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STAT1802 2001 Semester2

# STAT1802 2001 Semester2 - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1802 FINANCIAL MATHEMATICS May 22, 2001 9:30 a.rn. - 12:30 p.m. Candidates taking examinations that permit the use of calculators may use any cal— culator which fulﬁls the following criteria: (a) it should be selﬁcontained, silent, batteryuoperated and pocket-sized; and (b) it should have numeral-display facilities only and should be used only for the purpose of calculation. It is the candidate ’3 responsibility to ensure that the calculator operates satisfactorily and the candidate rnust record the name and type of the calculator on the front page of the ewarnination scripts. Lists of permitted/prohibited calculators will not be made available to candidates for reference, and the onus will be on the candidate to ensure that the calculator used will not be in violation of the criteria listed above. Answer ALL TEN questions. Marks are shown in square brackets. 1. An investment fund was worth \$100,000 at time 0. At time 1, the fund value has increased to \$108,000 and a deposit of \$30,000 was made. At time 2, the fund value declined to \$115,000 and \$43,000 was withdrawn. The investment fund was worth \$90,000 at time 3. Compute the annual effective yield rate by the time—weighted method. [3 marks] 2. (a) A loan is being repaid by 20 annual instalments of \$100 each of which are used to pay interest on the loan at 6% per annum effective and to build up a sinking fund at 4% effective. The sinking fund will accumulate to the original loan amount at the end of 20 years. Find the amount of the loan. [3 marks] (b) A loan of \$100,000 was issued and was repaid at par after two years. Interest on the loan was made at the end of each year at a rate of 15% per annum. At the date of issue, the value of the consumer price index (CPI) was 120. The CPI then increased by 10 each year. Calculate the real rate of interest earned on the loan. [3 marks] S&AS: STAT1802 Financial Mathematics 2 3. The force of interest is given by 6__ 0.03t+0.06 0§t<6 ‘ 0.12 631:. Calculate the present value at time 0 of a continuous payment stream from t = 8 to t = 16 if the rates of payment are \$50 per annum for the ﬁrst four years and \$100 per annum for the last four years. [7 marks] 4. A loan has been issued which is repaid by a twenty-year decreasing annuity- immediate. The loan is calculated at a rate of interest of 5% per annum effective. The ﬁrst payment is \$2,000 and payments decrease by \$100 per annum. Determine the amount of interest paid in years 9-12 inclusive. [7 marks] 5. An investor purchases a one-year zero-coupon bend redeemable at 106 and a par value two-year bond with 6% annual coupons. The two-year bond is bought to yield 6.5% per annum effective. Assuming no arbitrage, calculate the two-year spot rate and the one-year forward rate over the second year. [8 marks] 6. A loan of nominal amount of \$1,350,000 bears interest of 8% per annum payable in arrears, and is to be redeemed at par at the end of the 15th year. Calculate the purchase price of the loan to obtain a net effective yield of 8% per annum if the investor is liable to tax at 30% both on interest and on capital game. [8 marks] 7. Let I, be the annual yield on an insurance company’s fund for the tth year. The yields in different years are identically and independently distributed. (a) Suppose that I, has a uniform distribution over the interval [0.06, 0.09]. Find the mean and the variance of the accumulated value after ten years of a single investment of \$1 at time 0. [4 marks] (b) Suppose that the mean and the variance of I, are 0.06 and 0.001 re- spectively, and that 1 + It follows a lognormal distribution. Find a 95% conﬁdence interval for the accumulated value of an investment of \$1 at the end of three years. [7 marks] SScAS: STAT1802 Financial Mathematics . 3 8. Find the present value of a perpetuity-due with annual payments where the kth payment is equal to (k + 1)3, at an effective rate of 6% per annum. [12 marks] 9. A business venture requires initial investment of three payments, each of 0.8 10. million. The ﬁrst is due at the start of the project. The second is due one year after the start of the project, and the third is due one year after the second payment. After 15 years it is assumed that a further outlay of 1.2 million will be required to continue the project. The project is eXpected to provide no income before the end of the fourth year. The ﬁrst income payment of \$200,000 will be collected at the end of the fourth year. The second income payment collected one year later will increase to \$300,000. Thereafter the income will be received continuously and is expected to grow continuously at a rate of 5% per annum effective. It is assumed that the income will cease at the end of the 30th year from the start of the project. Calculate the discounted payback period for the project using an effective interest rate of 10.25% per annum . [16 marks] (a) State the three conditions for Redington’s immunization. EXplain why a fund can be immunized against small movements in the rate of interest under these conditions. [5 marks] An investor has to pay a lump sum of \$1,000,000 at the end of ten years from now and an annuity of \$100,000 per annum for 30 years. The ﬁrst annuity payment will be made one year from now. The investor currently holds an amount of cash equal to the present value of these two liabilities valued at an effective rate of interest of 6%per annum. The investor wishes to immunize his fund against small changes in the rate of interest by investing the cash in two zero-coupon bonds, namely Bond A and Bond B. The market prices of both bonds are calculated at an effective rate of interest of 6% per annum. The investor has decided to invest an amount in Bond A so that he will receive an amount of \$700,000 at maturity in one year’s time. The remainder of the cash is invested in Bond B. Determine the term needed for Bond B and the amount payable at ma- turity so that his fund can be immunized. [17 marks] ...
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