ࡱ> @B789:;<=>?xyXQ [ Ibjbj Q]jjrlddd$ZZZPfDx $ "   kD$$ dqN  < d J5|d:;   ОZ@J:B:;\0:GG:;Chapter Nine Interest Rate Risk II Chapter Outline Introduction Duration A General Formula for Duration The Duration of Interest Bearing Bonds The Duration of a Zero-Coupon Bond The Duration of a Consol Bond (Perpetuities) Features of Duration Duration and Maturity Duration and Yield Duration and Coupon Interest The Economic Meaning of Duration Semiannual Coupon Bonds Duration and Immunization Duration and Immunizing Future Payments Immunizing the Whole Balance Sheet of an FI Immunization and Regulatory Considerations Summary Appendix 9A: Difficulties in Applying the Duration Model to Real-World FI Balance Sheets Duration Matching can be Costly Immunization is a Dynamic Problem Large Interest Rate Changes and Convexity The Problem of the Flat Term Structure Floating-Rate Loans and Bonds Demand Deposits and Passbook Savings Mortgages and Mortgage-Backed Securities Futures, Options, Swaps, Caps, and Other Contingent Claims Solutions for End-of-Chapter Questions and Problems: Chapter Nine 1. 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 measures 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 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 markets. 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 spreadsheet program to assist in the calculations.) Coupon BondPar value =$1,000Coupon =0.10Annual paymentsYTM =0.08Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T1$100.00 0.92593$92.59 $92.59 2$1,100.00 0.85734$943.07 $1,886.15 Price =$1,035.67 Numerator =$1,978.74 Duration =1.9106= Numerator/PriceYTM =0.10TimeCash FlowPVIFPV of CFPV*CF*T1$100.00 0.90909$90.91 $90.91 2$1,100.00 0.82645$909.09 $1,818.18 Price =$1,000.00 Numerator =$1,909.09 Duration =1.9091= Numerator/PriceYTM =0.12TimeCash FlowPVIFPV of CFPV*CF*T1$100.00 0.89286$89.29 $89.29 2$1,100.00 0.79719$876.91 $1,753.83 Price =$966.20 Numerator =$1,843.11 Duration =1.9076= Numerator/Price b. How does the change in the current YTM affect the duration of this coupon bond? Increasing the yield-to-maturity decrease 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 =$1,000 Coupon =0.00YTM =0.08Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T1$0.00 0.92593$0.00 $0.00 2$1,000.00 0.85734$857.34 $1,714.68 Price =$857.34 Numerator =$1,714.68 Duration =2.0000= Numerator/PriceYTM =0.10TimeCash FlowPVIFPV of CFPV*CF*T1$0.00 0.90909$0.00 $0.00 2$1,000.00 0.82645$826.45 $1,652.89 Price =$826.45 Numerator =$1,652.89 Duration =2.0000= Numerator/PriceYTM =0.12TimeCash FlowPVIFPV of CFPV*CF*T1$0.00 0.89286$0.00 $0.00 2$1,000.00 0.79719$797.19 $1,594.39 Price =$797.19 Numerator =$1,594.39 Duration =2.0000= Numerator/Price d. How does the change in the current YTM affect the duration of the zero-coupon 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 initial 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 6 months. The remaining principal and accrued interest are due at 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. Time Cash Flow PVIF CF*PVIF T*CF*CVIF 1 $56,000 0.943396 $52,830.19 $52,830.19 2 $53,000 0.889996 $47,169.81 $94,339.62 Price = $100,000.00 $147,169.81 = Numerator  EMBED Equation.3 years b. 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 = $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 CF d. What proportion of the total present value of 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 + 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 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 =$1,000 Coupon =0.10Semiannual paymentsYTM =0.10Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T0.5$50.00 0.95238$47.62 $23.81 PVIF = 1/(1+YTM/2)^(Time*2)1$50.00 0.90703$45.35 $45.35 1.5$50.00 0.86384$43.19 $64.79 2$50.00 0.8227$41.14 $82.27 2.5$50.00 0.78353$39.18 $97.94 3$50.00 0.74622$37.31 $111.93 3.5$50.00 0.71068$35.53 $124.37 4$50.00 0.67684$33.84 $135.37 4.5$50.00 0.64461$32.23 $145.04 5$1,050.00 0.61391$644.61 $3,223.04 Price =$1,000.00 Numerator =$4,053.91 Duration =4.0539= Numerator/PriceFive-year Treasury BondPar value =$1,000 Coupon =0.10Semiannual paymentsYTM =0.12Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T0.5$50.00 0.9434$47.17 $23.58 DurationYTM1$50.00 0.89$44.50 $44.50 4.0539 0.101.5$50.00 0.83962$41.98 $62.97 4.0113 0.122$50.00 0.79209$39.60 $79.21 3.9676 0.142.5$50.00 0.74726$37.36 $93.41 3$50.00 0.70496$35.25 $105.74 3.5$50.00 0.66506$33.25 $116.38 4$50.00 0.62741$31.37 $125.48 4.5$50.00 0.5919$29.59 $133.18 5$1,050.00 0.55839$586.31 $2,931.57 .Price =$926.40 Numerator =$3,716.03 Duration =4.0113= Numerator/Price Five-year Treasury BondPar value =$1,000 Coupon =0.10Semiannual paymentsYTM =0.14Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T0.5$50.00 0.93458$46.73 $23.36 1$50.00 0.87344$43.67 $43.67 1.5$50.00 0.8163$40.81 $61.22 2$50.00 0.7629$38.14 $76.29 2.5$50.00 0.71299$35.65 $89.12 3$50.00 0.66634$33.32 $99.95 3.5$50.00 0.62275$31.14 $108.98 4$50.00 0.58201$29.10 $116.40 4.5$50.00 0.54393$27.20 $122.39 5$1,050.00 0.50835$533.77 $2,668.83 Price =$859.53 Numerator =$3,410.22 Duration =3.9676= Numerator/Price 5. Consider three Treasury bonds which each have 10 percent semiannual coupons and trade 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. a.Four-year Treasury Bond Par value =$1,000 Coupon =0.10Semiannual paymentsYTM =0.10Maturity =4TimeCash FlowPVIFPV of CFPV*CF*T0.5$50.00 0.952381$47.62 $23.81  PVIF = 1/(1+YTM/2)^(Time*2)1$50.00 0.907029$45.35 $45.35 1.5$50.00 0.863838$43.19 $64.79 2$50.00 0.822702$41.14 $82.27 2.5$50.00 0.783526$39.18 $97.94 3$50.00 0.746215$37.31 $111.93 3.5$50.00 0.710681$35.53 $124.37 4$1,050.00 0.676839$710.68 $2,842.73 Price =$1,000.00 Numerator =$3,393.19 Duration =3.3932= Numerator/PriceThree-year Treasury Bond Par value =$1,000 Coupon =0.10Semiannual paymentsYTM =0.10Maturity =3TimeCash FlowPVIFPV of CFPV*CF*T0.5$50.00 0.952381$47.62 $23.81  PVIF = 1/(1+YTM/2)^(Time*2)1$50.00 0.907029$45.35 $45.35 1.5$50.00 0.863838$43.19 $64.79 2$50.00 0.822702$41.14 $82.27 2.5$50.00 0.783526$39.18 $97.94 3$1,050.00 0.746215$783.53 $2,350.58 Price =$1,000.00 Numerator =$2,664.74 Duration=2.6647= Numerator/PriceTwo-year Treasury BondPar value =$1,000 Coupon =0.10Semiannual paymentsYTM =0.10Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T0.5$50.00 0.952381$47.62 $23.81  PVIF = 1/(1+YTM/2)^(Time*2)1$50.00 0.907029$45.35 $45.35 1.5$50.00 0.863838$43.19 $64.79 2$1,050.00 0.822702$863.84 $1,727.68 Price =$1,000.00 Numerator =$1,861.62 Duration=1.8616= Numerator/Price b. What conclusions can you reach about the relationship of duration and the time to maturity? Plot the relationship.  As maturity decreases, duration decreases at a decreasing rate. Although the graph below does not illustrate with great precision, the change in duration is less than the change in time to maturity.  6. 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 semiannually? What is the relationship of duration to the relative frequency of interest payments? Six-year CDPar value =$10,000 Coupon =0.06Annual paymentsYTM =0.06Maturity =6TimeCash FlowPVIFPV of CFPV*CF*T1$600.00 0.94340$566.04 $566.04  PVIF = 1/(1+YTM)^(Time)2$600.00 0.89000$534.00 $1,068.00 3$600.00 0.83962$503.77 $1,511.31 4$600.00 0.79209$475.26 $1,901.02 5$600.00 0.74726$448.35 $2,241.77 6$10,600 0.70496$7,472.58 $44,835.49 Price =$10,000.00 Numerator =$52,123.64 Duration=5.2124= Numerator/Price Six-year CDPar value =$10,000 Coupon =0.06Semiannual paymentsYTM =0.06Maturity =6TimeCash FlowPVIFPV of CFPV*CF*T0.5$300.00 0.970874$291.26 $145.63  PVIF = 1/(1+YTM/2)^(Time*2)1$300.00 0.942596$282.78 $282.78 1.5$300.00 0.915142$274.54 $411.81 2$300.00 0.888487$266.55 $533.09 2.5$300.00 0.862609$258.78 $646.96 3$300.00 0.837484$251.25 $753.74 3.5$300.00 0.813092$243.93 $853.75 4$300.00 0.789409$236.82 $947.29 4.5$300.00 0.766417$229.93 $1,034.66 5$300.00 0.744094$223.23 $1,116.14 5.5$300.00 0.722421$216.73 $1,192.00 6$10,300 0.701380$7,224.21 $43,345.28 Price =$10,000.00 Numerator =$51,263.12 Duration=5.1263= Numerator/Price Duration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being received 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-year zero-coupon bond trading at the same YTM? Why? A consol is a bond that pays a fixed coupon each year forever. A consol Consol Bond trading at a YTM of 10 percent has a duration of 11 years, while a zero- YTM D = 1 + 1/R coupon bond trading at a YTM of 10 percent, or any other YTM, has a 0.08 13.50 years duration of 20 years because no cash flows occur before the twentieth 0.10 11.00 years year. 0.12 9.33 years 8. 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 percent. 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 =$1,000 Coupon =0.07Annual paymentsYTM =0.09Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T1$70.00 0.917431$64.22 $64.22  PVIF = 1/(1+YTM)^(Time)2$70.00 0.841680$58.92 $117.84 3$70.00 0.772183$54.05 $162.16 4$70.00 0.708425$49.59 $198.36 5$1,070.00 0.649931$695.43 $3,477.13 Price =$922.21 Numerator =$4,019.71 Duration=4.3588= Numerator/PriceFive-year BondPar value =$1,000 Coupon =0.09Annual paymentsYTM =0.09Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T1$90.00 0.917431$82.57 $82.57  PVIF = 1/(1+YTM)^(Time)2$90.00 0.841680$75.75 $151.50 3$90.00 0.772183$69.50 $208.49 4$90.00 0.708425$63.76 $255.03 5$1,090.00 0.649931$708.43 $3,542.13 Price =$1,000.00 Numerator =$4,239.72 Duration=4.2397= Numerator/PriceFive-year BondPar value =$1,000 Coupon =0.11Annual paymentsYTM =0.09Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T1$110.00 0.917431$100.92 $100.92  PVIF = 1/(1+YTM)^(Time)2$110.00 0.841680$92.58 $185.17 3$110.00 0.772183$84.94 $254.82 4$110.00 0.708425$77.93 $311.71 5$1,110.00 0.649931$721.42 $3,607.12 Price =$1,077.79 Numerator =$4,459.73 Duration=4.1378= Numerator/Price b. What is the relationship between duration and the amount of coupon interest that is paid? Plot the relationship.  9. An insurance company is analyzing three bonds and is using duration as the measure of interest rate risk. The three bonds all 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 duration for each five-year bond? Five-year BondPar value =$10,000 Coupon =0.08Annual paymentsYTM =0.10Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T1$800.00 0.909091$727.27 $727.27  PVIF = 1/(1+YTM)^(Time)2$800.00 0.826446$661.16 $1,322.31 3$800.00 0.751315$601.05 $1,803.16 4$800.00 0.683013$546.41 $2,185.64 5$10,800.00 0.620921$6,705.95 $33,529.75 Price =$9,241.84 Numerator =$39,568.14 Duration=4.2814= Numerator/PriceFive-year BondPar value =$10,000 Coupon =0.10Annual paymentsYTM =0.10Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T1$1,000.00 0.909091$909.09 $909.09  PVIF = 1/(1+YTM)^(Time)2$1,000.00 0.826446$826.45 $1,652.89 3$1,000.00 0.751315$751.31 $2,253.94 4$1,000.00 0.683013$683.01 $2,732.05 5$11,000.00 0.620921$6,830.13 $34,150.67 Price =$10,000.00 Numerator =$41,698.65 Duration=4.1699= Numerator/Price Five-year BondPar value =$10,000 Coupon =0.12Annual paymentsYTM =0.10Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T1$1,200.00 0.909091$1,090.91 $1,090.91  PVIF = 1/(1+YTM)^(Time)2$1,200.00 0.826446$991.74 $1,983.47 3$1,200.00 0.751315$901.58 $2,704.73 4$1,200.00 0.683013$819.62 $3,278.46 5$11,200.00 0.620921$6,954.32 $34,771.59 Price =$10,758.16 Numerator =$43,829.17 Duration=4.0740= Numerator/Priceb. What is the relationship between duration and the amount of coupon interest that is paid? 10. 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, that is, paying interest and principal in equal payments each year. The loan is priced at par. a. What is the duration of the loan under both methods of payment? Two-year loan: Principal and interest at end of year two.Par value =100,000 Coupon =0.00No annual paymentsYTM =0.10Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T1$0.00 0.90909$0.00 $0.00 PVIF = 1/(1+YTM)^(Time)2$121,000 0.82645$100,000.0 200,000.00Price =$100,000.0 Numerator =200,000.00Duration=2.0000= Numerator/PriceTwo-year loan: Interest at end of year one, P & I at end of year two. Par value =100,000Coupon =0.10Annual paymentsYTM =0.10Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T1$10,000 0.909091$9,090.91 $9,090.91  PVIF = 1/(1+YTM)^(Time)2$110,000 0.826446$90,909.09 181,818.18Price =$100,000.0 Numerator =190,909.09Duration=1.9091= Numerator/PriceTwo-year loan: Amortized over two years.Amortized payment of $57.619.05Par value =100,000 Coupon =0.10YTM =0.10Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T1$57,619.05 0.909091$52,380.95 $52,380.95  PVIF = 1/(1+YTM)^(Time)2$57,619.05 0.826446$47,619.05 $95,238.10 Price =$100,000.0 Numerator =147,619.05 Duration=1.4762= Numerator/Price b. Explain the difference in the two results?  11. How is duration related to the interest elasticity of a fixed-income security? What is the relationship between duration and the price of the fixed-income security? Taking the first derivative of a bonds (or any fixed-income security) price (P) with respect to the yield to maturity (R) provides the following:  EMBED Equation.3  The economic interpretation is that D is a measure of the percentage change in price of a bond for a given percentage change in yield to maturity (interest elasticity). This equation can be rewritten to provide a practical application:  EMBED Equation.3  In other words, if duration is known, then the change in the price of a bond due to small changes in interest rates, R, can be estimated using the above formula. 12. You have discovered that the price of a bond rose from $975 to $995 when the YTM fell from 9.75 percent to 9.25 percent. What is the duration of the bond? We know  EMBED Equation.3  13. Calculate the duration of a 2-year, $1,000 bond that pays an annual coupon of 10 percent and trades at a yield of 14 percent. What is the expected change in the price of the bond if interest rates decline by 0.50 percent (50 basis points)? Two-year BondPar value =$1,000 Coupon =0.10Annual paymentsYTM =0.14Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T1$100.00 0.87719$87.72 $87.72  PVIF = 1/(1+YTM)^(Time)2$1,100.00 0.76947$846.41 $1,692.83 Price =$934.13 Numerator =$1,780.55 Duration=1.9061= Numerator/Price Expected change in price =  EMBED Equation.3 . This implies a new price of $941.94. The actual price using conventional bond price discounting would be $941.99. The difference of $0.05 is due to convexity, which was not considered in this solution. 