ACTUARIAL SCIENCE 211
TUTORIAL 8
1.
Two business projects, each of which takes two years to complete, produce the
following income and expenditure:
Project (i)
:
initial income of R1 000
after one year, expenditure of R2 000
after two years, income of R2 000.
Project (ii)
:
initial expenditure of R4 000
after one year, income of R7 000
after two years, expenditure of R1 500.
For each project find the rates of interest, if any, which make the present value of the
income equal to the present value of the expenditure.
For what range of positive interest rates does the net present value of Project (i)
exceed the net present value of Project (ii)?
( i  23,166% per annum )
2.
An investor is considering the purchase of an interest in a warehouse for R250 000.
The interest entitles the purchaser to receive rent for a five year period at a rate of
R64 000 per annum, payable quarterly in advance.
Maintenance and management costs are also fixed for the five year period at a rate of
R7 000 per annum payable by the investor at the beginning of each year. At the end
of the five year period the purchaser neither makes nor receives any further payments.
(a) Calculate the annual effective rate of return
(5,704%) (3)
(b) Calculate the annual effective real rate of return if inflation is 2% per annum for
the five years.
(3,63%) (2)
[Total 5]
3.
A fund is exactly sufficient to make specified payments annually in arrears for the
next 30 years. The last 15 payments are all of amount R8 000, and the first 15 are
16% of the fund size at the start of the year.
The fund will earn interest at the rate of 12% per annum while it is R100 000 or
larger, 10% while it is R50 000 or larger but less than R100 000, and 8% while it is
less than R50 000. Find the initial amount of the fund.
(Hint: Start your calculations at time t = 30 )
(R134 782,08) [10]
2
4.
(a) Explain briefly what is meant by discounted payback period in connection with a
discounted cash flow calculation.
(b) A company is about to set up a manufacturing operation to produce 5 000
televisions a year for 20 years. The costs of setting up the operation are all
incurred at outset and amount to R1 000 000. The annual running costs of the
operation are R150 000 in the first year. The price of the televisions and the
annual running costs each year will be 5% greater than the previous year. The
proceeds from the sale of the sets and the annual running costs are assumed to
occur halfway through each year. At the end of 20 years the manufacturing
operation will be worthless. The company has insufficient funds to finance the
venture but may borrow the initial outlay of R1 000 000 from a bank which will
charge an effective interest rate of 10% per annum. The bank loan is not for a
fixed term but may be reduced by repayments at any time. When the company
has funds to invest it will receive an effective interest rate of 8% per annum on its
deposits.
(i)
Calculate the minimum price of each television in the first year such that the
venture will just be viable.
( P = R45, 75)
(ii) The company decides that the price of each television in the first year will be
R50. Calculate the discounted payback period and the accumulated profit at
the end of 20 years.
(13½ years; R1 716 259,98)
[22]
5.
(a) What is meant by the Internal Rate of Return of an investment project?
(b) An investor is considering two mutually exclusive projects, A and B. Each has an
initial cost and produces income at the end of each of the next four years, as
shown below:
Year 1
Year 2
Year 3
Year 4
Project A
R10 000
R30 000
R30 000
R20 000
Project B
R20 000
R10 000
R30 000
R30 000
The initial cost of Project A is R76 785, and the internal rate of return of Project B is
6% per annum.
(i) What is the initial cost of Project B?
(R76 719)
(ii) What is the internal rate of return of Project A?
(6,2%)
(iii) Let NPVA (i ) , NPVB (i ) denote the net present values of Projects A and B
respectively, at the rate of interest i per annum. Show that, in the range
0 ≤ i ≤ 0,19 , there is exactly one value of i , i' , such that
NPVA (i ) < NPVB (i ) for i < i' and
NPVA (i ) > NPVB (i ) for i > i'
Find i' to the nearest 14 % .
(0,75%) [19]
3
6.
An investor is considering investment in either or both of the following projects.
Project A: For an initial outlay of R100 000, the investor receives income of R20 000
per annum in arrear for ten years, and R100 000 at the end of the tenth year.
Project B: For an initial outlay of R100 000, the investor receives the following cash
flows at the end of years 1 to 10 respectively: R20 000, R20 000, R20 000, R10 000,
R20 000, R20 000, R50 000, R20 000, R20 000 and R98 000.
The investor may borrow or lend money throughout at the same rate of interest.
(a) (i) For which positive rates of interest is project A profitable?
(ii) For which positive rates of interest is project B profitable?
( 0 ≤ i < 20% )
( 0 ≤ i < 19,999% )
(b) For which positive rates of interest is project A more profitable than B?
( 0,19898 < i < 0, 08475)
7. A new company is being set up, and has approached an investor for a loan of
R1 000 000. The loan is repayable at 120% at the end of 20 years if the new
company survives that long. However, if the new company fails, the loan is repayable
at par at the end of the year of failure. Interest is due at the rate of 20% per annum,
payable annually in arrear if the company survives to the payment date; no interest is
payable if the company does not survive to the payment date.
The investor assesses the chances of the new company failing as 40% in the first year,
25% in the second year, and nil thereafter.
(a) Calculate the expected values of the investor’s cash flows in each of the 20 years
of the contract.
