Chap009金融机构管理课后题答案

1、Chapter Nine Interest Rate Risk IIChapter OutlineIntroduction DurationA General Formula for DurationThe Duration of Interest Bearing BondsThe Duration of a Zero-Coupon BondThe Duration of a Consol Bond (Perpetuities)Features of DurationDuration and MaturityDuration and YieldDuration and Coupon Inter

2、estThe Economic Meaning of DurationSemiannual Coupon BondsDuration and ImmunizationDuration and Immunizing Future PaymentsImmunizing the Whole Balance Sheet of an FIImmunization and Regulatory Considerations Difficulties in Applying the Duration ModelDuration Matching can be CostlyImmunization is a

3、Dynamic ProblemLarge Interest Rate Changes and ConvexitySummaryAppendix 9A: Incorporating Convexity into the Duration ModelThe Problem of the Flat Term StructureThe Problem of Default RiskFloating-Rate Loans and BondsDemand Deposits and Passbook SavingsMortgages and Mortgage-Backed SecuritiesFutures

4、, Options, Swaps, Caps, and Other Contingent ClaimsSolutions for End-of-Chapter Questions and Problems: Chapter Nine1.What are the two different general interpretations of the concept of duration, and what is the technical definition of this term? How does duration differ from maturity?Duration meas

5、ures the average life of an asset or liability in economic terms. As such, duration has economic meaning as the interest sensitivity (or interest elasticity) of an assets value to changes in the interest rate. Duration differs from maturity as a measure of interest rate sensitivity because duration

6、takes into account the time of arrival and the rate of reinvestment of all cash flows during the assets life. Technically, duration is the weighted-average time to maturity using the relative present values of the cash flows as the weights.2.Two bonds are available for purchase in the financial mark

7、ets. The first bond is a 2-year,$1,000 bond that pays an annual coupon of 10 percent. The second bond is a 2-year,$1,000, zero-coupon bond.a. What is the duration of the coupon bond if the current yield-to-maturity (YTM) is 8 percent? 10 percent? 12 percent? (Hint: You may wish to create a spreadshe

8、et program to assist in the calculations.)Coupon BondPar value =$1,000Coupon = 0.10Annual paymentsYTM =0.08Maturity = 2TimeCash FlowPVIFPV of CFPV*CF*T1$100.000.92593$92.59$92.592$1,100.000.85734$943.07$1,886.15Price =$1,035.67Numerator =$1,978.74Duration =1.9106= Numerator/PriceYTM =0.10TimeCash Fl

9、owPVIFPV of CFPV*CF*T1$100.000.90909$90.91$90.912$1,100.000.82645$909.09$1,818.18Price =$1,000.00Numerator =$1,909.09Duration =1.9091= Numerator/PriceYTM =0.12TimeCash FlowPVIFPV of CFPV*CF*T1$100.000.89286$89.29$89.292$1,100.000.79719$876.91$1,753.83Price =$966.20Numerator =$1,843.11Duration =1.907

10、6= Numerator/Priceb. How does the change in the current YTM affect the duration of this coupon bond?Increasing the yield-to-maturity decreases the duration of the bond.c. Calculate the duration of the zero-coupon bond with a YTM of 8 percent, 10 percent, and 12 percent.Zero Coupon BondPar value =$1,

11、000Coupon = 0.00YTM =0.08Maturity = 2TimeCash FlowPVIFPV of CFPV*CF*T1$0.000.92593$0.00$0.002$1,000.000.85734$857.34$1,714.68Price =$857.34Numerator =$1,714.68Duration =2.0000= Numerator/PriceYTM =0.10TimeCash FlowPVIFPV of CFPV*CF*T1$0.000.90909$0.00$0.002$1,000.000.82645$826.45$1,652.89Price =$826

12、.45Numerator =$1,652.89Duration =2.0000= Numerator/PriceYTM =0.12TimeCash FlowPVIFPV of CFPV*CF*T1$0.000.89286$0.00$0.002$1,000.000.79719$797.19$1,594.39Price =$797.19Numerator =$1,594.39Duration =2.0000= Numerator/Priced. How does the change in the current YTM affect the duration of the zero-coupon

13、 bond?Changing the yield-to-maturity does not affect the duration of the zero coupon bond.e. Why does the change in the YTM affect the coupon bond differently than the zero- coupon bond?Increasing the YTM on the coupon bond allows for a higher reinvestment income that more quickly recovers the initi

14、al investment. The zero-coupon bond has no cash flow until maturity.3.A one-year, $100,000 loan carries a market interest rate of 12 percent. The loan requires payment of accrued interest and one-half of the principal at the end of six months. The remaining principal and accrued interest are due at