14. The duration of an 11-year, $1,000 Treasury bond paying a 10 percent semiannual coupon and selling at par has been estimated at 6.9 years. a. What is the modified duration of the bond (Modified Duration = D/(1 + R))? MD = 6.9/(1 + .10/2) = 6.57 years b. What will be the estimated price change of the bond if market interest rates increase 0.10 percent (10 basis points)? If rates decrease 0.20 percent (20 basis points)? Estimated change in price = -MD x (R x P = -6.57 x 0.001 x $1,000 = -$6.57. Estimated change in price = -MD x (R x P = -6.57 x -0.002 x $1,000 = $13.14. c. What would be the actual price of the bond under each rate change situation in part (b) using the traditional present value bond pricing techniques? What is the amount of error in each case? Rate Price Actual Change Estimated Price Error + 0.001 $993.43 $993.45 $0.02 - 0.002 $1,013.14 $1,013.28 -$0.14 15. Suppose you purchase a five-year, 13.76 percent bond that is priced to yield 10 percent. a. Show that the duration of this annual payment bond is equal to four years. Five-year BondPar value =$1,000 Coupon =0.1376Annual paymentsYTM =0.10Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T1$137.60 0.909091$125.09 $125.09  PVIF = 1/(1+YTM)^(Time)2$137.60 0.826446$113.72 $227.44 3$137.60 0.751315$103.38 $310.14 4$137.60 0.683013$93.98 $375.93 5$1,137.60 0.620921$706.36 $3,531.80 Price =$1,142.53 Numerator =$4,570.40 Duration=4.0002= Numerator/Price b. Show that, if interest rates rise to 11 percent within the next year and that if your investment horizon is four years from today, you will still earn a 10 percent yield on your investment. Value of bond at end of year four: PV = ($137.60 + $1,000) ( 1.11 = $1,024.86. Future value of interest payments at end of year four: $137.60*FVIFn=4, i=11% = $648.06. Future value of all cash flows at n = 4: Coupon interest payments over four years $550.40 Interest on interest at 11 percent 97.66 Value of bond at end of year four $1,024.86 Total future value of investment $1,672.92 Yield on purchase of asset at $1,142.53 = $1,672.92*PVIVn=4, i=?% ( i = 10.002332%. c. Show that a 10 percent yield also will be earned if interest rates fall next year to 9 percent. Value of bond at end of year four: PV = ($137.60 + $1,000) ( 1.09 = $1,043.67. Future value of interest payments at end of year four: $137.60*FVIFn=4, i=9% = $629.26. Future value of all cash flows at n = 4: Coupon interest payments over four years $550.40 Interest on interest at 9 percent 78.86 Value of bond at end of year four $1,043.67 Total future value of investment $1,672.93 Yield on purchase of asset at $1,142.53 = $1,672.93*PVIVn=4, i=?% ( i = 10.0025 percent. 16. Consider the case where an investor holds a bond for a period of time longer than the duration of the bond, that is, longer than the original investment horizon. a. If market interest rates rise, will the return that is earned exceed or fall short of the original required rate of return? Explain. In this case the actual return earned would exceed the yield expected at the time of purchase. The benefits from a higher reinvestment rate would exceed the price reduction effect if the investor holds the bond for a sufficient length of time. b. What will happen to the realized return if market interest rates decrease? Explain. If market rates decrease, the realized yield on the bond will be less than the expected yield because the decrease in reinvestment earnings will be greater than the gain in bond value. c. Recalculate parts (b) and (c) of problem 15 above, assuming that the bond is held for all five years, to verify your answers to parts (a) and (b) of this problem. The case where interest rates rise to 11 percent, n = five years: Future value of interest payments at end of year five: $137.60*FVIFn=5, i=11% = $856.95. Future value of all cash flows at n = 5: Coupon interest payments over five years $688.00 Interest on interest at 11 percent 168.95 Value of bond at end of year five $1,000.00 Total future value of investment $1,856.95 Yield on purchase of asset at $1,142.53 = $1,856.95*PVIFn=5, i=?% ( i = 10.2012 percent. The case where interest rates fall to 9 percent, n = five years: Future value of interest payments at end of year five: $137.60*FVIFn=5, i=9% = $823.50. Future value of all cash flows at n = 5: Coupon interest payments over five years $688.00 Interest on interest at 9 percent 135.50 Value of bond at end of year five $1,000.00 Total future value of investment $1,823.50 Yield on purchase of asset at $1,142.53 = $1,823.50*PVIVn=5, i=?% ( i = 9.8013 percent. d. If either calculation in part (c) is greater than the original required rate of return, why would an investor ever try to match the duration of an asset with his investment horizon? The answer has to do with the ability to forecast interest rates. Forecasting interest rates is a very difficult task, one that most financial institution money managers are unwilling to do. For most managers, betting that rates would rise to 11 percent to provide a realized yield of 10.20 percent over five years is not a sufficient return to offset the possibility that rates could fall to 9 percent and thus give a yield of only 9.8 percent over five years. 17. Two banks are being examined by the regulators to determine the interest rate sensitivity of their balance sheets. Bank A has assets composed solely of a 10-year, 12 percent, $1 million loan. The loan is financed with a 10-year, 10 percent, $1 million CD. Bank B has assets composed solely of a 7-year, 12 percent zero-coupon bond with a current (market) value of $894,006.20 and a maturity (principal) value of $1,976,362.88. The bond is financed with a 10-year, 8.275 percent coupon, $1,000,000 face value CD with a YTM of 10 percent. The loan and the CDs pay interest annually, with principal due at maturity. a. If market interest rates increase 1 percent (100 basis points), how do the market values of the assets and liabilities of each bank change? That is, what will be the net affect on the market value of the equity for each bank? For Bank A, an increase of 100 basis points in interest rate will cause the market values of assets and liabilities to decrease as follows: Loan: $120*PVIVAn=10,i=13% + $1,000*PVIVn=10,i=13% = $945,737.57. CD: $100*PVIVAn=10,i=11% + $1,000*PVIVn=10,i=11% = $941,107.68. Therefore, the decrease in value of the asset was $4,629.89 less than the liability. For Bank B: Bond: $1,976,362.88*PVIVn=7,i=13% = $840,074.08. CD: $82.75*PVIVAn=10,i=11% + $1,000*PVIVn=10,i=11% = $839,518.43. The bond value decreased $53,932.12, and the CD value fell $54,487.79. Therefore, the decrease in value of the asset was $555.67 less than the liability. b. What accounts for the differences in the changes of the market value of equity between the two banks? The assets and liabilities of Bank A change in value by different amounts because the durations of the assets and liabilities are not the same, even though the face values and maturities are the same. For Bank B, the maturities of the assets and liabilities are different, but the current market values and durations are the same. Thus the change in interest rates causes the same (approximate) change in value for both liabilities and assets. c. Verify your results above by calculating the duration for the assets and liabilities of each bank, and estimate the changes in value for the expected change in interest rates. Summarize your results. Ten-year CD:Bank B(Calculation in millions)Par value =$1,000 Coupon =0.08Annual paymentsYTM =0.10Maturity =10TimeCash FlowPVIFPV of CFPV*CF*T1$82.75 0.909091$75.23 $75.23  PVIF = 1/(1+YTM)^(Time)2$82.75 0.826446$68.39 $136.78 3$82.75 0.751315$62.17 $186.51 4$82.75 0.683013$56.52 $226.08 5$82.75 0.620921$51.38 $256.91 6$82.75 0.564474$46.71 $280.26 7$82.75 0.513158$42.46 $297.25 8$82.75 0.466507$38.60 $308.83 9$82.75 0.424098$35.09 $315.85 10$1,082.75 0.385543$417.45 $4,174.47 Price =$894.006Numerator =$6,258.15 Duration=7.0001= Numerator/Price The duration for the CD of Bank B is calculated above to be 7.001 years. Since the bond is a zero-coupon, the duration is equal to the maturity of 7 years. Using the duration formula to estimate the change in value: Bond: (Value =  EMBED Equation.3  CD: (Value =  EMBED Equation.3  The difference in the change in value of the assets and liabilities for Bank B is $1,024.04 using the duration estimation model. The small difference in this estimate and the estimate found in part a above is due to the convexity of the two financial assets. The duration estimates for the loan and CD for Bank A are presented below: Ten-year Loan: Bank A(Calculation in millions)Par value =$1,000 Coupon =0.12Annual paymentsYTM =0.12Maturity =10TimeCash FlowPVIFPV of CFPV*CF*T1$120.00 0.892857$107.14 $107.14  PVIF = 1/(1+YTM)^(Time)2$120.00 0.797194$95.66 $191.33 3$120.00 0.711780$85.41 $256.24 4$120.00 0.635518$76.26 $305.05 5$120.00 0.567427$68.09 $340.46 6$120.00 0.506631$60.80 $364.77 7$120.00 0.452349$54.28 $379.97 8$120.00 0.403883$48.47 $387.73 9$120.00 0.360610$43.27 $389.46 10$1,120.00 0.321973$360.61 $3,606.10 Price =$1,000.00 Numerator =$6,328.25 Duration=6.3282= Numerator/Price Ten-year CD: Bank A(Calculation in millions)Par value =$1,000 Coupon =0.10Annual paymentsYTM =0.10Maturity =10TimeCash FlowPVIFPV of CFPV*CF*T1$100.00 0.909091$90.91 $90.91  PVIF = 1/(1+YTM)^(Time)2$100.00 0.826446$82.64 $165.29 3$100.00 0.751315$75.13 $225.39 4$100.00 0.683013$68.30 $273.21 5$100.00 0.620921$62.09 $310.46 6$100.00 0.564474$56.45 $338.68 7$100.00 0.513158$51.32 $359.21 8$100.00 0.466507$46.65 $373.21 9$100.00 0.424098$42.41 $381.69 10$1,100.00 0.385543$424.10 $4,240.98 Price =$1,000.00 Numerator =$6,759.02 Duration=6.7590= Numerator/Price Using the duration formula to estimate the change in value: Loan: (Value =  EMBED Equation.3  CD: (Value =  EMBED Equation.3  The difference in the change in value of the assets and liabilities for Bank A is $4,943.66 using the duration estimation model. The small difference in this estimate and the estimate found in part a above is due to the convexity of the two financial assets. The reason the change in asset values for Bank A is considerably larger than for Bank B is because of the difference in the durations of the loan and CD for Bank A. 18. If you use duration only to immunize your portfolio, what three factors affect changes in the net worth of a financial institution when interest rates change? The change in net worth for a given change in interest rates is given by the following equation:  EMBED Equation.3  Thus, three factors are important in determining (E. 1) [DA - D L k] or the leveraged adjusted duration gap. The larger this gap, the more exposed is the FI to changes in interest rates. 2) A, or the size of the FI. The larger is A, the larger is the exposure to interest rate changes. 3) R/1 + R, or interest rate shocks. The larger is the shock, the larger is the exposure. 19. Financial Institution XY has assets of $1 million invested in a 30-year, 10 percent semiannual coupon Treasury bond selling at par. The duration of this bond has been estimated at 9.94 years. The assets are financed with equity and a $900,000, 2-year, 7.25 percent semiannual coupon capital note selling at par. a. What is the leverage-adjusted duration gap of Financial Institution XY? The duration of the capital note is 1.8975 years. Two-year Capital Note Par value =$900 Coupon =0.0725Semiannual paymentsYTM =0.0725Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T0.5$32.63 0.965018$31.48 $15.74  PVIF = 1/(1+YTM/2)^(Time*2)1$32.63 0.931260$30.38 $30.38 1.5$32.63 0.898683$29.32 $43.98 2$932.63 0.867245$808.81 $1,617.63 Price =$900.00 Numerator =$1,707.73 Duration=1.8975= Numerator/Price The leverage-adjusted duration gap can be found as follows:  EMBED Equation.3  b. What is the impact on equity value if the relative change in all market interest rates is a decrease of 20 basis points? Note, the relative change in interest rates is (R/(1+R/2) = -0.0020. The change in net worth using leverage adjusted duration gap is given by:  EMBED Equation.3  c. Using the information that you calculated in parts (a) and (b), infer a general statement about the desired duration gap for a financial institution if interest rates are expected to increase or decrease. If the FI wishes to be immune from the effects of interest rate risk, that is, either positive or negative changes in interest rates, a desirable leverage-adjusted duration gap (LADG) is zero. If the FI is confident that interest rates will fall, a positive LADG will provide the greatest benefit. If the FI is confident that rates will increase, then negative LADG would be beneficial. d. Verify your inference by calculating the change in market value of equity assuming that the relative change in all market interest rates is an increase of 30 basis points.  EMBED Equation.3  e. What would the duration of the assets need to be to immunize the equity from changes in market interest rates? Immunizing the equity from changes in interest rates requires that the LADG be 0. Thus, (DA-DLk) = 0 ( DA = DLk, or DA = 0.9*1.8975 = 1.70775 years. 