(5)
(b) Show that the internal rate of return to the investor, using the expected cash flows
is 17,19% per annum
(2)
[Total 7]
8. A property developer is constructing a small block of flats. The flats will take six
months to build. The developer pays out R200 000 at the outset of the project
followed by R50 000 at the end of each month for the following six months during the
building period.
The expected rental income from the flats is R8 000 per month, receivable at the start
of each month beginning with the seventh month. The maintenance and management
costs, which are to be paid by the developer, are expected to be R15 000 per annum
payable monthly in arrears with the first payment at the end of the seventh month. The
entire block of flats is expected to be sold twenty years after the start of the project for
R50 000.
(a) Determine the discounted payback period for the project at an effective rate of
return of 12% per annum.
(You are given that the discounted payback period is greater than 11 years and 6
months).
(11 years 9 months) (7)
4
(b) If the block of flats is subsequently sold 5 years after the start of the project, what
price does the developer need in order to earn an internal rate of return of 17% per
annum on the project?
(R532 384,54) (4)
(c) The developer assumes that inflation will be 3% per annum over the first two
years from the start of the project and 5% per annum for the following three years.
Write down an equation of value from which the price needed by the developer in
(b) above may be derived, if he wishes to earn a real internal rate of return of 12%
per annum.
(3)
(You are not required to calculate the price.)
[Total 14]
9. A construction company issues fixed interest bonds and ordinary shares to an investor.
Under the terms of the issue of ordinary shares, the investor is to purchase 1 000 000
shares at a purchase price of 40 cents each. The purchase price is to be paid in two
equal instalments with the first instalment being payable immediately and the second
in one month’s time. No dividend is expected to be paid for 5 years. Exactly 5 years
and 6 months after the payment of the first instalment, the first dividend of 1 cent per
share is expected to be paid. Dividends will then be paid every 6 months in perpetuity.
The two dividend payments within any one year are expected to be equal, but the total
annual rate of dividend is expected to increase at a rate of 6% per annum, compound.
The investor is to purchase R1 000 000 nominal of bonds, at a price of R90%. The
purchase price is to be paid in 3 level monthly instalments, the first being due
immediately. The bonds will pay interest at the rate of 10% per annum, payable halfyearly in arrears for 10 years, and will be redeemed at par at the end of 10 years. At
that time, the investor will have the option of buying a further 500 000 ordinary shares
at a guaranteed price of R1 each. The first dividend on these shares is expected to be
paid 6 months after the date of purchase.
All ordinary shares pay the same rate of dividend at the same time.
Calculate the net present value of the combined investment at an effective rate of
interest of 10% per annum, assuming that the investor exercises the option to buy the
additional 500 000 shares after 10 years.
(-R18 921,73) [13]
10. A businesswoman wishes to undertake one of the following two projects and intends
to borrow money to do so.
Project A requires an initial outlay of R50 000 with three further payments of R8 000
at yearly intervals starting at the end of year 1. The expected income will be R12 000
payable continuously for each of years 4 to 15.
Project B requires an initial outlay of R150 000 and the expected income will be
R8 400 per annum payable monthly in arrears for the first year. Thereafter the
expected income will increase annually by R1 800 (starting at the end of year 1) for
the next 14 years, but will still be payable monthly.
(a)
(i) Calculate the internal rate of return for each project.
(A: 8,8% ; B: 9,237%)
5
(ii) Money can be borrowed and invested at 6% per annum effective. Which
project would you advise her to invest in to maximize her accumulated
profit at the end of 15 years?
(B) (11)
(b)
The businesswoman chooses to invest in project A. After seven years the
interest rate on borrowings increases to 7% per annum effective while the
interest rate on investments remains at 6% per annum effective. Assuming that
the borrowings can be repaid at any time, and that the income on the project to
date are as expected, show that a future income of R12 193,85 per annum is
required in order to achieve the same accumulated profit as before?
(8)
[Total 19]
11. Company A has agreed to construct a factor on a site which it owns and to lease it to
company B. Company A expects to pay construction costs of R100 000 immediately
(at time t = 0 ) , R82 500 in one year’s time and R50 000 in two year’s time.
Company B will occupy the factory when it is completed in 2 years’ time and will pay
rent quarterly in advance for 40 years from that time (i.e. from time t = 2 ) . The initial
rent payable will be at the rate of R15 000 per annum and this will increase by R2 000
at the end of each 2 year period for the first 20 years of the lease. (The last of these
increases taking place at time t = 22 ) . Thereafter the rent will increase at the end of
each 2 year period at the rate of 5% per annum. At the end of the lease (at time
t = 42 ) the factory will be obsolete and will have no value.
(a) In order to asses the viability of the project, company A wants to know the net
present value of this project at 10% per annum. Calculate this figure.
(-R5 195,91) (19)
(b) At the outset company A has difficulty in making staff available to work on the
project so company B offers to pay all costs of construction and maintenance of
the factory if it is granted an immediate loan by company A of R230 000. The
loan will be repaid over 42 years by annual payments in arrears. The annual
payment will be R24 000 for the first 22 years and R23 000 thereafter. No rent
will be paid for the factory by company B.
(i) Calculate the internal rate of return on the proposed loan for company A.
(10,2205%)
(ii) State whether or not the loan offers company A higher internal rate of return
than the original proposal.
(4)
[Total 23]
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