102_qp_1104_final

Actuarial Society of India

EXAMINATIONS

10th November 2004

Subject 102– Financial Mathematics

Time allowed: Three Hours (10.30 am – 13.30 pm)

INSTRUCTIONS TO THE CANDIDATES

1. Do not write your name anywhere on the answer scripts. You have only to write your

Candidate’s Number on each answer script.

2. Mark allocations are shown in brackets.

3. Attempt all questions, beginning your answer to each question on a separate sheet.

4. Fasten your answer sheets together in numerical order of questions. This, you may

complete immediately after expiry of the examination time.

5. In addition to this paper you should have available graph paper, Actuarial Tables and

an electronic calculator.

AT THE END OF THE EXAMINATION

Hand in both your answer scripts and this question paper to the supervisor.

ASI 102 1104

Q.1) A loan of Rs 20,000 was issued, and was repaid at par after three years. Interest was paid on

the loan at the rate of 5% per annum, payable annually in arrears. The value of the retail price

index at various times was as follows:

At the date the loan was made 245.0

One year later 252.8

Two years later 264.1

Three years later 280.0

Calculate the real rate of return earned on the loan. [5]

Q.2) An investment which pays annual dividends is bought immediately after a dividend payment

has been made. Dividends are expected to grow at a compound rate of g per annum and price

inflation is expected to be at a rate of e per annum. If the dividend payment expected at the end of

the first year is d per unit invested and if it is assumed the investment is held indefinitely, show

that the expected real rate of return per annum per unit invested is given by

i = (d + g - e) / (1 + e).

[5]

Q.3) An insurance company calculates the single premium for a contract paying Rs 10,000 in

twenty five years’ time as the present value of the benefit payable, at the expected rate of interest it

will earn on its funds.

a) In any year the annual effective rate of interest is expected to be 7% with probability 0.3,

8% with probability 0.5 or 10% with probability 0.2.

Calculate the expected rate of interest and hence the single premium.

b) Consider the scenario where the annual effective rate of interest is expected to be the same

over each of the next twenty five years and is expected to be 7% with probability 0.3, 8%

with probability 0.5 or 10% with probability 0.2.

Calculate the expected profit at the end of the term of the contract. [4]

Q.4) A bond with an annual coupon of 8% per annum payable in arrear has just been issued with a

gross redemption yield of 6% per annum effective. The gross redemption yield on bonds of all

terms to maturity is 6% per annum effective.

The bond is redeemable at par at the option of the borrower on any coupon payment date from

the tenth anniversary of issue to the twentieth anniversary of issue. You may assume that no

taxes apply.

a) State with reasons when the bond is expected to be redeemed. [3]

b) Ten years after issue, the gross redemption yield on bonds of all terms to maturity is 10%

per annum effective.

i) When is the bond expected to be redeemed?

ii) Will the gross redemption yield for the buyer of the bond be greater than that assumed

at the time of issue?

[3]

Total [6]

of 5

ASI 102 1104

Q.5) On 1 April 2004 an investor, not liable to tax, had the choice of purchasing either:

Stock (A): 7½% Treasury Stock at a price of Rs 107 per Rs 100 nominal redeemable at

par in 2013.

Stock (B): 2% Index Linked Treasury Stock 2013 at a price of Rs 203 per Rs 100

nominal, redeemable at par.

The RPI base figure for indexing was 69.5 and the index applicable to the next coupon (coupon

payable half yearly) was 158.5. (This may also be taken as the latest known value of the RPI

on 1 April).

Assume that both stocks are redeemable on 31st March 2013; that both coupons are paid half

yearly in arrear, and that a coupon has just been paid.

Assuming that the RPI will grow continuously at a rate of 2.5% per annum from its latest

known value

a) Show that the real yield from stock (A) as at 1 April 2004 is 4% per annum effective.

[5]

b) By considering the series of future receipts from stock (B), derive a formula for the present

real value of Rs 100 nominal of that stock.

[6]

c) Calculate the real yield as at 1 April 2004 from stock (B).

[2]

Total [13]

Q.6) A loan has been issued which is repaid by a ten year decreasing immediate annuity. The loan is

calculated at a rate of interest of 4% per annum effective.