20. The balance sheet for Gotbucks Bank, Inc. (GBI) is presented below ($ millions): Assets Liabilities and Equity Cash $30 Core deposits $20 Federal funds 20 Federal funds 50 Loans (floating) 105 Euro CDs 130 Loans (fixed) 65 Equity 20 Total assets $220 Total liabilities & equity $220 NOTES TO THE BALANCE SHEET: The Fed funds rate is 8.5 percent, the floating loan rate is LIBOR + 4 percent, and currently LIBOR is 11 percent. Fixed rate loans have five-year maturities, are priced at par, and pay 12 percent annual interest. Core deposits are fixed-rate for 2 years at 8 percent paid annually. Euros currently yield 9 percent. a. What is the duration of the fixed-rate loan portfolio of Gotbucks Bank? Five-year Loan Par value =$1,000 Coupon =0.1200Annual paymentsYTM =0.12Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T1$120.00 0.892857$107.14 $107.14  PVIF = 1/(1+YTM)^(Time)2$120.00 0.797194$95.66 $191.33 3$120.00 0.711780$85.41 $256.24 4$120.00 0.635518$76.26 $305.05 5$1,120.00 0.567427$635.52 $3,177.59 Price =$1,000.00 Numerator =$4,037.35 Duration=4.0373= Numerator/Price The duration is 4.037 years. b. If the duration of the floating-rate loans and fed funds is 0.36 years, what is the duration of GBIs assets? DA = [30(0) + 65(4.037) + 125(.36)]/220 = 1.397 years c. What is the duration of the core deposits if they are priced at par? Two-year Core DepositsPar value =$1,000 Coupon =0.08Annual paymentsYTM =0.08Maturity =2TimeCash FlowPVIFPV of CFPV*CF*T1$80.00 0.92593$74.07 $74.07  PVIF = 1/(1+YTM)^(Time)2$1,080.00 0.85734$925.93 $1,851.85 Price =$1,000.00 Numerator =$1,925.93 Duration=1.9259= Numerator/Price The duration of the core deposits is 1.926 years. d. If the duration of the Euro CDs and Fed funds liabilities is 0.401 years, what is the duration of GBIs liabilities? DL = [20*(1.926) + 180*(.401)]/200 = .5535 years e. What is GBIs duration gap? What is its interest rate risk exposure? GBIs leveraged adjusted duration gap is: 1.397 - 200/220 * (.5535) = .8938 years f. What is the impact on the market value of equity if the relative change in all market interest rates is an increase of 1 percent (100 basis points)? Note, the relative change in interest rates is ((R/(1+R)) = 0.01. Since GBIs duration gap is positive, an increase in interest rates will lead to a decline in net worth. For a 1 percent increase, the change in net worth is: E = -0.8938 * (0.01) * $220 = -$1,966,360 (new net worth will be $18,033,640). g. What is the impact on the market value of equity if the relative change in all market interest rates is a decrease of 0.5 percent (-50 basis points)? Since GBIs duration gap is positive, an decrease in interest rates will lead to an increase in net worth. For a 0.5 percent decrease, the change in net worth is: E = -0.8938 * (-0.005) * $220 = $983,180 (new net worth will be $20,983,180). f. What variables are available to GBI to immunize the bank? How much would each variable need to change to get DGAP equal to 0? Immunization requires the bank to have a leverage-adjusted duration gap of 0.0. Therefore, GBI could reduce the duration of its assets to 0.5535 years by using more fed funds and floating rate loans. Or GBI could use a combination of reducing asset duration and increasing liability duration in such a manner that LADG is 0.0. 21. Hands Insurance Company issued a $90 million, 1-year, zero-coupon note at 8 percent add-on annual interest (paying one coupon at the end of the year). The proceeds were used to fund a $100 million, 2-year commercial loan at 10 percent annual interest. Immediately after these transactions were simultaneously closed, all market interest rates increased 1.5 percent (150 basis points). a. What is the true market value of the loan investment and the liability after the change in interest rates? The market value of the loan declined by $2,551,830.92 million, to $97.448 million. MVA = $10,000,000*PVIFAn=2, i=11.5% + $100,000,000* PVIFn=2, i=11.5% = $97,448,169.08 The market value of the note declined $1,232,876.71 to $88.767 million. MVL = $97,200,000* PVIFn=1, i=9.5% = $88,767,123.29 b. What impact did these changes in market value have on the market value of the equity? (E = (A - (L = -$2,551,830.92 (-$1,232,876.71) = -$1,313,954.21. The increase in interest rates caused the asset to decrease in value more than the liability which caused the value of the net worth to decrease by $1,313,954.21. c. What was the duration of the loan investment and the liability at the time of issuance? The duration of the loan investment is 1.909 years. Note: The calculation for this loan is shown in problem 2, second example. The duration of the liability is one year since it is a zero-coupon note. d. Use these duration values to calculate the expected change in the value of the loan and the liability for the predicted increase of 1.5 percent in interest rates. The approximate change in the market value of the loan for a 150 basis points change is:  EMBED Equation.3 . The expected market value of the loan using the above formula is $97,396,818.18, or $97.400 million. The approximate change in the market value of the note for a 150 basis points change is:  EMBED Equation.3 . The expected market value of the note using the above formula is $88,750,000, or $88.750 million. e. What was the duration gap of Hands Insurance Company after the issuance of the asset and note? The leverage-adjusted duration gap was [1.909 (0.9)1.0] = 1.009 years. f. What was the change in equity value forecast by this duration gap for the predicted increase in interest rates of 1.5 percent? (MVE = -1.009*[0.015/(1.10)]*$100,000,000 = -$1,375,909. Note that this calculation assumes that the change in interest rates is relative to the rate on the loan. Further, this estimated change in net worth compares with the estimates above in part (d) as follows: (MVE = (MVA - (MVL = -$2,603,182 (-$1,250,000) = -$1,353,182. g. If the interest rate prediction had been available during the time period in which the loan and the liability were being negotiated, what suggestions would you offer to reduce the possible effect on the equity of the company? What are the difficulties in implementing your ideas? Obviously the duration of the loan could be shortened relative to the liability, or the liability duration could be lengthened relative to the loan, or some combination of both. Shortening the loan duration would mean the possible use of variable rates, or some earlier payment of principal as was demonstrated in problem 10. The duration of the liability can not be lengthened without extending the maturity life of the note. In either case, the loan officer may have been up against market or competitive constraints in that the borrower or investor may have had other options. Other methods to reduce the interest rate risk under conditions of this nature include using derivatives such as options, futures, and swaps. 22. The following balance sheet information is available (amounts in $ thousands and duration in years) for a financial institution: Amount Duration T-bills $90 0.50 T-notes 55 0.90 T-bonds 176 x Loans 2,274 7.00 Deposits 2,092 1.00 Federal funds 238 0.01 Equity 715 Treasury bonds are 5-year maturities paying 6 percent semiannually and selling at par. a. What is the duration of the T-bond portfolio? 4.393 years as shown below. Treasury BondPar value =$176 Coupon =0.06Semiannual paymentsYTM =0.06Maturity =5TimeCash FlowPVIFPV of CFPV*CF*T0.5$5.28 0.97087$5.13 $2.56 1$5.28 0.94260$4.98 $4.98 1.5$5.28 0.91514$4.83 $7.25 2$5.28 0.88849$4.69 $9.38 2.5$5.28 0.86261$4.55 $11.39 3$5.28 0.83748$4.42 $13.27 3.5$5.28 0.81309$4.29 $15.03 4$5.28 0.78941$4.17 $16.67 4.5$5.28 0.76642$4.05 $18.21 5$181.28 0.74409$134.89 $674.45 .Price =$176.00 Numerator =$773.18 Duration=4.3931= Numerator/Price b. What is the average duration of all the assets? [(.5)(90) + (.9)(55) + (4.393)(176) + (7)(2724)]/3045 = 6.55 years c. What is the average duration of all the liabilities? [(1)(2092) + (0.01)(238)]/2330 = 0.90 years d. What is the leverage-adjusted duration gap? What is the interest rate risk exposure? DG = DA - kDL = 6.55 - (2330/3045)(0.90) = 5.86 years The duration gap is positive, indicating that an increase in interest rates will lead to a decline in net worth. e. What is the forecast impact on the market value of equity caused by a relative upward shift in the entire yield curve of 0.5 percent [i.e., (R/(1+R) = 0.0050]? The market value of the equity will change by the following: MVE = -DG * (A) * R/(1 + R) = -5.86(3045)(0.0050) = -$89.22. The loss in equity of $89,220 will reduce the equity (net worth) to $625,780. f. If the yield curve shifted downward by 0.25 percent (i.e., (R/(1+R) = -0.0025), what is the forecasted impact on the market value of equity? The change in the value of equity is MVE = -5.86(3045)(-0.0025) = $44,610. Thus, the market value of equity (net worth) will increase by $44,610, to $759,610. g. What variables are available to the financial institution to immunize the balance sheet? How much would each variable need to change to get DGAP equal to 0? Immunization requires the bank to have a leverage-adjusted duration gap of 0.0. Therefore, the FI could reduce the duration of its assets to 0.6887 years by using more T-bills and floating rate loans. Or the FI could try to increase the duration of its deposits possibly by using fixed-rate CDs with a maturity of 3 or 4 years. Finally, the FI could use a combination of reducing asset duration and increasing liability duration in such a manner that LADG is 0.0. This duration gap of 5.86 years is quite large and it is not likely that the FI will be able to reduce it to zero by using only balance sheet adjustments. For example, even if the FI moved all of its loans into T-bills, the duration of the assets still would exceed the duration of the liabilities after adjusting for leverage. This adjustment in asset mix would imply foregoing a large yield advantage from the loan portfolio relative to the T-bill yields in most economic environments. 23. Assume that a goal of the regulatory agencies of financial institutions is to immunize the ratio of equity to total assets, that is, ((E/A) = 0. Explain how this goal changes the desired duration gap for the institution. Why does this differ from the duration gap necessary to immunize the total equity? How would your answers change to part (h) in problem 20, or part (g) in problem 22, if immunizing equity to total assets was the goal? In this case the duration of the assets and liabilities should be equal. Thus if (E = (A, then by definition the leveraged adjusted duration gap is positive, since (E would exceed k(A by the amount of (1 k), and the FI would face the risk of increases in interest rates. In reference to problems 20 and 22, the adjustments on the asset side of the balance sheet would not need to be as strong, although the difference likely would not be large if the FI in question is a depository institution such as a bank or S&L. The following questions and problems are based on material in the appendix to the chapter. 24. Identify and discuss three criticisms of using the duration model to immunize the portfolio of a financial institution. The three criticisms are: a Immunization is a dynamic problem because duration changes over time. Thus, it is necessary to rebalance the portfolio as the duration of the assets and liabilities change over time. b Duration matching can be costly because it is not easy to restructure the balance sheet periodically, especially for large FIs. c Duration is not an appropriate tool for immunizing portfolios when the expected interest rate changes are large because of the existence of convexity. Convexity exists because the relationship between bond price changes and interest rate changes is not linear, which is assumed in the estimation of duration. Using convexity to immunize a portfolio will reduce the problem. 25. In general, what changes have occurred in the financial markets that allow financial institutions to more rapidly and efficiently restructure their balance sheets to meet desired goals? Why is it critical for an investment manager who has a portfolio immunized to match a desired investment horizon to rebalance the portfolio periodically? Why is convexity a desirable feature to be captured in a portfolio of assets? What is convexity? The growth of purchased funds markets, asset securitization, and loan sales markets have increased considerably the speed of major balance sheet restructurings. Further, as these markets have developed, the cost of the necessary transactions has also decreased. Finally, the growth and development of the derivative markets provides significant alternatives to managing the risk of interest rate movements only with on-balance sheet adjustments. Assets approach maturity at a different rate of speed than the duration of the same assets approaches zero. Thus, after a period of time, a portfolio or asset that was immunized against interest rate risk will no longer be immunized. In fact, portfolio duration will exceed the remaining time in the investment or target horizon, and changes in interest rates could prove costly to the institution. Convexity is a property of fixed-rate assets that reflects nonlinearity in the reflection of price-rate relationships. This characteristic is similar to buying insurance to cover part of the interest rate risk faced by the FI. The more convex is a given asset, the more insurance against interest rate changes is purchased. 26. A financial institution has an investment horizon of 2 years, 9.5 months. The institution has converted all assets into a portfolio of 8 percent, $1,000, 3-year bonds that are trading at a YTM of 10 percent. The bonds pay interest annually. The portfolio manager believes that the assets are immunized against interest rate changes. a. Is the portfolio immunized at the time of bond purchase? What is the duration of the bonds? Three-year BondsPar value =$1,000 Coupon =0.08Annual paymentsYTM =0.10Maturity =3TimeCash FlowPVIFPV of CFPV*CF*T1$80.00 0.90909$72.73 $72.73  PVIF = 1/(1+YTM)^(Time)2$80.00 0.82645$66.12 $132.23 3$1,080 0.75131$811.42 $2,434.26 Price =$950.26 Numerator =$2,639.22 Duration=2.7774= Numerator/Price The bonds have a duration of 2.7774 years, which is 33.33 months. For practical purposes, the bond investment horizon was immunized at the time of purchase. b. Will the portfolio be immunized one year later? After one year, the investment horizon will be 1 year, 9.5 months. At this time, the bonds will have a duration of 1.9247 years, or 1 year, 11+ months. Thus the bonds will no longer be immunized. Two-year BondsPar value =$1,000 Coupon =0.08Annual paymentsYTM =0.10Maturity =3TimeCash FlowPVIFPV of CFPV*CF*T1$80.00 0.90909$72.73 $72.73  PVIF = 1/(1+YTM)^(Time)2$1,080 0.82645$892.56 $1,785.12 Price =$965.29 Numerator =$1,857.85 Duration=1.9247= Numerator/Price c. Assume that one-year, 8 percent zero-coupon bonds are available in one year. What proportion of the original portfolio should be placed in zeros to rebalance the portfolio? The investment horizon is 1 year, 9.5 months, or 21.5 months. Thus, the proportion of bonds that should be placed in the zeros can be determined by the following analysis: 21.5 months = X*12 months + (1-X)*23 months ( X = 13.6 percent Thus 13.6 percent of the bond portfolio should be placed in the zeros after one year. 27. MLK Bank has an asset portfolio that consists of $100 million of 30-year, 8 percent coupon, $1,000 bonds that sell at par. a. What will be the bonds new prices if market yields change immediately by ( 0.10 percent? What will be the new prices if market yields change immediately by ( 2.00 percent? At +0.10%: Price = $80*PVIFAn=30, i=8.1% + $1,000* PVIFn=30, i=8.1% = $988.85 At 0.10%: Price = $80*PVIFAn=30, i=7.9% + $1,000* PVIFn=30, i=7.9% = $1,011.36 At +2.0%: Price = $80*PVIFAn=30, i=10% + $1,000* PVIFn=30, i=10% = $811.46 At 2.0%: Price = $80*PVIFAn=30, i=6.0% + $1,000* PVIFn=30, i=6.0% = $1,275.30 b. The duration of these bonds is 12.1608 years. What are the predicted bond prices in each of the four cases using the duration rule? What is the amount of error between the duration prediction and the actual market values? (P = -D*[(R/(1+R)]*P At +0.10%: (P = -12.1608*0.001/1.08*$1,000 = -$11.26 ( P = $988.74 At -0.10%: (P = -12.1608*-0.001/1.08*$1,000 = $11.26 ( P = $1,011.26 At +2.0%: (P = -12.1608*0.02/1.08*$1,000 = -$225.20 ( P = $774.80 At -2.0%: (P = -12.1608*-0.02/1.08*$1,000 = $225.20 ( P = $1,225.20 Price Price Market Duration Amount Determined Estimation of Error At +0.10%: $988.85 $988.74 $0.11 At -0.10%: $1,011.36 $1,011.26 $0.10 At +2.0%: $811.46 $774.80 $36.66 At -2.0%: $1,275.30 $1,225.20 $50.10 c. Given that convexity is 212.4, what are the bond price predictions in each of the four cases using the duration plus convexity relationship? What is the amount of error in these predictions? (P = {-D*[(R/(1+R)] + *CX*((R)2}*P At +0.10%: (P = {-12.1608*0.001/1.08 + 0.5*212.4*(0.001)2}*$1,000 = -$11.15 At -0.10%: (P = {-12.1608*-0.001/1.08 + 0.5*212.4*(-0.001)2}*$1,000 = $11.366 At +2.0%: (P = {-12.1608*0.02/1.08 + 0.5*212.4*(0.02)2}*$1,000 = -$182.72 At -2.0%: (P = {-12.1608*-0.02/1.08 + 0.5*212.4*(-0.02)2}*$1,000 = $267.68 (Price Price Price Duration & Duration & Market Convexity Convexity Amount Determined Estimation Estimation of Error At +0.10%: $988.85 -$11.15 $988.85 $0.00 At -0.10%: $1,011.36 $11.37 $1,011.37 $0.01 At +2.0%: $811.46 -$182.72 $817.28 $5.82 At -2.0%: $1,275.30 $267.68 $1,267.68 $7.62 d. Diagram and label clearly the results in parts (a), (b) and (c).  