257
Hence NPV = 968.2421 + 51.5257 - 935.8297 - 62.7013
= 21.2368 (£21,237)
(ii) If interest > 11% then 1
1+ i
decreases.
? PV of both income and outgo ?
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 8
However, PV of outgo is dominated by initial outlay of £500k at time 0 which
is unaffected.
? PV of income decreases by more than decrease in PV of outgo
? NPV = PV of income – PV of outgo
would reduce (and possibly become negative)
9 (i) ( ) ( ) 10 7
7 5
pv =1,000*exp ???-? 0.01t - 0.04 dt??? *exp ???-? 0.10 - 0.01t dt???
2 10 2 7
7 5
1000 exp 0.01 0.04 exp 0.10 0.01
2 2
t t t t
? ? ? ? ? ? ? ?
= * ?- ? - ? ?* ?- ? - ? ? ? ? ? ? ? ? ? ? ? ? ? ?
1000 exp 0.01 51 0.04 3 exp 0.10 2 0.01 24
2 2
? ? * ? ? ? ? * ? ? = * ?- ? - × ? ?* ?- ? * - ? ? ? ? ? ? ? ? ? ?
=1000*exp(-0.255+ 0.12 - 0.20 + 0.12)
=1000*exp(-0.215)
= 806.54
(ii) Required interest rate p.a. convertible monthly is given by
(12) 12 5
806.54 1 1,000
12
i
× ? ?
?? + ?? =
? ?
?i(12) = 4.3077%p.a. convertible monthly
(iii) Accumulated amount = ( ) ( ) 4 7 12
4 7
4 0.02 0.06 0.10 0.01 0.01 0.04
0
100 t
e t e dr e r dr e r drdt ? × ? × ? - × ? -
[ ]
2 7 2 12 4 0.01 0.01
2 2
4 7
4 0.02 0.06 0.10 0.04
0
100
r r
t
t r r r e e e e dt
? - ? ? - ? = ? × × ?? ?? × ?? ??
4 0.02 (0.24 0.06 ) (0.30 0.165) (0.475 0.200)
0
100 e te t e e dt = ? - - -
= 0.24 0.135 0.275 4 0.04
0
100e e e e- tdt ?
=
0.04 4
0.65
0
100
0.04
e t e
?- - ?
? ?
? ?
= 2,500e0.65 (1- e-0.16 )
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 9
= 2,500 * 1.915540829 * 0.1478562
= 708.06
10 (i) Option A:
(1+ it ) ~ Lognormal (µ,s2 )
ln (1+ it ) ~ N (µ,s2 )
ln (1+ it )10 = ln (1+ it ) + ln (1+ it ) +…+ ln (1+ it ) ~ N (10µ,10s2 )
since it ' s are independent
(1+ it )10 ~ Lognormal (10µ,10s2 )
( )
( ) ( ) ( )
( )
2
2 2 2
2
2 2
2
1 exp 1.055
2
1 exp 2 exp 1 0.07
0.07 exp 1 0.0043928
1.055
t
t
E i
Var i
? s ?
+ = ??µ + ?? =
? ?
+ = µ + s ? s - ? =
? ?
= ? s - ??s =
? ?
exp 0.0043928 1.055 ln1.055 0.0043928 0.051344
2 2
?µ + ? = ?µ = - = ? ?
? ?
10µ = 0.51344 ,10s2 = 0.043928
Let S10 be the accumulation of one unit after 10 years:
( 10 )
exp 0.51344 0.043928 1.70814
2
E S = ?? + ?? =
? ?
Accumulated sum is 100E (S10 ) = £170.81
Option B:
The accumulated sum at the end of five years is:
100×1.065 = 100×1.33823 = £133.823
Subject CT1 (Financial Mathematics Core Technical) — April 2008 — Examiners’ Report
Page 10
The expected value of the accumulated sum at the end of ten years is:
133.823(0.2×1.015 + 0.3×1.035 + 0.2×1.065 + 0.3×1.085 )
133.823(0.2 1.05101 0.3 1.15927 0.2 1.33823 0.3 1.46933)
£169.48
= × + × + × + ×
=
Option A:
( 10 ) exp(2 0.51344 0.043928) exp(0.043928) 1
2.91776 0.04491 0.13103
Var S = × + ?? - ??
= × =
Therefore standard deviation of £100 is 100 0.13103 = £36.20
Option B:
Here we need to find the expected value of the square of the accumulation as
follows:
133.8232 (0.2 1.051012+0.3 1.159272+0.2 1.338232+0.3 1.469332 )
= 29,189.86
× × × ×
The variance of the accumulation is therefore:
29,189.86 -169.482 = £2467.54
and the standard deviation is £21.62
(ii) For option A we require P[S10 <1.15]
P[ln S10 < ln1.15] where ln S10~N(0.51344,0.043928)
(0,1) ln1.15 0.51344
0.043928
P N ? - ?
? ? < ?
? ?
? P ??N (0,1) < -1.7829?? = 0.0373 ˜ 4%
For option B we first examine the lowest payout possible.
There is a probability of 0.2 that the amount will be 100×1.065 ×1.015 or less
which equals133.823×1.05101 = £140.65 . Therefore the probability of a
payment of less than £115 is zero.
(iii) Option A is riskier both from the perspective of having a higher standard
deviation of return and also a higher probability of a very low value.
END OF EXAMINERS’ REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
23 September 2008 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 12 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
© Faculty of Actuaries
CT1 S2008 © Institute of Actuaries
CT1 S2008—2
1 A 91-day government bill is purchased for £95 at the time of issue and is redeemed at
the maturity date for £100. Over the 91 days, an index of consumer prices rises from
220 to 222.
Calculate the effective real rate of return per annum. [3]
2 (i) State the strengths and weaknesses of using the money-weighted rate of return
as opposed to the time-weighted rate of return as a measure of an investment
manager’s skill. [3]
(ii) An investor had savings totalling £41,000 in an account on 1 January 2006.
He invested a further £12,000 in this account on 1 August 2006. The total
value of the account was £45,000 on 31 July 2006 and was £72,000 on 31
December 2007.
Assuming that the investor made no further deposits or withdrawals in relation
to this account, calculate the annual effective time-weighted rate of return for
the period 1 January 2006 to 31 December 2007. [2]
[Total 5]
3 (i) A forward contract with a settlement date at time T is issued based on an
underlying asset with a current market price of B.
The annualised risk-free force of interest applying over the term of the forward
contract is d and the underlying asset pays no income. Show that the
theoretical forward price is given by K = Bed T , assuming no arbitrage. [3]
(ii) An asset has a current market price of 200p, and will pay an income of 10p in
exactly three months’ time.
Calculate the price of a forward contract to be settled in exactly six months,
assuming a risk-free rate of interest of 8% per annum convertible quarterly. [3]
[Total 6]
4 Describe the characteristics of commercial property (i.e. commercial real estate) as an
investment. [5]
5 A bank offers two repayment alternatives for a loan that is to be repaid over ten years.
The first requires the borrower to pay £1,200 per annum quarterly in advance and the
second requires the borrower to make payments at an annual rate of £1,260 every
second year in arrears.
Determine which terms would provide the best deal for the borrower at a rate of
interest of 4% per annum effective. [5]
CT1 S2008—3 PLEASE TURN OVER
6 A pension fund holds an asset with current value £1 million. The investment return
on the asset in a given year is independent of returns in all other years. The annual
investment return in the next year will be 7% with probability 0.5 and 3% with
probability 0.5. In the second and subsequent years, annual investment returns will be
2%, 4% or 6% with probability 0.3, 0.4 and 0.3, respectively.
(i) Calculate the expected accumulated value of the asset after 10 years, showing
all steps in your calculations. [3]
(ii) Calculate the standard deviation of the accumulated value of the asset after 10
years, showing all steps in your calculations. [4]
(iii) Without doing any further calculations explain how the mean and variance of
the accumulation would be affected if the returns in years 2 to 10 were 1%,
4%, or 7%, with probability 0.3, 0.4 and 0.3 respectively. [2]
[Total 9]
7 The force of interest,d(t) , is a function of time and at any time t (measured in years)
is given by
0.05 0.02 for 0 5
( )
0.15 for 5
t t
t
t
? + = =
d = ? > ?
(i) Calculate the present value of £1,000 due at the end of 12 years. [5]
(ii) Calculate the annual effective rate of discount implied by the transaction in (i).
[2]
[Total 7]
8 A tax advisor is assisting a client in choosing between three types of investment. The
client pays tax at 40% on income and 40% on capital gains.
Investment A requires the investment of £1m and provides an income of £0.1m per
year in arrears for ten years. Income tax is deducted at source. At the end of the ten
years, the investment of £1m is returned.
In Investment B, the initial sum of £1m accumulates at the rate of 10% per annum
compound for ten years. At the end of the ten years, the accumulated value of the
investment is returned to the investor after deduction of capital gains tax.
Investment C is identical to Investment B except that the initial sum is deemed, for tax
purposes, to have increased in line with the index of consumer prices between the date
of the investment and the end of the ten-year period. The index of consumer prices is
expected to increase by 4% per annum compound over the period.
(i) Calculate the net rate of return expected from each of the investments. [7]
(ii) Explain why the expected rate of return is higher for Investment C than for
Investment B and is higher for Investment B than for Investment A. [3]
[Total 10]
CT1 S2008—4
9 Three bonds, paying annual coupons in arrears of 6%, are redeemable at £105 per
£100 nominal and reach their redemption dates in exactly one, two and three years’
time respectively. The price of each of the bonds is £103 per £100 nominal.
(i) Calculate the gross redemption yield of the three-year bond. [3]
(ii) Calculate to three decimal places all possible spot rates, implied by the
information given, as annual effective rates of interest. [4]
(iii) Calculate to three decimal places all possible forward rates, implied by the
information given, as annual effective rates of interest. [4]
[Total 11]
10 An insurance company is considering two possible investment options.
The first investment option involves setting up a branch in a foreign country. This will
involve an immediate outlay of £0.25m, followed by investments of £0.1m at the end
of one year, £0.2m at the end of two years, £0.3m at the end of three years and so on
until a final investment is made of £1m in ten years’ time. The investment will
provide annual payments of £0.5m for twenty years with the first payment at the end
of the eighth year. There will be an additional incoming cash flow of £5m at the end
of the 27th year.
The second investment option involves the purchase of 1 million shares in a bank at a
price of £4.20 per share. The shares are expected to provide a dividend of 21p per
share in exactly one year, 22.05p per share in two years and so on, increasing by 5%
per annum compound. The shares are expected to be sold at the end of ten years, just
after a dividend has been paid, for £5.64 per share.
(i) Determine which of the options has the higher net present value at a rate of
interest of 7% per annum effective. [9]
(ii) Without doing any further calculations, determine which option has the higher
discounted mean term at a rate of interest of 7% per annum effective. [2]
[Total 11]
CT1 S2008—5 PLEASE TURN OVER
11 A company has a liability of £400,000 due in ten years’ time.
The company has exactly enough funds to cover the liability on the basis of an
effective interest rate of 8% per annum. This is also the interest rate on which current
market prices are calculated and the interest rate earned on cash.
The company wishes to hold 10% of its funds in cash, and to invest the balance in the
following securities:
• a zero-coupon bond redeemable at par in twelve years’ time
• a fixed-interest stock which is redeemable at 110% in sixteen years’ time bearing
interest at 8% per annum payable annually in arrear
(i) Calculate the nominal amounts of the zero-coupon bond and the fixed-interest
stock which should be purchased to satisfy Redington’s first two conditions
for immunisation. [10]
(ii) Calculate the amount which should be invested in each of the assets mentioned
in (i). [2]
(iii) Explain whether the company would be immunised against small changes in
the rate of interest if the quantities of stock in part (i) are purchased. [2]
[Total 14]
CT1 S2008—6
12 An individual takes out a 25-year bank loan of £300,000 to purchase a house.
The individual agrees to pay only the interest payments, monthly in arrear, for the first
15 years whereupon he repays half of the capital as a lump sum. He then pays only
the interest for the remaining 10 years, quarterly in arrear, and repays the other half of
the capital as a lump sum at the end of the term.
(i) Calculate the total amount of interest paid by the individual, assuming an
effective rate of interest of 8½% p.a. [5]
(ii) The individual believes that he can earn a nominal rate of interest convertible
half-yearly of 9% p.a. from a separate savings account.
Calculate the level contribution he must make monthly in advance to the
savings account in order to repay half the capital after 15 years. [4]
(iii) The individual made the monthly contributions calculated in (ii) to the savings
account. However, over the first 15 years, the effective rate of return earned
on the savings account was 10% per annum.
The individual used the proceeds at that time to repay as much of the loan as
possible and then decided to repay the remainder of the loan by level
instalments of interest and capital. After the first 15 years, the effective rate of
interest changed to 7% per annum.
Calculate the level payment he must make, payable monthly in arrear, to repay
the loan over the final 10 years of the loan. [5]
[Total 14]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
Subject CT1 — Financial Mathematics
Core Technical
EXAMINERS’ REPORT
September 2008
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
R D Muckart
Chairman of the Board of Examiners
November 2008
© Faculty of Actuaries
© Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 2
Comments
Please note that different answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this. However, candidates may be penalised
where excessive rounding has been used or where insufficient working is shown.
Candidates appeared to be less well prepared than in previous recent diets. As has often been
the case when words rather than numbers have been required, Q4 was answered relatively
poorly despite only involving bookwork with a wide range of available points that could be
made. Many candidates also struggled with the first part of Q2 where explanation rather
than calculation was required. The remainder of the shorter questions were answered well
with candidates scoring particularly highly on Q7.
The more application styled questions (especially Qs 8, 11 and 12) tended to act as a clear
discriminator between stronger and weaker candidates with a significant minority of
candidates scoring very few marks on these questions. By contrast, Q9 on spot and forward
yields was answered relatively well compared to questions in previous diets on this topic.
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 3
1 If j = real rate of return then equation of value in real terms is:
95(1 )91/365 100 220
222
+ j =
(1+ j)91/365 =1.04315
therefore j = 18.465%
2 (i) MWRR
• Requires less information compared to TWRR
But
• Affected by amount and timing of net cashflows, which may not be in the
manager’s control and less fair measure than TWRR
• More difficult equation to solve than TWRR
• Also: equation may not have unique (or any) solution
(ii) Let TWRR = i
Then
(1 )2 45 72
41 57
1.386392811
17.745% p.a.
i
i
+ = ×
=
? =
3 (i) Consider two portfolios A and B at time 0.
Portfolio A: - buy forward at price of K
- deposit Ke-dT in risk-free asset
Portfolio B: - buy asset at price of B
Then, at maturity, both portfolios have the same value (i.e. hold the underlying
asset).
Thus, by the no-arbitrage principle, both portfolios must have same value at
time 0.
? Ke-dT = B ? K = BedT
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 4
(ii) i = 2% per quarter
? K = 200×(1.02)2 -10×1.02 =197.88
(using K Be T Ce (T t1) ) = d - d -
4 Main characteristics of commercial property investments:
• Many different types of properties available for investment, e.g. offices, shops and
industrial properties.
• Return comes from rental income and from the proceeds on sale.
• Total expected return higher than for gilts
• Rents and capital values are expected to increase broadly with inflation in the long
term
• Neither rental income nor capital values are guaranteed – capital values in
particular can fluctuate in the short term…
• …but rental income more secure than dividends
• Rents and capital values expected to increase when the price level rises (though
the relationship is far from perfect).
• Rental terms are specified in lease agreements. Typically, rents increase every
three to five years, Some leases have clauses which specify upward-only
adjustments of rents.
• Large unit sizes, leading to less flexibility than investment in shares
• Each property is unique…
• …. so can be difficult to value.
• Valuation is expensive, because of the need to employ an experienced surveyor
• Marketability and liquidity are poor because of uniqueness …
• …and because buying and selling incurs high costs.
• Rental income received gross of tax.
• Net rental income may be reduced by maintenance expenses
• There may be periods when the property is unoccupied, and no income is
received.
• The running yield from property investments will normally be higher than that for
ordinary shares.
5 Present value in first case is
(4) 10 1,200 i a 1200 1.024877 8.1109 = £9,975.210
d
× × = × ×
Present value in second case is:
( )
( )
10
2 4 10 2
2
1
2,520 ( ) 2,520
1
v
v v v v
v
-
× + +…+ = × ×
-
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 5
( )
( )
1 0.67556
2,520 0.92456 £10,020.01
1 0.92456
-
= × × =
-
Therefore first option is better for the borrower.
6 (i) Let it = investment return for year t
Then, the expected value of the accumulation (S10 ) is given by (in £ millions):
E (S10 ) = ( )
10
1
1 t
t
E i
=
? ?
?? + ??
? ?
?
( )
10
1
1 t
t
E i
=
=? + using independence
( ( )) 10
1
1 t
t
E i
=
=? +
Now, E (i1 ) = 0.5×(0.07 + 0.03) = 0.05
and for t ? 1, E (it ) = (0.3×0.02 + 0.4×0.04 + 0.3×0.06)
= 0.04
So the expected value of the accumulation is
1.05×1.049 =1.494477 (i.e. £1,494,477)
(ii) The variance of the accumulation is
2 ( ( 2 ) ( )2 )
1,000,000 × E S10 - E S10
where ( ) ( )
10
2 2
10
1
1 t
t
E S E i
=
? ?
= ?? + ??
? ?
?
= ( ) 10
2
1
1 2t t
t
E i i
=
? ?
?? + + ??
? ?
?
= ( ( ) ( )) 10
2
1
1 2 t t
t
E i E i
=
? + + from independence
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 6
Now ( 2 ) ( 2 2 )
E i1 = 0.5× 0.07 + 0.03 = 0.0029
for t ? 1, E (it2 ) = 0.3×0.022 + 0.4×0.042 + 0.3×0.062
= 0.00184
Hence,
( 2 ) ( ) ( )9
E S10 = 1+ 0.1+ 0.0029 × 1+ 0.08 + 0.00184
= 2.238739
Standard deviation of the accumulation is
( )1
1,000,000× 2.238739 -1.4944772 2 = £72,646
(iii) The mean would remain unchanged as the expected rate of return in years 2-10
is unchanged. The variance of the rate in years 2-10 has increased and this will
lead to an increase in the variance of the 10 year accumulation.
7 (i) Discounting from t = 12 to t = 5
( )
[ ]
12
5
12 1.05
5
12,5 exp 0.15
exp 0.15 0.34994
v ds
s e-
= ?- ? ? ?
? ?
= - = =
?
Discounting from t = 5 to t = 0
( ) 5
0
2 5 0.5
0
5,0 exp 0.05 0.02
exp 0.05 0.01 0.60653
v sds
s s e-
= ?- + ? ? ?
? ?
= ?- - ? = = ? ?
?
Hence present value of £1,000 at time t = 12
=1,000v (12,5)v (5,0) =1,000×0.34994×0.60653 = £212.25
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 7
(ii) The annual effective rate of discount is d such that:
( )
1
12
1000 1 12 212.25
1 0.21225 12.117%
d
d
- =
? = - =
8 (i) Investment A: the gross rate of return per annum effective is clearly 10%. The
net return is therefore (1-0.4)×10% = 6%per annum effective.