15、the end of the year.a. What is the duration of this loan?Cash flow in 6 months = $100,000 x .12 x .5 + $50,000 = $56,000 interest and principal. Cash flow in 1 year = $50,000 x 1.06 = $53,000 interest and principal.TimeCash FlowPVIFCF*PVIFT*CF*CVIF1$56,0000.943396$52,830.19$52,830.192$53,0000.889996

16、$47,169.81$94,339.62D = $147,169.81$100,000.00Price = $100,000.00$147,169.81 = Numeratorx 1 = 0.735849 years2b. What will be the cash flows at the end of 6 months and at the end of the year?Cash flow in 6 months = $100,000 x .12 x .5 + $50,000 = $56,000 interest and principal. Cash flow in 1 year =

17、$50,000 x 1.06 = $53,000 interest and principal.c. What is the present value of each cash flow discounted at the market rate? What is the total present value?$56,000 1.06= $52,830.19 = PVCF1$53,000 (1.06)2= $47,169.81 = PVCF2=$100,000.00 = PV Total CFd. What proportion of the total present value of

18、cash flows occurs at the end of 6 months? What proportion occurs at the end of the year?Proportiont=.5 = $52,830.19 $100,000 x 100 = 52.830 percent. Proportiont=1 = $47,169.81 $100,000 x 100 = 47.169 percent.e. What is the weighted-average life of the cash flows on the loan?D = 0.5283 x 0.5 years +

19、0.47169 x 1.0 years = 0.26415 + 0.47169 = 0.73584 years.f.How does this weighted-average life compare to the duration calculated in part (a) above?The two values are the same.4.What is the duration of a five-year, $1,000 Treasury bond with a 10 percent semiannual coupon selling at par? Selling with

20、a YTM of 12 percent? 14 percent? What can you conclude about the relationship between duration and yield to maturity? Plot the relationship. Why does this relationship exist?Five-year Treasury BondPar value =$1,000Coupon = 0.10Semiannual paymentsYTM =0.10Maturity = 5TimeCash FlowPVIFPV of CFPV*CF*T0

21、.5$50.000.95238$47.62$23.81PVIF = 1/(1+YTM/2)(Time*2)1$50.000.90703$45.35$45.351.5$50.000.86384$43.19$64.792$50.000.8227$41.14$82.272.5$50.000.78353$39.18$97.943$50.000.74622$37.31$111.933.5$50.000.71068$35.53$124.374$50.000.67684$33.84$135.374.5$50.000.64461$32.23$145.045$1,050.000.61391$644.61$3,2

22、23.04Price =$1,000.00Par value =$1,000Coupon = 0.10SemiannualpaymentsYTM =0.12Maturity = 5TimeCash FlowPVIFPV of CFPV*CF*T0.5$50.000.9434$47.17$23.58DurationYTM1$50.000.89$44.50$44.504.05390.101.5$50.000.83962$41.98$62.974.01130.122$50.000.79209$39.60$79.213.96760.142.5$50.000.74726$37.36$93.413$50.

23、000.70496$35.25$105.743.5$50.000.66506$33.25$116.384$50.000.62741$31.37$125.484.5$50.000.5919$29.59$133.185$1,050.000.55839$586.31$2,931.57 .Price =$926.40Numerator =$3,716.03 Duration =4.0113=Numerator/PriceNumerator =$4,053.91 Duration =4.0539= Numerator/Price Five-year Treasury BondFive-year Trea

24、sury BondPar value =$1,000Coupon = 0.10Semiannual paymentsYTM =0.14Maturity = 5TimeCash FlowPVIFPV of CFPV*CF*T0.5$50.000.93458$46.73$23.361$50.000.87344$43.67$43.671.5$50.000.8163$40.81$61.222$50.000.7629$38.14$76.292.5$50.000.71299$35.65$89.123$50.000.66634$33.32$99.953.5$50.000.62275$31.14$108.98

25、4$50.000.58201$29.10$116.404.5$50.000.54393$27.20$122.395$1,050.000.50835$533.77$2,668.83Price =$859.53Numerator =$3,410.22 Duration =3.9676= Numerator/PriceDuration and YTM4.084.05394.044.01134.003.96763.963.920.100.12Yield to Maturity0.14YearsAs the yield to maturity increases, duration decreases

26、because of the reinvestment of interim cash flows at higher rates.5.Consider three Treasury bonds each of which has a 10 percent semiannual coupon and trades at par.a. Calculate the duration for a bond that has a maturity of 4 years, 3 years, and 2 years?Please see the calculations on the next page.