The profiles for the estimates based on only ( 0.10 percent changes in rates are very close together and do not show clearly in a graph. However, the profile relationship would be similar to that shown above for the ( 2.0 percent changes in market rates. 28. Estimate the convexity for each of the following three bonds which all trade at YTM of 8 percent and have face values of $1,000. A 7-year, zero-coupon bond. A 7-year, 10 percent annual coupon bond. A 10-year, 10 percent annual coupon bond that has a duration value of 6.994 (( 7) years. (Market Value (Market Value Capital Loss + Capital Gain at 8.01 percent at 7.99 percent Divided by Original Price 7-year zero -0.37804819 0.37832833 0.00000048 7-year coupon -0.55606169 0.55643682 0.00000034 10-year coupon -0.73121585 0.73186329 0.00000057 Convexity = 108 * (Capital Loss + Capital Gain) Original Price at 8.00 percent 7-year zero CX = 100,000,000*0.00000048 = 48 7-year coupon CX = 100,000,000*0.00000034 = 34 10-year coupon CX = 100,000,000*0.00000057 = 57 An alternative method of calculating convexity for these three bonds using the following equation is illustrated at the end of this problem and onto the following page.  EMBED Equation.3  Rank the bonds in terms of convexity, and express the convexity relationship between zeros and coupon bonds in terms of maturity and duration equivalencies. Ranking, from least to most convexity: 7-year coupon bond, 7-year zero, 10-year coupon Convexity relationships: Given the same yield-to-maturity, a zero-coupon bond with the same maturity as a coupon bond will have more convexity. Given the same yield-to-maturity, a zero-coupon bond with the same duration as a coupon bond will have less convexity. Zero Coupon BondPar value =$1,000 Coupon =0YTM =0.08Maturity =7TimeCash FlowPVIFPV of CFPV*CF*T*(1+T)*(1+R)^21$0.00 0.92593$0.00 $0.00 $0.00 2$0.00 0.85734$0.00 $0.00 $0.00 3$0.00 0.79383$0.00 $0.00 $0.00 4$0.00 0.73503$0.00 $0.00 $0.00 5$0.00 0.68058$0.00 $0.00 $0.00 6$0.00 0.63017$0.00 $0.00 $0.00 71,000.00 0.58349$583.49 4,084.4332,675.46Price =$583.49 32,675.46680.58Numerator=$4,084.43 Duration=7.0000Convexity=48.0117-year Coupon BondPar value =$1,000 Coupon =0.1YTM =0.08Maturity =7TimeCash FlowPVIFPV of CFPV*CF*T*(1+T)*(1+R)^21$100.00 0.925926$92.59 $92.59 185.19 2$100.00 0.85734$85.73 $171.47 514.40 3$100.00 0.79383$79.38 $238.15 952.60 4$100.00 0.73503$73.50 $294.01 1,470.06 5$100.00 0.68058$68.06 $340.29 2,041.75 6$100.00 0.63017$63.02 $378.10 2,646.71 71,100.00 0.58349$641.84 $4,492.88 35,943.01 Price =$1,104.13 43,753.72 1287.9Numerator=$6,007.49 Duration=5.4409Convexity=33.97410-year Coupon BondPar value =$1,000 Coupon =0.1YTM =0.08Maturity =10TimeCash FlowPVIFPV of CFPV*CF*T*(1+T)*(1+R)^21$100.00 0.925926$92.59 $92.59 185.192$100.00 0.857339$85.73 $171.47 514.403$100.00 0.793832$79.38 $238.15 952.604$100.00 0.735030$73.50 $294.01 1470.065$100.00 0.680583$68.06 $340.29 2041.756$100.00 0.630170$63.02 $378.10 2646.717$100.00 0.583490$58.35 $408.44 3267.558$100.00 0.540269$54.03 $432.22 3889.949$100.00 0.500249$50.02 $450.22 4502.2410$1,100.0 0.463193$509.51 5,095.13 56046.41Price =1,134.20 75516.841322.9Numerator=7,900.63 Duration=6.9658Convexity=57.08329. A 10-year, 10 percent annual coupon, $1,000 bond trades at a YTM of 8 percent. The bond has a duration of 6.994 years. What is the modified duration of this bond? What is the practical value of calculating modified duration? Does modified duration change the result in using the duration relationship to estimate price sensitivity? Modified duration = Duration/(1+ R) = 6.994/1.08 = 6.4759. Some practitioners find this value easier to use because the percentage change in value can be estimated simply by multiplying the existing value times the basis point change in interest rates rather than by the relative change in interest rates. Using modified duration will not change the estimated price sensitivity of the asset. Additional Examples for Chapter 7 This example is to estimate both the duration and convexity of a 6-year bond paying 5 percent coupon annually and the annual yield to maturity is 6 percent. 6-year Coupon BondPar value =$1,000 Coupon =0.05YTM =0.06Maturity =6TimeCash FlowPVIFPV of CFPV*CF*T*(1+T)*(1+R)^21$50.00 0.94340$47.17 $47.17 $94.34 2$50.00 0.89000$44.50 $89.00 $267.00 3$50.00 0.83962$41.98 $125.94 $503.77 4$50.00 0.79209$39.60 $158.42 $792.09 5$50.00 0.74726$37.36 $186.81 1,120.89 6$1,050.00 0.70496$740.21 $4,441.25 31,088.76 Price =$950.83 33,866.85 1068.3Numerator =$5,048.60 Duration=5.3097Convexity=31.7 Using the textbook method: CX = 108 [(950.3506-950.8268)/950.8268 + (951.3032-950.8268)/950.8268] = 108[-0.0005007559 + 0.0005501073] = 31.70 What is the effect of a 2 percent increase in interest rates, from 6 percent to 8 percent? Using Present Values, the percentage change is: = ($950.8268 - $861.3136)/ $950.8268 = -9.41% Using the duration formula: MVA = -D*R/(1 + R) + 0.5CX(R)2 = -5.3097*[(0.02)/1.06] + 0.5(31.7)(0.02)2 = -0.1002 + .0063 = -9.38% Adding convexity adds more precision. Duration alone would have given the answer of -10.02%. PAGE  PAGE 84  EMBED Excel.Sheet.8   EMBED Excel.Sheet.8   EMBED Excel.Sheet.8   EMBED Excel.Sheet.8   EMBED Excel.Sheet.8   EMBED Excel.Sheet.8  As the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows at higher rates. Change inDurationMaturityDuration1.8616 22.6647 30.80313.3932 40.7285 Duration decreases as the amount of coupon interest increases. Change in Duration Coupon Duration 7% 4.2397 9% -0.1191 4.1378 11% -0.1019 Duration decreases as the amount of coupon interest increases. Change in Duration Coupon Duration 4.2814 7% 4.1699 9% -0.1115 4.0740 11% -0.0959 Duration decreases dramatically when a portion of the principal is repaid at the end of year one. Duration often is described as the weighted-average maturity of an asset. If more weight is given to early payments, the effective maturity of the asset is reduced. 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Change in1Although the graph does not illustrate with great2precision, the change in duration is less than thechange in time to maturity. Problem 6 Six-year CDPVIF = 1/(1+YTM)^(Time)oDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being greceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. Problem 7Semiannual paymentsAnnual payments Consol Bond D = 1 + 1/RlA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percentnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM,Qhas a duration of 20 years because no cash flows occur before the twentieth year. Problem 8Five-year Bond*Duration decreases as the amount of couponinterest increases. 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" &# %" $ $#  ( &" &#   " |Chart3&Sheet12Sheet23Sheet3bZ  3  @@   P%Solutions to end-of-chapter problems.Time Par valueCouponMaturity Cash FlowYTMPVIFPV of CFPrice =PV*CF*T Numerator = Duration =a. Coupon BondZero Coupon BondHb. Increasing the yield-to-maturity decreases the duration of the bond.Jd. Changing the YTM does not affect the duration of the zero coupon bond.\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.Ch. 9. Problem 2.= Numerator/Price Problem 4Five-year Treasury BondiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows at higher rates.Duration.PVIF = 1/(1+YTM/2)^(Time*2) Problem 5Four-year Treasury BondThree-year Treasury BondTwo-year Treasury Bondb. Change in Problem 6 Six-year CDPVIF = 1/(1+YTM)^(Time)oDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being greceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. Problem 7Semiannual paymentsAnnual payments Consol Bond D = 1 + 1/RlA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percentnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM,Qhas a duration of 20 years because no cash flows occur before the twentieth year. Problem 8Five-year Bond*Duration decreases as the amount of couponinterest increases. Problem 9 Problem 10No annual paymentsAmortized payment of $57.619.059Two-year loan: Principal and interest at end of year two.ETwo-year loan: Interest at end of year one, P & I at end of year two.(Two-year loan: Amortized over two years. Repayment ProvisionsP&I @ 2I@1Amortize5Duration decreases dramatically when a portion of the;principal is repaid at the end of year one. Duration often:is described as the weighted average maturity of an asset,7so if more weight is given to early payments, then the *effective maturity of the loan is reduced. Problem 13 Two-year BondCoupon = Maturity = Par value =YTM =2b. As maturity decreases, duration decreases at a decreasing rate.@Although the graph does not illustrate with great precision, the?change in duration is less than the change in time to maturity.JI t)cO,Dw:8*,A/*|)|HblAA?*|xbM_*|)|b`b,D00sT0b| ~Xj}bC* 0sT0bbXj}bgY)|* 0sT0bbbFI,D0@bXwbTbl`IS0bt0XwbblA_BA?JbXw)|XwLbblA_BA?J0bXw)|U)|UA?hbDs^4EJhb@XwXj}4Eb@8E * l}0sT0 b*W0sT04b`~.Xj}!Pv~ C0{T00sT0=0|8Pb8XbX0~0{T0dbtb0{T0Ub|b00sT00sT0%V00sT00{T0Ub0bbb00bb620b0b10bL05&b LP Pb?  I"PIFF CF??F*3`  '`  (`  *`  +`  ,`  -3d23 M NM4 3QQ ;Q ;Q3_4E4D$% M 3O&,Q4$% M 3O&+Q4FAy| 3O" 3 b#M43*??N(#M&-! M4% l GrM3O^&(Q $Time to Maturity'4% UEWMZ3O"&*Q Years'4523  NM43" 3_ M NM  MM< 444% YM:3O}&'Q .Duration and Maturity'4%  xrMj3O%Q'4%  DxrMj3O\%Q'4% xrMj3Og%Q'44e@@@eӼ?5;NQ@0L F% @e> 0L I  dMbP?_*+%m'?(?)?MHP LaserJet III@g,,@MSUDHP LaserJet IIId "d,,??U} }  }  }  } }  } } |  T0 b ;b ;| ; ; ; ;b ;b ; ; 0e ;18 ;00 ;@i ; 0 ;0 ;b ; 0 ;0 ;0 ;T0 ;} ;bO ;b }b ; ;}b ;O ;} ;v ;T0T ;Ch. 9..%Solutions to end-of-chapter problems.  Problem 2.  . Coupon Bond / 03 Par value =~ 1@@  3Coupon =~ 2$@  Annual payments  /  3YTM =~ 3 @  3 Maturity =~ 3@ /  4Time4 Cash Flow 4PVIF5PV of CF54PV*CF*T5 45/ ~ 6?!7Y@ DD* Kh/?D@D!7^B{ %W@ DD7'7^B{ %W@DDD7 7/ ~ 6@'70@DDD* 72YQo?D@D!7x@ DD7'7x@ DDD7 7/  3Price =#7C.@  % / 3 7 Numerator = 7# 7@ %5 3 Duration =  ! 85G#͑?  D D 9= Numerator/Price / : /   3YTM =~ 2$@  /  4Time 4 Cash Flow 4PVIF 5PV of CF   4PV*CF*T  / ~ 6?! 7Y@  DD*  ]tE? D @D ! 7袋.V@ D D  ' 7袋.V@D D D   / ~ 6@'70@DDD* Tx?r?D @D!7袋.h@ DD '7袋.h@DDD /  3Price =#7?@ %  /  37 Numerator =7#7tE]ԝ@  % 53 Duration = !8.袋? DD 9= Numerator/Price / : /  3YTM =~  (@ /  4Time4 Cash Flow 4PVIF5PV of CF 4PV*CF*T / ~ 6?!7Y@ DD* $I$I?D@D!7$I$IRV@ DD '7$I$IRV@DDD / ~ 6@'70@DDD* ?D@D!7r ^Ng@ DD '7r ^Ng@DDD /  3Price =#71@ % /  37 Numerator =7#7Sr̜@ %53 Duration = !8 F}? DD 9= Numerator/Price /QHb. Increasing the yield-to-maturity decreases the duration of the bond.  .Zero Coupon Bond /  3 Par value =~ ;@@  3Coupon =~ < /  3YTM =~ 6 @  3 Maturity =~ =@ /  4Time4 Cash Flow 4PVIF5PV of CF54PV*CF*T5 / 6?7* Kh/?D@D!7 DD7'7DDD7 / ~ 6@7@@D* 72YQo?D@D!7 kʊ@ DD7'7 kʚ@ DDD7 /  3Price =#7 kʊ@  % / Dla5>oa Da DasCso ;T0! ;b" ;b# ;|$ ;% ;& ;' ;b( ;b) ;* ; 0e+ ;18, ;00- ;@i. ; 0/ 00 b1 02 03 04 ;T05 ;}6 ;bO7 ;b8 ;}b9 ;: ;}b; ;O< ;}= ;v> ;T0T? ; 3 7 Numerator = 7# 7 kʚ@ %5 3 Duration =  ! 8@$ D D 9= Numerator/Price /! : / " "3YTM =~ ">$@" /  #4Time#4 Cash Flow #4PVIF#5PV of CF# #4PV*CF*T# / $6?7*$ ]tE?$D"@D$!$7% D$D$$7'$7%D$D$D$$ / ~ %6@%7@@%D*% Tx?r?%D"@D%!%7Ӊ@& D%D%%7'%7ә@'D%D%D%% / & &3Price =#&7Ӊ@' %$%& / ' 3'7 Numerator ='7#'7ә@$ %$%5'3 Duration =' !'8@+ D'D&' 9= Numerator/Price' /( : / ) )3YTM =~ )6(@) /  *4Time*4 Cash Flow *4PVIF*5PV of CF* *4PV*CF*T* / +6?7*+ $I$I?+D)@D+!+7, D+D++7'+7,D+D+D++ / ~ ,6@,7@@,D*, ?,D)@D,!,7}h@- D,D,,7',7}h@.D,D,D,, / - -3Price =#-7}h@. %+,- / . 3.7 Numerator =.7#.7}h@+ %+,5.3 Duration =. !.8@n = D.D-. 9= Numerator/Price. // S0Jd. Changing the YTM does not affect the duration of the zero coupon bond.0 e1\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more1 h2_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.2 3 Problem 43 4  4.Five-year Treasury Bond4 / 5 53 Par value =~ 5;@@ 5 53Coupon =~ 5<$@5 5 Semiannual payments 5 / 6 6FYTM =~ 6E$@ 6 63 Maturity =~ 6=@6 /  74Time74 Cash Flow 74PVIF75PV of CF7574PV*CF*T75 / ~ 86?+87I@8D5@D5@?98 yy?8#D6@?D8!87< D=D==7'=7r9r[@>D=D=D==7 / ~ >@ @+>7I@>D5@D5@?9> ?,?>#D6@?D>!>7J\A@? D>D>>7'>7~~_@?D>D>D>>7 / ~ ?@@+?7I@?D5@D5@?9? ??#D6@?D?!?7Y@@@ D?D??7'?7Y`@@D?D?D??7 / Dnl Da Da u4Jo@ ;T0A ;bB ;bC ;|D ;E ;F ;G ;bH ;bI ;J ; 0eK ;18L ;00M ;@iN ; 0O ;0P ;bQ ; 0R ;0S ;0T ;T0U ;}V ;bOW ;bX ;}bY ;Z ;}b[ ;O\ ;}] ;v^ ;T0T_ ;~ @6@+@7I@@D5@D5@?9@ ࢠ?@#D6@?D@!@7֢?@@A D@D@@7'@71'/!b@AD@D@D@@7 / ~ A6@1A7h@AD5@D5@?D59A ;Qh-?A#D6@?DA!AAKw$@B DADAA7'AAoU.@CDADADAA7 / B6 B3Price =#B7@@C %8AB / C 3C7 Numerator =C7#C7 UYҫ@8 %8AC3 Duration =C !C8ag47@ DCDBC 9= Numerator/PriceC /D / E  E.Five-year Treasury BondE / F F3 Par value =~ F;@@ F F3Coupon =~ F<$@F F Semiannual payments F / G GFYTM =~ GE(@ G G3 Maturity =~ G=@G /  H4TimeH4 Cash Flow H4PVIFH5PV of CFH5H4PV*CF*TH5 / ~ I6?