Investment B: the investment will accumulate to £1m×1.110 = £2.5937mat the
end of the ten years. The equation of value is:
( ) ( )( )
( )
( )
10 10
10
10
1 2.59374 1 0.4 2.59374 -1 1
1.95625 1
1 1.95625
6.94%
i i
i
i
i
- -
-
= + - +
= +
? + =
? =
Investment C: again the investment will accumulate to £2.5937m at the end of
ten years. However, the indexed purchase price is subtracted from the value of
the investment in this case. Thus the equation of value is:
( ) ( )( )
( ) ( ) ( )
( )
( )
10 10 10
10 10 10 10
10
10
1 2.59374 1 0.4 2.59374 -1 1.04 1
2.5937 1 0.4 2.59374 1 0.4 1.04 1
2.14834 1
1 2.14834
7.95%
i i
i i i
i
i
i
- -
- - -
-
= + - × +
= + - × + + × × +
= +
? + =
? =
(ii) All investments give a gross return of 10% per annum effective. Investment B
gives a higher return than A because the tax is deferred until the end of the
investment as capital gains tax is paid and not income tax. [However,
candidates might note that tax is paid on the interest earned by deferral of tax].
Investment C gives a higher return than investment B because the tax is only
paid on the real return over the ten year period which is lower than the
nominal return.
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 8
9 (i) 3
103 = 6a3 +105v
try i = 6%: 3
3 a = 2.6730 v = 0.83962
RHS = 104.1981
try i = 7%: 3
3 a = 2.6243 v = 0.81630
RHS = 101.4573
Using linear interpolation:
( )
( )
104.1981 103
0.06 0.01 0.06437 6.44%
104.1981 101.4573
i
-
= + × = =
-
(ii) Let in = spot yield for term n
Then
103(1+ i1 ) =111?i1 = 7.767%
( ) 1 ( ) 2
103 6 1.07767 111 1 i2 i2 6.736% - - = + + ? =
( ) 1 ( ) 2 ( ) 3
103 6 1.07767 6 1.06736 111 1 i3 i3 6.394% - - - = + + + ? =
(iii) First year forward rate is 7.767% (same as spot rate).
Forward rate from time one to time two is i such that:
1.07767(1+ i) = 1.067362 ?i = 5.715%
Forward rate from time two to time three is i such that:
1.067362 (1+ i) =1.063943 ?i = 5.713%
Forward rate from time one to time three is i such that:
1.07767(1+ i)2 =1.063943 ?i = 5.714%
Forward rate from time zero to two and from time zero to three are the same as
the respective spot rates (no additional marks for this point).
10 (i) NPV of first project in £m is:
( ) ( )
( )
27
27 7 10 0.5 5 0.1 0.25 at 7%
0.5 11.9867 5.3893 5 0.16093 0.1 34.7391 0.25
£0.379
a a v Ia
m
- + - -
= - + × - × -
=
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 9
The NPV of second project in £m is:
2 2 3 9 10 10
10 10
10
0.21 0.21(1.05) 0.21(1.05) 0.21(1.05) 5.64 4.2
0.21 1 1.05 5.64 4.2
1 1.05
v v v v v
v v v
v
+ + + + + -
? - ?
= ?? ?? + - ? - ?
…
v = 0.93458 v10 = 0.50835
Therefore NPV = 1.8055 + 5.64×0.50835- 4.2 = £0.473m
The second project has the higher net present value at 7% per annum effective.
(ii) The second project clearly has a discounted mean term of less then ten years.
However, the discounted mean term of the first project must be greater than
ten years because the undiscounted incoming cash flows are less than the
undiscounted outgoing cash flows after ten years.
11 (i) Working in ‘000s
Let X = Nominal amount of Zero Coupon Bond
Y = Nominal amount of 8% bond
400 10 185.2774 VL = v =
12
16 VA =18.52774 + Xv + 0.08Ya +1.1Yv16
Then, since VA =VL (1st condition)
?166.74966 = 0.39711 X + 0.08×8.8514 Y + 0.32108 Y
?166.74966 = 0.39711X +1.02919Y......(1)
2nd condition is ' '
VA =VL
' 4000 10 1852.7740 VL = v =
' 12 ( )
16 VA =12 X v + 0.08 Ia .Y +1.1*16 Y v16
= 4.76537 X + 0.08*61.1154 Y + 5.13727 Y
?1852.7740 = 4.76537 X +10.0265 Y.....(2)
?148.2429 = 2.32391Y
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 10
from (2) 4.76537 (1)
0.39711
? - * ? ?? ??
Hence Y = 63,790
X = 254,583
(ii) Amount invested in X is 254,583 v12
= 101,098
and amount invested in Y is:
185,277 - 18,528 - 101,098 = 65,651
(iii) The spread of the assets is clearly greater than the spread of the liability
(which is a single point).
Hence, Redington’s 3rd condition is satisfied and the fund is immunised.
12 (i) First 15 years:
Interest paid each month
(12) (12) 12
300,000 where1.085 1
12 12
i ? i ?
= × = ? + ?
? ?
? ?
(12)
0.0068215
12
? i =
? monthly interest = 0.0068215 × 300,000 = £2,046.45
After repayment of £150,000 after 15 years:
Interest paid each quarter
(4) (4) 4
150,000 where1.085 1
4 4
i ? i ?
= × = ? + ?
? ?
? ?
(4)
0.020604
4
? i =
? Quarterly interest = 0.020604 × 150,000 = £3,090.66
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 11
Total interest paid over the 25 years
= (2046.45 × 12 × 15) + (3090.66 × 4 × 10) = £491,987.40
(ii) 150,000 = (6)
30 X s???? @ 4½ %
where X = Amount paid in each 6 month period
( ) ( )
( )
30
6
30 6
1.045 1
s
d
-
???? =
where
1 (6) 6 1
1.045 6
? d ?
= ? - ?
? ?
? ?
?d(6) = 0.043856
Hence
(1.045)30 1
0.043856
150000 150000
62.5985
X
-
= =
? ?
? ?
? ?
= 2396.23
? Monthly contribution = 2396.23
6
= £399.37 per month
(iii) Savings proceeds after 15 years:
(12)
15
10%
12×399.37 s????
where ( )
( )
12
15 12 15
s i s
d
???? = ×
1.0533781 31.7725
33.46845
= ×
=
Hence, savings proceeds
= 4792.44 × 33.46845 = 160,395.56
? Loan o/s after 15 years
= 300,000 - 160,395.56 = 139,604.44
Subject CT1 (Financial Mathematics Core Technical) — September 2008 — Examiners’ Report
Page 12
Let Y = new monthly payment
139,604.44 = (12)
10
7%
12Y a
12 0.07 7.02358
0.06785
= Y ×
?Y = £1,605.50 per month
END OF EXAMINERS’ REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
21 April 2009 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 11 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
© Faculty of Actuaries
CT1 A2009 © Institute of Actuaries
CT1 A2009—2
1 Describe the characteristics of Government Bills. [3]
2 Describe the characteristics of:
(a) an interest-only loan (or mortgage); and
(b) a repayment loan (or mortgage). [4]
3 A loan is to be repaid by an annuity payable annually in arrear. The annuity starts at a
rate of £300 per annum and increases each year by £30 per annum. The annuity is to
be paid for 20 years.
Repayments are calculated using a rate of interest of 7% per annum effective.
Calculate:
(i) The amount of the loan. [3]
(ii) The capital outstanding immediately after the 5th payment has been made. [2]
(iii) The capital and interest components of the final payment. [2]
[Total 7]
4 (i) Explain what is meant by the “no arbitrage” assumption in financial
mathematics. [2]
An investor entered into a long forward contract for £100 nominal of a security eight
years ago and the contract is due to mature in four years’ time. The price per £100
nominal of the security was £94.50 eight years ago and is now £143.00. The risk-free
rate of interest can be assumed to be 5% per annum effective throughout the contract.
(ii) Calculate the value of the contract now if it were known from the outset that
the security will pay coupons of £9 two years from now and £10 three years
from now. You may assume no arbitrage. [5]
[Total 7]
CT1 A2009—3 PLEASE TURN OVER
5 A company’s required return for a particular investment project can be expressed as a
force of interest, d(t). This force of interest is a function of time and at any time t,
measured in years, is given by the formula:
( ) 0.05 0.002 0 5
( ) 0.06 5
t t t
t t
d = + = =
d = <
The expenditure required for this project is a payment of £100,000 at t = 0 and a
further payment of £80,000 at t = 2.
The income received from the project is a payment stream paid continuously from
t = 8 to t = 12 under which the annual rate of payment at time t is £100,000e0.001t .
Calculate the discounted payback period for this project. [8]
6 A pension fund purchased an office block nine months ago for £5 million.
The pension fund will spend a further £900,000 on refurbishment in two months time.
A company has agreed to occupy the office block six months from now. The lease
agreement states that the company will rent the office block for fifteen years and will
then purchase the property at the end of the fifteen year rental period for £6 million.
It is further agreed that rents will be paid quarterly in advance and will be increased
every three years at the rate of 4% per annum compound. The initial rent has been set
at £800,000 per annum with the first rental payment due immediately on the date of
occupation.
Calculate, as at the date of purchase of the office block, the net present value of the
project to the pension fund assuming an effective rate of interest of 8% per annum.
[8]
7 A fund had a value of £150,000 on 1 July 2006. A net cash flow of £30,000 was
received on 1 July 2007 and a further net cash flow of £40,000 was received on 1 July
2008. The fund had a value of £175,000 on 30 June 2007 and a value of £225,000 on
30 June 2008. The value of the fund on 1 January 2009 was £280,000.
(i) Calculate the time-weighted rate of return per annum earned on the fund
between 1 July 2006 and 1 January 2009. [3]
(ii) Calculate the money-weighted rate of return per annum earned on the fund
between 1 July 2006 and 1 January 2009. [4]
(iii) Explain why the time-weighted rate of return is more appropriate than the
money-weighted rate of return when comparing the performance of two
investment managers over the same period of time. [2]
[Total 9]
CT1 A2009—4
8 An insurance company has liabilities consisting of eleven annual payments of £1
million, with the first payment due to be made in 10 years’ time and the last payment
due to be made in 20 years’ time. The rate of interest is 6% per annum effective.
(i) Show that the discounted mean term of these liabilities, to four significant
figures, is 14.42 years. [3]
The insurance company holds two zero-coupon bonds, one paying £X in 10 years’
time and the other paying £Y in 20 years’ time.
(ii) Find values of X and Y such that Redington’s first two conditions for
immunisation from small changes in the rate of interest are satisfied. [6]
(iii) Explain, without making any further calculations, whether you would expect
Redington’s third condition for immunisation to be satisfied for the values of
X and Y calculated in (ii). [2]
[Total 11]
9 Two bonds paying annual coupons of 5% in arrear and redeemable at par have terms
to maturity of exactly one year and two years, respectively.
The gross redemption yield from the 1-year bond is 4.5% per annum effective; the
gross redemption yield from the 2-year bond is 5.3% per annum effective. You are
informed that the 3-year par yield is 5.6% per annum.
Calculate all zero-coupon yields and all one-year forward rates implied by the yields
given above. [12]
10 A loan pays coupons of 11% per annum quarterly on 1 January, 1 April, 1 July and
1 October each year. The loan will be redeemed at 115% on any 1 January from
1 January 2015 to 1 January 2020 inclusive, at the option of the borrower. In addition
to the redemption proceeds, the coupon then due is also paid.
An investor purchased a holding of the loan on 1 January 2005, immediately after the
payment of the coupon then due, at a price which gave him a net redemption yield of
at least 8% per annum effective. The investor pays tax at 30% on income and 25% on
capital gains.
On 1 January 2008 the investor sold the holding, immediately after the payment of the
coupon then due, to a fund which pays no tax. The sale price gave the fund a gross
redemption yield of at least 9% per annum effective.
Calculate the following:
(i) The price per £100 nominal at which the investor bought the loan. [6]
(ii) The price per £100 nominal at which the investor sold the loan. [4]
(iii) The net yield per annum convertible quarterly that was actually obtained by
the investor during the period of ownership of the loan. [5]
[Total 15]
CT1 A2009—5
11 An individual wishes to receive an annuity which is payable monthly in arrears for 15
years. The annuity is to commence in exactly 10 years at an initial rate of £12,000 per
annum. The payments increase at each anniversary by 3% per annum. The individual
would like to buy the annuity with a single premium 10 years from now.
(i) Calculate the single premium required in 10 years’ time to purchase the
annuity assuming an interest rate of 6% per annum effective. [5]
The individual wishes to invest a lump sum immediately in an investment product
such that, over the next 10 years, it will have accumulated to the premium calculated
in (i). The annual effective returns from the investment product are independent and
(1+ it ) is lognormally distributed, where it is the return in the tth year. The expected
annual effective rate of return is 6% and the standard deviation of annual returns is
15%.
(ii) Calculate the lump sum which the individual should invest immediately in
order to have a probability of 0.98 that the proceeds will be sufficient to
purchase the annuity in 10 years’ time. [9]
(iii) Comment on your answer to (ii). [2]
[Total 16]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
Subject CT1 — Financial Mathematics
Core Technical
EXAMINERS’ REPORT
April 2009
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
R D Muckart
Chairman of the Board of Examiners
June 2009
© Faculty of Actuaries
© Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 2
Comments
Please note that different answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this. However, candidates may be penalised
where excessive rounding has been used or where insufficient working is shown.
There were some excellent performances and well-prepared candidates scored well across the
whole paper. However, the comments below on each question concentrate on areas where
candidates could have improved their performance.
Q1, Q2.
As has often been the case when words rather than numbers have been required, these
bookwork questions were answered relatively poorly (although Q2 was answered better than
Q1).
Q3.
Well answered.
Q4.
Defining an arbitrage profit correctly was also acceptable as an answer to (i) although a
description of both possible arbitrage scenarios was required for full marks. Many
candidates performed the calculations well although the methodology being used was not
always clear.
Q5.
The question required an ability to bring together two separate elements of the syllabus and
less well-prepared candidates seemed to struggle with this.
Q6.
This was another question where students scored relatively poorly with many candidates
having difficulty with the income calculation. A common error was to assume that the income
rose by 4% every three years.
Q7.
This was answered much better than questions on the same topic in previous exams.
However, some candidates did confuse the money-weighted and time-weighted rates of
return.
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 3
Q8.
It was particularly disappointing to see many candidates using the wrong formula for DMT
in part (i) but ending their proof with‘=14.42 QED’ in the final line. This suggests a lack of
professionalism, honesty and integrity which are key attributes of the actuarial profession.
Part (ii) was well-answered with various different methods leading to the correct answer.
Q9.
This was the worst-answered question on the paper although it was still possible to score
significant marks by calculating forward rates using the correct formula even if the spot rates
had been calculated incorrectly.
Q10.
Part (i) was answered well but many candidates lost marks in part (ii) by not realising that a
separate test was required to ascertain the worst time to redemption. Many candidates
calculated the annual effective yield rather than the yield per annum convertible quarterly in
part (iii).
Q11.
Many candidates seemed confused as to what to calculate in part (i) and failed to distinguish
between the premium needed in 10 years’ time and the present value of that premium. Part
(ii) was answered well (although some candidates appeared to be short of time at this stage).
Part (iii) was answered very poorly with many candidates not appreciating the effects of the
high variance.
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 4
1 Characteristics of government bills:
• short-dated securities issued by governments to fund their short-term spending
requirements.
• issued at a discount and redeemed at par with no coupon.
• mostly denominated in the domestic currency, although issues can be made in
other currencies.
• yield is typically quoted as a simple rate of discount for the term of the bill
• absolutely secure
• often highly marketable despite being unquoted.
• often used as a benchmark risk-free short-term investment.
2 (a) An interest-only loan requires the borrower only to pay interest on the entire
loan in each time period. The loan does not reduce over time so the interest
remains constant. A separate investment or savings account can be established
in which payments are made to extinguish the whole loan at the end of the
term.
(b) A repayment loan involves level repayments of capital and interest. The first
part of the payment is used to pay interest on any remaining capital. The
remaining part of the payment is then used to repay capital so that the capital
gradually reduces over the term of the loan.
3 (i) 300a20 + 30v (?a)19 at 7%
= ( ) 1
1.07 300 10.594 + 30× ×82.9347 = 5503.47
(ii) Capital outstanding after 5 payments:
( ) 15 15 420a + 30 ?a
= 420×9.1079 + 30×61.5540 = 5671.94
(iii) Cap o/s after 19 payments = 870v @ 7% = £813.08
= Capital in the final payment
Interest in the final payment = 870 – 813.08 = £56.92
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 5
4 (i) The “no arbitrage” assumption means that neither of the following applies:
(a) an investor can make a deal that would give her or him an immediate
profit, with no risk of future loss;
nor
(b) an investor can make a deal that has zero initial cost, no risk of future
loss, and a non-zero probability of a future profit.
(ii) The forward price at the outset of the contract was:
( 10 11 ) ( )12
94.5 - 9v5% -10v5% × 1.05 =149.29
The forward price that should be offered now is:
( 2 3 ) ( )4
143- 9v5% -10v5% × 1.05 =153.39
Hence the value of the contract now is:
( ) 4
153.39 -149.29 v5% = 3.37
Note:
This result can also be obtained directly from:
143- 94.5×(1.05)8 = 3.38
since the coupons are irrelevant in this calculation.
5 Working in £000’s
PV of outgo = 2( )
0 0.05 0.002 100 80 t dt e+ -? +
=
2 2
0
0.05 0.001
100 80
t t
e
-? + ? + ? ?
= 100 +80e-0.104 =172.10
DPP is value of T for which:
PV (income paid up to T) = PV (outgo)
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 6
Where
PV (income paid up to T) = 0.001 ( )
8
100
? T e t v t dt
and v(t)
5( )
0 5 0.05 0.002 0.06 t t dt dt
e
-? + + ? ??? ? ??
=
( )
2 5
0
0.05 0.001 0.06 0.30 .
t t t e e
-? + ? = ? ? - -
= e-0.275. e-0.06t e0.30
= e0.025 e-0.06t
( ) 0.001 0.025 0.06
8
income paid up to 100
PV T T e t e e- tdt ? =?
0.025 0.059
8
100
T e e- tdt = ?
100 0.025 0.059 0.059 8
0.059
= e ?e- T - e- × ? - ? ?
= -1737.8222 e-0.059T +1083.97
?DPP is T such that
172.10 = -1737.8222e-0.059T +1083.97
0.059 0.52472
0.059 (0.52472) 10.93 years
e T
T Ln T
? - =
? - = ? =
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 7
6 Working in 000’s
PV of costs = 11
5000 + 900v 12 at 8%
= 5838.695
PV of income = 3 ( ( ) ( ) ( ) ( ) )
1 12 4 3 3 4 12 12 4
3 3 3 800v a???? +1.04 v a???? +??+ 1.04 v a????
= 3 ( ) ( ( ) ( ) )
1 12 4 3 12
3 800v a???? 1+ 1.04v +?? 1.04v
=
( )
( )
1.04 15
1.08
1.04 3
1.08
1
800 0.908281 1.049519 2.5771
1
? - ? × × × ×? ? ?? ?? ? - ?
