UNIVERSITY OF CAPE TOWN
BUS2016H
DIVISION OF ACTUARIAL SCIENCE
SCHOOL OF MANAGEMENT STUDIES
BUS2016H-FINANCIAL MATHEMATICS TEST 2
13 October 2020
Time : 1 12 hours
Instructions for candidates:
• Candidates should answer all questions showing clearly the working.
• The marks for each question or part thereof, are given in brackets [ ]. Total marks for
this test is [50]
• All working must be clearly shown including values of any level annuities obtained
from calculators. Not showing your working will result in loss of marks.
• All yield calculations must be done by at least 2 interpolation steps with the aid of
your financial calculator where possible.
Q1. Lusani (Pvt) Ltd, a small company just formed, sets up a new venture on 1 January
2020. The initial investment on 1 January 2020 is R2 million with a further R1.5 million
required on 1 August 2020. The running costs for the venture start immediately in
2020 at R150 000 per annum payable quarterly in arrears. The running costs will
increase at an effective rate of 4% per annum compounded. The increases will occur
every second year with the first increase taking place at the start of the third year.
It is expected that on 1 January 2022, income will begin at the rate of R300 000
per annum and that the rate will increase by R100 000 per annum on 1 January of
each subsequent year. It is assumed that the income will be received continuously
throughout the project.
The company expects to sell the business on 1 January 2032 for R3 million.
page 1 of 3
BUS2016H
(a) Calculate the net present value of the venture on 1 January 2020 at a rate of
interest of 6% per annum, convertible half-yearly.
[16]
(b) The original effective internal rate of return on the project was 9.55% per annum.
However, it is now assumed that the following two additional cashflows will be
incurred:
• the project will require a further investment of R1 million on 1 January 2021
and
• due to increased popularity of Lusani (Pvt)Ltd farming techniques, the income in the final year of the project will increase by R1 million.
Discuss, without further calculations, the combined effect of the additional cashflows on the effective internal rate of return, both in terms of the magnitude and
direction of change.
[3]
Q2. An investor purchased a bond with exactly 20 years to redemption. The bond, redeemable at par, has a gross redemption yield of 6%. It pays annual coupons, in
arrears, of 5%. The investor does not pay tax.
(a) Calculate the purchase price paid for the bond
[3]
(b) After exactly ten years, immediately after payment of the coupon then due, this
investor sells the bond to another investor. That investor pays income and capital
gains tax at a rate of 30%. The bond is purchased by the second investor to provide
a net rate of return of 6.5% per annum.
(i) Calculate the price paid by the second investor.
[6]
(ii) Calculate the annual effective rate of return earned by the first investor during
the period for which the bond was held.
[4]
Q3. A BEE company has a portfolio of annuity contracts under which it expects to pay
R1 million at the end of each of the next 20 years, followed by R0.5 million at the
end of each of the following 20 years. The government bond with the longest duration
in which it can invest its funds pays a coupon of 10% per annum in arrears and is
redeemed at par in 15 years time. The yield to maturity of the government bond is
6% per annum effective and a coupon payment has just been made.
(a) (i) Calculate the duration of the company’s liabilities at a rate of interest of 6%
per annum effective.
[5]
(ii) Calculate the duration of the company’s assets at a rate of interest of 6% per
annum effective, if all the company’s funds are invested in the government
bond with the longest duration.
[4]
(b) (i) Explain why the BEE company cannot immunize its liabilities by purchasing
government bonds.
[1]
(ii) Without any further calculations, state the circumstances under which the
BEE company would make a loss if there were a uniform change in interest
rates. Explain why a loss would be made.
[2]
page 2 of 3
BUS2016H
Q4. The government 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.
Calculate the one-year, two-year and three-year spot rates of interest at the date of
issue.
[6]
END OF QUESTION PAPER
page 3 of 3