+I7I@ID5@D5@?9I sHM0?I#DG@?DI!I7琚`G@J DIDII7'I7琚`7@JDIDIDI I7 IBDuration I 5YTMI /~ J6?+J7I@JD5@D5@?9J \mz?K#DG@?DJ!J7p*?F@K DJDJJ7'J7p*?F@KDJDJDJ J//JCŏ17@~ J 2$@J /~ K6?+K7I@KD5@D5@?9K 6B)?L#DG@?DK!K7޺;D@L DKDKK7'K7&MYX|O@LDKDKDK K//KCz): @~ K (@K /~ L6@+L7I@LD5@D5@?9L <_X?M#DG@?DL!L7DgBfC@M DLDLL7'L7DgBfS@MDLDLDL L//LCv@~ L ,@L /~ M6@+M7I@MD5@D5@?9M ?N#DG@?DM!M75YsB@N DMDMM7'M7nZW@NDMDMDMM/// / ~ N6@+N7I@ND5@D5@?9N Eh ?O#DG@?DN!N7uYA@O DNDNN7'N7d0oZ@ODNDNDNNC / ~ O@ @+O7I@OD5@D5@?9O \%H?P#DG@?DO!O70]@@P DODOO7'O7]@PDODODOOC / ~ P@@+P7I@PD5@D5@?9P B?Q#DG@?DP!P7^?@Q DPDPP7'P7^_@QDPDPDPPC / ~ Q6@+Q7I@QD5@D5@?9Q [@ ?R#DG@?DQ!Q7L=@R DQDQQ7'Q7{?`@RDQDQDQQC / ~ R6@1R7h@RD5@D5@?DF9R q#^?I#DG@?DR!RA< R@S DRDRR7'RAK )%@TDRDRDR RC.R / S6 S3Price =#S7j1@T %IRS / T 3T7 Numerator =T7#T7Zt@I %IRT3 Duration =T !T8ߖه @8 DTDST 9= Numerator/PriceT /U / V  V.Five-year Treasury BondV / W W3 Par value =~ W1@@ W W3Coupon =~ W<$@W W Semiannual payments W / X XFYTM =~ X?,@ X X3 Maturity =~ X=@X /  Y4TimeY4 Cash Flow Y4PVIFY5PV of CFY5Y4PV*CF*TY5 / ~ Z6?+Z7I@ZD5@D5@?9Z +J#?Z#DX@?DZ!Z7 N]G@[ DZDZZ7'Z7 N]7@[DZDZDZZ7 / ~ [6?+[7I@[D5@D5@?9[ ȟ5?\#DX@?D[![7,E@\ D[D[[7'[7,E@\D[D[D[[B / ~ \6?+\7I@\D5@D5@?9\ OK?]#DX@?D\!\7qNhD@] D\D\\7'\7AuN@]D\D\D\\C / ~ ]6@+]7I@]D5@D5@?9] @8i?^#DX@?D]!]7GzC@^ D]D]]7']7GzS@^D]D]D]]C / ~ ^6@+^7I@^D5@D5@?9^ %[@aD`D`D``B / ~ a@@+a7I@aD5@D5@?9a >Žџ?b#DX@?Da!a7 o=@b DaDaa7'a7 o]@bDaDaDaaD / ~ b6@+b7I@bD5@D5@?9b rg?c#DX@?Db!b7l}sZ2;@c DbDbb7'b7AY^@cDbDbDbbD / ~ c6@1c7h@cD5@D5@?DW9c E:eD?Z#DX@?Dc!cA@[Q"@d DcDcc7'cAP٤@eDcDcDccD / d6 d3Price =#d7:܊@e %Zcd / e 3e7 Numerator =e7#e76s@Z %Zce3 Duration =e !e8nݰ@J DeDde 9= Numerator/Pricee /f rgiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows g hat higher rates.h i j k l Problem 5l  m a. m.Four-year Treasury Bondm / n n3 Par value =~ n;@@ n n3Coupon =~ n<$@n n Semiannual payments n / o oFYTM =~ oE$@ o o3 Maturity =~ o=@o /  p4Timep4 Cash Flow p4PVIFp5PV of CFp5p4PV*CF*Tp5 / ~ q6?+q7I@qD5@D5@?9qHyy?q#D6@?Dq!q7<aDTT  T0 b b |    b b  0e 18 00 @i 0 0 b 0 0 0 T0 } bO b }b  }b O } v T0T     Problem 6  Six-year CD  Par value~ @ Coupon~ @Annual payments  #YTM~ $@ Maturity~ @   Time Cash Flow PVIFPV of CFPV*CF*T ~ ?!@ D@D@+sHM0?D@D!sHM@ DD'sHM@DDD  PVIF = 1/(1+YTM)^(Time) ~ @!@ D@D@+\mz?D@D!i @ DD'i @DDD ~ @!@ D@D@+6B)?D@D!&MYX|@ DD'92CB@DDD ~ @!@ D@D@+<_X?D@D!c}@ DD'c@DDD ~ @!@ D@D@+?D@D!Ѕ|@ DD'@DDD ~ @'@D@D@D+Eh ?D@D!(Jd+0@ DD'(7 o@DDD Price =#@ %  Numerator =#\DPits@ % Duration =!" pgu@ DD != Numerator/Price  Six-year CD  Par value~ @ Coupon~ @Semiannual payments  #YTM~ $@ Maturity~ @   Time Cash Flow PVIFPV of CFPV*CF*T ~ ?+r@D@D@?9'H%e?#D@?D!k@ DD'GA+@DDD Dl2<i}_<i} T0 b b |    b b  0e 18 00 @i 0 0 b 0 0 0 T0 } bO b }b  }b O } v T0T ~ @1@D@D@?D9' 7q?#D@?D!(w68@ DD'((*@DDD Price =#@ %  Numerator =# @ % Duration =!"?W@ DD != Numerator/Price xoDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being  pgreceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows.    Problem 7  Consol Bond  YTM D = 1 + 1/R ~  @#)+@ D ~ $@#)&@ D ~ (@#*"@ D  ulA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percent wnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM, ZQhas a duration of 20 years because no cash flows occur before the twentieth year.            Problem 8  a.Five-year Bond  Par value~ @@ Coupon~ @Annual payments  #YTM~ $"@ Maturity~ @  D l_2<BOOOz2D T0 b b |    b b  0e 18 00 @i 0 0 b 0 0 0 T0 } bO b }b  }b O } v T0T  Time Cash Flow PVIFPV of CFPV*CF*T ~ ?!Q@ D@D@+[?D@D! z|P@ DD' z|P@DDD  PVIF = 1/(1+YTM)^(Time) ~ @!Q@ D@D@+'B ?D@D!H5suM@ DD'H5su]@DDD ~ @!Q@ D@D@++!?D@D!HK@ DD'6Ed@DDD ~ @!Q@ D@D@+7%Yk?D@D!Ċi}H@ DD'Ċi}h@DDD ~ @'@D@D@D+cs D*@DDD Price =#zi^ь@ %  Numerator =#lMjig@ % Duration =!" ufo@ DD != Numerator/Price Five-year Bond  Par value~ @@ Coupon~ "@Annual payments  #YTM~ $"@ Maturity~ @   Time Cash Flow PVIFPV of CFPV*CF*T ~ ?!V@ DD+[?D@D!F*VgT@ DD'F*VgT@DDD  PVIF = 1/(1+YTM)^(Time) ~ @!V@ DD+'B ?D@D!}R@ DD'}b@DDD ~ @!V@ DD++!?D@D!R_Q@ DD']Nj@DDD ~ @!V@ DD+7%Yk?D@D!Ei]O@ DD'Ei]o@DDD ~ @'@DDD+csI@ DD'>I@DDD ~ @'@DDD++d?DD!(w$F1@ DD'( 8_@DDD Price =# @ %  Numerator =#L`R@ % Duration =!""g}* @ DD != Numerator/Price > 0I]BRR[D2Di}_  T0! b" b# |$ % & ' b( b) * 0e+ 18, 00- @i. 0/ 00 b1 02 03 04 T05 }6 bO7 b8 }b9 : }b; O< }= ;v> ;T0T? ;  Five-year Bond  !! Par value~ !@ !!Coupon~ !$@!!Annual payments! " "#YTM~ "$$@ ""Maturity~ "@ "  #Time# Cash Flow #PVIF#PV of CF##PV*CF*T# ~ $?!$@@$ D!D!+$]tE?0D"D$!$袋.h@% D$D$$'$袋.h@%D$D$D$ $ $PVIF = 1/(1+YTM)^(Time)$ ~ %@!%@@% D!D!+%'Tx?r?$D"D%!%Ӊ@& D%D%%'%ә@&D%D%D%% ~ &@!&@@& D!D!+&[V ?%D"D&!&7Uz@' D&D&&'&)V㛡@'D&D&D&& ~ '@!'@@' D!D!+'}??&D"D'!'x|X@( D'D''''x|X@(D'D'D'' ~ (@'(|@(D!D!D!+(+d?'D"D(!((Ur"@) D(D(('((IQլ@*D(D(D(( ))Price =#)@* %$() ** Numerator =*#*Pl]T\@$ %$(* Duration =*!*")#5@ D*D)* != Numerator/Price+ ,,Five-year Bond, -- Par value~ -@ --Coupon~ -(@--Annual payments- . .#YTM~ .$$@ ..Maturity~ .@ .  /Time/ Cash Flow /PVIF/PV of CF//PV*CF*T/ ~ 0?!0@0 D-D-+0]tE?D.D0!0. @1 D0D00'0. @1D0D0D0 0 0PVIF = 1/(1+YTM)^(Time)0 ~ 1@!1@1 D-D-+1'Tx?r?0D.D1!1,Ra@2 D1D11'1,Ra@2D1D1D11 ~ 2@!2@2 D-D-+2[V ?1D.D2!23A,@3 D2D22'2pw!@3D2D2D22 ~ 3@!3@3 D-D-+3}??2D.D3!3/휉@4 D3D33'3/휩@4D3D3D33 ~ 4@'4@4D-D-D-+4+d?3D.D4!4(Q*@5 D4D44'4(s@6D4D4D44 55Price =#5,$@6 %045 66 Numerator =6#6Ӌ f@0 %046 Duration =6!6"ZK@$ D6D56 != Numerator/Price7 8  9b. 9&9 Change in9 ::&Duration:Coupon:Duration: ;;%`TR' @~ ;+ @; <<%\C@~ <+$@<!lV}< = =%K@~ =+(@=_vO= > ? 3? *Duration decreases as the amount of coupon? Dl?i}_?i}_I]BRR@ ;T0A ;bB ;bC ;|D ;E ;J K bL bM N 0eO 18P 00Q @iR 0T 0U bV 0W 0X 0Y T0Z }[ bO] b^ }b_ @ @ interest increases.@ A B C D E J Problem 10J KBK9Two-year loan: Principal and interest at end of year two.K LL Par value~ Lj@ LLCoupon~ LLLNo annual paymentsL M M#YTM~ M$$@ MMMaturity~ M@ M  NTimeN Cash Flow NPVIFNPV of CFNNPV*CF*TN ~ O?!OO DLDL+O]tE?ODMDO!OP DODOO'OPDODODO O OPVIF = 1/(1+YTM)^(Time)O ~ P@*P@aDL?+PTx?r?PDMDP!P-j@Q DPDPP'P(jARDPDPDPP QQPrice =#Q,j@R %OPQ RR Numerator =R#RjAO %OPR Duration =R!R"@ DRDQR != Numerator/PriceTNTETwo-year loan: Interest at end of year one, P & I at end of year two.T UU Par value~ Uj@ UUCoupon~ U$@UUAnnual paymentsU V V#YTM~ V$$@ VVMaturity~ V@ V  WTimeW Cash Flow WPVIFWPV of CFWWPV*CF*TW ~ X?!X@X DUDU+X]tE?XDVDX!XE]t@Y DXDXX'XE]t@YDXDXDX X XPVIF = 1/(1+YTM)^(Time)X ~ Y@'Y@YDUDUDU+YTx?r?YDVDY!Y(E]t1@Z DYDYY'Y(E]t1A[DYDYDYY ZZPrice =#Z,i@[ %XYZ [[ Numerator =[#[.MAX %XY[ Duration =[![".袋?O D[DZ[ != Numerator/Price]1](Two-year loan: Amortized over two years.] ^^ Par value~ ^j@ ^^Coupon~ ^$@^(^Amortized payment of $57.619.05^ _ _#YTM~ _$$@ __Maturity~ _@ _ 8D3ji}_vi}_Y` T0a bb bc |d g h i bj bk ;l ; 0em ;18n ;00o ;@ip ; 0q ;0s bt 0u 0v 0w T0x }y bOz b{ }b `Time` Cash Flow `PVIF`PV of CF``PV*CF*T` a?ǭ_+a]tE?aD_Da!a袋@b DaDaa'a袋@bDaDaDa a aPVIF = 1/(1+YTM)^(Time)a b@ǭ_+bTx?r?bD_Db!b(7a@@c DbDbb'b(7a@@dDbDbDbb ccPrice =#c,lj@d %abc dd Numerator =d#d=;nAa %abd Duration =d!d"yy?X DdDcd != Numerator/Price gb.g&g Repaymentg Change ing hh&Durationh ProvisionshDurationh i~ i%@i+P&I @ 2i jj% q? j+I@1j48Ej k k%%䃞?k+Amortizek3ۿk l >m 5Duration decreases dramatically when a portion of them Dn ;principal is repaid at the end of year one. Duration oftenn Co :is described as the weighted average maturity of an asset,o @p 7so if more weight is given to early payments, then the p 3q *effective maturity of the loan is reduced.q s Problem 13s tt Two-year Bondt uu Par value~ u@@ uuCoupon~ u$@uuAnnual paymentsu v v#YTM~ v$,@ vvMaturity~ v@ v  wTimew Cash Flow wPVIFwPV of CFwwPV*CF*Tw ~ x?!xY@x DuDu+x}?xDvDx!x>U@y DxDxx'x>U@yDxDxDx x xPVIF = 1/(1+YTM)^(Time)x ~ y@'y0@yDuDuDu+y]z?yDvDy!y(&*rPs@z DyDyy'y(&*rPs@{DyDyDyy zzPrice =#zr)1@{ %xyz {{ Numerator ={#{Nq1қ@x %xy{ Duration ={!{" ^?P D{Dz{ != Numerator/Price6 }_[aDTY^dc`S3>i}_(  p  6NMM?cY e]`  IMHP LaserJet III@g,,@MSUDHP LaserJet IIId "d,,??3`  `  `   `   `   `  PH 0(  ?3d23 M NM4 3QQ ;JLQi ;JL Q3_4E4D$% M 3O& Q4$% M 3O& Q4FA4l e3O 3 b#M43*{Gz?N(#M&! M4%  [M3O]& Q &Yield to Maturity'4% 12MZ3O"& Q Years'4523  NM43" 3_ M NM  MM< 444% $X|M:3Ou& Q $Duration and YTM'4%  T[M3O:)Q'4% Y[M3O:)Q'4% [M3OQ)Q'44eee ~v  <NMM?U ]`h  I"h??3`  `  `  `  `  `  0@п3d23 M NM4 3QQ ;Q ;Q3_4E4D$% M 3O&Q4$% M 3O&Q4FA0h l3O 3 b#M43*??N(#M&! M4%  ZM3O^&Q $Time to Maturity'4% 00MZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% p{M:3O&Q .Duration and Maturity'4% ZM3Og)Q'4% p ^ZM3O\)Q'4%  ZM3O)Q'44eee ~v  <NMM?H ]`L&  I"L&??3`  `  `  `  ` | ` | 3d23 M NM4 3QQ ;Q ;Q3_4E4D$% M 3O&Q4$% M 3O&Q4FA, 3OE 3 bo#M43*@@??N(#M&! M4% 9 M3OM&Q  Coupon Rates '4% #/MZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444%  R M:3O&Q 6Duration and Coupon Rates'4% 4 jM3O~)Q'4% M3Ox)Q'4% wM3Og)Q'44eee ~v  <NMM?H9 C]`0  I"0??3` | ` | ` | ` | ` | 3d23 M NM4 3QQ ;;=Q ;;=Q3_4E4D$% M 3O&Q4$% M 3O&Q4FA, 3OE 3 bo#M43*@@??N(#M&! M4% 9 M3OM&Q  Coupon Rates '4% #/MZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444%  R M:3O&Q 6Duration and Coupon Rates'4% M3Og)Q'4% M3Ox)Q'4% 4 JM3O~)Q'44eee ~v  <NMM?Hg q]`3  I"3??3` |  ` | !` | "` | #` | )3d23 M NM4 3QQ ;ikQQ3_4E4D$% M 3O&"Q4$% M 3O&!Q4FA,b 3O y 3 bo#M43*?@??N(#M&#! M4%  M3O& Q 2Repayment Alternatives '4% L/MZ3O"& Q Years'4523  NM43" 3_ M NM  MM< 444% >R% M:3O&)Q >Duration and Repayment Choice'4% 4 M3O~)Q'4% M3Ox)Q'4% M3Og)Q'44eee >@ I  dMbP?_*+%"??eU>@ I  dMbP?_*+%"??eU>@ 3j Sheet1Sheet2Sheet3Chart3  WorksheetsCharts 6> _PID_GUIDAN{A81F3840-6ACB-11D3-AF7B-B0D2013B3FD9} FMicrosoft Excel ChartBiff8Ole CompObjdgbObjInfoWorkbookfh5Excel.Sheet.89q Oh+'0@H`x Ernest W. SwiftErnest W. SwiftMicrosoft Excel@Gn@% ՜.+,D՜.+,@ PXd lt|      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSVWZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ IBa=  =x 8X1Arial1Arial1Arial1Arial1.Times New Roman1.Times New Roman1.Times New Roman1.Times New Roman1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1.Times New Roman1Arial1.Times New Roman1Arial1Arial1Arial1Arial1Arial1Arial1Arial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)% "$"#,##0.0_);[Red]\("$"#,##0.0\)0.0 0.00000000 0.0000000 0.000000 0.00000 0.0000% #,##0.0000_);[Red]\(#,##0.0000\) 0.000                + ) , *         # " # "     "   #  #   " "  A"   $  %      # $" $# "  "   " ! ! " &# %" $ $#  ( &" &#  " ! ! $!  " " # qChart4%Sheet13Sheet24Sheet3bZ  3  @@  f O%Solutions to end-of-chapter problems.Time Par valueCouponMaturity Cash FlowYTMPVIFPV of CFPrice =PV*CF*T Numerator = Duration =a. Coupon BondZero Coupon BondHb. Increasing the yield-to-maturity decreases the duration of the bond.Jd. Changing the YTM does not affect the duration of the zero coupon bond.\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.Ch. 9. Problem 2.= Numerator/Price Problem 4Five-year Treasury BondiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows at higher rates.Duration.PVIF = 1/(1+YTM/2)^(Time*2) Problem 5Two-year Treasury Bondb. Change in Problem 6 Six-year CDPVIF = 1/(1+YTM)^(Time)oDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being greceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. Problem 7Semiannual paymentsAnnual payments Consol Bond D = 1 + 1/RlA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percentnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM,Qhas a duration of 20 years because no cash flows occur before the twentieth year. Problem 8Five-year Bond*Duration decreases as the amount of couponinterest increases. Problem 9 Problem 10No annual paymentsAmortized payment of $57.619.059Two-year loan: Principal and interest at end of year two.ETwo-year loan: Interest at end of year one, P & I at end of year two.(Two-year loan: Amortized over two years. Repayment ProvisionsP&I @ 2I@1Amortize5Duration decreases dramatically when a portion of the;principal is repaid at the end of year one. Duration often:is described as the weighted average maturity of an asset,7so if more weight is given to early payments, then the *effective maturity of the loan is reduced. Problem 13 Two-year BondCoupon = Maturity = Par value =YTM == PVIF = 1/(1+YTM)^(Time) PVIF = 1/(1+YTM/2)^(Time*2)Four-year Treasury Bond Three-year Treasury Bond J )tc NA0{ӔlsA?XbM_xb_b,D88EbblAblAA?bM_b_b,D80sT0b| ~Xj},bCJ0sT0bbXj}0bgYJ0sT0b$$blAA?_b\b_bk8TbXwXbblA_BA?J "uq I  dMbP?_*+%m'?(?)?MHP LaserJet III@g,,@MSUDHP LaserJet IIId "d,,??U} }  }  }  } }  } I } } } |  T0 b ;b ;_ ; ; ; ;b ;b ; ;b ; ;00 ;@n ;s ;0 ;b ; 0 ;0 ;0 ;T0 ;b ;} ;}b  ;}} ;}O ;} ; ;}} ;v ;Ch. 9..%Solutions to end-of-chapter problems.  Problem 2. - Coupon Bond. /2 Par value =~ 0@@ 2Coupon =~ 1$@Annual payments  . 2YTM =~ 2 @ 2 Maturity =~ 2@.  3Time3 Cash Flow 3PVIF4PV of CF43PV*CF*T434. ~ 5?!6Y@ DD*Kh/?D@D!6^B{ %W@ DD6'6^B{ %W@DDD66. ~ 5@'60@DDD*72YQo?D@D!6x@ DD6'6x@ DDD66. 2Price =#6C.@  %. 2 6 Numerator = 6# 6@ %5 2 Duration = ! 75G#͑?  D D 8= Numerator/Price . 9.   2YTM =~ 1$@ .  3Time 3 Cash Flow 3PVIF 4PV of CF  3PV*CF*T . ~ 5?! 6Y@  DD* ]tE? D @D ! 6袋.V@ D D ' 6袋.V@D D D  . ~ 5@'60@DDD*Tx?r?D @D!6袋.h@ DD'6袋.h@DDD. 2Price =#6?@ % . 26 Numerator =6#6tE]ԝ@  % 52 Duration =!7.袋? DD 8= Numerator/Price .9. 2YTM =~ (@.  3Time3 Cash Flow 3PVIF4PV of CF3PV*CF*T. ~ 5?!6Y@ DD*$I$I?D@D!6$I$IRV@ DD'6$I$IRV@DDD. ~ 5@'60@DDD*?D@D!6r ^Ng@ DD'6r ^Ng@DDD. 2Price =#61@ %. 26 Numerator =6#6Sr̜@ %52 Duration =!7 F}? DD 8= Numerator/Price .QHb. Increasing the yield-to-maturity decreases the duration of the bond. -Zero Coupon Bond. 2 Par value =~ :@@ 2Coupon =~ ;. 2YTM =~ 5 @ 2 Maturity =~ <@.  3Time3 Cash Flow 3PVIF4PV of CF43PV*CF*T4. 5?6*Kh/?D@D!6 DD6'6DDD6. ~ 5@6@@D*72YQo?D@D!6 kʊ@ DD6'6 kʚ@ DDD6. 2Price =#6 kʊ@  %. Dla5>oa Da DasCso ;T0! ;b" ;b# ;_$ ;% ;& ;' ;b( ;b) ;* ;b+ ;, ;00- ;@n. ;s/ 00 b1 02 03 04 ;T05 ;b6 ;}7 ;}b8 ;9 ;}}: ;}O; ;}< ;= ;}}> ;v? ; 2 6 Numerator = 6# 6 kʚ@ %5 2 Duration = ! 7@$ D D 8= Numerator/Price .!9. ""2YTM =~ "=$@".  #3Time#3 Cash Flow #3PVIF#4PV of CF##3PV*CF*T#. $5?6*$]tE?$D"@D$!$6% D$D$$6'$6%D$D$D$$. ~ %5@%6@@%D*%Tx?r?%D"@D%!%6Ӊ@& D%D%%6'%6ә@'D%D%D%%. &&2Price =#&6Ӊ@' %$%&. '2'6 Numerator ='6#'6ә@$ %$%5'2 Duration ='!'7@+ D'D&' 8= Numerator/Price' .(9. ))2YTM =~ )5(@).  *3Time*3 Cash Flow *3PVIF*4PV of CF**3PV*CF*T*. +5?6*+$I$I?+D)@D+!+6, D+D++6'+6,D+D+D++. ~ ,5@,6@@,D*,?,D)@D,!,6}h@- D,D,,6',6}h@.D,D,D,,. --2Price =#-6}h@. %+,-. .2.6 Numerator =.6#.6}h@+ %+,5.2 Duration =.!.7@n = D.D-. 8= Numerator/Price. ./ S0Jd. Changing the YTM does not affect the duration of the zero coupon bond.0 e1\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more1 h2_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.2 3 Problem 43 4 4-Five-year Treasury Bond4. 552 Par value =~ 5:@@ 552Coupon =~ 5;$@55Semiannual payments 5 . 