= 1965.3133× 4.038121
= 7936.173
PV of proceeds from sale 3
= 6000v16 12 =1717.969
NPV of project = 7936.173+1717.969 – 5838.695
= 3815.447 (i.e. £3,815,447)
7 Working in 000’s
(i) TWRR is i such that
( ) 1
1+ i 2 2 = 175 225 280
150 175 30 225 40
× ×
+ +
175 225 280 1.352968
150 205 265
= × × =
? i =12.85%p.a.
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 8
(ii) MWRR is i such that
( ) ( ) ( ) 1 1 1
150 1+ i 2 2 + 30 1+ i 1 2 + 40 1+ i 2 = 280
Try: i = 12%, LHS = 277.02
i = 12.5%, LHS = 279.58
i = 13%, LHS = 282.16
(28 27.958)
12.5% 0.5%
(28.216 27.958)
i
-
? = + ×
-
= 12.58% p.a.
(iii) The TWRR is better for comparing 2 investment manager’s performances as it
is not sensitive to cash flow amounts and timing of payments. The MWRR is
sensitive to both.
8 (i) Working in £m
Discounted mean term =
10 11 12 20
10 11 12 20
10 11 12 ............... 20
.................
v v v v
v v v v
+ + + +
+ + + +
2 3 11
2 3 11
10 11 12 ............ 20
................
v v v v
v v v v
+ + + +
=
+ + + +
( ) ( ) 11 11 11
11 11
9
9
a a a
a a
+ ? ?
= = + at 6%
( )11 42.7571
9 42.7571 14.42128
7.8869
to 4 significent figures DMT = 14.42
a
DMT
? =
? = + =
?
(ii) First condition: pv assets = pv liabilities
10 20 9
11 ? Xv +Yv = v a *1 at 6%.
X *0.55839 +Y *0.31180 = 0.59190*7.8869 (using tables)
= 4.668256 ………….(1)
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 9
2nd condition: DMT assets = DMT liabilities
10 20
10 20
X *10v Y *20v 14.42128
Xv Yv
+
? =
+
(use of 14.42 from (i) will be
accepted)
? X *5.5839 +Y *6.236 = 14.42128*(Xv10 +Yv20 )
= 14.42128*4.668256 from (1)
= 67.3222 (or 67.3163 if DMT of 14.42 is used)…………(2)
Equn (2) – 10* Equ n (1) ?
Y *6.236 -Y *3.1180 = 67.3222 -10*4.668256
20.639667 6.6195 (or 6.6176 if DMT of 14.42 is used)
3.1180
?Y = =
[or ' '
A L V =V (differentiating with respect to i)
( ( ) )
11 21 11 12 21
10
11 11
10 20 10 11 20
9
Xv Yv v v v
v a Ia
+ = + + +
= +
…
?5.2679X + 5.8831Y = 63.5112 ………….(2)
Equ n (2) – 5.2679
5.8831
× Equ n (1)
? 2.94155Y =19.4711?Y = 6.6193 ]
Equn (1) ? X * 0.55839 = 4.668256 – 6.6195 * 0.31180
? X = 4.6639 (or 4.6650 if DMT of 14.42 is used)
[check, in equ n (2). 4.6639 * 5.5839 + 6.6195 * 6.236 = 67.3222]
(iii) For the third condition to be satisfied, it is necessary for the spread of the
assets to exceed the spread of the liabilities. This appears to be the case given
that the liabilities occur in equal annual amounts at durations from 10 years to
20 years, whereas the assets are concentrated in two lumps at the two most
extreme durations, 10 years and 20 years.
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 10
9 Let the 1-year and 2-year zero-coupon yields (spot rates) be ii and i2 respectively.
1
105 105 @ 4.5%
1
v
i
=
+
?i1 = 0.045
For the 2-year spot rate:
( )
2
2 2 5.3% 5.3%
1 2
5 105 5 100
1 1
a v
i i
+ = +
+ +
( )
2
2 2
2
1 1 5 105 5 1.053 100
1.045 1 i 0.053 1.053
?? - ??
+ = ? ? +
+
= 9.257681 + 90.186858
= 99.444539
( )2
2
105 99.444539 5
1 i 1.045
= -
+
( )2
2
1 105
94.659850
? + i =
?i2 = 5.3202% p.a.
For the 3-year spot rate:
The 3-year par yield is 5.6% p.a.
( )2 ( )3 ( )3
1 2 3 3
1 0.056 1 1 1 1
1 i 1 i 1 i 1 i
? ?
? = ? + + ? +
? + + + ? + ? ?
( )3 ( )2
3
1.056 1 0.056 0.056
1 i 1.045 1.053202
? = - -
+
( )3
3
1 1.056
0.895926
? + i =
?i3 = 5.6324% p.a.
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 11
1-year forward rates:
f0 = i1 = 4.5% p.a.
( )( ) ( )2
1+ i1 1+ f1 = 1+ i2
2
1
1 1.053202
1.045
? + f =
? f1 = 6.1468% p.a.
( )2 ( ) ( )3
1+ i2 1+ f2 = 1+ i3
( )
( )
3
2 2
1.056324
1
1.053202
? + f =
? f2 = 6.2596% p.a.
10 (i) check for capital gain:
( 1 ) ( )
1 0.11 1 0.3
1.15
g - t = * -
= 0.06696
i = 8% ? i(4) = 0.077706
(4) ( )
?i > g 1- t1
? There’s a capital gain and thus loan should be assumed to be redeemed at
the latest possible date.
Let P be price at which the investor bought the loan.
Then
(4) 15 ( ) 15
15 P =11×0.7a +115v - 0.25 115 - P v at 8%
7.7 1.029519 8.5595 0.75 115 0.31524
1 0.25 0.31524
P × × + × ×
? =
- ×
= £103.17 per £100 nominal
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 12
(ii) check for capital gain:
( 1)
1 0.11 0.095652
1.15
g -t = =
( )
( ) ( )
4
4
1
9% 0.087113
1
i i
i g t
= ? =
? < -
? There’s no capital gain and thus loan should be assumed to be redeemed at
the earliest possible date.
Let P' be the price at which the investor sold the loan. Then
(4) 7
7 P' =11a +115v at 9%
=11×1.033144×5.033+115×0.54703
= £120.1064 per £100 nominal
(iii) Let j be the yield per quarter. Then
12
12
103.17 11 0.7 120.1064
4
= × a + v -0.25(120.1064 -103.17)v12 at j %
12
12 ?103.17 =1.925 a +115.8723 v
Try
j = 3%: RHS = 100.4319638
j = 2.5%: RHS = 105.9042724
Linear interpolation:
( )
( )
103.17 105.9042724
0.025 0.005
100.4319638 105.9042724
j
-
= + ×
-
= 0.02749828
Hence, net yield is 11% p.a. (or 10.99931% p.a.) payable quarterly.
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 13
11 (i) In 10 years’ time the single premium P is
P = ( (12) (12) ( )2 (12) 2 ( )14 14 (12) )
1 1 1 1 12000 a +1.03a v + 1.03 a v +...+ 1.03 v a
= ( ) 2 14
12
1
12000 1 1.03 1.03 ... 1.03
1.06 1.06 1.06
a
? ? ? ? ? ? ? + + ? ? + + ? ? ? ? ? ? ? ? ? ? ?
= ( )
15
12
1
1 1.03
12000 1.06 1 1.03
1.06
a
? ? ? ? ? - ? ? ?
? ? ? ?
? ?
? - ? ? ?
? ?
where ( )
( )
12
1 12
a i v
i
=
1.027211 0.969067
1.06
= =
12000 0.969067 0.3499146
0.0283019
? P = × ×
= 143,774.45
(ii) ( ) 2
2 1 1.06 t E i es + = = µ+
( ) ( ) ( ) 1 0.15 2 2 2 . 2 1 Var it e e + = = µ+s s -
Then
( )
2 2
2
0.15 1
1.06
= es -
?s2 = 0.01982706
1.06 0.01982706
2
?µ = ??n -
= 0.04835538
? S10 ~ LN (0.4835538,0.1982706)
Let X be the amount to be invested at time 0
Subject CT1 (Financial Mathematics Core Technical) — April 2009 — Examiners’ Report
Page 14
We want Pr ( X.S10 =143,774.45) = 0.98
so 10
Pr S 143,774.45 0.98
X
? = ? = ? ?
? ?
so
143774.45
2
10
1
10
Ln X ? - µ ?
-F?? ?? ? s ?
= 0.02
143774.45
2
10
2.0537
10
Ln X - µ
? =-
s
So Ln143774.45 2.0537 0.1982706
X
= - × + 0.4835538
= - 0.430909
143774.45 0.6499179
X
? =
? X = £221,219.41
(iii) It might seem odd that the initial investment needs to be substantially higher
than the single premium required in 10 years’ time to have a 98% probability
of accumulating to the single premium.
This strange result is explained by the fact that the variance of the interest rate
is so high relative to the mean. There is therefore a significant risk that the
investment will decrease in value over the next 10 years.
END OF EXAMINERS’ REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
30 September 2009 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 10 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is not required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
© Faculty of Actuaries
CT1 S2009 © Institute of Actuaries
CT1 S2009—2
1 A 182-day government bill, redeemable at £100, was purchased for £96 at the time of
issue and was later sold to another investor for £97.89. The rate of return received by
the initial purchaser was 5% per annum effective.
(a) Calculate the length of time in days for which the initial purchaser held the
bill.
(b) Calculate the annual simple rate of return achieved by the second investor.
[4]
2 List the characteristics of an equity investment. [4]
3 An investor bought a number of shares at 78 pence each on 31 December 2005. She
received dividends on her holding on 31 December 2006, 2007 and 2008. The rate of
dividend per share is given in the table below:
Date Rate of dividend per share Retail price index
31.12.2005
31.12.2006
31.12.2007
31.12.2008
- - - - - -
4.1 pence
4.6 pence
5.1 pence
147.7
153.4
158.6
165.1
On 31 December 2008, she sold her shares at a price of 93 pence per share.
Calculate, using the retail price index values shown in the table, the effective annual
real rate of return achieved by the investor [7]
4 A fixed-interest security has just been issued. The security pays half-yearly coupons
of 5% per annum in arrear and is redeemable at par 20 years after issue.
(i) Calculate the price to provide an investor with a net redemption yield of 6%
per annum effective. The investor pays tax at a rate of 20% on income and is
not subject to capital gains tax. [3]
(ii) Determine the annual effective gross redemption yield of this security
assuming the price calculated in (i) is paid. [5]
(iii) Determine the real annual effective gross redemption yield of this security if
the rate of inflation is constant over the twenty years at 3% per annum. [2]
[Total 10]
CT1 S2009—3 PLEASE TURN OVER
5 The force of interest d(t) at time t is a + bt2where a and b are constants. An amount
of £100 invested at time t = 0 accumulates to £130 at time t = 5 and £200 at time
t = 10.
(i) Calculate the values of a and b. [6]
(ii) Calculate the constant rate of interest per annum convertible monthly that
would give rise to the same accumulation from time t = 0 to time t = 5. [2]
(iii) Calculate the constant force of interest that would give rise to the same
accumulation from time t = 5 to time t = 10. [2]
[Total 10]
6 (i) Distinguish between a future and an option. [2]
An investor wishes to purchase a one year forward contract on a risk-free bond
which has a current market price of £97 per £100 nominal. The bond will pay
coupons at a rate of 7% per annum half yearly. The next coupon payment is
due in exactly six months and the following coupon payment is due just before
the forward contract matures. The six-month risk-free spot interest rate is 5%
per annum effective and the 12-month risk-free spot interest rate is 6% per
annum effective.
(ii) Stating all necessary assumptions:
(a) Calculate the forward price of the bond.
(b) Calculate the six-month forward rate for an investment made in six
months’ time.
(c) Calculate the purchase price of a risk-free bond with exactly one year
to maturity which is redeemed at par and which pays coupons of 4%
per annum half-yearly in arrears.
(d) Calculate the gross redemption yield from the bond in (c).
(e) Comment on why your answer in (d) is close to the one-year spot rate.
[10]
[Total 12]
CT1 S2009—4
7 A member of a pensions savings scheme invests £1,200 per annum in monthly
instalments, in advance, for 20 years from his 25th birthday. From the age of 45, the
member increases his investment to £2,400 per annum. At each birthday thereafter
the annual rate of investment is further increased by £100 per annum. The
investments continue to be made monthly in advance for 20 years until the
individual’s 65th birthday.
(i) Calculate the accumulation of the investment at the age of 65 using a rate of
interest of 6% per annum effective. [6]
At the age of 65, the scheme member uses his accumulated investment to purchase an
annuity with a term of 20 years to be paid half-yearly in arrear. At this time the
interest rate is 5% per annum convertible half-yearly.
(ii) Calculate the annual rate of payment of the annuity. [3]
(iii) Calculate the discounted mean term of the annuity, in years, at the time of
purchase. [3]
[Total 12]
8 A bank offers a customer two different repayment options on a loan of £50,000 as
follows:
Option 1 – level instalments of capital and interest are paid annually in arrear over a
period of 20 years.
Option 2 – over the 20-year term the customer pays only interest on the loan, annually
in arrear at a rate of 5.5% per annum with the whole of the capital amount payable at
the end of the term. The customer will take out a separate savings policy which
involves making monthly payments in advance such that the proceeds will be
sufficient to repay the loan at the end of its term. The payments into the savings
policy accumulate at a rate of interest of 4% per annum effective.
(i) Determine the effective rate of interest per annum that would be paid by the
customer on the loan under Option 1, given that the level annual instalment on
this loan is £4,012.13. [3]
(ii) Determine the annual effective rate of interest paid by a customer under
Option 2. [7]
[Total 10]
CT1 S2009—5 PLEASE TURN OVER
9 A life insurance company is issuing a single premium policy which will pay out
£20,000 in twenty years time. The interest rate the company will earn on the invested
funds over the first ten years of the policy will be 4% per annum with a probability of
0.3 and 6% per annum with a probability of 0.7. Over the second ten years the
interest rate earned will be 5% per annum with probability 0.5 and 6% per annum
with probability 0.5.
(i) Calculate the premium that the company would charge if it calculates the
premium using the expected annual rate of interest in each ten year period. [2]
(ii) Calculate the expected profit to the company if the premium is calculated as in
(i). The rate of interest in the second ten year period is independent of that in
the first ten year period. [3]
(iii) Explain why, despite the company using the expected rate of interest to
calculate the premium, there is a positive expected profit. [2]
(iv) By considering each possible outcome in (ii):
(a) Find the range of possible profits.
(b) Calculate the standard deviation of the profit to the company. [7]
[Total 14]
CT1 S2009—6
10 A group of experts is analysing options to try to avert problems caused by climate
change. They agree on the following expected costs and benefits of climate change
over the next 50 years, starting from the current time. All figures are given in 2009
dollars.
Costs of climate change:
?? Serious events will occur once every three years, in arrear, each giving rise to
costs of $30bn, incurred immediately on the date of the event.
?? Communities affected by climate change will incur costs of $20bn per annum
incurred continuously, increasing at a continuous rate of 1% per annum.
?? Other costs, assumed to be $40bn per annum, will be incurred annually in
arrear.
Benefits arising from climate change:
?? Benefits from higher crop yields and lower heating costs are assumed to be
$10bn per annum, incurred annually in arrear.
The experts are considering whether to recommend investment in a carbon storing
technology which, it is believed, will reduce all the costs and benefits listed above to
zero. The technology requires a one-off investment immediately of $440bn. Costs
are then assumed to be $50bn per annum incurred annually in arrear for 50 years.
The experts do not agree about the appropriate rate of interest at which to evaluate the
options available. One group believes that the net present value of using the carbon
storage technology should be evaluated at a real rate of return of 4% per annum
effective. A second group believe that it should be evaluated at a real rate of return of
1% per annum effective.
(i) Define what is meant by the discounted payback period of an investment and
indicate its main disadvantage as an investment decision criterion. [3]
(ii) Explain why the project must have a discounted payback period when the
interest rate is 1.5% and the internal rate of return is higher than 1.5%. [2]
(iii) Calculate the net present value of the carbon storing technology at a real rate
of interest of 1% per annum effective. [5]
(iv) Calculate the net present value of the carbon storing technology at a real rate
of interest of 4% per annum effective. [5]
`
(v) Comment on whether the investment in the carbon storing technology should
go ahead. [2]
[Total 17]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
Faculty of Actuaries
Institute of Actuaries
Subject CT1 — Financial Mathematics.
Core Technical.
September 2009 examinations
EXAMINERS REPORT
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
R D Muckart
Chairman of the Board of Examiners
December 2009
Comments for individual questions are given with the solutions that follow.
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 2
Please note that different answers may be obtained to those shown in these solutions depending
on whether figures obtained from tables or from calculators are used in the calculations but
candidates are not penalised for this. However, candidates may be penalised where excessive
rounding has been used or where insufficient working is shown.
Well-prepared candidates scored well across the whole paper. However, the comments below on
each question concentrate on areas where candidates could have improved their performance.
1
a.
97.89
96 1.05 97.89 1.05
96
t t
97.89
96 ln
0.400 years or 146 days
ln 1.05
t
b. Second investor held the bill for 36 days. Therefore
36 365 100
97.89 1 100 1 21.854%
365 36 97.89
i i
This was answered well except by the very weakest candidates.
2
Issued by corporations.
Holders entitled to a distribution (dividend) declared from profits.
Potential for high returns relative to other asset classes.
Commensurate risk of capital losses.
Lowest ranking finance issued by companies.
Initial running yield low but has potential to increase with dividend growth.
Dividends and capital values have the potential to grow in nominal terms during times of inflation.
Return made up of income return and capital gains.
Marketability depends on the size of the issue.
Ordinary shareholders receive voting rights in proportion to their holding.
This question was not answered as well as the examiners would have expected given that
the topic is standard bookwork.
3
We convert all cash flow to amounts in time 0 values:
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 3
Dividend paid at
147.7
1:10000 0.041 394.77
153.4
t
Dividend paid at
147.7
2:10000 0.046 428.39
158.6
t
Dividend paid at
147.7
3:10000 0.051 456.25
165.1
t
Sale proceeds at
147.7
3:10000 0.93 8319.87
165.1
t
Equation of value involving v where
1
1
v
r
and r = real rate of return:
2 3 7800 394.77 v 428.39v 8776.17v ..... (1)
[To estimate r:
Approx nominal rate of return is
93 78
4.6 / 78 12.3% p.a.
3
Average inflation over 3 year period comes from
1
165.1 3
1 3.8%
147.7
p.a.
Approx real return:
1.123
1 8.2%p.a.
1.038
]
Try r 8%, RHS of (1) 7699.61
r 7%, RHS of (1) 7907.09
7907.09 7800
7% 1%
7907.69 7699.61
r
= 7.52 % p.a.
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 4
Some candidates seemed to struggle to derive the equation of value based on a real rate
of return and multiplied (rather than divided) the payments by the increase in the
inflation index.
