DUBLIN CITY UNIVERSITY
SEMESTER ONE EXAMINATIONS 2018/19
MODULE:
Financial Mathematics
MS318
QUALIFICATIONS:
BSc in Actuarial Mathematics,
Common Entry, Actuarial, Financial Maths
YEAR OF STUDY:
3
EXAMINERS:
Ms. Diane Cadden
Prof. M. Tretyakov,
Dr. S. Balagopalan (ext. 6568)
TIME ALLOWED:
3 hours
INSTRUCTIONS:
Attempt all TWELVE questions.
REQUIREMENTS:
Log Tables, Actuarial Tables
PLEASE DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO.
The use of programmable or text storing calculators is expressly forbidden.
MS318, Semester One 2018/19
Page 1 of 5
QUESTION 1
[TOTAL MARKS: 5]
Suppose that the force of interest per annum at time t years is
δ(t) =
1
.
1+t
(a) Calculate the present value of 1 due at time t.
[2 marks]
(b) What is the rate of payment ρ(t) required so that the accumulation at
time t = T of a payment stream, paid continuously from time 0 to T is equal
to T for all T > 0.
[3 marks]
QUESTION 2
[TOTAL MARKS: 11]
Calculate the time in years for e10, 000 to accumulate to e11, 000 at
(a) a simple rate of interest of 4% per annum,
[2 marks]
(b) a compound interest rate of 4% per annum convertible quarterly,
[2 marks]
(c) an effective rate of discount of 4% per annum,
[2 marks]
(d) a force of interest of 4% per annum.
[2 marks]
(e) Let n be the time in years taken for e10, 000 to accumulate to e11, 000
at a compound interest rate of 4% per annum effective. Without doing
further calculations, state which two of the above periods n lies between,
and briefly justify your answer.
[3 marks]
QUESTION 3
[TOTAL MARKS: 6]
Annual deposits are made into a fund at the beginning of each year for 10
years. The first five deposits are of e1, 000 each and deposits increase at
3.8461% per year thereafter. If the fund earns interest at 8% pa effective,
find the accumulated value of the fund after 10 years.
[6 marks]
QUESTION 4
[TOTAL MARKS: 8]
¯ n
(a) Define and briefly explain the difference between the quantities (Iā)
and (Iā) n .
[4 marks]
(b) Derive the formula
(Iā) n =
from the definition of (Iā) n .
MS318, Semester One 2018/19
ā n − nv n
δ
[4 marks]
Page 2 of 5
QUESTION 5
[TOTAL MARKS: 14]
A loan of e16, 000 was issued to be repaid by a level annuity-certain payable
annually in arrears over 10 years and calculated on the basis of an interest
rate of 8% per annum. The terms of the loan provided that at any time
the lender could alter the rate of interest, in which case the amount of the
annual repayment would be revised appropriately.
(a) Find the initial amount of the annual repayment.
[3 marks]
(b) Immediately after the fourth repayment was made, the annual rate of
interest was increased to 10%. Find the revised amount of the level annual
repayment.
[3 marks]
(c) Immediately after the seventh repayment was made, the annual rate of
interest was reduced to 9%. There was no further change to the rate of
interest. Find
(i) the final amount of the level annual repayment,
(ii) the effective rate of interest i paid by the borrower on the completed
transaction, and
(iii) the level payment corresponding to a uniform interest rate i throughout
the period of the loan.
[8 marks]
QUESTION 6
[TOTAL MARKS: 4]
(a) Explain briefly what the “no-arbitrage assumption” means.
[2 marks]
(b) State the law of one price.
[2 marks]
QUESTION 7
[TOTAL MARKS: 10]
An investor has recieved offers from two companies, which want to be acquired. Company A’s price is e2 million. Company B’s price is e3 million.
After analysis, the investor estimates that Company A’s profitability is consistent with a perpetuity of e300, 000 (in arrears) a year and that Company
B’s prospects are consistent with a perpetuity (in arrears) of e435, 000 a
year. The investor has a budget that limits acquisitions to a maximum
purchase cost of e4 million. Their opportunity cost of capital relative to
undertaking either project is 12% per annum.
(a) Determine which company or companies (if any) they should purchase
according to the NPV measure.
[4 marks]
(b) Determine which company or companies (if any) they should purchase
according the IRR measure.
[4 marks]
(c) State which company or companies (if any) they should purchase. Justify
your answer.
[2 marks]
MS318, Semester One 2018/19
Page 3 of 5
QUESTION 8
[TOTAL MARKS: 8]
A project requires an initial investment of e9 million payable on 1 January
2019 and e12 million payable continuously during 2020. From 1 January
2021, to 31 December 2033, the project will receive income half-yearly in
arrears at a rate of e5 million per annum.
(a) Calculate the discounted payback period at an effective rate of interest
of 9% per annum.
[5 marks]
(b) Without doing any further calculations, explain whether the discounted
payback period would be greater than, less than, or equal to that given in
part (a) if the effective interest rate were substantially greater than 9% per
annum.
[3 marks]
QUESTION 9
[TOTAL MARKS: 6]
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 arrears at a rate of 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.
[6 marks]
QUESTION 10
[TOTAL MARKS: 12]
A fund had a value of e21, 000 on 1 July 2018. A net cash flow of e5, 000
was received on 1 July 2019 and a further cash flow of e8, 000 was received
on 1 July 2020. Immediately before receipt of the first net cash flow, the
fund had a value of e24, 000, and immediately before receipt of the second
net cash flow the fund had a value of e32, 000. The value of the fund on 1
July 2021 was e38, 000.
(a) Calculate the annual effective money weighted rate of return earned on
the fund over the period 1 July 2018 to 1 July 2021.
[5 marks]
(b) Calculate the annual effective time weighted rate of return earned on
the fund over the period 1 July 2018 to 1 July 2021.
[4 marks]
(c) Explain why the values in (a) and (b) differ.
MS318, Semester One 2018/19
[3 marks]
Page 4 of 5
QUESTION 11
[TOTAL MARKS: 8]
(a) Explain what is meant by a forward contract. Your answer should include
reference to the terms short forward position and long forward position.
[3 marks]
(b) An investor entered into a long forward contract for e100 nominal of a
security seven years ago and the contract is due to mature in three years’
time. The price per e100 nominal of the security was e96 seven years ago
and is now e148. The risk-free rate of interest can be assumed to be 4%
per annum effective during the contract. Calculate the value of the contract
now if the security will pay a single coupon of e7 in two years’ time and
this was known from the outset. You should assume no arbitrage.
[5 marks]
QUESTION 12
[TOTAL MARKS: 8]
A company considers that, on average, it will earn interest on its funds
at the rate of 4% per annum. However, the investment policy is such that
in any one year the yield on the company’s funds is equally likely to take
any value between 2% and 6%. For both single and annual premium capital
redemption policies with a term of three years and premium e1, find the
mean accumulation and the standard deviation of the accumulation at the
maturity date. You may ignore all expenses.
[8 marks]
MS318, Semester One 2018/19
Page 5 of 5