STAT1802 2005 Semester 2

# STAT1802 2005 Semester 2 - 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 20, 2005 Time: 2:30 p.m. — 5:30 p.m. Candidates taking examinations that permit the use of calculators may use any calculator which fulﬁls the following criteria: ( a ) it should be self-contained, silent, battery-operated and pocket-sized and ( b ) it should have numeral—display facilities only and should be used only for the purposes of calculation. It is the candidate’s responsibility to ensure that the calculator operates satisfactorily and the candidate must record the name and type of the calculator on the frontlpage of the examination 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 SEVENTEEN questions. Marks are shown in square brackets. 1. A principal of \$100,000 is deposited into a fund at time 0. Find the amount of interest earned during the fourth period if the effective rate of interest per period is in = 0.005n + 0.004n2 for n = 1, . . .,6. [5 marks] 2. A principal of \$10,000 is deposited into a fund at time 0. Find the accumulated amount of the fund at the end of the second year if the nominal annual rate of discount is 6% convertible once every four years. [5 marks] 3. Deposits of \$1,000 are placed into a fund at the beginning of each month for the next ten years. At the beginning of the 13th year, quarterly Withdrawals begin and continue forever. Find the amount of each Withdrawal if the annual effective rates of interest are 7% for the ﬁrst 5 years, 8% for the second 5 years, and 9% thereafter. [5 marks] S&AS: STAT1802 Financial Mathematics 2 Find the accumulated value (at the end of the 11th year) of a 11-payment annuity- due under which payments are made at the beginning Of each year as follows: \$1, \$4, \$7, \$10, \$13, \$16, \$19, \$22, \$25, \$28 and \$31. The annual effective interest rate is 5%. [5 marks] Annuities X and Y provide the following payments: End of Year Annuity X Annuity Y 1-10 1 0.5K 11«20 2 K 21-30 1 1.5K Annuities X and Y have equal present values at an annual effective interest rate 2' such that v10 = 0.5. Determine K. [5 marks] Jeff deposits \$1,000 at the beginning of each year for 13 years into Fund X. Donald deposits only \$10,000 at the beginning of the ﬁrst year into Fund Y. Fund X earns an annual effective rate of 15% for the ﬁrst 5 years and 5% thereafter while Fund Y earns an annual effective rate of 2'. At the end of 13 years, the accumulated values of Fund X and Fund Y are equal to each other. Find 2‘. [5 marks] Smith deposits 1000 in Bank A on January 1, 2005. Bank A credits interest at annual rate 2' = 15%. If Smith closes his account within a year, he receives simple interest up to the time of withdrawal. Smith visits Bank B across the street and is told that he can open an account anytime that year and receive simple interest at annual rate 2' = 14.5%, paid from the date of deposit to December 31. Smith consults his ﬁnancial mathematics textbook and realizes that if he chooses the right day to close his account at Bank A and immediately re-deposits the proceeds in a new account in Bank B he will maximize the return on his 1000 over the year of 2005. What is that day? [5 marks] Sam buys a TV from the ABC Store for \$480 by paying \$50 in cash, \$100 every three months foreone year (four payments of \$100), and a ﬁnal payment in 15 months (three months after the ﬁnal quarterly payment). Find the amount of the ﬁnal payment if Sam earns a 3-month effective compound interest rate of 3%. [5 marks] A magazine offers a one—year subscription at a cost of £ 15.00 with renewal the following year at £ 16.50. Also offered is a two-year subscription at a cost of .6 28.00. What is the effective annual interest rate that makes the two—year subscription equivalent to two successive one—year subscriptions? [5 marks] S&AS: 10. The table below presents an amortization schedule for a six-year mortgage with annual payments. STAT1802 Financial Mathematics Beginning Ending Year Balance Payment Interest Principal Balance 1 \$80,000.00 \$17,833.58 \$7,200.00 \$10,633.58 \$69,366.42 2 \$69,366.42 \$17,833.58 \$6,242.98 \$11,590.60 \$57,775.82 3 \$57,775.82 \$17,833.58 \$5,199.82 \$12,633.76 \$45,142.06 4 \$45,142.06 \$17,833.58 \$4,062.79 \$13,770.79 \$31,371.27 5 \$31,371.27 \$17,833.58 \$2,823.41 \$15,010.17 \$16,361.10 6 \$16,361.10 \$17,833.58 \$1,472.50 \$16,361.08 -0« (a) Determine the value of the following two mortgage-backed securities under the assumption that the discount rate for valuing these'securities is i = 10% and that the mortgage is paid off at the end of year 6, assuming that the above mortgage is the only mortgage in the pool. All the cash ﬂows are ﬁxed and certain (i.e., Prepayments are not allowed). (1) An interest-only (IO) strip — A mortgage derivative created by “strip— ping” the interest component from a pool of mortgage—backed securities. The holder of these instruments receives the interest portion of the yearly payments from the underlying mortgages. (ii) A principal-only (PO) strip —— A mortgage derivative created by “strip— ping” the principal component from a pool of mortgage—backed securities. The investor in these instruments receives the principal portion of the yearly payments from the underlying mortgages. (b) Calculate the Macaulay’s Duration (D) of the above IO strip. (c) Calculate the Convexity (C) of the above PO strip. [Total: 15 marks] 11. The proceeds of a life insurance policy are left on deposit, with interest credited at the end of each year. The beneﬁciary makes withdrawals from the fund at the end of each year t, t = 1,2,... , 10. At the minimum effective interest rate of 3% per year guaranteed in the policy, the equal annual withdrawal would be \$1000. However, the insurer credits annual effective interest rate at 4% for the ﬁrst four years and 5% of the next six years. The actual amount withdrawn at the end of the year t is Ft Wt = T_-—a all—t 0.03 where F, is the amount of the fund, including interest, prior to the withdrawal. Calculate W10. [Total: 10 marks] S&AS: 12. 13. 14. 15. 16. 17. STAT1802 Financial Mathematics ' 4 Find the present value at time 0 of an 72-year continuous annuity based on force ofinterest +< s > t p l-l—rest ’ where 19,7‘ and s are constants. [5 marks] A loan of amount L is to be paid with n periodic payments of amount K each at periodic effective interest rate 2' > 0, where n is even. The same loan can, be repaid by (n/ 2) payments if the periodic payment is increased to an amount larger than K. Determine whether the new payment is exactly double, more than double, or less than double the value of K. Show your derivations. [5 marks] You are given an n—year annuity—due of 1 per year plus a ﬁnal payment at time (n + k —— 1), for 0 < k < 1. The present value of the payments can be simpliﬁed to 1 _ vn+k d Determine the ﬁnal payment. [5 marks] A loan of \$10,000 is amortized by equal annual payments for 30 years at an effective annual interest rate of 5%. Determine the year in which the interest portion of the payment is most nearly equal to one—third of the payment. [5 marks] A loan is to be amortized by n level annual payments of X, where n > 5. You are given: 0 The amount of interest in the ﬁrst payment is 604.00. 0 The amount of interest in the third payment is 593.75. 0 The amount of interest in the ﬁfth payment is 582.45. Calculate the value of X. [5 marks] The present value of a series of payments of 2 at the end of every eight years, forever, is equal to 5. Calculate the annual effective rate of interest. [5 marks] ...
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## This note was uploaded on 03/18/2012 for the course STAT 1802 taught by Professor Dr.k.c.yuen during the Spring '08 term at HKU.

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STAT1802 2005 Semester 2 - THE UNIVERSITY OF HONG KONG...

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