4
(i) Let required price = P:
2 20
20
P 1 0.2 5a 100v at 6%
2 20
20 2 20
0.06
= 11.4699 11.6394; 0.311805
0.059126
i
a a v
i
Therefore
1 0.2 5 11.6394 100 0.311805
46.5576 31.1805 77.7381
P
(ii) The equation of value for the gross rate of return is:
2 20
20
77.7381 5a 100v
If i = 8%
2 20
20 2 20
= 1.019615 9.8181 10.0107; 0.21455
i
a a v
i
RHS = 50.0534 + 21.4550 = 71.5084
If i = 7%
2 20
20 2 20
= 1.017204 10.5940 10.7763; 0.25842
i
a a v
i
RHS = 53.8813 + 25.8420 = 79.7233
Interpolating gives
79.7233 77.7381
0.07 0.01 7.24% 7.2% say
79.7233 71.5084
i
(iii) If the nominal rate of return is 7.2% per annum effective and inflation is 3% per
annum effective, then the real rate of return is calculated from:
1.072
1 4.1%
1.03
This question was answered very well.
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 5
5
(i)
5
5
2 1 3
3 0
0
130 100exp a bt dt 100exp at bt 100exp 5a 41.667b
10
10
2 1 3
3 0
0
200 100exp a bt dt 100exp at bt 100exp 10a 333.333b
ln 1.3 5a 41.667b
ln 2 10a 333.333b
The second expression less twice times the first expression gives:
ln(2) 2ln(1.3) 250b b 0.0006737
ln(2) 333.333 0.0006737
0.04686
10
a
(ii)
1
60
60
12
12 130 12
100 1 130 12 1 5.259% p.a.
12 100
i
i i
(iii) 5 200
130 200 5 ln 8.616% p.a.
130
e
This question was answered very well.
6
(i) A future is a contract which obliges the parties to deliver/take delivery of a
particular quantity of a particular asset at a particular time at a fixed price.
An option is the right to buy or sell a particular quantity of a particular asset at (or
before) a particular time at a given price.
(ii) Assume no arbitrage
a. Buying the forward is exactly the same as buying the bond except that the
forward will not pay coupons and the forward does not require immediate
settlement.
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 6
Let the forward price = F. The equation of value is:
1
2
1.06
97 1.06 3.5 3.5
1.05
102.82 3.62059 3.5 95.6994
F
b. Let six month forward interest rate 1
2
0.5,0.5
1.06
1 3.4454%
1.05
f
This does not have to be expressed as a rate of interest per annum
effective, though it could be.
c.
0.5 1
2 1.05 102 P 1.06 1.9518 96.2264 98.1782
d. Gross redemption yield is i such that
0.5 1
98.1782 2 1 i 102 1 i
Using the formula for solving a quadratic (interpolation will do):
0.5
1 i 0.97133 . Therefore, i ˜ 6% (in fact 5.99%).
e. Answer is very close to 6% (the one-year spot rate) because the payments
from the bond are so heavily weighted towards the redemption time in one
year.
This was generally well-answered apart from part (e). A common error in parts (c) and
(d) was to assume that the coupon payments were 4% per half-year.
7 .
(i) The accumulation is
12 20 12 12 20
20 20 20
1200s?? 1.06 2300s?? 100 Ia?? 1.06
20 20
12 20 20 20
1200 1.06 2300 100 1.06
1,200 36.7856 3.20714 2,300 36.7856
1.032211
100 98.7004 3.20714
1.032211 141,571.88 84,606.88 31,654.60
266,138
i
s s Ia
d
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 7
(ii) Let half-yearly payment = X
40
Xa 266,138 at 2.5%
266,138
10,601.94
25.1028
X
Therefore, annual rate of payment = £21,203.88
(iii) Work in half-years. Discounted mean term is:
2 40 10,601.94 v + 2v +?+40v /266,138
Numerator =
40
10,601.94 Ia at 2.5% per half year effective.
10,601.94 433.3248 4,584,075
Therefore DMT = 17.26 half years or 8.63 years.
In part (i), many candidates developed the correct formula although calculation errors
were common. In such cases, candidates also lost marks for not showing and explaining
their working fully. Part (ii) was answered well but many candidates surprisingly had
trouble calculating the DMT in part (iii). In this part, candidates often lost marks for not
showing the units properly at the end of the answer; indeed, in many cases, showing the
units may well have alerted candidates to possible mistakes.
2
(i) The equation of value for the borrower is
20
4,012.13a 50,000 .
Therefore
20
50,000
a = = 12.4622
4,012.13
From inspection of tables, i = 5%
(ii) The second customer pays interest of 0.055 50,000 = £2,750 per annum, annually in arrear.
The annual rate of monthly payments in advance from the savings policy is X such that:
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 8
12
20
20 12
=50,000 at 4%
50,000
50,000
£1,643.69
29.7781 1.021537
Xs
i
Xs
d
X
??
The equation of value for this borrower is:
12
20 20
20 12 20
50,000 2,750 1,643.686
2,750 1,643.686
a a
i
a a
d
??
Try i = 6%: RHS = 51,002.41
Try i = 7%: RHS = 47,200.14
By interpolation i = 6.3%
Part (i) was well answered but weaker candidates failed to recognise the need to
calculate separately the payments into the savings policy in part (ii).
3
(i) The expected annual interest rate in the first ten years is 0.3 0.04 + 0.7 0.06 =
0.054. The expected interest rate in the second ten years is clearly 5.5%.
If the premium is calculated on the basis of these interest rates, then the premium will be P such
that:
10 10
20,000 1.054 1.055
20,000 2.89022 6,919.89
P
P P
(ii) The expected accumulation factor in the first ten years is:
10 10 0.3 1.04 0.7 1.06 1.69767
The expected accumulation factor in the second ten years is:
10 10
0.5 1.05 1.06 1.70987
As they are independent, we can multiply the accumulation factors together and multiply by the
premium to give an expected accumulation of: 6,919.89 1.69767 1.70987 = 20,087.04.
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 9
The expected profit is 87.04.
(iii) There is an expected profit because (in general) the accumulation of a sum of money at the
expected interest rate is not equal to the expected accumulation when the interest rate is a random
variable.
(iv) The highest possible outcome for the accumulation factor is:
10 10 1.06 1.06 = 3.20714 with probability 0.7 0.5 = 0.35
The lowest possible outcome is:
10 10 1.04 1.05 = 2.41116 with probability 0.3 0.5 = 0.15.
The range is therefore: 6,919.89 (3.20714 – 2.41116) = 5,508.05.
The other two possible outcomes are:
10 10 1.06 1.05 = 2.91710 with probability 0.7 0.5 = 0.35
and
10 10 1.04 1.06 = 2.65089 with probability 0.3 0.5 = 0.15
The mean accumulation factor is: 1.69767 1.70987 = 2.90280
The variance of the accumulation from one unit of investment is:
0.35(3.20714-2.90280)2 + 0.15(2.41116-2.90280)2
+ 0.35 (2.91710-2.90280)2 +0.15 (2.65089-2.90280)2
= 0.03241 + 0.03626 + 0.00007 + 0.00952 = 0.07826.
Standard deviation is 0.07826 = 0.27976.
Standard deviation of the accumulation of the whole premium is: 6,919.89 0.27976 = £1,935.88
which is also the standard deviation of the profit.
This was the worst answered question on the paper with many candidates not recognising
that the accumulation of a sum of money at the expected interest rate is not equal to the
expected accumulation when the interest rate is a random variable. The calculation of the
standard deviation of the accumulation was generally only calculated correctly by the
strongest candidates.
4
(i) The discounted payback period is the first time at which the accumulated profit
from/net present value of the cash flows from a project is positive at a given
interest rate.
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 10
It is an inappropriate decision criterion because it does not tell us anything about
the overall profitability of the project.
(ii) If the internal rate of return were greater than 1.5% then the net present value of
the project at 1.5% must be greater than zero. As such, there must be a discounted
payback period as the discounted payback period is the first time at which the net
present value is greater than zero: such a time must exist.
(iii) Returns are real rates of return and figures are in 2009 dollar terms so we are
automatically working with real rather than nominal values. All figures below are
in $bn.
The net benefits from using the technology are the $30 every three years; $20 incurred
continuously increasing at 1% per annum and $30 per annum incurred annually in arrears.
The costs of the technology are $440 incurred immediately and $50 incurred annually in arrears.
The net present value of the project at 1% per annum effective is:
3 6 48
50 50
30 v v v 50 20 30a 440 50a
The 20 does not need to be discounted because the cash flows are growing at the same rate as they
are being discounted.
48
3
3 50
1
30 560 20
1
v
v a
v
calculated at 1%
1 0.62026
30 0.97059 560 20 39.1961
1 0.97059
= 375.967 560 783.922
152.045
(iv) The net present value of the project at 4% per annum effective is:
3 6 48 '
50 50 50
30 v v v 20a 30a 440 50a
All are calculated at 4% except
'
50
a which is calculated at
1.04
- 1 2.97%
1.01
i
48
3 '
3 50 50
1
30 20 440 20
1
v i
v a a
v
1 0.15219
30 0.88900 20 1.014779 25.8755 440 20 21.4822
1 0.88900
Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report
Page 11
203.704 525.158 440 429.644
140.790
(v) Whether the investment should go ahead would depend on the choice of the interest rate – it is
clearly a crucial assumption (students could make a choice themselves and indicate whether it
should go ahead on the basis of that rate but there must be some justification for the choice).
This question was also poorly answered possibly because project appraisal using real
interest rates has rarely been examined in the past (and also possibly because of time
pressure). Whilst some parts of the question were challenging (e.g. the treatment of the
increasing costs of climate change), it was disappointing that many candidates failed to
recognise that the costs of climate change no longer incurred would be a benefit of the
carbon storing technology project and so failed to score many marks.
END OF EXAMINERS’ REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
27 April 2010 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 11 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is NOT required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
© Faculty of Actuaries
CT1 A2010 © Institute of Actuaries
CT1 A2010—2
1 (i) Explain the difference
(a) between options and futures
(b) between call options and put options
[4]
A security is priced at £60. Coupons are paid half-yearly. The next coupon is due in
two months’ time and will be £2.80. The risk-free force of interest is 6% per annum.
(ii) Calculate the forward price an investor should agree to pay for the security in
three months’ time assuming no arbitrage. [3]
[Total 7]
2 In January 2008, the government of a country issued an index-linked bond with a term
of two years. Coupons were payable half-yearly in arrear, and the annual nominal
coupon rate was 4%. Interest and capital payments were indexed by reference to the
value of an inflation index with a time lag of six months.
A tax-exempt investor purchased £100,000 nominal at issue and held it to redemption.
The issue price was £98 per £100 nominal.
The inflation index was as follows:
Date Inflation Index
July 2007 110.5
January 2008 112.1
July 2008 115.7
January 2009 119.1
July 2009 123.2
(i) Calculate the investor’s cashflows from this investment and state the month
when each cashflow occurs. [3]
(ii) Calculate the annual effective money yield obtained by the investor to the
nearest 0.1% per annum. [3]
[Total 6]
CT1 A2010—3 PLEASE TURN OVER
3 A company issues ordinary shares to an investor who is subject to income tax at 20%.
Under the terms of the ordinary share issue, the investor is to purchase 1,000,000
shares at a purchase price of 45p each on 1 January 2011.
No dividend is expected to be paid for 2 years. The first dividend payable on
1 January 2013 is expected to be 5p per share. Dividends will then be paid every 6
months in perpetuity. The two dividend payments in any calendar year are expected
to be the same, but the dividend payment is expected to increase at the end of each
year at a rate of 3% per annum compound.
Calculate the net present value of the investment on 1 January 2011 at an effective
rate of interest of 8% per annum. [5]
4 An investor is considering purchasing a fixed interest bond at issue which pays halfyearly
coupons at a rate of 6% per annum. The bond will be redeemed at £105 per
£100 nominal in 10 years’ time. The investor is subject to income tax at 20% and
capital gains tax at 25%.
The inflation rate is assumed to be constant at 2.8571% per annum.
Calculate the price per £100 nominal if the investor is to obtain a net real yield of 5%
per annum. [7]
5 Let ft denote the one-year forward rate of interest over the year from time t to time
(t +1) .
The current forward rates in the market are:
time, t 0 1 2 3
one-year forward rate, ft 4.4% p.a. 4.7% p.a. 4.9% p.a. 5.0% p.a.
A fixed-interest security pays coupons annually in arrear at the rate of 7% per annum
and is redeemable at par in exactly four years.
(i) Calculate the price per £100 nominal of the security assuming no arbitrage. [3]
(ii) Calculate the gross redemption yield of the security. [3]
(iii) Explain, without doing any further calculations, how your answer to part (ii)
would change if the annual coupon rate on the security were 9% per annum
(rather than 7% per annum). [2]
[Total 8]
CT1 A2010—4
6 The annual returns, i, on a fund are independent and identically distributed. Each
year, the distribution of 1 + i is lognormal with parameters µ = 0.05 and s2 = 0.004,
where i denotes the annual return on the fund.
(i) Calculate the expected accumulation in 25 years’ time if £3,000 is invested in
the fund at the beginning of each of the next 25 years. [5]
(ii) Calculate the probability that the accumulation of a single investment of £1
will be greater than its expected value 20 years later. [5]
[Total 10]
7 A pension fund has to pay out benefits at the end of each of the next 40 years. The
benefits payable at the end of the first year total £1 million. Thereafter, the benefits
are expected to increase at a fixed rate of 3.8835% per annum compound.
(i) Calculate the discounted mean term of the liabilities using a rate of interest of
7% per annum effective. [5]
The pension fund can invest in both coupon-paying and zero-coupon bonds with a
range of terms to redemption. The longest-dated bond currently available in the
market is a zero-coupon bond redeemed in exactly 15 years.
(ii) Explain why it will not be possible to immunise this pension fund against
small changes in the rate of interest. [2]
(iii) Describe the other practical problems for an institutional investor who is
attempting to implement an immunisation strategy. [3]
[Total 10]
8 A loan is repayable by annual instalments paid in arrear for 20 years. The first
instalment is £4,650 and each subsequent instalment is £150 greater than the previous
instalment.
Calculate the following, using an interest rate of 9% per annum effective:
(i) the amount of the original loan [3]
(ii) the capital repayment in the tenth instalment [4]
(iii) the interest element in the last instalment [2]
(iv) the total interest paid over the whole 20 years [2]
[Total 11]
CT1 A2010—5 PLEASE TURN OVER
9 A company is undertaking a new project. The project requires an investment of £5m
at the outset, followed by £3m three months later.
It is expected that the investment will provide income over a 15 year period starting
from the beginning of the third year. Net income from the project will be received
continuously at a rate of £1.7m per annum. At the end of this 15 year period there
will be no further income from the investment.
Calculate at an effective rate of interest of 10% per annum:
(i) the net present value of the project [3]
(ii) the discounted payback period [4]
A bank has offered to loan the funds required to the company at an effective rate of
interest of 10% per annum. Funds will be drawn from the bank when required and the
loan can be repaid at any time. Once the loan is paid off, the company can earn
interest on funds from the venture at an effective rate of interest of 7% per annum.
(iii) Calculate the accumulated profit at the end of the 17 years. [4]
[Total 11]
10 A pension fund’s assets were invested with two fund managers.
On 1 January 2007 Manager A was given £120,000 and Manager B was given
£100,000. A further £10,000 was invested with each manager on 1 January 2008 and
again on 1 January 2009.
The values of the funds were:
31 December 2007 31 December 2008 31 December 2009
Manager A £130,000 £135,000 £180,000
Manager B £140,000 £145,000 £150,000
(i) Calculate the time-weighted rates of return earned by Manager A and Manager
B over the period 1 January 2007 to 31 December 2009. [4]
(ii) Show that the money-weighted rate of return earned by Manager A over the
period 1 January 2007 to 31 December 2009 is approximately 9.4% per
annum. [2]
(iii) Explain, without performing further calculations, whether the money-weighted
rate of return earned by Manager B over the period 1 January 2007 to
31 December 2009 was higher than, lower than or equal to that earned by
Manager A. [3]
(iv) Discuss the relative performance of the two fund managers. [3]
[Total 12]
CT1 A2010—6
11 The force of interest d(t) is a function of time and at any time t, measured in years, is
given by the formula
0.04 0.02 0 5
( )
0.05 5
t t
t
t
? + = <
d = ? = ?
.
(i) Derive and simplify as far as possible expressions for v(t), where for v(t) is the
present value of a unit sum of money due at time t. [5]
(ii) (a) Calculate the present value of £1000 due at the end of 17 years.
(b) Calculate the rate of interest per annum convertible monthly implied
by the transaction in part (ii)(a). [4]
A continuous payment stream is received at a rate of 10e0.01t units per annum between
t = 6 and t = 10.
(iii) Calculate the present value of the payment stream. [4]
[Total 13]
END OF PAPER
Faculty of Actuaries Institute of Actuaries
EXAMINERS’ REPORT
April 2010 Examinations
Subject CT1 — Financial Mathematics
Core Technical
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
R D Muckart
Chairman of the Board of Examiners
July 2010
© Faculty of Actuaries
© Institute of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 2
Comments
Please note that different answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this. However, candidates may be penalised
where excessive rounding has been used or where insufficient working is shown.
Well-prepared candidates scored well across the whole paper and the examiners were
pleased with the general standard of answers. However, questions that required an element
of explanation or analysis were less well answered than those which just involved
calculation. The comments below concentrate on areas where candidates could have
improved their performance.
Q2.
A common error was to divide the nominal payments by the increase in the index factor
(rather than multiplying).
Q3.
Many candidates made calculation errors in this question but may have scored more marks if
their working had been clearer.
Q6.
Many candidates assumed that the accumulation in part (i) was for a single payment.
Q7.
The calculation was often performed well. In part (ii), many explanations were unclear and
some candidates seemed confused between DMT and convexity although a correct
explanation could involve either of these concepts.
Q9.
A common error was to assume that income only started after three years rather than
‘starting from the beginning of the third year’.
Q10.
This question was answered well but examiners were surprised by the large number of
candidates who used interpolation or other trial and error methods in part (ii) when the
answer had been given in the question. The examiners recommend that students pay attention
to the details given in the solutions to parts (iii) and (iv). For such questions, candidates
should be looking critically at the figures given/calculated and making points specific to the
scenario rather than just making general statements taken from the Core Reading.
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 3
1 (i) (a) Options – holder has the right but not the obligation to trade
Futures – both parties have agreed to the trade and are obliged to do so.
(b) Call Option – right but not the obligation to BUY specified asset at
specified price at specified future date.
Put Option – right but not the obligation to SELL specified asset at
specified price at specified future date.
(ii) 3 1
K = 60e0.06× 12 - 2.80e0.06× 12 = 60.90678 - 2.81404 = £58.09
2 (i) Cash flows:
Issue price: Jan 08 -0.98×100,000 = –£98,000
Interest payments: July 08 0.02 100,000 112.1
110.5
× × = £2,028.96
Jan 09 0.02 100,000 115.7
110.5
× × = £2,094.12
July 09 0.02 100,000 119.1
110.5
× × = £2,155.66
Jan 10 0.02 100,000 123.2
110.5
× × = £2,229.86
Capital redeemed: Jan 10 100,000 123.2
110.5
× = £111,493.21
(ii) Equation of value is:
1 1
2 298000 = 2028.96v + 2094.12v + 2155.66v1 + 2229.86v2 +111493.21v2
At 11%, RHS = 97955.85 ˜ 98000
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 4
3 Purchase price = 0.45 × 1,000,000 = £450,000
PV of dividends = ( ) 1 1 1
50000× 1- 0.2 × ???? v2 + v22 ?? +1.03?? v3 + v32 ?? +1.032 ?? v4 + v42 ?? + ?? ?? ? ? ? ? ? ?
