Summer 2012 Examination

ST226Actuarial Investigations: FinancialSuitable for all students

Instructions to candidates

Time allowed: 3 hours

This paper contains 10 questions. Answer All questions.

You are supplied with: Formulae and Tables for Actuarial Examinations(Institute of Actuaries)Murdoch and Barnes Statistical Tables

You may also use: Electronic calculator (as prescribed by the examina-tion regulations).

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Taxes and expenses should be ignored unless otherwise indicated.

You may find the following information useful:If Y ∼ N(µ, σ2), i.e. Y is normally distributed with mean µ and varianceσ2, we have

E(

esY)

= eµs+1

2σ2s2 .

1. Consider the function

b (i, p, n) = a(p)n − a

(p)n .

Investigate whether b (i, p, n) is increasing or decreasing in each one ofthe variables i, p and n (keeping the other two constant).

(1+1+1=3 marks)

2. What is a forward contract? What is a European call option? Whatis a European put option?

(2+2+2=6 marks)

3. The force of interest at time t is given by

δ (t) =1

10 + t.

An increasing annuity pays an amount of 11 at the end of the year 1,12 at the end of year 2 and so on till the final payment which is 20 atthe end of year 10. Calculate the present value of the annuity at time0 and the accumulation of the annuity at time 10.

(4+2=6 marks)

4. An investor invested 100 in a fund on each one of the 1st of Januaryof 2009, 2010 and 2011. The following table gives the price of a unitin the fund on the 1st of January of each year:

Year Unit Price2009 1002010 1052011 1032012 107

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Show that the money weighted rate of return for the period 1.1.2009-1.1.2012 is approximately 2.1% and calculate the time weighted rateof return.

(3+3=6 marks)

5. Two companies A and B are competing for the same contract thatwill be awarded in one year’s time. The current share price of bothcompanies is 100. If company A wins the contract, its share price willrise to 104 and the share of company B will fall to 96. If company Bwins the contract, its share price will rise to 108 and the share priceof A will become Y . The risk free rate of interest is 2% per annum.Calculate Y assuming there are no arbitrage opportunities.

(6 marks)

.

6. Calculate the present value of 3 per annum payable annually in arrearsfor 10 years followed by 2 per annum payable annually in arrears for afurther 10 years assuming that the effective rate of discount is 2% perannum for the first 5 years, the effective interest rate is 3% for the next10 years and the force of interest is 4% for the final 5 years. Calculatealso the accumulated value of this series of payments at time 20.

(8+2=10 marks)

7. A bond having nominal value of 200 is repayable in two equal instal-ments at par. The first half is redeemable after 5 years and the secondhalf after 10 years. It is also subject to a coupon of 4% per annumpayable half yearly in arrears.

(a) An investor is subject to income tax at 30%. Find the price sheshould pay to realise a yield of 3%.

(6 marks)

(b) Suppose now that in addition to income tax at 30%, she is alsosubject to capital gains tax at 30%. In calculating her capitalgain the investor is allowed to inflate her cost of acquisition at1% per annum to the time of the capital gain. Find the price sheshould pay to realise a yield of 3% in this case.

(4 marks)

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8. A loan is repayable over 10 years by a special decreasing annuitypayable annually in arrears, calculated on the basis of an effectiveinterest rate of 8% per annum. The annuity payment each year isdivided into an interest payment (equal to 8% of the loan outstandingat start of the year) and a capital payment, which is used to reducethe amount of the loan outstanding. The annuity decreases in a sucha way, that if income tax of 50% were to be deducted from the interestcontent of each annuity payment then the capital repayment plus theinterest payment net of tax would be 1000 each year. There are twoinvestors that are considering purchasing the loan. Investor A is liableto income tax at 50% and investor B is liable to income tax at 20%.

(a) Calculate the price investor A should pay to purchase the loan toachieve a net effective yield of 4%.

(3 marks)

(b) What is the amount of the loan?

(2 marks)

(c) Calculate the price investor A should pay to to purchase the loanto achieve a net effective yield of 6%.

(3 marks)

(d) Calculate the price investor B should pay to to purchase the loanto achieve a net effective yield of 6%.

(6 marks)

9. Consider three bonds all with nominal amount 100 redeemable at par.

� Bond A: A one year zero coupon bond with price 96.15.

� Bond B: A three year zero coupon bond with price 83.96.

� Bond C: A two year bond paying coupons annually in arrears ata coupon rate of 7% with price 103.78.

(a) Use this information to find the one year, two year and three yearspot rates.

(4 marks)

(b) Derive the two year forward rate in one year’s time.

(3 marks)

(c) Suppose now that the spot rate curve is flat and equal to 6%.Compute the new price for each bond.

(3 marks)

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(d) Continuing from (c), suppose that an investor has a liability of10000 in two years time and wants to hedge against the risk ofinterest rate changes by investing in bonds of type A and B. UseRedington’s theory to calculate how many bonds of each type heshould hold.

(6 marks)

(e) Based on the questions above explain the limitation of Reding-ton’s theory.

(2 marks)

10. The yield of a fund for year t where t = 1, 2, 3 is represented by therandom variable Zt. Zt is such that

ln (1 + Z1) = Y1 + Y2,

ln (1 + Z2) = Y2 + Y3,

andln (1 + Z3) = Y3 + Y4,

where Y1, Y2, . . . is a sequence of independent and identically distributedrandom variables with a normal distribution with mean µ and varianceσ2.

(a) An amount 1 is invested at time 0. Find expressions in terms of µand σ2 for the expected value and the variance of its accumulatedvalue at time 3.

(4 marks)

(b) Suppose now that 1 is invested at time 0 as well as 2 at time 1.Find expressions in terms of µ and σ2 for the expected value andthe variance of the accumulated value at time 3.

(4+4=8marks)

(c) Suppose now that E (Zt) = 0.05 and V ar (Zt) = 0.04 Calculatethe probability that the accumulation at time 3 of the investmentin part (a) is larger than 1.

(7 marks)

(d) What difficulty would you encounter if you wanted to calculatethe probability the accumulation in (b) will exceed any givenvalue at time 3.

(2 marks)

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