27、a.Four-year Treasury BondPar value =$1,000Coupon = 0.10Semiannual paymentsTimeYTM =0.10Cash FlowPVIFPV of CFMaturity = 4PV*CF*T0.5$50.000.952381$47.62$23.81PVIF = 1/(1+YTM/2)(Time*2)1$50.000.907029$45.35$45.351.5$50.000.863838$43.19$64.792$50.000.822702$41.14$82.272.5$50.000.783526$39.18$97.943$50.0

28、00.746215$37.31$111.933.5$50.000.710681$35.53$124.374$1,050.000.676839$710.68$2,842.73Price =$1,000.00Numerator =$3,393.19 Duration = 3.3932 = Numerator/PriceThree-year Treasury BondPar value =$1,000Coupon = 0.10Semiannual paymentsTimeYTM =Cash Flow0.10PVIFPV of CFMaturity = 3PV*CF*T0.5$50.000.95238

29、1$47.62$23.81PVIF =1/(1+YTM/2)(Time*2)1$50.000.907029$45.35$45.351.5$50.000.863838$43.19$64.792$50.000.822702$41.14$82.272.5$50.000.783526$39.18$97.943$1,050.000.746215$783.53$2,350.58Price =$1,000.00Numerator =$2,664.74 Duration= 2.6647 = Numerator/PriceTwo-year Treasury BondPar value =$1,000Coupon

30、 = 0.10Semiannual paymentsTimeYTM =Cash Flow0.10PVIFPV of CFMaturity = 2PV*CF*T0.5$50.000.952381$47.62$23.81PVIF =1/(1+YTM/2)(Time*2)1$50.000.907029$45.35$45.351.5$50.000.863838$43.19$64.792$1,050.000.822702$863.84$1,727.68Price =$1,000.00Numerator =$1,861.62 Duration= 1.8616 = Numerator/Priceb. Wha

31、t conclusions can you reach about the relationship of duration and the time to maturity? Plot the relationship.Duration and Maturity4.003.39323.002.66472.001.86161.000.0023Tim e to Maturity4YearsAs maturity decreases, duration decreases at a decreasing rate. Although the graph below does not illustr

32、ate with great precision, the change in duration is less than the change in time to maturity.DurationMaturityChange in Duration1.861622.664730.80313.393240.72856.A six-year, $10,000 CD pays 6 percent interest annually. What is the duration of the CD? What would be the duration if interest were paid

33、semiannually? What is the relationship of duration to the relative frequency of interest payments?Six-year CD Par value =$10,000Coupon = 0.06Annual paymentsYTM =0.06Maturity = 6TimeCash FlowPVIFPV of CFPV*CF*T1$600.000.94340$566.04$566.04PVIF =1/(1+YTM)(Time)2$600.000.89000$534.00$1,068.003$600.000.

34、83962$503.77$1,511.314$600.000.79209$475.26$1,901.025$600.000.74726$448.35$2,241.776$10,6000.70496$7,472.58$44,835.49Price =$10,000.00Numerator =$52,123.64 Duration= 5.2124 = Numerator/PriceSix-year CD Par value =$10,000Coupon =0.06Semiannual paymentsYTM =0.06Maturity =6TimeCash FlowPVIFPV of CFPV*C

35、F*T0.5$300.000.970874$291.26$145.63PVIF = 1/(1+YTM/2)(Time*2)1$300.000.942596$282.78$282.781.5$300.000.915142$274.54$411.812$300.000.888487$266.55$533.092.5$300.000.862609$258.78$646.963$300.000.837484$251.25$753.743.5$300.000.813092$243.93$853.754$300.000.789409$236.82$947.294.5$300.000.766417$229.

36、93$1,034.665$300.000.744094$223.23$1,116.145.5$300.000.722421$216.73$1,192.006$10,3000.701380$7,224.21$43,345.28Price =$10,000.00Numerator =$51,263.12Duration = 5.1263 = Numerator/PriceDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being recei

37、ved more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows.7.What is the duration of a consol bond that sells at a YTM of 8 percent? 10 percent? 12 percent? What is a consol bond? Would a consol trading at a YTM of 10 percent have a greater duration than a 20-y

38、ear zero-coupon bond trading at the same YTM? Why?A consol is a bond that pays a fixed coupon each year forever. A consolConsol Bondtrading at a YTM of 10 percent has a duration of 11 years, while a zero-YTMD = 1 + 1/Rcoupon bond trading at a YTM of 10 percent, or any other YTM, has a0.0813.50 years

39、duration of 20 years because no cash flows occur before the twentieth0.1011.00 yearsyear.0.129.33 years8.Maximum Pension Fund is attempting to balance one of the bond portfolios under its management. The fund has identified three bonds which have five-year maturities and which trade at a YTM of 9 pe