??
=
1
2 40000? v2 + v2 ? ?1+1.03v +1.032v2 + ? ? ? ? ? ? ?
?? @ 8%
= 40000 1.68231 1 1,453,516
1 1.03 1.08
? ?
× ×? ? = ? - ?
? NPV = 1,453,516 – 450,000 = £1,003,516
4 Let i = money yield
?1+ i =1.0285714×1.05 =1.08 ?i = 8%p.a.
Check whether CGT is payable: compare i(2) with (1-t ) g
(1 ) 0.8 6 0.04571
105
- t g = × =
From tables, i(2) = 7.8461% ?i(2) > (1-t ) g
? CGT is payable
P (2) 10 10
10 = 0.8×6a +105v - 0.25(105 - P)v @ 8%
(2) 10
10
10
0.8 6 0.75 105
1 0.25
a v
v
× + ×
=
-
4.8 1.019615 6.7101 78.75 0.46319
1 0.25 0.46319
× × + ×
=
- ×
= £78.39
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 5
5 (i) Let P denote the current price (per £100 nominal) of the security.
Then, we have:
7 7 7 107 108.0872
1.044 1.044 1.047 1.044 1.047 1.049 1.044 1.047 1.049 1.05
P= + + + =
× × × × × ×
(ii) The gross redemption yield, i , is given by:
% 4
4 % 108.09 7 i 100
= ×a + ×vi
Then, we have:
5% 107.0919 0.045 (0.05 0.045) 108.0872 108.9688 0.0473
4.5% 108.9688 107.0919 108.9688
i RHS
i
i RHS
= ? = ? ? - ? ?? ˜ + - ×? ? = = ? = ? ? - ?
(iii) The gross redemption yield represents a weighted average of the forward rates
at each duration, weighted by the cash flow received at that time.
Thus, increasing the coupon rate will increase the weight applied to the cash
flows at the early durations and, as the forward rates are lower at early
durations, the gross redemption yield on a security with a higher coupon rate
will be lower than above.
Note to markers: no marks for simply plugging 9% pa in, and providing no
explanation for result.
6 (i) ( ) 1 2
E 1+ i = eµ+ 2s
1
= e0.05+ 2 × 0.004
=1.0533757
?E[i] = 0.0533757 since E(1+ i) =1+ E(i)
Let A be the accumulation at the end of 25 years of £3,000 paid annually in
advance for 25 years.
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 6
Then E[A] = 3000S????25 at rate j = 0.0533757
(( ) ) ( )
1 25 1
3000 1
j
j
j
+ -
= × +
(1.053375725 1)
3000 1.0533757
0.0533757
-
= ×
= £158,036.43
(ii) Let the accumulation be S20
S20 has a log-normal distribution with parameters 20µ and 20s2
[ ] 1 2
20 2 20
E S20 e ? = µ+ × s
{or (1+ j)20}
= exp(20×0.05+10×0.004)
= e1.04 = 2.829217
In ( 2 )
S20 ~ N 20µ, 20s
?In S20 ~ N (1, 0.08)
Pr (S20 > 2.829217) = Pr (1n S20 >1n 2.829217)
Pr 1n 2.829217-1 where (0,1)
0.08
N ? ?
= ? ? > ? ?
? ?
~
= Pr ( Z > 0.14) =1-F(0.14)
= 1 – 0.55567
= 0.44433 i.e. 44.4%
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 7
7 (i) DMT of liabilities is given by:
( ) ( ) ( )
( ) ( ) ( )
2 2 3 39 40
7% 7% 7% 7%
2 2 3 39 40
7% 7% 7% 7%
1 1 2 1.038835 3 1.038835 40 1.038835
1 1.038835 1.038835 1.038835
v v v v
v v v v
× × + × × + × × + + × ×
× + × + × + + ×
…
…
=
( )
( )
2 3 40
1
2 3 40
1
1.038835 1.038835 2 1.038835 3 1.038835 40 1.038835
1.07 1.07 1.07 1.07
1.038835 1.038835 1.038835 1.038835 1.038835
1.07 1.07 1.07 1.07
-
-
?? ? ? ? ? ? ? ? ? × ?? ? + ×? ? + ×? ? + + ×? ? ?
?? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? × ?? ? + ? ? + ? ? + + ? ?
?? ? ? ? ? ? ? ?
…
…
???
= * * * *
* * * *
2 3 40
2 3 40
2 3 40 i i i i
i i i i
v v v v
v v v v
+ × + × + + ×
+ + + +
…
…
=
( ) *
*
40
40
i
i
Ia
a
where *
*
*
1 1.038835 1.07 1 0.07 0.038835 0.03
i 1 1.07 1.038835 1.038835 v i
i
-
= = ? = - = =
+
.
Hence, DMT of liabilities is:
( )3%
40
3%
40
384.8647 16.65
23.1148
Ia
a
= = years
(Alternative method for DMT formula
2 2 39 39 3% 3% ( )3%
40 40 40
2 2 39 39 3% 3% 3%
40 40 40
(1 2 3 40 ) ( ) ( )
(1 )
v gv g v g v v Ia Ia Ia DMT
v gv gv g v va a a
+ + + +
= = = =
+ + + +
?? ???? ????
?? ???? ????
where g = 1.038835.)
(ii) Even if the fund manager invested entirely in the 15-year zero-coupon bond,
the DMT of the assets will be only 15 years (and, indeed, any other portfolio
of securities will result in a lower DMT).
Thus, it is not possible to satisfy the second condition required for
immunisation (i.e. DMT of assets = DMT of liabilities).
Hence, the fund cannot be immunised against small changes in the rate of
interest.
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 8
(iii) The other problems with implementing an immunisation strategy in practice
include:
• the approach requires a continuous re-structuring of the asset portfolio to
ensure that the volatility of the assets remains equal to that of the liabilities
over time
• for most institutional investors, the amounts and timings of the cash flows
in respect of the liabilities are unlikely to be known with certainty
• institutional investor is only immunised for small changes in the rate of
interest
• the yield curve is unlikely to be flat at all durations
• changes in the term structure of interest rates will not necessarily be in the
form of a parallel shift in the curve (e.g. the shape of the curve can also
change from time to time)
8 (i) Loan = 4500a20 +150(Ia)20 at 9%
?Loan = 4500×9.1285+150×70.9055
= 41,078.25 +10,635.83 = 51,714.08
(ii) Loan o/s after 9th year = ( ) ( ) 11 11 4500 +1350 a +150 Ia at 9%
Loan o/s = 5,850×6.8052 +150×35.0533
= 39,810.42 + 5258.00 = 45,068.42
Repayment = 6000 - 45,068.42× 0.09 = £1,943.84
(Alternative solution to (ii)
(ii) Loan o/s after 9th year = ( ) ( ) 11 11 4500 +1350 a +150 Ia at 9%
= 5,850×6.8052 +150×35.0533 = 45,068.42 as before
Loan o/s after 10th year = ( ) ( ) 10 10 4500 +1500 a +150 Ia at 9%
= 6,000× 6.4177 +150×30.7904 = 43,124.76
Repayment = 45,068.42 - 43,124.76 = £1,943.66 )
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 9
(iii) Last instalment = 4650 +19×150 = 7500
Loan o/s = 1 7500a = 7500v
Interest = 7500×0.91743×0.09 = £619.27
(iv) Total payments 20 4650 1 19 20 150
2
= × + × × ×
= 93,000 + 28,500 =121,500
Total interest = 121,500 – 51,714.08 = £69,785.92
9 (i) NPV = 1
4 2
15 -5 -3v +1.7a v @10%
NPV = 15 -5 - 3×0.976454 +1.7×0.82645× i a @10%
d
= -5- 2.929362 +1.404965×1.049206×7.6061
= -7.929362 +11.21213458
= 3.282772575
NPV = £3.283m
(ii) DPP is t + 2 such that
1
1.7 2 5 3 4 1.474097708 7.929362 @10% t t a v = + v ? a =
1 1.1 5.379129 1 1.1 0.5379129
0.1
t
t
-
- -
= ? - =
?0.4620871=1.1-t ?1n 0.4620871= -t 1n 1.1
?t = 8.100
?DPP =10.1 years
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 10
(iii) Accumulated profit 17 years from start of project:
( 6.9 )
6.9 7%
1.07 1
1.7s 1.7 @7%
-
= = ×
d
(1.076.9 1)
1.7
0.067659
-
= ×
=1.7×8.79346
= £14.95m
10 (i) The values of the funds before and after the cash injections are:
Manager A Manager B
1 January 2007 120,000 100,000
31 December 2007 130,000 140,000 140,000 150,000
31 December 2008 135,000 145,000 145,000 155,000
31 December 2009 180,000 150,000
Thus, TWRR for Manager A is given by:
(1 )3 130 135 180 0.0905 or 9.05%
120 140 145
+ i = × × ?i =
And, TWRR for Manager B is given by:
(1 )3 140 145 150 0.0941 or 9.41%
100 150 155
+ i = × × ?i =
(ii) MWRR for Manager A is given by:
120×(1+ i)3 +10×(1×i)2 +10×(1+ i) =180
Then, putting i = 0.094 gives LHS =180.03 which is close enough to 180.
(iii) Both funds increased by 50% over the three year period and received the same
cashflows at the same times.
Since the initial amount in fund B was lower, the cash inflows received
represent a larger proportion of fund B and hence the money weighted return
earned by fund B over the period will be lower, particularly since the returns
were negative for the 2nd and 3rd years.
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 11
[Could also note that for fund B:
100×(1+ i)3 +10×(1×i)2 +10×(1+ i) =150
So by a proportional argument 120×(1+ i)3 +12×(1×i)2 +12×(1+ i) =180
which when compared with the equation for fund A in (ii) clearly shows that
the return for B is lower.]
(iv) The money weighted rate of return is higher for fund A, whilst the time
weighted return is higher for fund B.
When comparing the performance of investment managers, the time weighted
rate of return is generally better because it ignores the effects of cash inflows
or outflows being made which are beyond the manager’s control.
In this case, Manager A’s best performance is in the final year, when the fund
was at its largest, whilst Manager B’s best performance was in the first year,
where his fund was at its lowest.
Overall, it may be argued that Manager B has performed slightly better than
Manager A since Manager B achieved the higher time weighted return.
11 (i) t < 5
( ) ( ) 0 0.04 0.02 t s ds
v t e
-? + =
=
2
0
0.04 0.01
t
s s
e
-? + ? ? ?
=
0.04t 0.01t2 e
-? + ? ? ?
t = 5
v (t ) =
{ 5( ) }
0 5 0.04 0.02 0.05 t s ds ds
e
- ? + +?
= ( ) 0.05( 5) 5 t v e× -?? - ??
= e 0.45 e 0.05(t 5) e [0.05t 0.2] - × -?? - ?? = - +
Subject CT1 (Financial Mathematics Core Technical) — April 2010 — Examiners’ Report
Page 12
(ii) (a) PV 1,000e [0.05 17 0.2] e 1.05 = - × + = -
= 349.94
(b)
(12) 204
1000 1 349.94
12
i
- ? ?
? + ? =
? ?
? ?
?i(12) = 6.1924%
(iii) [ ] 10 0.45 0.05 0.25 0.01
6
PV = ? e- e- t- 10e tdt
=
0.2 10 0.04
6
10e- e- tdt ?
=
0.04 10
0.2
6
10
0.04
e t e
-
- ? ?
?- ?
?? ??
= 8.18733×2.90769
= 23.806
(Alternative Solution to (iii)
Accumulated value at time t = 10
( [ ] )
10 0.01 10
6
10 0.01 10
6
10 0.01 0.5 0.05 10 0.5 0.04
6 6
0.5 0.04 10
6
10 exp 0.05
10 exp 0.05
10 10
10 276.293 324.233 47.940
0.04
t
t
t
t
t t t
t
e ds dt
e s dt
e e dt e dt
e
- -
-
= ? ? ? ?
? ?
=
= =
? ?
= ? ? = - + =
?? - ??
? ?
?
? ?
Present value = v (10) 47.940 0.63763e [0.05 10 0.25] 47.940 23.806 × = - × - × =
END OF EXAMINERS’ REPORT
Faculty of Actuaries Institute of Actuaries
EXAMINATION
7 October 2010 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 10 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is NOT required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
© Faculty of Actuaries
CT1 S2010 © Institute of Actuaries
CT1 S2010—2
1 A bond pays coupons in perpetuity on 1 June and 1 December each year. The annual
coupon rate is 3.5% per annum. An investor purchases a quantity of this bond on 20
August 2009.
Calculate the price per £100 nominal to provide the investor with an effective rate of
return per annum of 10%. [3]
2 A bond is redeemed at £110 per £100 nominal in exactly four years’ time. It pays
coupons of 4% per annum half-yearly in arrear and the next coupon is due in exactly
six months’ time. The current price is £110 per £100 nominal.
(i) (a) Calculate the gross rate of return per annum convertible half-yearly
from the bond.
(b) Calculate the gross effective rate of return per annum from the bond.
[2]
(ii) Calculate the net effective rate of return per annum from the bond for an
investor who pays income tax at 25%. [2]
[Total 4]
3 The annual rates of return from an asset are independently and identically distributed.
The expected accumulation after 20 years of £1 invested in this asset is £2 and the
standard deviation of the accumulation is £0.60.
(a) Calculate the expected effective rate of return per annum from the asset,
showing all the steps in your working.
(b) Calculate the variance of the effective rate of return per annum.
[6]
4 A six-month forward contract was issued on 1 April 2009 on a share with a price of
700p at that date. It was known that a dividend of 20p per share would be paid on
1 May 2009. The one-month spot, risk-free rate of interest at the time of issue was
5% per annum effective and the forward rate of interest from 1 May to 30 September
was 3% per annum effective.
(i) Calculate the forward price at issue, assuming no arbitrage, explaining your
working. [3]
It has been suggested that the forward price cannot be calculated without making a
judgement about the expected price of the share when the forward contract matures.
(ii) Explain why this statement is not correct. [2]
(iii) Comment on whether the method used in part (i) would still be valid if it was
not known with certainty that the dividend due on 1 May 2009 would be paid.
[1]
[Total 6]
CT1 S2010—3 PLEASE TURN OVER
5 (a) Describe the characteristics of Eurobonds.
(b) Describe the characteristics of convertible bonds.
[6]
6 On 1 January 2001 the government of a particular country bought 200 million shares
in a particular bank for a total price of £2,000 million. The shares paid no dividends
for three years. On 30 June 2004 the shares paid dividends of 10 pence per share. On
31 December 2004, they paid dividends of 20 pence per share. Each year, until the
end of 2009, the dividend payable every 30 June rose by 10% per annum compound
and the dividend payable every 31 December rose by 10% per annum compound. On
1 January 2010, the shares were sold for their market price of £3,500 million.
(i) Calculate the net present value on 1 January 2001 of the government’s
investment in the bank at a rate of interest of 8% per annum effective. [5]
(ii) Calculate the accumulated profit from the government’s investment in the
bank on the date the shares are sold using a rate of interest of 8% per annum
effective. [1]
[Total 6]
7 (i) State the three conditions that are necessary for a fund to be immunised from
small, uniform changes in the rate of interest. [2]
(ii) A pension fund has liabilities of £10m to meet at the end of each of the next
ten years. It is able to invest in two zero-coupon bonds with a term to
redemption of three years and 12 years respectively. The rate of interest is 4%
per annum effective.
Calculate:
(a) the present value of the liabilities of the pension fund
(b) the duration of the liabilities of the pension fund
(c) the nominal amount that should be invested in the zero-coupon bonds
to ensure that the present values and durations of the assets and
liabilities is the same
[7]
(iii) One year later, just before the pension payment then due, the rate of interest is
5% per annum effective.
(a) Determine whether the duration of the assets and the liabilities are still
equal.
(b) Comment on the practical usefulness of the theory of immunisation in
the context of the above result.
[6]
[Total 15]
CT1 S2010—4
8 The force of interest, d(t), is a function of time and at any time t, measured in years, is
given by the formula
( ) 0.05 0.001 0 20
0.05 20
t t
t
t
? + = =
d = ? > ?
(i) Derive and simplify as far as possible expressions for v(t), where v(t) is the
present value of a unit sum of money due at time t. [5]
(ii) (a) Calculate the present value of £100 due at the end of 25 years.
(b) Calculate the rate of discount per annum convertible quarterly implied
by the transaction in part (ii)(a). [4]
(iii) A continuous payment stream is received at rate 30e-0.015t units per annum
between t = 20 and t = 25. Calculate the accumulated value of the payment
stream at time t = 25. [4]
[Total 13]
9 The government of a particular country has just issued three bonds with terms to
redemption of exactly one, two and three years respectively. Each bond is redeemed
at par and pays coupons of 8% annually in arrear. The annual effective gross
redemption yields from the one, two and three year bonds are 4%, 3% and 3%
respectively.
(i) Calculate the one-year, two-year and three-year spot rates of interest at the
date of issue. [8]
(ii) Calculate all possible forward rates of interest from the above spot rates of
interest. [4]
An index of retail prices has a current value of 100.
(iii) Calculate the expected level of the retail prices index in one year, two years’
and three years’ time if the expected real spot rates of interest are 2% per
annum effective for all terms. [5]
(iv) Calculate the expected rate of inflation per annum in each of the next three
years. [2]
[Total 19]
CT1 S2010—5
10 On 1 April 2003 a company issued securities that paid no interest and that were to be
redeemed for £70 after five years. The issue price of the securities was £64. The
securities were traded in the market and the market prices at various different dates
are shown in the table below.
Date Market price
of securities (£)
1 April 2003 64
1 April 2004 65
1 April 2005 60
1 April 2006 65
1 April 2007 68
1 April 2008 70
(i) Explain why the price of the securities might have fallen between 1 April 2004
and 1 April 2005. [1]
Two investors bought the securities at various dates. Investor X bought 100 securities
on 1 April 2003 and 1,000 securities on 1 April 2005. Investor Y bought 100
securities every year on 1 April from 2003 to 2007 inclusive. Both investors held the
securities until maturity.
(ii) Construct a table showing the nominal amount of the securities held and the
market value of the holdings for X and Y on 1 April each year, just before any
purchases of securities. [5]
(iii) (a) Calculate the effective money weighted rate of return per annum for X
for the period from 1 April 2003 to 1 April 2008.
(b) Calculate the effective time weighted rate of return per annum for X
for the period from 1 April 2003 to 1 April 2008.
[6]
(iv) (a) Determine whether the effective money weighted rate of return for Y is
lower or higher than that for X for the period from 1 April 2003 to
1 April 2008.
(b) Determine the effective time weighted rate of return per annum for Y
for the period from 1 April 2003 to 1 April 2008.