The first repayment is Rs 100 and repayments decrease by Rs 10 per annum. Calculate

a) The initial loan amount

b) Interest content of the fifth instalment

c) Capital paid in the 6th, 7th and the 8th instalment.

[7]

Q.7) A loan of nominal amount Rs 10,000,000 is to be issued bearing a coupon of 8% per annum

payable quarterly in arrears. The loan is to be repaid at the end of fifteen years at 110% of the

nominal value. An institution not subject to either income or capital gains tax bought the whole

issue at a price to yield 9% per annum effective.

a) Calculate the price per Rs 100 nominal which the institution paid. [3]

b) Exactly five years later, immediately after the coupon payment, the institution sold the

entire issue of stock to an investor who pays both income and capital gains tax at a rate of

20%.

of 5

ASI 102 1104

i) Calculate the price per Rs 100 nominal which the investor pays the institution to earn a

net redemption yield of 7% per annum effective.

[5]

ii) Calculate the annual effective rate of return earned by the institution on the whole

transaction.

[4]

Total [12]

Q.8) The following n-year spot rates were observed at time t = 0.

1 year spot rate of interest 4 per cent




5 year spot rate of interest 7½ per cent


a) Define what is meant by an n-year spot rate of interest. [2]

b) Calculate the two year forward rate of interest at time t = 3. [2]

c) Using the above n-year spot rates calculate the 6-year par yield at time t = 0. [4]

Total [8]

Q.9) An investor borrows Rs 10,000 at an effective rate of interest of 15% per annum to finance a

project. Income from the project is received at a level rate of Rs 1,800 per annum, payable

quarterly in arrear, for 20 years. Calculate the discounted payback period.

[4]

Q.10) Letk t C denote a series of cash flow payments at time k t for k = 1, 2, ..., n, and )(i P denote

the present value of these payments at an effective interest rate i , so that ∑=

+=n

k

t

t k

k iC i P

1

)1()( .

a) Define the discounted mean term (or Macaulay duration) of the cash flows in terms of k t ,

k t C and v .

[1]

b) Define the volatility (or effective duration) of the cash flows in terms of )(i P .

[1]

Total [2]

Q.11) a) In an annuity the rate of payment per unit of time is continuously increasing so that at time

t it is t .

State the standard notation and derive a formula for calculating present value of such an

annuity when rate of interest is j p.a. payable continuously.

[3]

b) A financial institution is considering the purchase of a tyre factory which has recently

ceased production. The institution forecasts that:

− the cost of reopening the factory will be Rs 500,000, and this will be incurred

continuously throughout the first twelve months

of 5

ASI 102 1104

− after the first twelve months the revenue from sales of tyres, less costs of sale and

production, will grow continuously from zero to Rs 1,000,000 at a constant rate of Rs

100,000 per annum

− when the revenue from sales of tyres, less costs of sale and production, reaches Rs

1,000,000 it will then decline continuously at a constant rate of Rs 50,000 per annum

until it reaches Rs 100,000

− when the revenue declines to Rs 100,000 production will stop and the factory will have

zero value

Additional costs are expected to be constant throughout at Rs 80,000 p.a., excluding the

first year. These are also incurred continuously.

What price should the institution pay to earn an internal rate of return (IRR) of 20% p.a.

effective?

[9]

Total [12]

Q.12) State the main differences between equity and convertible debentures issued by a company.

[3]

Q.13) In the context of a derivatives market define the terms

− call option

− put option

− long party

[3]

Q.14) a) Define “no arbitrage theory”. [2]

b) Using “no arbitrage theory”, explain a method for calculating price of a forward contract

with no income, using risk free rate of return.

[2]

Total [4]

Q.15) In any year the yield on funds invested with a given insurance company has mean value j

and standard deviation s, and it is independent of the yields in all previous years.

a) Derive formulae for the mean and the variance of the accumulated value after n years of a

single investment of 1 at time 0.

[6]

b) Let )(t i be the rate of interest earned in the t th year.

Each year the value of ( ))(1 t i+ has a lognormal distribution with parameters µ = 0.08 andσ = 0.04.

Calculate the probability that a single investment of Rs 1,000 will accumulate over 16

years to more than Rs 4,250.

[6]

Total [12]

*****************

of 5

102_qp_1104_final

Documents