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Core Technical 1 Financial Mathematics Question Paper
Core Technical 1 Financial Mathematics
Course:
Institution: question papers
Exam Year:2012
INSTITUTE AND FACULTY OF ACTUARIES
EXAMINATION
3 October 2012 (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 S2012 © Institute and Faculty of Actuaries
CT1 S2012–2
1 An investor is considering two investments. One is a 91-day deposit which pays a
rate of interest of 4% per annum effective. The second is a treasury bill.
Calculate the annual simple rate of discount from the treasury bill if both investments
are to provide the same effective rate of return. [3]
2 The nominal rate of discount per annum convertible quarterly is 8%.
(i) Calculate the equivalent force of interest. [1]
(ii) Calculate the equivalent effective rate of interest per annum. [1]
(iii) Calculate the equivalent nominal rate of discount per annum convertible
monthly. [2]
[Total 4]
3 An investment fund is valued at £120m on 1 January 2010 and at £140m on 1 January
2011. Immediately after the valuation on 1 January 2011, £200m is paid into the
fund. On 1 July 2012, the value of the fund is £600m.
(i) Calculate the annual effective time-weighted rate of return over the two-and-a
half year period. [3]
(ii) Explain why the money-weighted rate of return would be higher than the timeweighted
rate of return. [2]
[Total 5]
4 A ten-month forward contract was issued on 1 September 2012 for a share with a
price of £10 at that date. Dividends of £1 per share are expected on 1 December
2012, 1 March 2013 and 1 June 2013.
(i) Calculate the forward price assuming a risk-free rate of interest of 8% per
annum convertible half-yearly and no arbitrage. [4]
(ii) Explain why it is not necessary to use the expected price of the share at the
time the forward matures in the calculation of the forward price. [2]
[Total 6]
CT1 S2012–3 PLEASE TURN OVER
5 (i) State the characteristics of a Eurobond [4]
(ii) (a) State the characteristics of a certificate of deposit.
(b) Two certificates of deposit issued by a given bank are being traded. A
one-month certificate of deposit provides a rate of return of 12 per cent
per annum convertible monthly. A two-month certificate of deposit
provides a rate of return of 24 per cent per annum convertible monthly.
Calculate the forward rate of interest per annum convertible monthly in
the second month, assuming no arbitrage. [4]
[Total 8]
6 A loan is to be repaid by an increasing annuity. The first repayment will be £200 and
the repayments will increase by £100 per annum. Repayments will be made annually
in arrear for ten years. The repayments are calculated using a rate of interest of 6%
per annum effective.
(i) Calculate the amount of the loan [2]
(ii) (a) Calculate the interest component of the seventh repayment.
(b) Calculate the capital component of the seventh repayment.
[4]
(iii) Immediately after the seventh repayment, the borrower asks to have the
original term of the loan extended to fifteen years and wishes to repay the
outstanding loan using level annual repayments. The lender agrees but
changes the interest rate at the time of the alteration to 8% per annum
effective.
Calculate the revised annual repayment. [3]
[Total 9]
CT1 S2012–4
7 An individual wishes to make an investment that will pay out £200,000 in twenty
years’ time. The interest rate he will earn on the invested funds in the first ten years
will be either 4% per annum with probability of 0.3 or 6% per annum with probability
0.7. The interest rate he will earn on the invested funds in the second ten years will
also be either 4% per annum with probability of 0.3 or 6% per annum with probability
0.7. However, the interest rate in the second ten year period will be independent of
that in the first ten year period.
(i) Calculate the amount the individual should invest if he calculates the
investment using the expected annual interest rate in each ten year period. [2]
(ii) Calculate the expected value of the investment in excess of £200,000 if the
amount calculated in part (i) is invested. [3]
(iii) Calculate the range of the accumulated amount of the investment assuming the
amount calculated in part (i) is invested. [2]
[Total 7]
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
0.03 0.01 for 0 9
( )
0.06 for 9
t t
t
t
? + = =
d = ? < ?
(i) Derive, and simplify as far as possible, expressions for ?(t) where ?(t) is the
present value of a unit sum of money due at time t. [5]
(ii) (a) Calculate the present value of £5,000 due at the end of 15 years.
(b) Calculate the constant force of interest implied by the transaction in
part (a). [4]
A continuous payment stream is received at rate 100e-0.02t units per annum between
t = 11 and t = 15.
(iii) Calculate the present value of the payment stream. [4]
[Total 13]
CT1 S2012–5 PLEASE TURN OVER
9 (i) Describe three theories that have been put forward to explain the shape of the
yield curve. [7]
The government of a particular country has just issued five bonds with terms to
redemption of one, two, three, four and five years respectively. The bonds are
redeemed at par and have coupon rates of 4% per annum payable annually in arrear.
(ii) Calculate the duration of the one-year, three-year and five-year bonds at a
gross redemption yield of 5% per annum effective. [6]
(iii) Explain why a five-year bond with a coupon rate of 8% per annum would have
a lower duration than a five-year bond with a coupon rate of 4% per annum.
[2]
Four years after issue, immediately after the coupon payment then due the
government is anticipating problems servicing its remaining debt. The government
offers two options to the holders of the bond with an original term of five years:
Option 1: the bond is repaid at 79% of its nominal value at the scheduled time with no
final coupon payment being paid.
Option 2: the redemption of the bond is deferred for seven years from the original
redemption date and the coupon rate reduced to 1% per annum for the remainder of
the existing term and the whole of the extended term.
Assume the bonds were issued at a price of £95 per £100 nominal.
(iv) Calculate the effective rate of return per annum from Options 1 and 2 over the
total life of the bond and determine which would provide the higher rate of
return. [6]
(v) Suggest two other considerations that bond holders may wish to take into
account when deciding which options to accept. [2]
[Total 23]
CT1 S2012–6
10 Two investment projects are being considered.
(i) Explain why comparing the two discounted payback periods or comparing the
two payback periods are not generally appropriate ways to choose between
two investment projects. [3]
The two projects each involve an initial investment of £3m. The incoming cash flows
from the two projects are as follows:
Project A
In the first year, Project A generates cash flows of £0.5m. In the second year it will
generate cash flows of £0.55m. The cash flows generated by the project will continue
to increase by 10% per annum until the end of the sixth year and will then cease.
Assume that all cash flows are received in the middle of the year.
Project B
Project B generates cash flows of £0.64m per annum for six years. Assume that all
cash flows are received continuously throughout the year.
(ii) (a) Calculate the payback period from Project B.
(b) Calculate the discounted payback period from Project B at a rate of
interest of 4% per annum effective.
[5]
(iii) Show that there is at least one “cross-over point” for Projects A and B between
0% per annum effective and 4% per annum effective where the cross-over
point is defined as the rate of interest at which the net present value of the two
projects is equal. [6]
(iv) Calculate the duration of the incoming cash flows from Projects A and B at a
rate of interest of 4% per annum effective. [6]
(v) Explain why the net present value of Project A appears to fall more rapidly
than the net present value of Project B as the rate of interest increases. [2]
[Total 22]
END OF PAPER
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