[7]
(v) Discuss the relationship between the different rates of return that have been
calculated. [3]
[Total 22]
END OF PAPER
INSTITUTE AND FACULTY OF ACTUARIES
EXAMINERS’ REPORT
September 2010 Examinations
Subject CT1 — Financial Mathematics
Core Technical
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
T J Birse
Chairman of the Board of Examiners
December 2010
© Institute and Faculty of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 2
Comments
Please note that different answers may be obtained from those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this. However, candidates may be penalised
where excessive rounding has been used or where insufficient working is shown. Candidates
also lose marks for not showing their working in a methodical manner which the examiner
can follow. This can particularly affect candidates on the pass/fail borderline when the
examiners have to make a judgement as to whether they can be sure that the candidate has
communicated a sufficient command of the syllabus to be awarded a pass.
The general standard of answers was noticeably lower than in previous sessions and there
were a significant number of very ill-prepared candidates. As in previous exams, questions
that required an element of explanation or analysis were less well answered than those which
just involved calculation.
Comments on individual questions, where relevant, can be found after the solution to each
question. These comments concentrate on areas where candidates could have improved their
performance.
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 3
1 Working in half years:
The present value of the security on 1st June would have been (2)
3.5
i
20 August is 80 days later so the present value is ( ) ( )80
365
2
3.5 1 i
i
+
Hence the price per £100 nominal is ( )80
365 3.5 1.1 £36.611
0.097618
=
2 (i) (a) Gross rate of return convertible half yearly is simply 4/110 = 0.03636
or 3.636%.
(b) Gross effective rate of return is
0.03636 2 1 1
2
? + ? - = ? ?
? ?
0.03669 or 3.669%
(ii) The net effective rate of return per half year is 0.75 0.03636 0.013635
2
× = .
The net effective rate of return per annum is therefore:
(1.013635)2 -1= 0.02746 or 2.746%.
A common error was to divide the nominal payments by the increase in the index factor
(rather than multiplying).
3 (a) Let S20 be the accumulation of the unit investment after 20 years:
E (S20 ) = E ??(1+ i1 )(1+ i2 )…(1+ i20 )??
E (S20 ) = E[1+ i1]E[1+ i2 ]…E[1+ i20 ] as {it}are independent
E[it ] = j ( ) ( )20
? E S20 = 1+ j = 2
1
? j = 2 20 -1 = 3.5265%
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 4
(b) The variance of the effective rate of return per annum is s2 where
[ ] (( ) ) ( ) 2 2 20 40 2 Var Sn = 1+ j + s - 1+ j = 0.6
(( ) ) ( )
( )
1
20
1 1 20 10
2 2 20 2 2
2 2
0.6 1 1
0.6 2 2 0.004628
s j j
? ?
= ? + + ? - +
? ?
= + - =
Many candidates made calculation errors in this question but may have scored more marks if
their working had been clearer.
4 (i) Assuming no arbitrage, buying the share is the same as buying the forward
except that the cash does not have to be paid today and a dividend will be
payable from the share.
Therefore, price of forward is:
( ) ( ) ( ) 1 5 5
700 1.05 12 1.03 12 - 20 1.03 12
= 711.562 – 20.248 = 691.314
(ii) The no arbitrage assumption means that we can compare the forward with the
asset from which the forward is derived and for which we know the market
price. As such we can calculate the price of the forward from this, without
knowing the expected price at the time of settlement. [It could also be
mentioned that the market price of the underlying asset does, of course,
already incorporate expectations].
(iii) If it was not known with certainty that the dividend would be received we
could not use a risk-free interest rate to link the cash flows involved with the
purchase of the forward with all the cash flows from the underlying asset.
5 (a) Eurobonds
• Medium-to-long-term borrowing.
• Pay regular coupon payments and a capital payment at maturity.
• Issued by large corporations, governments or supranational organisations.
• Yields to maturity depend on the risk of the issuer.
• Issued and traded internationally (not in core reading).
• Often have novel features.
• Usually unsecured
• Issued in any currency
• Normally large issue size
• Free from regulation of any one government
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 5
(b) Convertible Securities
• Generally unsecured loan stocks.
• Can be converted into ordinary shares of the issuing company.
• Pay interest/coupons until conversion.
• Provide levels of income between that of fixed-interest securities and equities.
• Risk characteristics vary as the final date for convertibility approaches.
• Generally less volatility than in the underlying share price before conversion.
• Combine lower risk of debt securities with the potential for gains from
equity investment.
• Security and marketability depend upon issuer
• Generally provide higher income than ordinary shares and lower income than
conventional loan stock or preference shares
6 (i) Net present value (all figures in £m)
( )
( )
3 0.5 1.5 2 2.5 5 5.5
3 2 2 3 5 6 9
2,000 0.1 200 1.1 1.1 1.1
0.2 200 1.1 1.1 1.1 3,500
v v v v v
v v v v v v
= - + × × + + + +
+ × × + + + + +
…
…
at 8% per annum effective.
( )( ( ) ( ) ( ) )
( )
2.5 3 2 3 6 9
2.5 3 ' 9
6
2,000 200 0.1 0.2 1.1 1.1 1.1 1.1 3,500
1.1
2,000 200 0.1 0.2 3,500
1.1
v v v v v v v
v v a v
= - + + + + + + +
= - + + +
…
where the annuity is evaluated at a rate of 0.08 0.1 1.818%
1 0.1
-
= -
+
per annum
effective.
( ) 6
'6
1 1 0.018181
6.4011
0.018181
a
- - -
= =
-
and so net present value is
2,000 200 (0.1 1.08 2.5 0.2 1.08 3 ) 6.4011 3,500 1.08 9 £31.66
1.1
- + × - + × - × + × - = m
(ii) Accumulated profit at the time of sale is 31.66×1.089 = £63.30m
Many candidates assumed that the accumulation in part (i) was for a single payment.
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 6
7 (i) The present value of the assets is equal to the present value of the liabilities.
The duration of the assets is equal to the duration of the liabilities.
The spread of the asset terms around the duration is greater than that for the
liability terms (or, equivalently, convexity of assets is greater).
(ii) (a) Present value of liabilities (in £m)
=10a10 at 4% =10×8.1109 = 81.109
(b) Duration is equal to
( )10
10
10 41.9922 at 4% 5.1773 years
10 8.1109
Ia
a
= =
(c) Let the amounts to be invested in the two zero coupon bonds be X and
Y.
3 12
3 12
81.109 (1)
3 12 419.922 (2)
Xv Yv
Xv Yv
+ =
+ =
(2) less 3 times (1) gives:
9 12 176.595
176.595 £31.415
9 0.62460
Yv
Y m
=
? = =
×
Substituting back into (1) gives:
(81.109 31.415 0.62460)
£69.164
0.88900
X m
- ×
= =
(iii) (a) In one year, the present value of the liabilities is:
9 10 +10a at 5% =10 +10×7.1078 = 81.078
Numerator of duration is ( )9 10×0 +10 Ia = 332.347
Duration of liabilities is therefore 332.347 4.0991 years
81.078
=
Present value of assets is:
69.164×v2 + 31.415×v11 = 69.164×0.90703+ 31.415×0.58468
= 81.101
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 7
Duration of assets will be:
2 69.164 2 11 31.415 11
81.101
2 69.164 0.90703 11 31.415 0.58468 4.0383 years
81.101
× ×v + × ×v
× × + × ×
= =
(b) One of the problems of immunisation is that there is a need to
continually adjust portfolios. In this example, a change in the interest
rate means that a portfolio that has a present value and duration equal
to that of the liabilities at the outset does not have a present value and
duration equal to that of the liabilities one year later.
The calculation was often performed well. In part (ii), many explanations were unclear and
some candidates seemed confused between DMT and convexity although a correct
explanation could involve either of these concepts.
8 (i) t = 20 :
( )
2
0
2
0
0.05 0.0005
exp 0.05 0.001
exp 0.05 0.001
2
t
t
t t
v t sds
s s
e- -
= ?- + ? ? ?
? ?
?? ? ? ?? = ?- ? + ? ?
?? ?? ?? ??
=
?
t > 20:
( ) ( )
( ) { [ ] }
20
0 20
20
1.2 1 0.05 0.2 0.05
exp 0.05
20 exp 0.05
t
t
t t
v t s ds ds
v s
e- e - e- -
= ?-? d + ?? ? ? ??
? ? ??
= -
= =
? ?
(ii) (a) PV = 100v (25) =100e-0.2-0.05×25
=100e-1.45 = £23.46
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 8
(b)
( )
( )
4 4 25
100 1 100 25 23.46
4
d v
× ? ?
? - ? = =
? ?
? ?
( ) ( 1 )
?d 4 = 4 1- 0.2346 100 = 0.05758
(iii) PV =
25 0.015 (0.2 0.05 )
20
30e- te- + t dt ?
( )
25 0.2 25 0.2 0.065 0.065
20 20
0.2
1.625 1.3
30 30
0.065
30 28.575
0.065
e e tdt e e t
e e e
-
- - -
-
- -
= = ? ? - ? ?
= - =
-
?
Accumulated value = ( )
28.575 28.575 0.2 0.05 25 28.575 1.45 121.82
25
e e
v
= + × = =
9 (i) The one-year spot rate of interest is simply 4% per annum effective.
For two-year spot rate of interest
First we need to find the price of the security, P:
2
2 P = 8a +100v at 3% per annum effective.
2
2 a =1.91347 v = 0.942596
? P = 8×1.91347 +100×0.942596 =109.5673
Let the t-year spot rate of interest be it.
We already know that i1= 4%. i2 is such that:
( )2
2
109.56736 8 108
1.04 1 i
= +
+
( ) 2
1 i2 0.943287 - ? + =
?i2 = 0.029623 or 2.9623%.
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 9
For three-year spot rate of interest we need to find the price of the security P:
3
P = 8a3 +100v at 3% per annum effective.
3
3 a = 2.8286 v = 0.91514
? P = 8× 2.8286 +100×0.91514 =114.1428
i3 is such that:
( )2 ( )3
3
114.1428 8 8 + 108
1.04 1.029623 1 i
= +
+
( )3
3
108 114.1428 15.23860 98.9042
1 i
? = - =
+
?i3 = 0.02976 or 2.976%.
(ii) The one year forward rate of interest beginning at the present time is clearly
4%.
The forward rate for one year beginning in one year is f1,1 such that:
( ) 2
1.04 1+ f1,1 = 1.029623 ? f1,1 = 0.01935 =1.935%.
The forward rate for one year beginning in two years is f2,1 such that:
2 ( ) 3
1.029623 1+ f2,1 = 1.02976 ? f2,1 = 0.03003 = 3.003%.
The forward rate for two years beginning in one year is f1,2 such that:
3 ( )2
1.02976 =1.04 1+ f1,2
? f1,2 = 0.02468 = 2.468%
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 10
(iii) Let the t-year “spot rate of inflation” be et
For each term
( )
( )
( )
t t
t
t
1 1.02 1 = 1
1 1.02
t t t
t
t
i i e
e
+ ? + ? = ? + ? ?
+ ? ?
( 1 ) 1
1 = 1.04 1.96%
1.02
+ e ?e =
and so the value of the retail price index after one year would be 101.96
( )
2
2
2 2
1 = 1.029623 0.943%
1.02
+ e ?? ?? ?e =
? ?
and so the value of the retail price index after two years would be
100(1.00943)2 =101.90
( )
3
3
3 3
1 = 1.02976 0.9569%
1.02
+ e ?? ?? ?e =
? ?
and so the value of the retail price index after three years would be
100(1.009569)3 =102.90
(iv) The “spot” rates of inflation or the price index values could be used.
Clearly the expected rate of inflation in the first year is 1.96%.
The expected rate of inflation in the second year is:
101.90 101.96 = 0.06%.
101.96
-
-
The expected rate of inflation in the third year is:
102.90 101.90 0.98%
101.90
-
=
A common error was to assume that income only started after three years rather than
“starting from the beginning of the third year”.
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 11
10 (i) The price of the securities might have fallen because interest rates have risen
or because their risk has increased (for example credit risk).
(ii)
Date Market
price of
securities
(£)
X Y
No of
securities
held
before
purchases
Market
value of
holdings
before
purchases
(£)
No of
securities
held
before
purchases
Market
value of
holdings
before
purchases
(£)
1 April 2003 64 – – – –
1 April 2004 65 100 6,500 100 6,500
1 April 2005 60 100 6,000 200 12,000
1 April 2006 65 1,100 71,500 300 19,500
1 April 2007 68 1,100 74,800 400 27,200
1 April 2008 70 1,100 77,000 500 35,000
(iii) (a) Money weighted rate of return is i where:
6, 400(1+ i)5 + 60,000(1+ i)3 = 77,000
try i = 5% LHS = 77,625.70
try i = 4% LHS = 75,278.42
interpolation implies that
0.05 0.01 77,625.70 77,000 4.73%
77,625.70 75, 278.42
i -
= - × =
-
(Note true answer is 4.736%)
(b) Time weighted rate of return is i where using figures in above table:
(1 )5 6,000 77,000 1.09375.
6,400 6,000 60,000
+ i = =
+
?i =1.808%
Subject CT1 (Financial Mathematics Core Technical) — September 2010 — Examiners’ Report
Page 12
(iv) (a) Money weighted rate of return is i where:
6,400(1 )5 6,500(1 )4 6,000(1 )3 6,500(1 )2 6,800(1 )
35,000
+ i + + i + + i + + i + + i
=
Put in i = 4.73%; LHS = 37,026.95
Therefore the money weighted rate of return for Y is less to make LHS
less.
(b) Time weighted rate of return for Y uses the figures in the above table:
(1+ i)5 = 6,500 12,000 19,500 27,200 35,000
6,400 6,500 + 6,500 12,000 + 6,000 19,500 + 6,500 27,200 + 6,800
= 1.09375.
?i =1.808%
(Student may reason that the TWRRs are the same and can be derived
from the security prices in which case, time would be saved.)
(v) The money weighted rate of return was higher for X than for Y because there
was a much greater amount invested when the fund was performing well than
when it was performing badly.
The money weighted rate of return for X (and probably for Y) was more than
the time weighted rate of return because the latter measures the rate of return
that would be achieved by having one unit of money in the fund from the
outset for five years: both X and Y has less in the fund in the years it
performed badly.
This question was answered well but examiners were surprised by the large number of
candidates who used interpolation or other trial and error methods in part (ii) when the
answer had been given in the question. The examiners recommend that students pay attention
to the details given in the solutions to parts (iii) and (iv). For such questions, candidates
should be looking critically at the figures given/calculated and making points specific to the
scenario rather than just making general statements taken from the Core Reading.
END OF EXAMINERS’ REPORT
INSTITUTE AND FACULTY OF ACTUARIES
EXAMINATION
19 April 2011 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 10 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is NOT required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
CT1 A2011 © Institute and Faculty of Actuaries
CT1 A2011—2
1 The force of interest, d(t), is a function of time and at any time t, measured in years, is
given by the formula
0.04 0.003 2 for 0 5 ( )
0.01 0.03 for 5
t t t
t t
?? + < = d = ?
?? + <
(i) Calculate the amount to which £1,000 will have accumulated at t = 7 if it is
invested at t = 3. [4]
(ii) Calculate the constant rate of discount per annum, convertible monthly, which
would lead to the same accumulation as that in (i) being obtained. [3]
[Total 7]
2 A one-year forward contract on a stock is entered into on 1 January 2011 when the
stock price is £68 and the risk-free force of interest is 14% per annum. The stock is
expected to pay an annual dividend of £2.50 with the next dividend due in eight
months’ time.
On 1 April 2011, the price of the stock is £71 and the risk-free force of interest is 12%
per annum. The dividend expectation is unchanged.
Calculate the value of the contract to the holder of the long forward position on
1 April 2011. [6]
3 An investment trust bought 1,000 shares at £135 each on 1 July 2005. The trust
received dividends on its holding on 30 June each year that it held the shares.
The rate of dividend per share was as given in the table below:
30 June
in year
Rate of dividend per
share (£)
Retail price
index
2005
2006
2007
2008
2009
2010
…
7.9
8.4
8.8
9.4
10.1
121.4
125.6
131.8
138.7
145.3
155.2
On 1 July 2010, the investment trust sold its entire holding of the shares at a price of
£151 per share.
(i) Using the retail price index values shown in the table, calculate the real rate of
return per annum effective achieved by the trust on its investment. [6]
(ii) Explain, without doing any further calculations, how your answer to (i) would
alter (if at all) if the retail price index for 30 June 2008 had been greater than
138.7 (with all other index values unchanged). [2]
[Total 8]
CT1 A2011—3 PLEASE TURN OVER
4 The n-year spot rate of interest yn , is given by:
0.03 for 1, 2, 3 and 4
n 1000
y = + n n =
(i) Calculate the implied one-year and two-year forward rates applicable at time
t = 2. [3]
(ii) Calculate, assuming no arbitrage:
(a) The price at time t = 0 per £100 nominal of a bond which pays annual
coupons of 4% in arrear and is redeemed at 115% after 3 years.
(b) The 3-year par yield.
[6]
[Total 9]
5 A loan of nominal amount £100,000 was issued on 1 April 2011 bearing interest
payable half-yearly in arrear at a rate of 6% per annum. The loan is to be redeemed
with a capital payment of £105 per £100 nominal on any coupon date between 20 and
25 years after the date of issue, inclusive, with the date of redemption being at the
option of the borrower.
An investor who is liable to income tax at 20% and capital gains tax of 35% wishes to
purchase the entire loan on 1 June 2011 at a price which ensures that the investor
achieves a net effective yield of at least 5% per annum.
(i) Determine whether the investor would make a capital gain if the investment is
held until redemption. [3]
(ii) Explain how your answer to (i) influences the assumptions made in calculating
the price the investor should pay. [2]
(iii) Calculate the maximum price the investor should pay. [5]
[Total 10]
CT1 A2011—4
6 The value of the assets held by a pension fund on 1 January 2010 was £10 million.
On 30 April 2010, the value of the assets had fallen to £8.5 million. On 1 May 2010,
the fund received a contribution payment of £7.5 million and paid out £2 million in
benefits. On 31 December 2010, the value of the fund was £17.1 million.
(i) Calculate the annual effective money-weighted rate of return (MWRR) for
2010. [3]
(ii) Calculate the annual effective time-weighted rate of return (TWRR) for 2010.
[3]
(iii) Explain why the MWRR is higher than the TWRR for 2010. [2]
The fund manager’s bonus for 2010 is based on the return achieved by the fund over
the year.
(iv) State, with reasons, which of the two rates of return calculated above would be
more appropriate for this purpose. [2]
[Total 10]
7 A loan of £60,000 was granted on 1 July 1998.
The loan is repayable by an annuity payable quarterly in arrear for 20 years. The
amount of the quarterly repayment increases by £100 after every four years. The
repayments were calculated using a rate of interest of 8% per annum convertible
quarterly.
(i) Show that the initial quarterly repayment is £1,370.41. [5]
(ii) Calculate the amount of capital repaid that was included in the payment made
on 1 January 1999. [3]
(iii) Calculate the amount of capital outstanding after the quarterly repayment due
on 1 July 2011 has been made. [4]
[Total 12]
8 A company has liabilities of £10 million due in three years’ time and £20 million due
in six years’ time. The investment manager for the company is able to buy zerocoupon
bonds for whatever term he requires and has adequate monies at his disposal.