66EYTM =~ 6D$@ 662 Maturity =~ 6<@6.  73Time73 Cash Flow 73PVIF74PV of CF7473PV*CF*T74. ~ 85?+86I@8D5@D5@?98yy?8#D6@?D8!86< D=D==6'=6r9r[@>D=D=D==6. ~ >? @+>6I@>D5@D5@?9>?,?>#D6@?D>!>6J\A@? D>D>>6'>6~~_@?D>D>D>>6. ~ ??@+?6I@?D5@D5@?9???#D6@?D?!?6Y@@@ D?D??6'?6Y`@@D?D?D??6. Dnl Da Da u4Jo@ ;T0A ;bB ;bC ;_D ;E ;F ;G ;bH ;bI ;J ;bK ;L ;00M ;@nN ;sO ;0P ;bQ ; 0R ;0S ;0T ;T0U ;bV ;}W ;}bX ;Y ;}}Z ;}O[ ;}\ ;] ;}}^ ;v_ ;~ @5@+@6I@@D5@D5@?9@ࢠ?@#D6@?D@!@6֢?@@A D@D@@6'@61'/!b@AD@D@D@@6. ~ A5@1A6h@AD5@D5@?D59A;Qh-?A#D6@?DA!A@Kw$@B DADAA6'A@oU.@CDADADAA6. B5B2Price =#B6@@C %8AB. C2C6 Numerator =C6#C6 UYҫ@8 %8AC2 Duration =C!C7ag47@ DCDBC 8= Numerator/PriceC .D. E E-Five-year Treasury BondE. FF2 Par value =~ F:@@ FF2Coupon =~ F;$@FFSemiannual payments F . GGEYTM =~ GD(@ GG2 Maturity =~ G<@G.  H3TimeH3 Cash Flow H3PVIFH4PV of CFH4H3PV*CF*TH4. ~ I5?+I6I@ID5@D5@?9IsHM0?I#DG@?DI!I6琚`G@J DIDII6'I6琚`7@JDIDIDI I6IADuration I 4YTMI .~ J5?+J6I@JD5@D5@?9J\mz?K#DG@?DJ!J6p*?F@K DJDJJ6'J6p*?F@KDJDJDJ J..JBŏ17@~ J 1$@J .~ K5?+K6I@KD5@D5@?9K6B)?L#DG@?DK!K6޺;D@L DKDKK6'K6&MYX|O@LDKDKDK K..KBz): @~ K (@K .~ L5@+L6I@LD5@D5@?9L<_X?M#DG@?DL!L6DgBfC@M DLDLL6'L6DgBfS@MDLDLDL L..LBv@~ L ,@L .~ M5@+M6I@MD5@D5@?9M?N#DG@?DM!M65YsB@N DMDMM6'M6nZW@NDMDMDMM.... ~ N5@+N6I@ND5@D5@?9NEh ?O#DG@?DN!N6uYA@O DNDNN6'N6d0oZ@ODNDNDNNB. ~ O? @+O6I@OD5@D5@?9O\%H?P#DG@?DO!O60]@@P DODOO6'O6]@PDODODOOB. ~ P?@+P6I@PD5@D5@?9PB?Q#DG@?DP!P6^?@Q DPDPP6'P6^_@QDPDPDPPB. ~ Q5@+Q6I@QD5@D5@?9Q[@ ?R#DG@?DQ!Q6L=@R DQDQQ6'Q6{?`@RDQDQDQQB. ~ R5@1R6h@RD5@D5@?DF9Rq#^?I#DG@?DR!R@< R@S DRDRR6'R@K )%@TDRDRDR RB.R. S5S2Price =#S6j1@T %IRS. T2T6 Numerator =T6#T6Zt@I %IRT2 Duration =T!T7ߖه @8 DTDST 8= Numerator/PriceT .U. V V-Five-year Treasury BondV. WW2 Par value =~ W0@@ WW2Coupon =~ W;$@WWSemiannual payments W . XXEYTM =~ X>,@ XX2 Maturity =~ X<@X.  Y3TimeY3 Cash Flow Y3PVIFY4PV of CFY4Y3PV*CF*TY4. ~ Z5?+Z6I@ZD5@D5@?9Z+J#?Z#DX@?DZ!Z6 N]G@[ DZDZZ6'Z6 N]7@[DZDZDZZ6. ~ [5?+[6I@[D5@D5@?9[ȟ5?\#DX@?D[![6,E@\ D[D[[6'[6,E@\D[D[D[[A. ~ \5?+\6I@\D5@D5@?9\OK?]#DX@?D\!\6qNhD@] D\D\\6'\6AuN@]D\D\D\\B. ~ ]5@+]6I@]D5@D5@?9]@8i?^#DX@?D]!]6GzC@^ D]D]]6']6GzS@^D]D]D]]B. ~ ^5@+^6I@^D5@D5@?9^%[@aD`D`D``A. ~ a?@+a6I@aD5@D5@?9a>Žџ?b#DX@?Da!a6 o=@b DaDaa6'a6 o]@bDaDaDaaC. ~ b5@+b6I@bD5@D5@?9brg?c#DX@?Db!b6l}sZ2;@c DbDbb6'b6AY^@cDbDbDbbC. ~ c5@1c6h@cD5@D5@?DW9cE:eD?Z#DX@?Dc!c@@[Q"@d DcDcc6'c@P٤@eDcDcDccC. d5d2Price =#d6:܊@e %Zcd. e2e6 Numerator =e6#e66s@Z %Zce2 Duration =e!e7nݰ@J DeDde 8= Numerator/Pricee .f rgiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows g hat higher rates.h i j k l Problem 5l  ma.$mJFour-year Treasury Bond mJJ. nn2 Par value =~ n:@@ nn2Coupon =~ n;$@nnSemiannual payments n . ooEYTM =~ oD$@ oo2 Maturity =~ o<@o.  p3Timep3 Cash Flow p3PVIFp4PV of CFp4p3PV*CF*Tp4. ~ q5?+q6I@qD5@D5@?9qFyy?q#D6@?Dq!q6<aDTT  T0 b b _ ; ; ; ;b ;b ; ;b ; ;00 ;@n ;s ;0 ;b ; 0 ;0 ;0 ;T0 ;b ;} ;}b ; ;}} ;}O ;} ; ;}} ;v ;    Problem 6 J Six-year CDJ. 2 Par value =~ :@ 2Coupon =~ ;@Annual payments . EYTM =~ D@ 2 Maturity =~ <@.  3Time3 Cash Flow 3PVIF4PV of CF43PV*CF*T4. ~ 5?!6@ D@D@+KsHM0?D@D!6sHM@ DD6'6sHM@DDD$ PVIF = 1/(1+YTM)^(Time). ~ 5@!6@ D@D@+K\mz?D@D!6i @ DD6'6i @DDD6. ~ 5@!6@ D@D@+6B)?D@D!6&MYX|@ DD6'692CB@DDD6. ~ 5@!6@ D@D@+<_X?D@D!6c}@ DD6'6c@DDD6. ~ 5@!6@ D@D@+?D@D!6Ѕ|@ DD6'6@DDD6. ~ 5@'0@D@D@D+Eh ?D@D!@Jd+0@ DD6'@7 o@DDD6. 52Price =#6@ %. 26 Numerator =6#6\DPits@ %5Duration =!7 pgu@ DD 8= Numerator/Price .. - Six-year CD. 2 Par value =~ :@ 2Coupon =~ ;@Semiannual payments . EYTM =~ D@ 2 Maturity =~ <@.  3Time3 Cash Flow 3PVIF4PV of CF43PV*CF*T4. ~ 5?+6r@D@D@?9FH%e?#D@?D!6k@ DD6'6GA+@DDD6. Dl 4>oa >o ;T0 ;b ;b ;_    b b  b  00 @n s 0 b 0 0 0 T0 b } }b  }} }O }  }} v ~ 5@10@D@D@?D9F 7q?#D@?D!@w68@ DD6'@(*@DDD6. 52Price =#6@ %. 26 Numerator =6#6 @ %2Duration =!7?W@ DD 8= Numerator/Price .. xoDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being  pgreceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows.    Problem 7  Consol Bond  YTM D = 1 + 1/R ~  @#(+@ D ~ $@#(&@ D ~ (@#)"@ D  ulA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percent wnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM, ZQhas a duration of 20 years because no cash flows occur before the twentieth year.            Problem 8 Five-year Bond  Par value =~ L@@ Coupon =~ H@Annual payments NYTM =~ M"@  Maturity =~ I@  D0 la 4>DQQQ z2? T0 b b _    b b  b  00 @n s 0 b 0 0 0 T0 b } }b  }} }O }  }} v  Time Cash Flow PVIFPV of CFPV*CF*T ~ ?!Q@ D@D@+[?D@D! z|P@ DD' z|P@DDD$ PVIF = 1/(1+YTM)^(Time)  ~ @!Q@ D@D@+&B ?D@D!H5suM@ DD'H5su]@DDD ~ @!Q@ D@D@++!?D@D!HK@ DD'6Ed@DDD ~ @!Q@ D@D@+7%Yk?D@D!Ċi}H@ DD'Ċi}h@DDD ~ @'@D@D@D+cs D*@DDD Price =#zi^ь@ %  Numerator =#lMjig@ %Duration =!! ufo@ DD = Numerator/Price Five-year Bond  Par value =~ L@@ Coupon =~ H"@Annual payments NYTM =~ M"@  Maturity =~ I@   Time Cash Flow PVIFPV of CFPV*CF*T ~ ?!V@ DD+[?D@D!F*VgT@ DD'F*VgT@DDD$ PVIF = 1/(1+YTM)^(Time)  ~ @!V@ DD+&B ?D@D!}R@ DD'}b@DDD ~ @!V@ DD++!?D@D!R_Q@ DD']Nj@DDD ~ @!V@ DD+7%Yk?D@D!Ei]O@ DD'Ei]o@DDD ~ @'@DDD+csI@ DD'>I@DDD ~ @'@DDD++d?DD!'w$F1@ DD'' 8_@DDD Price =# @ %  Numerator =#L`R@ % Duration =!!"g}* @ DD = Numerator/Price > 0I]BRR[D2Di}_  T0! b" b# _$ % & ' b( b) * b+ , 00- @n. s/ 00 b1 02 03 04 T05 b6 }7 }b8 9 }}: }O; }< = ;}}> ;v? ;  Five-year Bond  !! Par value~ !@ !!Coupon~ !$@!!Annual payments! " ""YTM~ "#$@ ""Maturity~ "@ "  #Time# Cash Flow #PVIF#PV of CF##PV*CF*T# ~ $?!$@@$ D!D!+$]tE?0D"D$!$袋.h@% D$D$$'$袋.h@%D$D$D$ $ $PVIF = 1/(1+YTM)^(Time)$ ~ %@!%@@% D!D!+%&Tx?r?$D"D%!%Ӊ@& D%D%%'%ә@&D%D%D%% ~ &@!&@@& D!D!+&[V ?%D"D&!&7Uz@' D&D&&'&)V㛡@'D&D&D&& ~ '@!'@@' D!D!+'}??&D"D'!'x|X@( D'D''''x|X@(D'D'D'' ~ (@'(|@(D!D!D!+(+d?'D"D(!('Ur"@) D(D(('('IQլ@*D(D(D(( ))Price =#)@* %$() ** Numerator =*#*Pl]T\@$ %$(* Duration =*!*!)#5@ D*D)* = Numerator/Price+ ,,Five-year Bond, -- Par value~ -@ --Coupon~ -(@--Annual payments- . ."YTM~ .#$@ ..Maturity~ .@ .  /Time/ Cash Flow /PVIF/PV of CF//PV*CF*T/ ~ 0?!0@0 D-D-+0]tE?D.D0!0. @1 D0D00'0. @1D0D0D0 0 0PVIF = 1/(1+YTM)^(Time)0 ~ 1@!1@1 D-D-+1&Tx?r?0D.D1!1,Ra@2 D1D11'1,Ra@2D1D1D11 ~ 2@!2@2 D-D-+2[V ?1D.D2!23A,@3 D2D22'2pw!@3D2D2D22 ~ 3@!3@3 D-D-+3}??2D.D3!3/휉@4 D3D33'3/휩@4D3D3D33 ~ 4@'4@4D-D-D-+4+d?3D.D4!4'Q*@5 D4D44'4's@6D4D4D44 55Price =#5,$@6 %045 66 Numerator =6#6Ӌ f@0 %046 Duration =6!6!ZK@$ D6D56 = Numerator/Price7 8  9b. 9%9 Change in9 ::%Duration:Coupon:Duration: ;;$`TR' @~ ;* @; <<$\C@~ <*$@<!lV}< ==$K@~ =*(@=_vO= > ?3?*Duration decreases as the amount of coupon? Dl?i}_?i}_I]BRR@ ;T0A ;bB ;bC ;_D ;E ;J K bL bM N bO P 00Q @nR sT 0U bV 0W 0X 0Y T0Z b[ }] }b^ _ }}@@interest increases.@ A B C D E J Problem 10J KBK9Two-year loan: Principal and interest at end of year two.K LL Par value~ Lj@ LLCoupon~ LLLNo annual paymentsL M M"YTM~ M#$@ MMMaturity~ M@ M  NTimeN Cash Flow NPVIFNPV of CFNNPV*CF*TN ~ O?!OO DLDL+O]tE?ODMDO!OP DODOO'OPDODODO O OPVIF = 1/(1+YTM)^(Time)O ~ P@*P@aDL?+PTx?r?PDMDP!P,j@Q DPDPP'P'jARDPDPDPP QQPrice =#Q+j@R %OPQ RR Numerator =R#RjAO %OPR Duration =R!R!@ DRDQR = Numerator/PriceTNTETwo-year loan: Interest at end of year one, P & I at end of year two.T UU Par value~ Uj@ UUCoupon~ U$@UUAnnual paymentsU V V"YTM~ V#$@ VVMaturity~ V@ V  WTimeW Cash Flow WPVIFWPV of CFWWPV*CF*TW ~ X?!X@X DUDU+X]tE?XDVDX!XE]t@Y DXDXX'XE]t@YDXDXDX X XPVIF = 1/(1+YTM)^(Time)X ~ Y@'Y@YDUDUDU+YTx?r?YDVDY!Y'E]t1@Z DYDYY'Y'E]t1A[DYDYDYY ZZPrice =#Z+i@[ %XYZ [[ Numerator =[#[.MAX %XY[ Duration =[![!.袋?O D[DZ[ = Numerator/Price]1](Two-year loan: Amortized over two years.] ^^ Par value~ ^j@ ^^Coupon~ ^$@^(^Amortized payment of $57.619.05^ _ _"YTM~ _#$@ __Maturity~ _@ _ 8D3ji}_vi}_Y` T0a bb bc _d g h i bj bk ;l ;bm ;n ;00o ;@np ;sq ;0s bt 0u 0v 0w T0x by }z }b{  `Time` Cash Flow `PVIF`PV of CF``PV*CF*T` a?ǭ_+a]tE?aD_Da!a袋@b DaDaa'a袋@bDaDaDa a aPVIF = 1/(1+YTM)^(Time)a b@ǭ_+bTx?r?bD_Db!b'7a@@c DbDbb'b'7a@@dDbDbDbb ccPrice =#c+lj@d %abc dd Numerator =d#d=;nAa %abd Duration =d!d!yy?X DdDcd = Numerator/Price gb.g%g Repaymentg Change ing hh%Durationh ProvisionshDurationh i~ i$@i*P&I @ 2i jj$ q? j*I@1j48Ej kk$%䃞?k*Amortizek3ۿk l >m5Duration decreases dramatically when a portion of them Dn;principal is repaid at the end of year one. Duration oftenn Co:is described as the weighted average maturity of an asset,o @p7so if more weight is given to early payments, then the p 3q*effective maturity of the loan is reduced.q s Problem 13s tt Two-year Bondt uu Par value~ u@@ uuCoupon~ u$@uuAnnual paymentsu v v"YTM~ v#,@ vvMaturity~ v@ v  wTimew Cash Flow wPVIFwPV of CFwwPV*CF*Tw ~ x?!xY@x DuDu+x}?xDvDx!x>U@y DxDxx'x>U@yDxDxDx x xPVIF = 1/(1+YTM)^(Time)x ~ y@'y0@yDuDuDu+y]z?yDvDy!y'&*rPs@z DyDyy'y'&*rPs@{DyDyDyy zzPrice =#zr)1@{ %xyz {{ Numerator ={#{Nq1қ@x %xy{ Duration ={!{! ^?P D{Dz{ = Numerator/Price6 }_[aDTY^dc`S3>i}_(  p  6NMM?Y e]`  IMHP LaserJet III@g,,@MSUDHP LaserJet IIId "d,,??3`  `  `   `   `   `  PH 0(  迷3d23 M NM4 3QQ ;JLQi ;JL Q3_4E4D$% M 3O& Q4$% M 3O& Q4FAt\ u3O  3 b#M43*{Gz?N(#M&! M4%  /[M3O]& Q &Yield to Maturity'4% uC2MZ3O"& Q Years'4523  NM43" 3_ M NM  MM< 444% clM:3Og& Q $Duration and YTM'4%  T[M3O:)Q'4% P[M3O:)Q'4% [M3OQ)Q'44eee ~v  <NMM? ]`h  I"h??3`  `  `  `  `  `  ?п3d23 M NM4 3QQ ;Q ;Q3_4E4D$% |M 3O&Q4$% |M 3O&Q4FApX }3O 3 b#M43*??N(#M&! M4%  9ZM3O^&Q $Time to Maturity'4% B0MZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% GjM:3O}&Q .Duration and Maturity'4% +ZM3Og)Q'4%  MZM3O\)Q'4%  yZM3O)Q'44eee ~v  <NMM?z ]`L&  I"L&??3`  `  `  `  ` | ` | ?3d23 M NM4 3QQ ;Q ;Q3_4E4D$% M 3O&Q4$% M 3O&Q4FAk 3O1 3 bo#M43*@@??N(#M&! M4% 0 M3OM&Q  Coupon Rates '4% AMZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% Rk M:3O&Q 6Duration and Coupon Rates'4% C jM3O~)Q'4% M3Ox)Q'4% wM3Og)Q'44eee ~v  <NMM?z9 C]`0  I"0??3` | ` | ` | ` | ` | ?3d23 M NM4 3QQ ;;=Q ;;=Q3_4E4D$% M 3O&Q4$% M 3O&Q4FAk 3O1 3 bo#M43*@@??N(#M&! M4% 0 M3OM&Q  Coupon Rates '4% AMZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% Rk M:3O&Q 6Duration and Coupon Rates'4% M3Og)Q'4% M3Ox)Q'4% C JM3O~)Q'44eee ~v  <NMM?zg q]`3  I"3??3` |  ` | !` | "` | #` | )?3d23 M NM4 3QQ ;ikQQ3_4E4D$% M 3O&"Q4$% M 3O&!Q4FAkN 3O 3 bo#M43*?@??N(#M&#! M4% h M3O& Q 2Repayment Alternatives '4% 8AMZ3O"& Q Years'4523  NM43" 3_ M NM  MM< 444% R2 M:3O&)Q >Duration and Repayment Choice'4% C M3O~)Q'4% M3Ox)Q'4% wM3Og)Q'44eee >@  mm I  dMbP?_*+%"Dm??eU>@ I  dMbP?_*+%"??eU>@ SummaryInformation(iDocumentSummaryInformation8_999005744cl FPt0yOle  4j Sheet1Sheet2Sheet3Chart4  WorksheetsCharts 6> _PID_GUIDAN{A81F3840-6ACB-11D3-AF7B-B0D2013B3FD9} FMicrosoft Excel ChartBiff8CompObjknbObjInfoWorkbookmoY5SummaryInformation(pExcel.Sheet.89q Oh+'0@H`x Ernest W. SwiftErnest W. SwiftMicrosoft Excel@Gn@% ՜.+,D՜.+,@ PXd lt| IBa=9C =x 8X1Arial1Arial1Arial1Arial1.Times New Roman1.Times New Roman1.Times New Roman1.Times New Roman1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1.Times New Roman1Arial1.Times New Roman1Arial1Arial1Arial1Arial1Arial1Arial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)% "$"#,##0.0_);[Red]\("$"#,##0.0\)0.0 0.00000000 0.0000000 0.000000 0.00000 0.0000% #,##0.0000_);[Red]\(#,##0.0000\) 0.000                + ) , *         # " # "     "   #  #  " "  A"   $  %      # $" $# "  "   " ! ! " &# %" $ $#  ( &" &#  " $!  Chart5?%Sheet13Sheet24Sheet3bZ  3  @@  f O%Solutions to end-of-chapter problems.Time Par valueCouponMaturity Cash FlowYTMPVIFPV of CFPrice =PV*CF*T Numerator = Duration =a. Coupon BondZero Coupon BondHb. Increasing the yield-to-maturity decreases the duration of the bond.Jd. Changing the YTM does not affect the duration of the zero coupon bond.\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.Ch. 9. Problem 2.= Numerator/Price Problem 4Five-year Treasury BondiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows at higher rates.Duration.PVIF = 1/(1+YTM/2)^(Time*2) Problem 5Two-year Treasury Bondb. Change in Problem 6 Six-year CDPVIF = 1/(1+YTM)^(Time)oDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being greceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. Problem 7Semiannual paymentsAnnual payments Consol Bond D = 1 + 1/RlA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percentnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM,Qhas a duration of 20 years because no cash flows occur before the twentieth year. Problem 8Five-year Bond*Duration decreases as the amount of couponinterest increases. Problem 9 Problem 10No annual paymentsAmortized payment of $57.619.059Two-year loan: Principal and interest at end of year two.ETwo-year loan: Interest at end of year one, P & I at end of year two.(Two-year loan: Amortized over two years. Repayment ProvisionsP&I @ 2I@1Amortize5Duration decreases dramatically when a portion of the;principal is repaid at the end of year one. Duration often:is described as the weighted average maturity of an asset,7so if more weight is given to early payments, then the *effective maturity of the loan is reduced. Problem 13 Two-year BondCoupon = Maturity = Par value =YTM == PVIF = 1/(1+YTM)^(Time) PVIF = 1/(1+YTM/2)^(Time*2)Four-year Treasury Bond Three-year Treasury Bond J {tjcUN,D{ H,AH߀HblAA?HxbM_H߀b_b,D80sT0b| ~Xj}bC0sT0bbXj}bgY߀0sT0b XcblAA?_Lb\bH߀_xbk8d}bXwbblA_BA?JbXw߀XwLbblA_BA?J0bXw߀>߀>A?hbDs^FJhb@Xw Xj}Fb@ F l}0sT0c b*W0sT0 4b`X2~Xj}!PvX2 C0{T00sT00|t{Pbt{XbX0X20{T0dbtb0{T0b|b00sT00sT0[V00sT00{T0b0bbb00bb620b0b10L03&b LO Pb  I"PIFF CF??F*3` | '` | (` | *` | +` | ,@3d23 M NM4 3QQ ;;=Q ;;=Q3_4E4D$% M 3O&+Q4$% M 3O&*Q4FAA. @3O@b | 3 bo#M43*@@??N(#M&,! M4% $ rM3OM&(Q  Coupon Rates '4% XMZ3O"&(Q Years'4523  NM43" 3_ M NM  MM< 444% 1Y, M:3O&'Q 6Duration and Coupon Rates'4% _ 1Mj3O~Q'4% Mj3Ox%Q'4% `bMj3Og%Q'44e{Gz??Q?e`TR' @\C@K@e>  I  dMbP?_*+%m'?(?)?MHP LaserJet III@g,,@MSUDHP LaserJet IIId "d,,??U} }  }  }  } }  } I } } } |  T0 b ;b ;b ; ; ; ;b ;b ; ; 0e ; ;} ; ;} ;} ; ; ;~ ;O ;O ;e| ; ;b Ib ; ; ; ;be ;b ;e ;Ch. 9..%Solutions to end-of-chapter problems.  Problem 2. , Coupon Bond- .1 Par value =~ /@@ 1Coupon =~ 0$@Annual payments  - 1YTM =~ 1 @ 1 Maturity =~ 1@-  2Time2 Cash Flow 2PVIF3PV of CF32PV*CF*T323- ~ 4?!5Y@ DD*Kh/?D@D!5^B{ %W@ DD5'5^B{ %W@DDD55- ~ 4@'50@DDD*72YQo?D@D!5x@ DD5'5x@ DDD55- 1Price =#5C.@  %- 1 5 Numerator = 5# 5@ %5 1 Duration = ! 65G#͑?  D D 7= Numerator/Price - 8-   1YTM =~ 0$@ -  2Time 2 Cash Flow 2PVIF 3PV of CF  2PV*CF*T - ~ 4?! 5Y@  DD* ]tE? D @D ! 5袋.V@ D D ' 5袋.V@D D D  - ~ 4@'50@DDD*Tx?r?D @D!5袋.h@ DD'5袋.h@DDD- 1Price =#5?@ % - 15 Numerator =5#5tE]ԝ@  % 51 Duration =!6.