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172A Ch 9 Q - 9.5 PROBLEMS{CHAPTER 9[hill Chapter 9 writing...

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Unformatted text preview: 9.5 PROBLEMS. {CHAPTER 9 [hill Chapter 9 writing problems [I 1 [following Section [9.17:1 C. L. Trowbridge classified risks as C 1, C2. C3, and C4 risks when he was president of the Society of Actuaries. C3 risks are ad— dressed by asset—liability management. Describe Trowhridge‘s classification. Include several examples of each type of risk. (2) Write a short essay on the history ol'cash-flow management and immunization. Names you might look for in your research include 'l'j'alling C. Keepmans and Frank M. Redlngten. {9.1} Overview (1] Leland Price mu st pay $5.333 six months from new and $13,331] one year from now. He vo'shes to purchase bends so that together they form a portfolio of assets that exactly match his liabilities. Available bonds are a six~month rere- coupon 31.31111 bond that has a 3.023% annual yield. and a one—year 31,1101} par—value 6% bond with semiannual coupons and a 4% nominal yield {conu vertible semiannually}. How much must Leland pay to purchase the bonds? Assume that he may buy any quantity he wishes of each bond. {2] Suppose that you need to pay 335.033 in three years and that you can finance this with acre-coupon bonds yielding 5.5% with terms of two years and six years. Imagine that you spend 322,354.36 purchasing a two—year bond and 37.45 I152 for a six-year bend, and these are each priced to yield 5.5%. Sap- pnsc also that. at the end of two years, no matter what the yield rate i may then be, you sell the remaining bond at a purchase price to yield f, combine the proceeds with the $24.33 I .52 front the redeemed bond. and use the total to buy a one-year zero—coupon bond. illustrate that this immunizes against inter— est rate risk by showing that it produces the needed $35.0le three years after your initial bond purchases if i = l3‘ll} [a high rate} err' = 1% [a low rate). {9.2} Macaulay and modified duration (1} Compute the Macaulay duration of a tenuyear 6% $1,331] bond having annual coupons and a redemption of 3 l .2lli'l if the yield to maturity is 3%. [2} The cement price of a bond having annual coupons is 31.312. The deriva- tive of the price function of the bond with respect to the yield to maturity is —3‘i‘.4—43.31 when evaluated at the current annual yield. which is 2%. Calcu— late the Macaulay duration i'thi'i. no] and the modified duration BLUE 1} of the bond (3]: a rare-coupon hond matures in eight years it is sold to yield 5% annually Find the modified donation Df. '35. l]. {4) The current price of a noncallab'le bond with annual coupons ls $1.12fl.53. and the current annual yield is 4. 25%. The modified duration Bf. [1425. 2} is 3 .53. Estimate the price of the bond iI the yield Increases to 4 4%. About what would the price he ifrlre discount rate decreases to an annual effective rate ol 3. 1%? (5) Calculate the Macaulay duration DIES. no} and the modified duration D{.fl5. 2] of a preferred stock that pays dividends forever of $511 each six months, with the next dividend in exactly six months. on Calculate tFe Macaulay duration ores, no] and the modified duration BLUE, 1] e a stock that pays annual dividends forever, assuming that the first dividend, payable in exactly one year, is $1311 and then. each subsequent divi— dend is 2% more than the previous one. [1} On January 1 . 19‘? 3, Jonathan Linden is obligated to pay annual level payments for sixteen years. beginning with a payment on January 1, 1929. His financial adviser tells him that this liability has Macaulay duration “1.39. Determine the annual effective interest rate that the adviser used to figure the duration. {3} Santoshinherits a perpetuity with annual payments. The first payment is 31 .403 and payments increase by 3% each year. lCtbtain a fennuia for the Macaulay duration flfiiice} of the remaining payments an instant after the fr—th pay— ment. (Ell Mustafa's portfolio consists of an annuity with monthly payments of $1,030 each month for five years and a $23,331} 3% eight—year par-value bond hearing semiannual coupons. Calculate the Macaulay duration of'the portfolio at 9%. {10] it thirty—year mortgage with no early repayment option is repaid with level rnenthl y payments. The interest rate on the mortgage is 6.3% nonrl nal convert— ible monthly. Calculate the Macaulay duration on the mortgage at the equiva- lent amrual effeeu‘ve interest rate. {9.3} Convexity [1 1| Calculate the Macaulay convexity of a tenryear' 6‘32 $1,333 bend having annual coupons and a redemption of 51233 if the yield to maturity is 3%. [This is the same hand considered in Problem {9.2.11.1 (2]: A 32.11111 1% threc~year bond has scrniannu al coupons and redemption amount 32, 225. it is purchased to provide the investor with a 5'3: arurual eli'oetive yield. i-ind the Macaulay convexities C(4'i'rh. no). Cffi'ih, no}, and Cfo'lh. co}. Also compute the modified convexity C {5%. 2]. {3} A bond has Macaulay duration Elli . do} = 5.3 and Macaulay convexity C(i. co} = 1.2. Determine (313.4) as a function of r'. {4] A perpetuity has a level payment at the end ot'each year. The annual effective interest rate is 5.2%. Calculate the Macaulay duration and the Macaulay corr— vexity. Use these to estimate the percentage decline in the market value of the perpetuity if the annual effmfive interest rate increases by 53 basis pehrts. an... rim—Huh. u _.:..n._nu.. .nrfl ...r_...:.:.:..q fl”. HE... H.323. main... ”.5... EEE: u gamma... a.n magma Gum... F”. man... :5. 1. a... 1.... mi u . + .. a... m3... 5.... In M AH... 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