(i) Explain whether it is possible for the investment manager to immunise the
fund against small changes in the rate of interest by purchasing a single zerocoupon
bond. [2]
The investment manager decides to purchase two zero-coupon bonds, one for a term
of four years and the other for a term of 20 years. The current interest rate is 4% per
annum effective.
(ii) Calculate the amount that must be invested in each bond in order that the
company is immunised against small changes in the rate of interest. You
should demonstrate that all three Redington conditions are met. [10]
[Total 12]
CT1 A2011—5
9 A company is considering investing in a project. The project requires an initial
investment of three payments, each of £105,000. The first is due at the start of the
project, the second six months later, and the third payment is due one year after the
start of the project.
After 15 years, it is assumed that a major refurbishment of the infrastructure will be
required, costing £200,000.
The project is expected to provide a continuous income stream as follows:
• £20,000 in the second year
• £23,000 in the third year
• £26,000 in the fourth year
• £29,000 in the fifth year
Thereafter the continuous income stream is expected to increase by 3% per annum
(compound) at the start of each year. The income stream is expected to cease at the
end of the 30th year from the start of the project.
(i) Show that the net present value of the project at a rate of interest of 8% per
annum effective is £4,000 (to the nearest £1,000). [7]
(ii) Calculate the discounted payback period for the project, assuming a rate of
interest of 8% per annum effective. [5]
[Total 12]
10 The annual rates of return from a particular investment, Investment A, are
independently and identically distributed. Each year, the distribution of (1+ it ),where
it is the rate of interest earned in year t , is log-normal with parameters µ and s2 .
The mean and standard deviation of it are 0.06 and 0.03 respectively.
(i) Calculate µ and s2. [5]
An insurance company has liabilities of £15m to meet in one year’s time. It currently
has assets of £14m. Assets can either be invested in Investment A, described above,
or in Investment B which has a guaranteed return of 4% per annum effective.
(ii) Calculate, to two decimal places, the probability that the insurance company
will be unable to meet its liabilities if:
(a) All assets are invested in Investment B.
(b) 75% of assets are invested in Investment A and 25% of assets are
invested in Investment B. [6]
(iii) Calculate the variance of return from each of the portfolios in (ii)(a) and
(ii)(b). [3]
[Total 14]
END OF PAPER
INSTITUTE AND FACULTY OF ACTUARIES
EXAMINERS’ REPORT
April 2011 examinations
Subject CT1 — Financial Mathematics
Core Technical
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
T J Birse
Chairman of the Board of Examiners
July 2011
© Institute and Faculty of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 2
General comments
Please note that different answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this. However, candidates may be penalised
where excessive rounding has been used or where insufficient working is shown.
The general performance was slightly worse than in April 2010 but well-prepared candidates
scored well across the whole paper. As in previous diets, questions that required an element
of explanation or analysis, such as Q3(ii) and Q6(iii) were less well answered than those that
just involved calculation. The comments that follow the questions concentrate on areas
where candidates could have improved their performance. Where no comment is made the
question was generally answered well by most candidates.
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 3
1 (i) We want
7 ( )
1000 3
s ds
e
? d
5( 2 ) 7 ( )
3 5 0.04 0.003 0.01 0.03
1000
s ds s ds
e
? + + + ? ??? ? ??
=
where ( ) 5 2 3 5
3 3
? 0.04 + 0.003s ds = ??0.04s + 0.001 s ??
= 0.325 – 0.147 = 0.178
and ( )
7 7 2
5 5
0.01 0.03 0.01 0.03
2
+ s ds = ?? s + s ?? ? ? ?
= 0.805 - 0.425 = 0.380
? accumulation at t = 7 is
1000e(0.178 0.380) 1000e0.558 1,747.17 + = =
(ii)
(12) 4 12
1747.17 1 1000
12
d
× ? ?
? - ? =
? ?
? ?
?d(12) = 0.138692
2 Forward price of the contract is ( ) ( ) 0.14 1
0 0 68 K = S - I edT = - I e ×
where I is the present value of income during the term of the contract = 8
2.5e-0.14× 12
( 8 )
0.14 12 0.14
K0 68 2.5e e 75.59919 ? = - - × =
Forward price a new contract issued at time r (3 months) is
( ) ( ) ( ) 9
* T r 71 0.12 12
Kr Sr I e I e = - d - = - * ×
(where I* is the present value of income during the term of the contract)
( ) 5
0.12 12 0.05 0.09
2.5e K0.25 71 2.5e e 75.08435 = - × ? = - - =
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 4
( ) ( )
( ) 9
12
0
0.12
Value of original contract
75.08435 75.59919
0.47053 47.053
T r
Kr K e
e
p
-d -
- ×
= -
= -
= - = -
Many candidates failed to incorporate the change in the value of d. Another common error
was in counting the number of months.
3 (i) 135,000 7,900 121.4 . 8,400 121.4 2 8,800 121.4 3
125.6 131.8 138.7
= × ?+ × ? + × ?
9,400 121.4 4 (10,100 151,000) 121.4 5
145.3 155.2
+ × ? + + × v
at i'%where i' = real yield
Approx yield:
135,000 = (7635.828 + 7737.178 + 7702.379 + 7853.820 +126015.077) ?5
?i'??3.1% p.a.
Try i' = 3%, RHS =137434.955
Try i' = 3.5%, RHS =134492.919
0.035 0.005 135000 134492.919
137434.955 134492.919
0.03414 (i.e. 3.4% p.a.)
i ' - = - ×
-
=
(ii) The term:
8,800 121.4
RPI (June 2008)
×
would have a lower value (i.e. the dividend paid on 30 June 2008 would have
a lower value when expressed in June 2005 money units). The real yield
would therefore be lower than 3.4% p.a.
The most common error on this question was incorrect use of the indices, e.g. many
candidates inverted them. Several candidates also had difficulty in setting up the equation of
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 5
value. The examiners noted that a large number of final answers were given to excessive
levels of accuracy given the approximate methods used.
4 (i) We can find forward rates f2,1 and f2,2 from:
( )3 ( )2 ( )
1+ y3 = 1+ y2 1+ f2,1 and
( )4 ( )2 ( )2
1+ y4 = 1+ y2 1+ f2,2
( )3 ( )2 ( )
? 1.033 = 1.032 1+ f2,1
? f2,1 = 3.50029% p.a.
and ( )4 ( )2 ( )2
1.034 = 1.032 1+ f2,2
? f2,2 = 3.60039% p.a.
(ii) (a) Price per £100 nominal
( 2 3 ) 3
3.1% 3.2% 3.3% 3.3%
4 v + v + v +115 v
= 4(0.969932 + 0.938946 + 0.907192) +115×0.907192
= 115.59
(b) Let yc3 = 3- year par yield
( 2 3 ) 3
3 3.1% 3.2% 3.3% 3.3%
1 = yc v + v + v + v
1 = yc3 (0.969932 + 0.938946 + 0.907192) + 0.907192
? yc3 = 0.032957
i.e. 3.2957% p.a.
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 6
5 (i)
( )
( )
2 2
1 1.05 2 4.939%
2
i i
? ?
? + ? = ? =
? ?
? ?
(or use tables)
g( 1 )
1 0.06 0.80 0.0457
1.05
- t = × =
So (2) ( )
i > g 1-t1 ?there is a capital gain on the contract
(ii) Since there is a capital gain, the loan is least valuable to the investor if the
repayment is made by the borrower at the latest possible date. Hence, we
assume redemption occurs 25 years after issue in order to calculate the
minimum yield achieved.
(iii) If A is the price per £100 of loan:
( ) ( ) ( ( )) 2 10
12 12 2 24
25 A =100×0.06×0.80 a 1.05 + 105 - 0.35 105 - A v at 5%
( ) ( ( )) 2
4.8 1.012348 14.0939 1.05 12 105 0.35 105 0.29771
Hence 69.0452 20.3187 99.759
1 0.35 0.29771
A
A
= × × × + - - ×
+
= =
- ×
? Price of loan = £99,759
The majority of this question was well-answered but most candidates struggled with the two
month adjustment. This adjustment needs to be directly incorporated into the equation of
value. Calculating the price first without adjustment and then multiplying by (1+i)1/6 will lead
to the wrong answer.
6 (i) MWRR is given by:
( ) ( )8
10.0× 1+ i + 5.5× 1+ i 12 =17.1
Try 11%, LHS = 16.996
Try 12%, LHS =17.132
MMRR 0.11 0.01 17.1 16.996 11.8%p.a.
17.132 16.996
-
= + × =
-
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 7
(ii) TWRR is given by:
8.5 17.1 1 3.821%p.a.
10.0 8.5 5.5
× = + i?i =
+
(iii) MWRR is higher since fund received a large (net) cash flow at a favourable
time (i.e. just before the investment returns increased).
(iv) TWRR is more appropriate. Cash flows into and out of the fund are outside
the control of the fund manager, and should not influence the level of bonus
payable. TWRR is not distorted by amount and/or timing of cash flows
whereas MWRR is.
The calculations in parts (i) and (ii) were generally well done but parts (iii) and (iv) were
poorly answered (or not answered at all) even by many of the stronger candidates. In (iii) for
example, candidates were expected to comment on the timing of the cashflows for this
particular year.
7 (i) Let initial quarterly amount be X. Work in time units of one quarter. The
effective rate of interest per time unit is
0.08 0.02
4
= (i.e.2% per quarter)
So
16 32 48 64
80 64 48 32 16 60,000 = X a +100v a +100v a +100v a +100v a at 2%
(where
64
2%
64
1 35.921415)
0.02
a v -
= =
= 39.7445X + 2,616.695465 +1,627.606705 + 907.1436682 + 382.3097071
60,000 5,533.756
39.7445
£1,370.41 per quarter
X -
? =
=
(ii) Interest paid at the end of the first quarter (i.e. on 1 October 1998) is
60,000×0.02 = £1,200
Hence, capital repaid on 1 October 1998 is
1370.41-1200 = £170.41
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 8
Therefore, interest paid on 1 January 1999 is
(60000 -170.41)×0.02 =1196.59
? capital repaid on 1 January 1999 is
1370.41-1196.59 =173.82
(iii) Loan outstanding at 1 July 2011 (after repayment of instalment)
12
12 16 =1670.41 a +1770.41? a at 2%
=1670.41×10.5753+1770.41×0.78849×13.5777
= £36,619
Candidates found this to be the most challenging question on the paper. The easiest method
was to work in quarters with an effective rate of 2% per quarter. Where candidates worked
using a year as the time period the most common error was to allow for an increase to
payments of £100 pa when the increases were £400pa when they occurred. In part (i), the
examiners were disappointed to see many attempts with incorrect and/or insufficient working
end with the numerical answer that had been given in the question. A candidate who claims
to have obtained a correct answer after making obvious errors in the working is not
demonstrating the required level of skill and judgement and, indeed, is behaving
unprofessionally.
Part (iii) was very poorly answered with surprisingly few candidates recognising the
remaining loan was simply the present value of the last 28 payments.
8 (i) No, because the spread (convexity) of the liabilities would always be greater
than the spread (convexity) of the assets then the 3rd Redington condition
would never be satisfied.
(ii) Work in £millions
Let proceeds from four-year bond = X
Let proceeds from 20-year bond = Y
Require PV Assets = PV Liabilities
X?4 +Y?20 =10?3 + 20?6 (1)
Require DMT Assets = DMT Liabilities
?4X?4 + 20Y?20 = 30?3 +120?6 (2)
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 9
(2) - 4 × (1)
? 16Y?20 = 40?6 - 10?3
6 3
20
40 10 31.61258 8.88996 £3.11175m
16 7.30219
Y ? - ? -
? = = =
?
From (1):
3 6 20
4
10 20 8.88996 15.80629 1.42016 £27.22973m
0.8548042
X Y
? + ? - ? + -
= = =
?
So amount to be invested in 4-year bond is
X?4=£23.27609m
And amount to be invested in 20-year bond is
Y?20 = £1.42016m
Require Convexity of Assets > Convexity of Liabilities
?20X?6 + 420Y?22 >120?5 + 840?8
LHS = 981.869 > 712.411 = RHS
Therefore condition is satisfied and so above strategy will immunise company
against small changes in interest rates.
Or state that spread of assets (t = 4 to t = 20) is greater than spread of
liabilities (t = 3 to t = 6).
Part (i) was poorly answered. In part (ii) many candidates correctly derived X and Y as the
proceeds from the two bonds. However, only the better candidates recognised that the
amounts to be invested (as required by the question) were therefore Xv4 and Yv20.
9 (i) PV of outgo (£000s)
1
105 1 v2 v 200v15 366.31
? ?
? + + ? + =
? ?
? ?
at 8%
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 10
PV of income
( ( ) ( ) ( ) )
( )
2 3 4
1 5 2 24
25 25
2 3 4 5
1
20 23 26 29
29 1.03 1 1.03 1.03 ... 1.03
1 1.03
20 23 26 29 29 1.03
1 1.03
v v v v
a
v v v v
v
a v v v v v
v
? + + + ?
? ?
? + + + + + ?
? ?
? ? - ?? = ? + + + + ×? ??
? ? - ?? ? ? ??
PV of income
{ } 1 = a 80.193+ 20.329×14.996 = 370.61
So NPV is 4.30 (=£4,300)
(ii) The NPV is very small. It is considerably less than the PV of the final year’s
income ( ( )25 29 )
1 29× 1.03 × a ×v = 6.272 ; therefore the DPP must fall in the
final year.
We know the DPP exists as the NPV > 0.
So DPP is 29 + r where
( )24 24
1
1 1.03
366.31 80.193 20.329
1 1.03
v
a
v
?? ? - ??? = ×? + ×? ?? ? ? - ?? ? ? ??
29 1.0325 29 r + × ×v × a at 8%
366.31 364.335 6.5169
0.3031
0.97668 0.307
r
r
r
a
a
v r
? = +
? =
? = ? =
So the DPP is 29.31.
This question tended to separate out the stronger and weaker candidates. The most common
errors in part (i) were discounting for an extra year, not including the one-year annuity
factor and incorrectly calculating the geometric progression. Many candidates also lost
marks through poorly presented or illegible methods that were therefore difficult for the
examiners to follow. Part (ii) was poorly attempted with few candidates completing the
question.
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 11
10 (i)
( )
( )
( )( )
2
2
2 2
2
2
1 1.06
1 0.03 0.0009
1.06 (1)
0.0009 1 (2)
t
t
E i
Var i
e
e e
??µ+s ??
? ?
µ+s s
+ =
+ = =
? =
= -
( )
( ) ( )
2
2 2
2 0.0009 1
1 1.06
? = = es -
( )
( )
2
2
0.0009 1
1.06
0.000800676 and =0.0282962
Ln
? ?
?s = ? + ?
? ?
? ?
= s
( )
0.000800676
1.06 2
1.06 0.000800676
2
0.0578686
e
Ln
?µ+ ? ? ?
? = ? ?
?µ = -
=
(ii) (a) Working in £m. Assets would accumulate to 14×1.04 =14.56 <15
? Probability = 1.00
(b) The guaranteed portion of the fund would accumulate to
0.25×14×1.04 = 3.64.
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, April 2011
Page 12
( )( )
( )
( ( ) )
( )
non-guaranteed portion needs to accumulate to
15 3.64 11.36
we require probability that
0.75 14 1 11.36
Pr 1 1.081905
Pr ln 1 ln 1.081905
ln 1 0.0578686 ln1.081905 0.0578686 Pr
0.0282962 0.0282962
t
t
t
t
i
i
i
i
?
- =
?
× + <
= + <
= + <
+ - -
= <
Pr ( 0.7370169) where (0,1).
0.77
Z Z N
? ?
? ?
? ?
= <
=
~
(iii) (a) Return is fixed (= 4% p.a.) ? variance of return = 0
(b) Return from portfolio = 0.25×0.04 + 0.75 it
?Variance of return = 0.752Var (it )
= 0.752 ×0.0009 = 0.00050625
[In monetary terms the variance of return for (iii)(b) will be
(£14m)2 ×0.00050625 = £299, 225m which is equivalent to a standard
deviation of £315,000]
This question was generally well answered by those candidates who had left enough time to
fully attempt the question. In part (i) the common errors were equating the mean to 0.06
instead of 1.06 and using 0.03 as the variance instead of 0.032. Part (ii) was also well
answered although many candidates quoted the probability of meeting liabilities when the
probability of not meeting the liabilities was asked for. Part (iii) a) was answered well by the
candidates who attempted it, while part b) was not answered well. In part (iii) answers given
in terms of the annual return and in terms of the monetary amounts were both fully
acceptable.
END OF EXAMINERS’ REPORT
INSTITUTE AND FACULTY OF ACTUARIES
EXAMINATION
27 September 2011 (am)
Subject CT1 — Financial Mathematics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 10 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is NOT required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
CT1 S2011 © Institute and Faculty of Actuaries
CT1 S2011—2
1 A 91-day treasury bill is issued by the government at a simple rate of discount of 8%
per annum.
Calculate the annual effective rate of return obtained by an investor who purchases
the bill at issue. [3]
2 State the characteristics of index-linked government bonds. [3]
3 An individual intends to retire on his 65th birthday in exactly four years’ time. The
government will pay a pension to the individual from age 68 of £5,000 per annum
monthly in advance. The individual would like to purchase an annuity certain so that
his income, including the government pension, is £8,000 per annum paid monthly in
advance from age 65 until his 78th birthday. He is to purchase the annuity by a series
of payments made over four years quarterly in advance starting immediately.
Calculate the quarterly payments the individual has to make if the present value of
these payments is equal to the present value of the annuity he wishes to purchase at a
rate of interest of 5% per annum effective. Mortality should be ignored. [6]
4 A pension fund makes the following investments (£m):
1 January 2009 1 July 2009 1 January 2010
1.5 6.0 4.0
The rates of return earned on money invested in the fund were as follows:
1 January 2009 to
30 June 2009
1 July 2009 to
31 December 2009
1 January 2010 to
31 December 2010
1% 2% 5%
Assume that 1 January to 30 June and 1 July to 31 December are precise half-year
periods.
(i) Calculate the time-weighted rate of return per annum effective over the two
years from 1 January 2009 to 31 December 2010. [3]
(ii) Calculate the money-weighted rate of return per annum effective over the two
years from 1 January 2009 to 31 December 2010. [3]
[Total 6]
CT1 S2011—3 PLEASE TURN OVER
5 A nine-month forward contract is issued on 1 March 2011 on a stock with a price of
£9.56 per share at that date. Dividends of 20 pence per share are expected on both
1 April 2011 and 1 October 2011.
(i) Calculate the forward price, assuming a risk-free rate of interest of 3% per
annum effective and no arbitrage. [4]
(ii) (a) Explain why the expected price of the share in nine months’ time is not
needed to calculate the forward price.
(b) Explain why the price of an option would be explicitly dependent on
the variance of the share price but the price of a forward would not be.
[4]
[Total 8]
6 The force of interest, d(t), is a function of time and at any time t, measured in years, is
a +bt where a and b are constants. An amount of £45 invested at time t = 0
accumulates to £55 at time t = 5 and £120 at time t = 10.
(i) Calculate the values of a and b. [5]
(ii) Calculate the constant force of interest per annum that would give rise to the
same accumulation from time t = 0 to time t = 10. [2]
[Total 7]
7 An investment manager is considering investing in the ordinary shares of a particular
company.