袋? DD 7= Numerator/Price -8- 1YTM =~ (@-  2Time2 Cash Flow 2PVIF3PV of CF2PV*CF*T- ~ 4?!5Y@ DD*$I$I?D@D!5$I$IRV@ DD'5$I$IRV@DDD- ~ 4@'50@DDD*?D@D!5r ^Ng@ DD'5r ^Ng@DDD- 1Price =#51@ %- 15 Numerator =5#5Sr̜@ %51 Duration =!6 F}? DD 7= Numerator/Price -QHb. Increasing the yield-to-maturity decreases the duration of the bond. ,Zero Coupon Bond- 1 Par value =~ 9@@ 1Coupon =~ :- 1YTM =~ 4 @ 1 Maturity =~ ;@-  2Time2 Cash Flow 2PVIF3PV of CF32PV*CF*T3- 4?5*Kh/?D@D!5 DD5'5DDD5- ~ 4@5@@D*72YQo?D@D!5 kʊ@ DD5'5 kʚ@ DDD5- 1Price =#5 kʊ@  %- Dla5>oa Da DasCso ;T0! ;b" ;b# ;b$ ;% ;& ;' ;b( ;b) ;* ; 0e+ ;, ;}- ;. ;}/ }0 1 2 ~3 O4 ;O5 ;e|6 ;7 ;b8 ;Ib9 ;: ;; ;< ;be= ;b> ;e? ; 1 5 Numerator = 5# 5 kʚ@ %5 1 Duration = ! 6@$ D D 7= Numerator/Price -!8- ""1YTM =~ "<$@"-  #2Time#2 Cash Flow #2PVIF#3PV of CF##2PV*CF*T#- $4?5*$]tE?$D"@D$!$5% D$D$$5'$5%D$D$D$$- ~ %4@%5@@%D*%Tx?r?%D"@D%!%5Ӊ@& D%D%%5'%5ә@'D%D%D%%- &&1Price =#&5Ӊ@' %$%&- '1'5 Numerator ='5#'5ә@$ %$%5'1 Duration ='!'6@+ D'D&' 7= Numerator/Price' -(8- ))1YTM =~ )4(@)-  *2Time*2 Cash Flow *2PVIF*3PV of CF**2PV*CF*T*- +4?5*+$I$I?+D)@D+!+5, D+D++5'+5,D+D+D++- ~ ,4@,5@@,D*,?,D)@D,!,5}h@- D,D,,5',5}h@.D,D,D,,- --1Price =#-5}h@. %+,-- .1.5 Numerator =.5#.5}h@+ %+,5.1 Duration =.!.6@n = D.D-. 7= Numerator/Price. -/ S0Jd. Changing the YTM does not affect the duration of the zero coupon bond.0 e1\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more1 h2_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.2 3 Problem 43 4 4,Five-year Treasury Bond4- 551 Par value =~ 59@@ 551Coupon =~ 5:$@55Semiannual payments 5 - 66DYTM =~ 6C$@ 661 Maturity =~ 6;@6-  72Time72 Cash Flow 72PVIF73PV of CF7372PV*CF*T73- ~ 84?+85I@8D5@D5@?98yy?8#D6@?D8!85< D=D==5'=5r9r[@>D=D=D==5- ~ >> @+>5I@>D5@D5@?9>?,?>#D6@?D>!>5J\A@? D>D>>5'>5~~_@?D>D>D>>5- ~ ?>@+?5I@?D5@D5@?9???#D6@?D?!?5Y@@@ D?D??5'?5Y`@@D?D?D??5- Dnl Da Da u4Jo@ ;T0A ;bB ;bC ;bD ;E ;F ;G ;bH ;bI ;J ; 0eK ;L ;}M ;N ;}O ;}P ;Q ;R ;~S ;OT ;OU ;e|V ;W ;bX ;IbY ;Z ;[ ;\ ;be] ;b^ ;e_ ;~ @4@+@5I@@D5@D5@?9@ࢠ?@#D6@?D@!@5֢?@@A D@D@@5'@51'/!b@AD@D@D@@5- ~ A4@1A5h@AD5@D5@?D59A;Qh-?A#D6@?DA!A?Kw$@B DADAA5'A?oU.@CDADADAA5- B4B1Price =#B5@@C %8AB- C1C5 Numerator =C5#C5 UYҫ@8 %8AC1 Duration =C!C6ag47@ DCDBC 7= Numerator/PriceC -D- E E,Five-year Treasury BondE- FF1 Par value =~ F9@@ FF1Coupon =~ F:$@FFSemiannual payments F - GGDYTM =~ GC(@ GG1 Maturity =~ G;@G-  H2TimeH2 Cash Flow H2PVIFH3PV of CFH3H2PV*CF*TH3- ~ I4?+I5I@ID5@D5@?9IsHM0?I#DG@?DI!I5琚`G@J DIDII5'I5琚`7@JDIDIDI I5I@Duration I 3YTMI -~ J4?+J5I@JD5@D5@?9J\mz?K#DG@?DJ!J5p*?F@K DJDJJ5'J5p*?F@KDJDJDJ J--JAŏ17@~ J 0$@J -~ K4?+K5I@KD5@D5@?9K6B)?L#DG@?DK!K5޺;D@L DKDKK5'K5&MYX|O@LDKDKDK K--KAz): @~ K (@K -~ L4@+L5I@LD5@D5@?9L<_X?M#DG@?DL!L5DgBfC@M DLDLL5'L5DgBfS@MDLDLDL L--LAv@~ L ,@L -~ M4@+M5I@MD5@D5@?9M?N#DG@?DM!M55YsB@N DMDMM5'M5nZW@NDMDMDMM---- ~ N4@+N5I@ND5@D5@?9NEh ?O#DG@?DN!N5uYA@O DNDNN5'N5d0oZ@ODNDNDNNA- ~ O> @+O5I@OD5@D5@?9O\%H?P#DG@?DO!O50]@@P DODOO5'O5]@PDODODOOA- ~ P>@+P5I@PD5@D5@?9PB?Q#DG@?DP!P5^?@Q DPDPP5'P5^_@QDPDPDPPA- ~ Q4@+Q5I@QD5@D5@?9Q[@ ?R#DG@?DQ!Q5L=@R DQDQQ5'Q5{?`@RDQDQDQQA- ~ R4@1R5h@RD5@D5@?DF9Rq#^?I#DG@?DR!R?< R@S DRDRR5'R?K )%@TDRDRDR RA.R- S4S1Price =#S5j1@T %IRS- T1T5 Numerator =T5#T5Zt@I %IRT1 Duration =T!T6ߖه @8 DTDST 7= Numerator/PriceT -U- V V,Five-year Treasury BondV- WW1 Par value =~ W/@@ WW1Coupon =~ W:$@WWSemiannual payments W - XXDYTM =~ X=,@ XX1 Maturity =~ X;@X-  Y2TimeY2 Cash Flow Y2PVIFY3PV of CFY3Y2PV*CF*TY3- ~ Z4?+Z5I@ZD5@D5@?9Z+J#?Z#DX@?DZ!Z5 N]G@[ DZDZZ5'Z5 N]7@[DZDZDZZ5- ~ [4?+[5I@[D5@D5@?9[ȟ5?\#DX@?D[![5,E@\ D[D[[5'[5,E@\D[D[D[[@- ~ \4?+\5I@\D5@D5@?9\OK?]#DX@?D\!\5qNhD@] D\D\\5'\5AuN@]D\D\D\\A- ~ ]4@+]5I@]D5@D5@?9]@8i?^#DX@?D]!]5GzC@^ D]D]]5']5GzS@^D]D]D]]A- ~ ^4@+^5I@^D5@D5@?9^% @+`5I@`D5@D5@?9`ݐ?a#DX@?D`!`5VZ2#?@a D`D``5'`5W>[@aD`D`D``@- ~ a>@+a5I@aD5@D5@?9a>Žџ?b#DX@?Da!a5 o=@b DaDaa5'a5 o]@bDaDaDaaB- ~ b4@+b5I@bD5@D5@?9brg?c#DX@?Db!b5l}sZ2;@c DbDbb5'b5AY^@cDbDbDbbB- ~ c4@1c5h@cD5@D5@?DW9cE:eD?Z#DX@?Dc!c?@[Q"@d DcDcc5'c?P٤@eDcDcDccB- d4d1Price =#d5:܊@e %Zcd- e1e5 Numerator =e5#e56s@Z %Zce1 Duration =e!e6nݰ@J DeDde 7= Numerator/Pricee -f rgiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows g hat higher rates.h i j k l Problem 5l  ma.$mGFour-year Treasury Bond mGG- nn1 Par value =~ n9@@ nn1Coupon =~ n:$@nnSemiannual payments n - ooDYTM =~ oC$@ oo1 Maturity =~ o;@o-  p2Timep2 Cash Flow p2PVIFp3PV of CFp3p2PV*CF*Tp3- ~ q4?+q5I@qD5@D5@?9qEyy?q#D6@?Dq!q5< @+w5I@wD5@D5@?9wE?,?w#D6@?Dw!w5J\A@x DwDww5'w5~~_@xDwDwDww5- ~ x>@1x5h@qD5@D5@?Dn9xE?x#D6@?Dx!x?66]s5@y DxDxx5'x?66]s5@zDxDxDxx5- y4y1Price =#y5@@z %qxy- z1z5 Numerator =z5#z5_@q %qxz1Duration z=!z6`a?% @[ DzDyz 7= Numerator/Pricez -{#{,Three-year Treasury Bond {- ||1 Par value =~ |9@@ ||1Coupon =~ |:$@||Semiannual payments | - }}DYTM =~ }C$@ }}1 Maturity =~ };@}-  ~2Time~2 Cash Flow ~2PVIF~3PV of CF~3~2PV*CF*T~3- ~ 4?+5I@D5@D5@?9Eyy?#D6@?D!5<aDTT  T0 b b b ; ; ; ;b ;b ; ; 0e ; ;} ; ;} ;} ; ; ;~ ;O ;O ;e| ; ;b ;Ib ; ; ; ;be ;b ;e ;    Problem 6 G Six-year CDG- 1 Par value =~ 9@ 1Coupon =~ :@Annual payments - DYTM =~ C@ 1 Maturity =~ ;@-  2Time2 Cash Flow 2PVIF3PV of CF32PV*CF*T3- ~ 4?!5@ D@D@+HsHM0?D@D!5sHM@ DD5'5sHM@DDD$ PVIF = 1/(1+YTM)^(Time)- ~ 4@!5@ D@D@+H\mz?D@D!5i @ DD5'5i @DDD5- ~ 4@!5@ D@D@+6B)?D@D!5&MYX|@ DD5'592CB@DDD5- ~ 4@!5@ D@D@+<_X?D@D!5c}@ DD5'5c@DDD5- ~ 4@!5@ D@D@+?D@D!5Ѕ|@ DD5'5@DDD5- ~ 4@'/@D@D@D+Eh ?D@D!?Jd+0@ DD5'?7 o@DDD5- 41Price =#5@ %- 15 Numerator =5#5\DPits@ %4Duration =!6 pgu@ DD 7= Numerator/Price -- , Six-year CD- 1 Par value =~ 9@ 1Coupon =~ :@Semiannual payments - DYTM =~ C@ 1 Maturity =~ ;@-  2Time2 Cash Flow 2PVIF3PV of CF32PV*CF*T3- ~ 4?+5r@D@D@?9EH%e?#D@?D!5 @+5r@D@D@?9ES;}?#D@?D!55O²}n@ DD5'5Oj@DDD5- ~ >@+5r@D@D@?9E{='B?#D@?D!5 4^"Tm@ DD5'5 4^"T@DDD5- ~ 4@+5r@D@D@?9Eb|?#D@?D!5h™l@ DD5'5}*@DDD5- ~ 4@+5r@D@D@?9EJ ?#D@?D!5F4Mk@ DD5'5@ @p@DDD5- ~ 4@+5r@D@D@?9Eࣝ9?#D@?D!5 >k@ DD5'5GA+@DDD5- Dl 4>oa >o ;T0 ;b ;b ;b    b b  0e  }  } }   ~ O O e|  b Ib    be ;b ;e ;~ 4@1/@D@D@?D9E 7q?#D@?D!?w68@ DD5'?(*@DDD5- 41Price =#5@ %- 15 Numerator =5#5 @ %1Duration =!6?W@ DD 7= Numerator/Price -- xoDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being  pgreceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows.    Problem 7  Consol Bond  YTM D = 1 + 1/R ~  @#'+@ D ~ $@#'&@ D ~ (@#("@ D  ulA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percent wnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM, ZQhas a duration of 20 years because no cash flows occur before the twentieth year.            Problem 8 ,Five-year Bond- 1 Par value =~ 9@@ 1Coupon =~ :@Annual payments - DYTM =~ C"@ 1 Maturity =~ ;@- DR la 4>DQQQ | 4A ;T0 ;b ;b ;b ; ; ; ;b ;b ; ; 0e ; ;} ; ;} ;} ; ; ;~ ;O ;O ;e| ; ;b ;Ib ; ; ; ;be ;b ;e ; 2Time2 Cash Flow 2PVIF3PV of CF32PV*CF*T3- ~ 4?!5Q@ D@D@+E[?D@D!5 z|P@ DD5'5 z|P@DDD$ PVIF = 1/(1+YTM)^(Time)-- ~ 4@!5Q@ D@D@+EB ?D@D!5H5suM@ DD5'5H5su]@DDD5- ~ 4@!5Q@ D@D@+E+!?D@D!5HK@ DD5'56Ed@DDD5- ~ 4@!5Q@ D@D@+E7%Yk?D@D!5Ċi}H@ DD5'5Ċi}h@DDD5- ~ 4@'5@D@D@D+Ecs D*@DDD5- 41Price =#5zi^ь@ %- 15 Numerator =5#5lMjig@ %1Duration =!6 ufo@ DD 7= Numerator/Price -- ,Five-year Bond- 1 Par value =~ 9@@ 1Coupon =~ :"@Annual payments - DYTM =~ C"@ 1 Maturity =~ ;@-  2Time2 Cash Flow 2PVIF3PV of CF32PV*CF*T3- ~ 4?!5V@ DD+E[?D@D!5F*VgT@ DD5'5F*VgT@DDD$ PVIF = 1/(1+YTM)^(Time)-- ~ 4@!5V@ DD+EB ?D@D!5}R@ DD5'5}b@DDD5- ~ 4@!5V@ DD+E+!?D@D!5R_Q@ DD5'5]Nj@DDD5- ~ 4@!5V@ DD+E7%Yk?D@D!5Ei]O@ DD5'5Ei]o@DDD5- ~ 4@'5@DDD+EcsI@ DD5'5>I@DDD5- ~ 4@'5@DDD+E+d?DD!?w$F1@ DD5'? 8_@DDD5- 41Price =#5 @ %- 15 Numerator =5#5L`R@ %1Duration =!6"g}* @ DD 7= Numerator/Price -- > 0I]BRR[D4Aoa  ;T0! ;b" ;b# ;b$ ;% ;& ;' ;b( ;b) ;* ; 0e+ ;, ;}- ;. ;}/ ;}0 ;1 ;2 ;~3 ;O4 ;O5 ;e|6 ;7 b8 Ib9 : ; < be= ;b> ;e? ;  ,Five-year Bond - !!1 Par value =~ !9@ !!1Coupon =~ !:$@!!Annual payments ! - ""DYTM =~ "C$@ ""1 Maturity =~ ";@"-  #2Time#2 Cash Flow #2PVIF#3PV of CF#3#2PV*CF*T#3- ~ $4?!$5@@$ D!D!+$E]tE?0D"D$!$5袋.h@% D$D$$5'$5袋.h@%D$D$D$$$ PVIF = 1/(1+YTM)^(Time)$-- ~ %4@!%5@@% D!D!+%ETx?r?$D"D%!%5Ӊ@& D%D%%5'%5ә@&D%D%D%%5- ~ &4@!&5@@& D!D!+&E[V ?%D"D&!&57Uz@' D&D&&5'&5)V㛡@'D&D&D&&5- ~ '4@!'5@@' D!D!+'E}??&D"D'!'5x|X@( D'D''5''5x|X@(D'D'D''5- ~ (4@'(5|@(D!D!D!+(E+d?'D"D(!(?Ur"@) D(D((5'(?IQլ@*D(D(D((5- )4)1Price =#)5@* %$()- *1*5 Numerator =*5#*5Pl]T\@$ %$(*1Duration *=!*6)#5@ D*D)* 7= Numerator/Price* -+- ,,,Five-year Bond,- --1 Par value =~ -9@ --1Coupon =~ -:(@--Annual payments - - ..DYTM =~ .C$@ ..1 Maturity =~ .;@.-  /2Time/2 Cash Flow /2PVIF/3PV of CF/3/2PV*CF*T/3- ~ 04?!05@0 D-D-+0E]tE?D.D0!05. @1 D0D005'05. @1D0D0D0$0 PVIF = 1/(1+YTM)^(Time)0-- ~ 14@!15@1 D-D-+1ETx?r?0D.D1!15,Ra@2 D1D115'15,Ra@2D1D1D115- ~ 24@!25@2 D-D-+2E[V ?1D.D2!253A,@3 D2D225'25pw!@3D2D2D225- ~ 34@!35@3 D-D-+3E}??2D.D3!35/휉@4 D3D335'35/휩@4D3D3D335- ~ 44@'45@4D-D-D-+4E+d?3D.D4!4?Q*@5 D4D445'4?s@6D4D4D445- 5451Price =#55,$@6 %045- 6165 Numerator =65#65Ӌ f@0 %0461Duration 6=!66ZK@$ D6D56 7= Numerator/Price6 -7 8  9b. 9%9 Change in9 ::%Duration:Coupon:Duration: ;;$`TR' @~ ;) @; <<$\C@~ <)$@<!lV}< ==$K@~ =)(@=_vO= > ?3?*Duration decreases as the amount of coupon? DGlAoa Aoa K_DTT @ ;T0A ;bB ;bC ;bD ;E ;J K bL bM N 0eO P }Q R }T }U V W ~X OY OZ e|[ ] b^ Ib_ @@interest increases.@ A B C D E J Problem 10J KBK9Two-year loan: Principal and interest at end of year two.K LL Par value~ Lj@ LLCoupon~ LLLNo annual paymentsL M M"YTM~ M#$@ MMMaturity~ M@ M  NTimeN Cash Flow NPVIFNPV of CFNNPV*CF*TN ~ O?!OO DLDL+O]tE?ODMDO!OP DODOO'OPDODODO O OPVIF = 1/(1+YTM)^(Time)O ~ P@*P@aDL?+PTx?r?PDMDP!P+j@Q DPDPP'P&jARDPDPDPP QQPrice =#Q*j@R %OPQ RR Numerator =R#RjAO %OPR Duration =R!R!@ DRDQR = Numerator/PriceTNTETwo-year loan: Interest at end of year one, P & I at end of year two.T UU Par value~ Uj@ UUCoupon~ U$@UUAnnual paymentsU V V"YTM~ V#$@ VVMaturity~ V@ V  WTimeW Cash Flow WPVIFWPV of CFWWPV*CF*TW ~ X?!X@X DUDU+X]tE?XDVDX!XE]t@Y DXDXX'XE]t@YDXDXDX X XPVIF = 1/(1+YTM)^(Time)X ~ Y@'Y@YDUDUDU+YTx?r?YDVDY!Y&E]t1@Z DYDYY'Y&E]t1A[DYDYDYY ZZPrice =#Z*i@[ %XYZ [[ Numerator =[#[.MAX %XY[ Duration =[![!.袋?O D[DZ[ = Numerator/Price]1](Two-year loan: Amortized over two years.] ^^ Par value~ ^j@ ^^Coupon~ ^$@^(^Amortized payment of $57.619.05^ _ _"YTM~ _#$@ __Maturity~ _@ _ 8D3ji}_vi}_Y` T0a bb bc bd g h i bj bk ;l ; 0em ;n ;}o ;p ;}q ;}s t u ~v Ow Ox e|y z b{ Ib `Time` Cash Flow `PVIF`PV of CF``PV*CF*T` a?ǭ_+a]tE?aD_Da!a袋@b DaDaa'a袋@bDaDaDa a aPVIF = 1/(1+YTM)^(Time)a b@ǭ_+bTx?r?bD_Db!b&7a@@c DbDbb'b&7a@@dDbDbDbb ccPrice =#c*lj@d %abc dd Numerator =d#d=;nAa %abd Duration =d!d!yy?X DdDcd = Numerator/Price gb.g%g Repaymentg Change ing hh%Durationh ProvisionshDurationh i~ i$@i)P&I @ 2i jj$ q? j)I@1j48Ej kk$%䃞?k)Amortizek3ۿk l >m5Duration decreases dramatically when a portion of them Dn;principal is repaid at the end of year one. Duration oftenn Co:is described as the weighted average maturity of an asset,o @p7so if more weight is given to early payments, then the p 3q*effective maturity of the loan is reduced.q s Problem 13s tt Two-year Bondt uu Par value~ u@@ uuCoupon~ u$@uuAnnual paymentsu v v"YTM~ v#,@ vvMaturity~ v@ v  wTimew Cash Flow wPVIFwPV of CFwwPV*CF*Tw ~ x?!xY@x DuDu+x}?xDvDx!x>U@y DxDxx'x>U@yDxDxDx x xPVIF = 1/(1+YTM)^(Time)x ~ y@'y0@yDuDuDu+y]z?yDvDy!y&&*rPs@z DyDyy'y&&*rPs@{DyDyDyy zzPrice =#zr)1@{ %xyz {{ Numerator ={#{Nq1қ@x %xy{ Duration ={!{! ^?P D{Dz{ = Numerator/Price6 }_[aDTY^dc`S3>i}_(  p  6NMM?Y e]`  IMHP LaserJet III@g,,@MSUDHP LaserJet IIId "d,,??3`  `  `   `   `   `  PH 0(  迷3d23 M NM4 3QQ ;JLQi ;JL Q3_4E4D$% M 3O& Q4$% M 3O& Q4FAt\ u3O  3 b#M43*{Gz?N(#M&! M4%  /[M3O]& Q &Yield to Maturity'4% uC2MZ3O"& Q Years'4523  NM43" 3_ M NM  MM< 444% clM:3Og& Q $Duration and YTM'4%  T[M3O:)Q'4% P[M3O:)Q'4% [M3OQ)Q'44eee ~v  <NMM? ]`h  I"h??3`  `  `  `  `  `  ?п3d23 M NM4 3QQ ;Q ;Q3_4E4D$% |M 3O&Q4$% |M 3O&Q4FApX }3O 3 b#M43*??N(#M&! M4%  9ZM3O^&Q $Time to Maturity'4% B0MZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% GjM:3O}&Q .Duration and Maturity'4% +ZM3Og)Q'4%  MZM3O\)Q'4%  yZM3O)Q'44eee ~v  <NMM?z ]`L&  I"L&??3`  `  `  `  ` | ` | ?3d23 M NM4 3QQ ;Q ;Q3_4E4D$% M 3O&Q4$% M 3O&Q4FAk 3O1 3 bo#M43*@@??N(#M&! M4% 0 M3OM&Q  Coupon Rates '4% AMZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% Rk M:3O&Q 6Duration and Coupon Rates'4% C jM3O~)Q'4% M3Ox)Q'4% wM3Og)Q'44eee ~v  <NMM?z9 C]`0  I"0??3` | ` | ` | ` | ` | ?3d23 M NM4 3QQ ;;=Q ;;=Q3_4E4D$% M 3O&Q4$% M 3O&Q4FAk 3O1 3 bo#M43*@@??N(#M&! M4% 0 M3OM&Q  Coupon Rates '4% AMZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% Rk M:3O&Q 6Duration and Coupon Rates'4% M3Og)Q'4% M3Ox)Q'4% C JM3O~)Q'44eee ~v  <NMM?zg q]`3  I"3??3` |  ` | !` | "` | #` | )?3d23 M NM4 3QQ ;ikQQ3_4E4D$% M 3O&"Q4$% M 3O&!Q4FAkN 3O 3 bo#M43*?@??N(#M&#! M4% h M3O& Q 2Repayment Alternatives '4% 8AMZ3O"& Q Years'4523  NM43" 3_ M NM  MM< 444% R2 M:3O&)Q >Duration and Repayment Choice'4% C M3O~)Q'4% M3Ox)Q'4% wM3Og)Q'44eee >4@7 77 mm I  dMbP?_*+%"Dm??eU>@ I  dMbP?_*+%"??eU>@ DocumentSummaryInformation8_999009327s F0[BOle CompObjrub 5j Sheet1Sheet2Sheet3Chart5  WorksheetsCharts 6> _PID_GUIDAN{A81F3840-6ACB-11D3-AF7B-B0D2013B3FD9} FMicrosoft Excel ChartBiff8Excel.Sheet.89q Oh+'0@H`x Ernest W. SwiftErnest W. 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" &# %" $ $#  ( &" &#  " $!  $ $  &" !" Chart6S&Sheet17Sheet28Sheet3bZ  3  @@   P%Solutions to end-of-chapter problems.Time Par valueCouponMaturity Cash FlowYTMPVIFPV of CFPrice =PV*CF*T Numerator = Duration =a. Coupon BondZero Coupon BondHb. Increasing the yield-to-maturity decreases the duration of the bond.Jd. Changing the YTM does not affect the duration of the zero coupon bond.\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.Ch. 9. Problem 2.= Numerator/Price Problem 4Five-year Treasury BondiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows at higher rates.Duration.PVIF = 1/(1+YTM/2)^(Time*2) Problem 5Two-year Treasury Bondb. Change in Problem 6 Six-year CDPVIF = 1/(1+YTM)^(Time)oDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being greceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. Problem 7Semiannual paymentsAnnual payments Consol Bond D = 1 + 1/RlA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percentnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM,Qhas a duration of 20 years because no cash flows occur before the twentieth year. Problem 8Five-year Bond*Duration decreases as the amount of couponinterest increases. Problem 9 Problem 10No annual paymentsAmortized payment of $57.619.059Two-year loan: Principal and interest at end of year two.ETwo-year loan: Interest at end of year one, P & I at end of year two.