40、rcent. The bonds differ only in that the coupons are 7 percent, 9 percent, and 11 percent.a. What is the duration for each bond?Five-year BondPar value =$1,000Coupon = 0.07Annual paymentsYTM =0.09Maturity = 5TimeCash FlowPVIFPV of CFPV*CF*T1$70.000.917431$64.22$64.22PVIF =1/(1+YTM)(Time)2$70.000.841

41、680$58.92$117.843$70.000.772183$54.05$162.164$70.000.708425$49.59$198.365$1,070.000.649931$695.43$3,477.13Price =$922.21Numerator =$4,019.71 Duration= 4.3588 = Numerator/PriceFive-year BondPar value =$1,000Coupon = 0.09Annual paymentsYTM =0.09Maturity = 5TimeCash FlowPVIFPV of CFPV*CF*T1$90.000.9174

42、31$82.57$82.57PVIF =1/(1+YTM)(Time)2$90.000.841680$75.75$151.503$90.000.772183$69.50$208.494$90.000.708425$63.76$255.035$1,090.000.649931$708.43$3,542.13Price =$1,000.00Numerator =$4,239.72 Duration = 4.2397 = Numerator/PriceFive-year BondPar value =$1,000Coupon = 0.11Annual paymentsYTM =0.09Maturit

43、y = 5TimeCash FlowPVIFPV of CFPV*CF*T1$110.000.917431$100.92$100.92PVIF =1/(1+YTM)(Time)2$110.000.841680$92.58$185.173$110.000.772183$84.94$254.824$110.000.708425$77.93$311.715$1,110.000.649931$721.42$3,607.12Price =$1,077.79Numerator =$4,459.73 Duration= 4.1378 = Numerator/Priceb. What is the relat

44、ionship between duration and the amount of coupon interest that is paid? Plot the relationship.Duration and Coupon Rates4.35884.23974.13784.007%9%Coupon Rates 11%Duration decreases as the amount of coupon interest increases.YearsChange in DurationCouponDuration 4.35887%4.23979%-0.11914.137811%-0.101

45、99.An insurance company is analyzing three bonds and is using duration as the measure of interest rate risk. All three bonds trade at a YTM of 10 percent and have $10,000 par values. The bonds differ only in the amount of annual coupon interest that they pay: 8, 10, or 12 percent.a. What is the dura

46、tion for each five-year bond?Five-year BondPar value =$10,000Coupon = 0.08Annual paymentsYTM =0.10Maturity = 5TimeCash FlowPVIFPV of CFPV*CF*T1$800.000.909091$727.27$727.27PVIF =1/(1+YTM)(Time)2$800.000.826446$661.16$1,322.313$800.000.751315$601.05$1,803.164$800.000.683013$546.41$2,185.645$10,800.00

47、0.620921$6,705.95$33,529.75Price =$9,241.84Numerator =$39,568.14 Duration = 4.2814 = Numerator/PriceFive-year BondPar value = $10,000Coupon = 0.10Annual payments YTM =0.10Maturity = 5Time Cash FlowPVIFPV of CFPV*CF*T1$1,000.00 0.909091$909.09$909.09PVIF = 1/(1+YTM)(Time)2$1,000.00 0.826446$826.45$1,

48、652.893$1,000.00 0.751315$751.31$2,253.944$1,000.00 0.683013$683.01$2,732.055$11,000.00 0.620921$6,830.13$34,150.67Price =$10,000.00Numerator =$41,698.65 Duration = 4.1699 = Numerator/PriceFive-year BondPar value = $10,000Coupon = 0.12Annual payments YTM =0.10Maturity = 5Time Cash FlowPVIFPV of CFPV

49、*CF*T1$1,200.00 0.909091$1,090.91$1,090.91PVIF = 1/(1+YTM)(Time)2$1,200.00 0.826446$991.74$1,983.473$1,200.00 0.751315$901.58$2,704.734$1,200.00 0.683013$819.62$3,278.465$11,200.00 0.620921$6,954.32$34,771.59Price =$10,758.16Numerator =$43,829.17 Duration = 4.0740 = Numerator/PriceDuration and Coupo

50、n Rates4.504.28144.16994.07404.008%10%Coupon Rates 12%b.What is the relationship between duration and the amount of coupon interest that is paid?Duration decreases as the amount of coupon interest increases.YearsChange in DurationCouponDuration4.28147%4.16999%-0.11154.074011%-0.095910.You can obtain a loan for $100,000 at a rate of 10 percent for two years. You have a choice of either paying the principal at the end of the second year or amortizing the loan,