The current price of the shares is 12 pence per share. It is highly unlikely that the
share will pay any dividends in the next five years. However, the investment manager
expects the company to pay a dividend of 2 pence per share in exactly six years’ time,
2.5 pence per share in exactly seven years’ time, with annual dividends increasing
thereafter by 1% per annum in perpetuity.
In five years’ time, the investment manager expects to sell the shares. The sale price
is expected to be equal to the present value of the expected dividends from the share at
that time at a rate of interest of 8% per annum effective.
(i) Calculate the effective gross rate of return per annum the investment manager
will obtain if he buys the share and then sells it at the expected price in five
years’ time.
[6]
(ii) Calculate the net effective rate of return per annum the investment manager
will obtain if he buys the share today and then sells it at the expected price in
five years’ time if capital gains tax is payable at 25% on any capital gains. [3]
(iii) Calculate the net effective real rate of return per annum the investment
manager will obtain if he buys the share and then sells it at the expected price
in five years’ time if capital gains tax is payable at 25% on any capital gains
and inflation is 4% per annum effective. There is no indexation allowance. [3]
[Total 12]
CT1 S2011—4
8 (i) State the conditions that are necessary for an insurance company to be
immunised from small, uniform changes in the rate of interest. [2]
An insurance company has liabilities to pay £100m annually in arrear for the next 40
years. In order to meet these liabilities, the insurance company can invest in zero
coupon bonds with terms to redemption of five years and 40 years.
(ii) (a) Calculate the present value of the liabilities at a rate of interest of 4%
per annum effective.
(b) Calculate the duration of the liabilities at a rate of interest of 4% per
annum effective. [5]
(iii) Calculate the nominal amount of each bond that the fund needs to hold so that
the first two conditions for immunisation are met at a rate of interest of 4% per
annum effective. [5]
(iv) (a) Estimate, using your calculations in (ii) (b), the revised present value
of the liabilities if there were a reduction in interest rates by 1.5% per
annum effective.
(b) Calculate the present value of the liabilities at a rate of interest of 2.5%
per annum effective.
(c) Comment on your results to (iv) (a) and (iv) (b). [6]
[Total 18]
9 (i) Describe the information that an investor can obtain from the following yield
curves for government bonds:
(a) A forward rate yield curve.
(b) A spot rate yield curve.
(c) A gross redemption yield curve. [6]
An investor is using the information from a government bond spot yield curve to
calculate the present value of a corporate eurobond with a term to redemption of
exactly five years. The investor will value each payment that is due from the bond at a
rate of interest equal to j = i + 0.01+ 0.001t where:
• t is the time in years at which the payment is due
• i is the annual t-year effective spot rate of interest from the government bond spot
yield curve and i = 0.02t for t = 5
The eurobond pays annual coupons of 10% of the nominal amount of the bond and is
redeemed at par.
(ii) Calculate the present value of the eurobond. [6]
(iii) Calculate the gross redemption yield from the eurobond. [3]
CT1 S2011—5
(iv) Explain why the investor might use such a formula for j to determine the
interest rates at which to value the payments from the corporate eurobond. [3]
[Total 18]
10 A country’s football association is considering whether to bid to host the World Cup
in 2026. Several countries aspiring to host the World Cup will be making bids.
Regardless of whether the bid is successful, the association will incur various costs.
For two years, starting on 1 January 2012, the association will incur costs at a rate of
£2m per annum, assumed to be paid continuously, to prepare the bid.
If the football association is successful, the following costs will be incurred from
1 January 2016 until 31 December 2025:
• One stadium will be built each year for ten years. The first stadium will be built in
2016 and is expected to cost £200m; the stadium built in 2017 is expected to cost
£210m; and so on, with the cost of each stadium rising by 5% each year. The costs
of building each stadium are assumed to be incurred halfway through the relevant
year.
• Administration costs at a rate of £100m per annum will be incurred, payable
monthly in advance from 1 January 2025 until 31 December 2026.
• Revenues from television, ticket receipts, advertising and so on are expected to be
£3,300m and are assumed to be received continuously throughout 2026.
(i) Explain why the payback period is not a good indicator of whether this project
is worthwhile. [3]
The football association decides to judge whether to go ahead with the bid by
calculating the net present value of the costs and revenues from a successful bid on
1 January 2012 at a rate of interest of 4% per annum effective.
(ii) Determine whether the association should make the bid. [13]
The football association is discussing how it might factor into its calculations the fact
that it is not certain to win the right to host the World Cup because other countries are
also bidding.
(iii) Explain how you might adjust the above calculations if the probability of
winning the right to host the World Cup is 0.1 and whether this adjustment
would make it more likely or less likely that the bid will go ahead. [3]
[Total 19]
END OF PAPER
INSTITUTE AND FACULTY OF ACTUARIES
EXAMINERS’ REPORT
September 2011 examinations
Subject CT1 — Financial Mathematics
Core Technical
Purpose of Examiners’ Reports
The Examiners’ Report is written by the Principal Examiner with the aim of helping
candidates, both those who are sitting the examination for the first time and who are using
past papers as a revision aid, and also those who have previously failed the subject. The
Examiners are charged by Council with examining the published syllabus. Although
Examiners have access to the Core Reading, which is designed to interpret the syllabus, the
Examiners are not required to examine the content of Core Reading. Notwithstanding that,
the questions set, and the following comments, will generally be based on Core Reading.
For numerical questions the Examiners’ preferred approach to the solution is reproduced in
this report. Other valid approaches are always given appropriate credit; where there is a
commonly used alternative approach, this is also noted in the report. For essay-style
questions, and particularly the open-ended questions in the later subjects, this report contains
all the points for which the Examiners awarded marks. This is much more than a model
solution – it would be impossible to write down all the points in the report in the time allowed
for the question.
T J Birse
Chairman of the Board of Examiners
December 2011
© Institute and Faculty of Actuaries
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
General comments on Subject CT1
CT1 provides a grounding in financial mathematics and its simple applications. It introduces
compound interest, the time value of money and discounted cashflow techniques which are
fundamental building blocks for most actuarial work.
Please note that different answers may be obtained to those shown in these solutions
depending on whether figures obtained from tables or from calculators are used in the
calculations but candidates are not penalised for this. However, candidates may be penalised
where excessive rounding has been used or where insufficient working is shown.
Comments on the September 2011 paper
The general performance was considerably better than in September 2010 and also slightly
better than in April 2011. Well-prepared candidates scored well across the whole paper.
As in previous diets, questions that required an element of explanation or analysis, such as
Q5(ii) and Q9(iv) were less well answered than those that just involved calculation. Marginal
candidates should note that it is important to explain and show understanding of the concepts
and not just mechanically go through calculations. The comments that follow the questions
concentrate on areas where candidates could have improved their performance. Where no
comment is made the question was generally answered well by most candidates.
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 3
1 ( ) ( ) 91
91 365
365 1 0.08 1i - - × = +
( ) 91
0.980055 1 i 365 - = +
1+ i =1.08416?i = 8.416%
2 Issued by the government
Pay regular interest
Redeemable at a given redemption date
Normally liquid/marketable
More or less risk-free relative to inflation
Low expected return
Low default risk
Coupon and capital payments linked to an index of prices…
… with a time lag.
This type of bookwork question is common in CT1 exam papers. As such, it was disappointing
that only about one-sixth of candidates obtained full marks here (which could be achieved by
listing six distinct features).
3 Let the annual rate of payment = X
Present value of the payments = (4)
4 Xa????
Present value of the payments needed from the annuity is:
(12) 4 (12) 7
3 10 8,000a???? v + 3,000a???? v
(4) (12) 4 (12) 7
4 3 10 Xa???? = 8,000a???? v + 3,000a???? v
3 (4) a 2.7232 i 1.031059
d
= =
4 a = 3.5460 10 (12) a 7.7217 i 1.026881
d
= = v4 = 0.82270 v7 = 0.71068
( ) ( )
4 7
(4) 4 12 3 12 10 X i a 8,000 i a v 3,000 i a v
d d d
= +
X ×1.031059×3.5460 = 8,000×1.026881×2.7232×0.82270
+3,000×1.026881×7.7217×0.71068
3.65614X =18,404.80 +16.905.51
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 4
X = £9,657.81
?Quarterly payment is: £2,414.45.
Many candidates struggled to allow correctly for the Government pension. In some cases,
candidates would have scored more marks if they had explained their methodology and their
workings more clearly.
4 (i) The fund value on 30 June 2009 will be:
1.5×1.01 =1.515
The fund value on 31 December 2009 will be:
(1.5×1.01+ 6)×1.02 = 7.6653
The fund value on 31 December 2010 will be:
??(1.5×1.01+ 6)×1.02 + 4??×1.05 =12.2486
TWRR is i such that:
1.515 7.6653 12.2486 (1 )2 1.0817
1.5 7.515 11.6653
× × = + i =
?i = 4.005%
(This can also be calculated directly from the rates of return for which no
marks would be lost).
(ii) The equation of value is:
( ) ( ) ( ) 1
1.5 1+ i 2 + 6.0 1+ i 1 2 + 4 1+ i =12.2486
Try i = 4% LHS = 12.146
Try i = 4.5% LHS = 12.22754
Try i = 5% LHS = 12.3094
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 5
Interpolate:
0.045 12.2486 12.22754 0.005
12.3094 12.22754
i -
= + ×
-
= 0.04629 or 4.63%
A common error was to assume that the 1% and 2% rates of return were annualised figures
rather than returns over a six-month period.
5 (i) Forward price is accumulated value of the share less the accumulated value of
the expected dividends:
( ) ( ) ( ) 9 8 2
F = 9.56 1.03 12 - 0.2 1.03 12 - 0.2 1.03 12
= 9.7743 – 0.20398 – 0.20099
= £9.3693
(ii) (a) Although the share will be bought in nine months, it is not necessary to
take into account the expected share price. The current share price
already makes an allowance for expected movements in the price and
the investor is simply buying an instrument that is (more or less)
identical to the underlying share but with deferred payment. As such,
under given assumptions, the forward can be priced from the underlying
share.
(b) An option does not have to be exercised. As such, movements in the
share price in one direction will benefit the holder whereas movements
in the other direction will not harm him. The more volatile is the
underlying share price, the more potential there is for gain for the
holder of the option (with limited risk of loss), compared with holding
the underlying share. This is not the case for a forward which has to be
exercised.
Part (i) was well-answered but part (ii) was very poorly answered. The examiners anticipated
that many candidates would find part (ii)(b) challenging but it was pleasing to see some of
the strongest candidates give some well-reasoned explanations for this part.
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 6
6 (i) 5( )
45 0 55 a bt dt e ? + = (1)
10( )
45 0 120 a bt dt e ? + = (2)
From (1)
2 5
0
45exp 55
2
at bt
? ?
? + ? =
?? ??
ln 55 5 12.5 0.2007
45
?? ?? = a + b =
? ?
(1a)
From (2)
2 10
0
45exp 120
2
at bt
? ?
? + ? =
?? ??
ln 120 10 50 0.98083
45
?? ?? = a + b =
? ?
From (1a)
10a = 0.4014 - 25b (2a)
Substituting into (2a)
0.4014 + 25b = 0.98083
0.98083 0.4014 0.02318
25
b
-
? = =
Substituting into (1a)
5a +12.5×0.02318 = 0.2007
0.2007 12.5 0.0231772 0.01781
5
a
- ×
? = =-
(ii) 45e10d =120
10 120 ;10 ln 120 0.98083
45 45
e d = d = ?? ?? =
? ?
?d = 0.09808 or 9.808 %
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 7
7 (i) Expected price of the shares in five years is:
X = 2v + 2.5v2 + 2.5×1.01×v3 + 2.5×1.012v4 +...
= 2v + 2.5v2 + 2.5v2 (1.01v +1.012v2 +....)
1.01v +1.012v2 +... at 8% 1
i '
=
where ' 1.08 1 0.069307
1.01
i = - =
2 0.92593 2.5 0.85734 2.5 0.85734
0.069307
X ×
= × + × +
= 3.9952 + 30.9254 = 34.9206
Equation of value for the investor is:
12(1+ i)5 = 34.9206
i = 0.23817 or 23.817%
(ii) 12(1+ i)5 = 34.9206 - (34.9206 -12)×0.25
where i is the net rate of return.
12(1+ i)5 = 29.1905
i = 0.1946 or 19.46%
(iii) The cash flow received in nominal terms is still the same: 29.190495
The equation of value expressed in real terms is:
( )
5
5
12 29.1905
1
v
f
=
+
where f = 0.04
( )5
5 12 1.04
0.50016
29.1905
v
×
= =
1
?v = 0.50016 5 = 0.87061
i = 14.86%
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 8
8 (i) The present value of the assets is equal to the present value of the liabilities at
the starting rate of interest.
The duration /discounted mean term/volatility of the assets is equal to that of
the liabilities.
The convexity of the assets (or the spread of the timings of the asset
cashflows) around the discounted mean term is greater than that of the liabilities.
(ii) (a) PV of liabilities is: 40 £100m a at 4%
= £100m×19.7928
= £1,979.28m
(b) The duration of the liabilities is:
40 40
1 1
100 / 100
t t
t t
t t
t v v
= =
= =
S S (working in £m)
( )
40
1 40
100
100
1,979.28 1,979.28
t
t
t
t v
a
=
= ?
= =
S
at 4%
100 306.3231 15.4765
1,979.28
×
= = years
(iii) Let x = nominal amount of five-year bond
y = nominal amount of 40-year bond.
working in £m
1,979.28 = xv5 + yv40 ? (1)
30,632.31 = 5xv5 + 40yv40 ? (2)
multiply equation (1) by 5.
9,896.4 = 5xv5 + 5yv40 ? (1a)
subtract (1a) from (2) to give
20735.91 = 35yv40
40
20,735.91
35
y
v
=
×
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 9
with v40 = 0.20829
y = 2,844.38
Substitute into (1) to give:
1,979.28 = Xv5 + 2,844.38×0.20829
v5 = 0.82193
1,979.28 2,844.38 0.20829 1,687.28
0.82193
x - ×
= =
Therefore £1,687.28m nominal of the five-year bond and £2,844.38m nominal
of the 40-year bond should be purchased.
(iv) (a) The duration of the liabilities is 15.4765
Therefore the volatility of the liabilities is 15.4765
1.04
=14.88125%
The value of the liabilities would therefore change by:
1.5×0.1488125×1,979.28m = £441.81m
and the revised present value of the liabilities will be £2,421.09m.
(b) PV of liabilities is: 40 £100m a at 2.5%
= £100m×
1 1.025 40
0.025
- -
= £2,510.28m.
(c) The PV of liabilities has increased by £531m. This is significantly
greater than that estimated in (iv) (a). This estimation will be less valid
for large changes in interest rates as in this case.
The first three parts were generally well-answered but, in part (iv), the examiners were
surprised that so few candidates were able to use the duration to estimate the change in the
value of the liability.
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 10
9 (i) (a) The theoretical rate of return that could be achieved over a given time
period in the future from investment in government bonds today.
(b) The theoretical rate of return that could be achieved between the
current time and a given future time from investment in government
bonds.
(c) The gross redemption yield that could be theoretically achieved by
investing in government bonds of different terms to redemption. The
yield curve represents a statistical average gross redemption yield.
(ii)
Time Government
bond yield
Valuation rate
of interest
P.V factor
1
2
3
4
5
0.02
0.04
0.06
0.08
0.1
0.031
0.052
0.073
0.094
0.115
0.96993
0.90358
0.80947
0.69812
0.58026
PV =10(0.96993+ 0.90358 + 0.80947 + 0.69812 + 0.58026) +100×0.58026
= 97.6396.
(iii) GRY is such that: 5
5 97.6396 =10a +100v
Try 11% 5
5 a = 3.69590 v = 0.59345 RHS = 96.30397
Try 10% 5
5 a = 3.7908 v = 0.62092 RHS = 100 [calculation not necessary]
Interpolate to find i:
97.6396 96.30397 0.01 0.11
100 96.30397
i -
= - × +
-
?i = 0.10639 or 10.64%
(iv) It is reasonable for the investor to price a corporate bond with reference to the
rates of return from government bonds which may be (more or less) risk free.
A risk premium will then need to be added.
It is also not unreasonable that this risk premium rises with term as the
uncertainty regarding credit risk rises.
This question proved to be the most difficult on the paper. The examiners had anticipated that
some candidates would have difficulty with part (i) but it was disappointing to see the number
of candidates who were unable to give even a basic description of a spot rate and a forward
rate. Part (iv) was also very poorly answered and whilst it had been anticipated that only the
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 11
strongest candidates would make all the relevant points, the examiners were surprised at how
many candidates failed to score any marks on this part.
10 (i) The payback period measures the earliest time at which the project breaks
even but takes no account either of interest on borrowings or on cash flows
received after the payback period. It is therefore a poor measure of ultimate
profitability.
(ii) The present value of preparation costs is (in £m):
2a2 @4% per annum effective.
2 2 = 2. i .a i =1.019869 a =1.8861
d d
= 2×1.019869×1.8861 = 3.847
The present value the stadium building costs is (in £m):
1 1 1 1
2 2 2 2 200v4 + 200×1.05v5 + 200×1.052v6 +...+ 200×1.059v13
( ) 1
2 200v4 1+1.05v +1.052v2 +...+1.059v9
1
2
4 1 1.0510 10 200
1 1.05
v v
v
? - ?
= ? ?
?? - ??
with
1
2 v = 0.96154 v10 = 0.67556 1.0510 =1.62889 v4 = 0.83820
200 0.83820 1 1.62889 0.67556
1 1.05 0.96154
? - × ? = × ×? ? ? - × ?
= £1,750.837
Present value of admin. costs is (£m):
(12) 13
2 100a???? v @ 4%
with (12) i 1.021537
d
= v13 = 0.60057 2 a =1.8861
=100×1.021537×1.8861×0.60057
=115.714
Subject CT1 (Financial Mathematics Core Technical) — Examiners’ Report, September 2011
Page 12
Present value of revenue (£m):
14
1 3,300a v with 14
1 i =1.019869 a = 0.9615 v = 0.57748
d
= 3,300×1.019869×0.9615×0.57748
= 1,868.781
NPV = 1,868.781 – 115.714 – 1,750.837 – 3.847 = –£1.617m.
Therefore should not make a bid.
(iii) One way of dealing with this would be to multiply the NPV of all the revenues
and costs that are only received if the bid is won by 0.1.
The costs of preparing the bid would be incurred for certain and therefore not
multiplied by 0.1. This adjustment would make it less likely the bid will go
ahead because the only certain item is a cost.
This question contained a potential ambiguity regarding the timing of the administration
costs. Although the examiners felt that the approach given in the model solution was the most
logical, candidates who assumed that the administration costs were only payable during
2025 were given full credit. This question was answered well and it was very pleasing to see
that (a) candidates managed their time efficiently and so left enough time to make a good
attempt at the question with the most marks and (b) candidates who made calculation errors
still clearly explained their method and so were able to pick up significant marks for their
working.
END OF EXAMINER’S REPORT

Core Technical 1 Financial Mathematics [Including Examiners Report(2005-2011)] Question Paper

More Question Papers

Popular Exams