(Two-year loan: Amortized over two years. Repayment ProvisionsP&I @ 2I@1Amortize5Duration decreases dramatically when a portion of the;principal is repaid at the end of year one. Duration often:is described as the weighted average maturity of an asset,7so if more weight is given to early payments, then the *effective maturity of the loan is reduced. Problem 13 Two-year BondCoupon = Maturity = Par value =YTM == PVIF = 1/(1+YTM)^(Time) PVIF = 1/(1+YTM/2)^(Time*2)Four-year Treasury Bond Three-year Treasury Bond HTwo-year loan: Interest at end of year one, P & I at end of year two. J itAXcCNp{$̳Pb,D8#F NI bxblAA?$bM_(Н0b_b,D80sT0b| ~Xj}bC^0sT0bazp0FIbbFI000 LblAA`IS0bt00$_Duration and Repayment Choice'4% `Mj3Og Q'4% Mj3Ox%Q'4% _ Mj3O~%Q'44ee@ q?%䃞?e> % I  dMbP?_*+%m'?(?)?MHP LaserJet III@g,,@MSUDHP LaserJet IIId "d,,??U} }  }  }  } }  } I } } } |  T0 b ;b ;T ; ; ; ;b ;b ;@ ; 0e ;14 ;00 ;@i ;hi ;0 ;b ; 0 ;} ;bO ;b ;}b ; ;b O ;} ;v ;T0T ;e ;b ;T0 ;Ch. 9..%Solutions to end-of-chapter problems.  Problem 2. * Coupon Bond+ ,/ Par value =~ -@@ /Coupon =~ .$@Annual payments  + /YTM =~ / @ / Maturity =~ /@+  0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T101+ ~ 2?!3Y@ DD*Kh/?D@D!3^B{ %W@ DD3'3^B{ %W@DDD33+ ~ 2@'30@DDD*72YQo?D@D!3x@ DD3'3x@ DDD33+ /Price =#3C.@  %+ / 3 Numerator = 3# 3@ %5 / Duration = ! 45G#͑?  D D 5= Numerator/Price + 6+   /YTM =~ .$@ +  0Time 0 Cash Flow 0PVIF 1PV of CF  0PV*CF*T + ~ 2?! 3Y@  DD* ]tE? D @D ! 3袋.V@ D D ' 3袋.V@D D D  + ~ 2@'30@DDD*Tx?r?D @D!3袋.h@ DD'3袋.h@DDD+ /Price =#3?@ % + /3 Numerator =3#3tE]ԝ@  % 5/ Duration =!4.袋? DD 5= Numerator/Price +6+ /YTM =~ (@+  0Time0 Cash Flow 0PVIF1PV of CF0PV*CF*T+ ~ 2?!3Y@ DD*$I$I?D@D!3$I$IRV@ DD'3$I$IRV@DDD+ ~ 2@'30@DDD*?D@D!3r ^Ng@ DD'3r ^Ng@DDD+ /Price =#31@ %+ /3 Numerator =3#3Sr̜@ %5/ Duration =!4 F}? DD 5= Numerator/Price +QHb. Increasing the yield-to-maturity decreases the duration of the bond. *Zero Coupon Bond+ / Par value =~ 8@@ /Coupon =~ 9+ /YTM =~ 2 @ / Maturity =~ :@+  0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T1+ 2?3*Kh/?D@D!3 DD3'3DDD3+ ~ 2@3@@D*72YQo?D@D!3 kʊ@ DD3'3 kʚ@ DDD3+ /Price =#3 kʊ@  %+ Dla5>oa Da DasCso ;T0! ;b" ;b# ;T$ ;% ;& ;' ;b( ;b) ;@* ; 0e+ ;14, ;00- ;@i. ;hi/ 00 b1 02 }3 bO4 ;b5 ;}b6 ;7 ;b8 ;O9 ;}: ;v; ;T0T< ;e= ;b> ;T0? ; / 3 Numerator = 3# 3 kʚ@ %5 / Duration = ! 4@$ D D 5= Numerator/Price +!6+ ""/YTM =~ ";$@"+  #0Time#0 Cash Flow #0PVIF#1PV of CF##0PV*CF*T#+ $2?3*$]tE?$D"@D$!$3% D$D$$3'$3%D$D$D$$+ ~ %2@%3@@%D*%Tx?r?%D"@D%!%3Ӊ@& D%D%%3'%3ә@'D%D%D%%+ &&/Price =#&3Ӊ@' %$%&+ '/'3 Numerator ='3#'3ә@$ %$%5'/ Duration ='!'4@+ D'D&' 5= Numerator/Price' +(6+ ))/YTM =~ )2(@)+  *0Time*0 Cash Flow *0PVIF*1PV of CF**0PV*CF*T*+ +2?3*+$I$I?+D)@D+!+3, D+D++3'+3,D+D+D+++ ~ ,2@,3@@,D*,?,D)@D,!,3}h@- D,D,,3',3}h@.D,D,D,,+ --/Price =#-3}h@. %+,-+ ./.3 Numerator =.3#.3}h@+ %+,5./ Duration =.!.4@n = D.D-. 5= Numerator/Price. +/ S0Jd. Changing the YTM does not affect the duration of the zero coupon bond.0 e1\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more1 h2_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.2 3 Problem 43 4 4*Five-year Treasury Bond4+ 55/ Par value =~ 58@@ 55/Coupon =~ 59$@55Semiannual payments 5 + 66CYTM =~ 6B$@ 66/ Maturity =~ 6:@6+  70Time70 Cash Flow 70PVIF71PV of CF7170PV*CF*T71+ ~ 82?+83I@8D5@D5@?98yy?8#D6@?D8!83< D=D==3'=3r9r[@>D=D=D==3+ ~ >= @+>3I@>D5@D5@?9>?,?>#D6@?D>!>3J\A@? D>D>>3'>3~~_@?D>D>D>>3+ ~ ?=@+?3I@?D5@D5@?9???#D6@?D?!?3Y@@@ D?D??3'?3Y`@@D?D?D??3+ Dnl Da Da u4Jo@ ;T0A ;bB ;bC ;TD ;E ;F ;G ;bH ;bI ;@J ; 0eK ;14L ;00M ;@iN ;hiO ;0P ;bQ ; 0R ;}S ;bOT ;bU ;}bV ;W ;bX ;OY ;}Z ;v[ ;T0T\ ;e] ;b^ ;T0_ ;~ @2@+@3I@@D5@D5@?9@ࢠ?@#D6@?D@!@3֢?@@A D@D@@3'@31'/!b@AD@D@D@@3+ ~ A2@1A3h@AD5@D5@?D59A;Qh-?A#D6@?DA!A>Kw$@B DADAA3'A>oU.@CDADADAA3+ B2B/Price =#B3@@C %8AB+ C/C3 Numerator =C3#C3 UYҫ@8 %8AC/ Duration =C!C4ag47@ DCDBC 5= Numerator/PriceC +D+ E E*Five-year Treasury BondE+ FF/ Par value =~ F8@@ FF/Coupon =~ F9$@FFSemiannual payments F + GGCYTM =~ GB(@ GG/ Maturity =~ G:@G+  H0TimeH0 Cash Flow H0PVIFH1PV of CFH1H0PV*CF*TH1+ ~ I2?+I3I@ID5@D5@?9IsHM0?I#DG@?DI!I3琚`G@J DIDII3'I3琚`7@JDIDIDI I3I?Duration I 1YTMI +~ J2?+J3I@JD5@D5@?9J\mz?K#DG@?DJ!J3p*?F@K DJDJJ3'J3p*?F@KDJDJDJ J++J@ŏ17@~ J .$@J +~ K2?+K3I@KD5@D5@?9K6B)?L#DG@?DK!K3޺;D@L DKDKK3'K3&MYX|O@LDKDKDK K++K@z): @~ K (@K +~ L2@+L3I@LD5@D5@?9L<_X?M#DG@?DL!L3DgBfC@M DLDLL3'L3DgBfS@MDLDLDL L++L@v@~ L ,@L +~ M2@+M3I@MD5@D5@?9M?N#DG@?DM!M35YsB@N DMDMM3'M3nZW@NDMDMDMM++++ ~ N2@+N3I@ND5@D5@?9NEh ?O#DG@?DN!N3uYA@O DNDNN3'N3d0oZ@ODNDNDNN@+ ~ O= @+O3I@OD5@D5@?9O\%H?P#DG@?DO!O30]@@P DODOO3'O3]@PDODODOO@+ ~ P=@+P3I@PD5@D5@?9PB?Q#DG@?DP!P3^?@Q DPDPP3'P3^_@QDPDPDPP@+ ~ Q2@+Q3I@QD5@D5@?9Q[@ ?R#DG@?DQ!Q3L=@R DQDQQ3'Q3{?`@RDQDQDQQ@+ ~ R2@1R3h@RD5@D5@?DF9Rq#^?I#DG@?DR!R>< R@S DRDRR3'R>K )%@TDRDRDR R@.R+ S2S/Price =#S3j1@T %IRS+ T/T3 Numerator =T3#T3Zt@I %IRT/ Duration =T!T4ߖه @8 DTDST 5= Numerator/PriceT +U+ V V*Five-year Treasury BondV+ WW/ Par value =~ W-@@ WW/Coupon =~ W9$@WWSemiannual payments W + XXCYTM =~ X<,@ XX/ Maturity =~ X:@X+  Y0TimeY0 Cash Flow Y0PVIFY1PV of CFY1Y0PV*CF*TY1+ ~ Z2?+Z3I@ZD5@D5@?9Z+J#?Z#DX@?DZ!Z3 N]G@[ DZDZZ3'Z3 N]7@[DZDZDZZ3+ ~ [2?+[3I@[D5@D5@?9[ȟ5?\#DX@?D[![3,E@\ D[D[[3'[3,E@\D[D[D[[?+ ~ \2?+\3I@\D5@D5@?9\OK?]#DX@?D\!\3qNhD@] D\D\\3'\3AuN@]D\D\D\\@+ ~ ]2@+]3I@]D5@D5@?9]@8i?^#DX@?D]!]3GzC@^ D]D]]3']3GzS@^D]D]D]]@+ ~ ^2@+^3I@^D5@D5@?9^%[@aD`D`D``?+ ~ a=@+a3I@aD5@D5@?9a>Žџ?b#DX@?Da!a3 o=@b DaDaa3'a3 o]@bDaDaDaaA+ ~ b2@+b3I@bD5@D5@?9brg?c#DX@?Db!b3l}sZ2;@c DbDbb3'b3AY^@cDbDbDbbA+ ~ c2@1c3h@cD5@D5@?DW9cE:eD?Z#DX@?Dc!c>@[Q"@d DcDcc3'c>P٤@eDcDcDccA+ d2d/Price =#d3:܊@e %Zcd+ e/e3 Numerator =e3#e36s@Z %Zce/ Duration =e!e4nݰ@J DeDde 5= Numerator/Pricee +f rgiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows g hat higher rates.h i j k l Problem 5l  ma.$mFFour-year Treasury Bond mFF+ nn/ Par value =~ n8@@ nn/Coupon =~ n9$@nnSemiannual payments n + ooCYTM =~ oB$@ oo/ Maturity =~ o:@o+  p0Timep0 Cash Flow p0PVIFp1PV of CFp1p0PV*CF*Tp1+ ~ q2?+q3I@qD5@D5@?9qDyy?q#D6@?Dq!q3<66]s5@y DxDxx3'x>66]s5@zDxDxDxx3+ y2y/Price =#y3@@z %qxy+ z/z3 Numerator =z3#z3_@q %qxz/Duration z=!z4`a?% @[ DzDyz 5= Numerator/Pricez +{#{*Three-year Treasury Bond {+ ||/ Par value =~ |8@@ ||/Coupon =~ |9$@||Semiannual payments | + }}CYTM =~ }B$@ }}/ Maturity =~ }:@}+  ~0Time~0 Cash Flow ~0PVIF~1PV of CF~1~0PV*CF*T~1+ ~ 2?+3I@D5@D5@?9Dyy?#D6@?D!3<DÖ5|@ DD3'>1(]@DDD3+ 2/Price =#3@@ %+ /3 Numerator =3#3U"zѤ@ %/Duration =!4 UbQ@x DD 5= Numerator/Price +*Two-year Treasury Bond+ / Par value =~ 8@@ /Coupon =~ 9$@Semiannual payments + CYTM =~ B$@ / Maturity =~ :@+  0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T1+ ~ 2?+3I@D@D@?9Dyy?#D6@?D!3<f@ DD3'>f@DDD3+ 2/Price =#3@@ %+ /3 Numerator =3#3M~@ %/Duration =!4YWHC6? DD 5= Numerator/Price +  ?12 Change in ?Duration1Maturity0Duration @Ӽ?~ E@2 @5;NQ@~ E@2\m? @0L F% @~ E@2Zd;O?         DlaIoa >aDTT  T0 b b T ; ; ; ;b ;b ;@ ; 0e ;14 ;00 ;@i ;hi ;0 ;b ; 0 ;} ;bO ;b ;}b ; ;b ;O ;} ;v ;T0T ;e ;b ;T0 ;    Problem 6 F Six-year CDF+ / Par value =~ 8@ /Coupon =~ 9@Annual payments + CYTM =~ B@ / Maturity =~ :@+  0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T1+ ~ 2?!3@ D@D@+GsHM0?D@D!3sHM@ DD3'3sHM@DDD$ PVIF = 1/(1+YTM)^(Time)+ ~ 2@!3@ D@D@+G\mz?D@D!3i @ DD3'3i @DDD3+ ~ 2@!3@ D@D@+6B)?D@D!3&MYX|@ DD3'392CB@DDD3+ ~ 2@!3@ D@D@+<_X?D@D!3c}@ DD3'3c@DDD3+ ~ 2@!3@ D@D@+?D@D!3Ѕ|@ DD3'3@DDD3+ ~ 2@'-@D@D@D+Eh ?D@D!>Jd+0@ DD3'>7 o@DDD3+ 2/Price =#3@ %+ /3 Numerator =3#3\DPits@ %2Duration =!4 pgu@ DD 5= Numerator/Price ++ * Six-year CD+ / Par value =~ 8@ /Coupon =~ 9@Semiannual payments + CYTM =~ B@ / Maturity =~ :@+  0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T1+ ~ 2?+3r@D@D@?9DH%e?#D@?D!3k@ DD3'3GA+@DDD3+ Dl 4>oa >o ;T0 ;b ;b ;T    b b @ 0e 14 00 @i hi 0 b 0 } bO b }b  b O } v T0T e ;b ;T0 ;~ 2@1-@D@D@?D9D 7q?#D@?D!>w68@ DD3'>(*@DDD3+ 2/Price =#3@ %+ /3 Numerator =3#3 @ %/Duration =!4?W@ DD 5= Numerator/Price ++ xoDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being  pgreceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows.    Problem 7  Consol Bond  YTM D = 1 + 1/R ~  @#'+@ D ~ $@#'&@ D ~ (@#("@ D  ulA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percent wnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM, ZQhas a duration of 20 years because no cash flows occur before the twentieth year.            Problem 8 *Five-year Bond+ / Par value =~ 8@@ /Coupon =~ 9@Annual payments + CYTM =~ B"@ / Maturity =~ :@+ DR la 4>DQQQ | 4A ;T0 ;b ;b ;T ; ; ; ;b ;b ;@ ; 0e ;14 ;00 ;@i ;hi ;0 ;b ; 0 ;} ;bO ;b ;}b ; ;b ;O ;} ;v ;T0T ;e ;b ;T0 ; 0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T1+ ~ 2?!3Q@ D@D@+D[?D@D!3 z|P@ DD3'3 z|P@DDD$ PVIF = 1/(1+YTM)^(Time)++ ~ 2@!3Q@ D@D@+DB ?D@D!3H5suM@ DD3'3H5su]@DDD3+ ~ 2@!3Q@ D@D@+D+!?D@D!3HK@ DD3'36Ed@DDD3+ ~ 2@!3Q@ D@D@+D7%Yk?D@D!3Ċi}H@ DD3'3Ċi}h@DDD3+ ~ 2@'3@D@D@D+Dcsi@ DD3'>> D*@DDD3+ 2/Price =#3zi^ь@ %+ /3 Numerator =3#3lMjig@ %/Duration =!4 ufo@ DD 5= Numerator/Price ++ *Five-year Bond+ / Par value =~ 8@@ /Coupon =~ 9"@Annual payments + CYTM =~ B"@ / Maturity =~ :@+  0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T1+ ~ 2?!3V@ DD+D[?D@D!3F*VgT@ DD3'3F*VgT@DDD$ PVIF = 1/(1+YTM)^(Time)++ ~ 2@!3V@ DD+DB ?D@D!3}R@ DD3'3}b@DDD3+ ~ 2@!3V@ DD+D+!?D@D!3R_Q@ DD3'3]Nj@DDD3+ ~ 2@!3V@ DD+D7%Yk?D@D!3Ei]O@ DD3'3Ei]o@DDD3+ ~ 2@'3@DDD+DcsSf#@ DD3'>(R@@DDD3+ 2/Price =#3?@ %+ /3 Numerator =3#3^I@ %/Duration =!4ܔ y@ DD 5= Numerator/Price ++ *Five-year Bond+ / Par value =~ -@@ /Coupon =~ 9&@Annual payments + CYTM =~ <"@ / Maturity =~ :@+  0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T1+ ~ 2?!3[@ DD+D[?D@D!33R1:Y@ DD3'33R1:Y@DDD$ PVIF = 1/(1+YTM)^(Time)++ ~ 2@!3[@ DD+DB ?D@D!3`Ym%W@ DD3'3`Ym%g@DDD3+ ~ 2@!3[@ DD+D+!?D@D!3ǁ+d@ DD3'>9|=.@DDD3+ 2/Price =#3@,א@ %+ /3 Numerator =3#3޻k@ %/Duration =!4n%@ DD 5= Numerator/Price +Dla Aoa Aoa T0 b b T   ; ;b ;b  ;@  ; 0e  ;14  ;00  ;@i ;hi ;0 b ; 0 ;} ;bO ;b ;}b ; ;b ;O ;} ;v ;T0T ;e   b. % Change in %DurationCouponDuration $8gDio@~ )@ $gs@~ )"@!lV} $"uq@~ )&@_vO  3*Duration decreases as the amount of coupon   interest increases.             Problem 9 *Five-year Bond+ / Par value =~ 8@ /Coupon =~ 9 @Annual payments  + CYTM =~ B$@ / Maturity =~ :@+  0Time0 Cash Flow 0PVIF1PV of CF10PV*CF*T1+ ~ 2?!3@4 DD+D]tE?(DD!3袋.@ DD3'3袋.@DDD$ PVIF = 1/(1+YTM)^(Time)++ ~ 2@!3@ DD+DTx?r?DD!37A@ DD3'37A@DDD3+ ~ 2@!3@ DD+D[V ?DD!3_w+jȂ@ DD3'33A,@DDD3+ ~ 2@!3@ DD+D}??DD!3>I@ DD3'3>I@DDD3+ ~ 2@'3@DDD+D+d?DD!>w$F1@ DD3'> 8_@DDD3+ 2/Price =#3 @ %+ /3 Numerator =3#3L`R@ %/Duration =!4"g}* @ DD 5= Numerator/Price ++ > 0I]BRR[D4Aoa  ;T0! ;b" ;b# ;T$ ;% ;& ;' ;b( ;b) ;@* ; 0e+ ;14, ;00- ;@i. ;hi/ ;00 ;b1 ; 02 ;}3 ;bO4 ;b5 ;}b6 ;7 b8 O9 }: v; T0T< e= ;b> ;T0? ;  *Five-year Bond + !!/ Par value =~ !8@ !!/Coupon =~ !9$@!!Annual payments ! + ""CYTM =~ "B$@ ""/ Maturity =~ ":@"+  #0Time#0 Cash Flow #0PVIF#1PV of CF#1#0PV*CF*T#1+ ~ $2?!$3@@$ D!D!+$D]tE?0D"D$!$3袋.h@% D$D$$3'$3袋.h@%D$D$D$$$ PVIF = 1/(1+YTM)^(Time)$++ ~ %2@!%3@@% D!D!+%DTx?r?$D"D%!%3Ӊ@& D%D%%3'%3ә@&D%D%D%%3+ ~ &2@!&3@@& D!D!+&D[V ?%D"D&!&37Uz@' D&D&&3'&3)V㛡@'D&D&D&&3+ ~ '2@!'3@@' D!D!+'D}??&D"D'!'3x|X@( D'D''3''3x|X@(D'D'D''3+ ~ (2@'(3|@(D!D!D!+(D+d?'D"D(!(>Ur"@) D(D((3'(>IQլ@*D(D(D((3+ )2)/Price =#)3@* %$()+ */*3 Numerator =*3#*3Pl]T\@$ %$(*/Duration *=!*4)#5@ D*D)* 5= Numerator/Price* +++ ,,*Five-year Bond,+ --/ Par value =~ -8@ --/Coupon =~ -9(@--Annual payments - + ..CYTM =~ .B$@ ../ Maturity =~ .:@.+  /0Time/0 Cash Flow /0PVIF/1PV of CF/1/0PV*CF*T/1+ ~ 02?!03@0 D-D-+0D]tE?D.D0!03. @1 D0D003'03. @1D0D0D0$0 PVIF = 1/(1+YTM)^(Time)0++ ~ 12@!13@1 D-D-+1DTx?r?0D.D1!13,Ra@2 D1D113'13,Ra@2D1D1D113+ ~ 22@!23@2 D-D-+2D[V ?1D.D2!233A,@3 D2D223'23pw!@3D2D2D223+ ~ 32@!33@3 D-D-+3D}??2D.D3!33/휉@4 D3D333'33/휩@4D3D3D333+ ~ 42@'43@4D-D-D-+4D+d?3D.D4!4>Q*@5 D4D443'4>s@6D4D4D443+ 525/Price =#53,$@6 %045+ 6/63 Numerator =63#63Ӌ f@0 %046/Duration 6=!64ZK@$ D6D56 5= Numerator/Price6 +7 8  9b. 9%9 Change in9 ::%Duration:Coupon:Duration: ;;$`TR' @~ ;) @; <<$\C@~ <)$@<!lV}< ==$K@~ =)(@=_vO= > ?3?*Duration decreases as the amount of coupon? DGlAoa Aoa K_DTT @ ;T0A ;bB ;bC ;TD ;E ;J K ;bL ;bM ;@N ; 0eO ;14P ;00Q ;@iR ;hiS ,0T ;bU ; 0V ;}W ;bOX ;bY ;}bZ ;[ ;b\ ,O] ;}^ ;v_ ;T0T@@interest increases.@ A B C D E J Problem 10J KBK*9Two-year loan: Principal and interest at end of year two.K+ LL/ Par value =~ LKj@ LL/Coupon =~ L9LLNo annual payments L + MMCYTM =~ MB$@ MM/ Maturity =~ M:@M+  N0TimeN0 Cash Flow N0PVIFN1PV of CFN1N0PV*CF*TN1+ ~ O2?!O3O DLDL+O]tE?ODMDO!O3P DODOO3'O3PDODODO O3 OPVIF = 1/(1+YTM)^(Time) O + ~ P2@*P-@aDL?+PTx?r?PDMDP!PHj@Q DPDPP3'PIjARDPDPDPP3+ Q2Q/Price =#Q7j@R %OPQJ+ R/R3 Numerator =R3#RJjAO %OPR/Duration R=!R4@ DRDQR 5= Numerator/PriceR +S+++++++++++ TQT*HTwo-year loan: Interest at end of year one, P & I at end of year two. T+ UU/ Par value =~ ULj@ UU/Coupon =~ U9$@UUAnnual payments U + VVCYTM =~ VB$@ VV/ Maturity =~ V:@V+  W0TimeW0 Cash Flow W0PVIFW1PV of CFW1W0PV*CF*TW1+ ~ X2?!X-@X DUDU+XD]tE?XDVDX!X3E]t@Y DXDXX3'X3E]t@YDXDXDX$X PVIF = 1/(1+YTM)^(Time)X+ ~ Y2@'Y-@YDUDUDU+YDTx?r?YDVDY!Y>E]t1@Z DYDYY3'YIE]t1A[DYDYDYY3+ Z2Z/Price =#Z7i@[ %XYZJ+ [/[3 Numerator =[3#[J.MAX %XY[/Duration [=![4.袋?O D[DZ[ 5= Numerator/Price[ +\+++++++++++ ]1]*(Two-year loan: Amortized over two years.](]Amortized payment of $57.619.05]+ ^^/ Par value =~ ^Kj@ ^^/Coupon =~ ^9$@^+ __CYTM =~ _B$@ __/ Maturity =~ _:@_+ <F 5loa {oa s` ;T0a ;bb ;bc ;Td ;g h i bj bk ;@l ; 0em ;14n ;00o ;@ip ;hiq ;0s bt 0u }v bOw bx }by z b{ O `0Time`0 Cash Flow `0PVIF`1PV of CF`1`0PV*CF*T`1+ a2?3ǭ_+aD]tE?aD_Da!a3袋@b DaDaa3'a3袋@bDaDaDa$a PVIF = 1/(1+YTM)^(Time)a+ b2@3ǭ_+bDTx?r?bD_Db!b>7a@@c DbDbb3'b>7a@@dDbDbDbb3+ c2c/Price =#c7lj@d %abc+ d/d3 Numerator =d3#dA=;nAa %abd/Duration d=!d4yy?X DdDcd 5= Numerator/Priced + gb.g%g Repaymentg Change ing hh%Durationh ProvisionshDurationh i~ i$@i)P&I @ 2i jj$ q? j)I@1j48Ej kk$%䃞?k)Amortizek3ۿk l >m5Duration decreases dramatically when a portion of them Dn;principal is repaid at the end of year one. Duration oftenn Co:is described as the weighted average maturity of an asset,o @p7so if more weight is given to early payments, then the p 3q*effective maturity of the loan is reduced.q s Problem 13s tt Two-year Bondt uu Par value~ u@@ uuCoupon~ u$@uuAnnual paymentsu v v"YTM~ v#,@ vvMaturity~ v@ v  wTimew Cash Flow wPVIFwPV of CFwwPV*CF*Tw ~ x?!xY@x DuDu+x}?xDvDx!x>U@y DxDxx'x>U@yDxDxDx x xPVIF = 1/(1+YTM)^(Time)x ~ y@'y0@yDuDuDu+y]z?yDvDy!y&&*rPs@z DyDyy'y&&*rPs@{DyDyDyy zzPrice =#zr)1@{ %xyz {{ Numerator ={#{Nq1қ@x %xy{ Duration ={!{! ^?P D{Dz{ = Numerator/Price6 a]cFV[ `fe`S3>i}_(  p  6NMM?Y e]`  IMHP LaserJet III@g,,@MSUDHP LaserJet IIId "d,,??3`  `  `   `   `   `  PH 0(  迷3d23 M NM4 3QQ ;JLQi ;JL Q3_4E4D$% M 3O& Q4$% M 3O& Q4FAt\ u3O  3 b#M43*{Gz?N(#M&! M4%  /[M3O]& Q &Yield to Maturity'4% uC2MZ3O"& Q Years'4523  NM43" 3_ M NM  MM< 444% clM:3Og& Q $Duration and YTM'4%  Tv[M3O:%Q'4% rv[M3O:%Q'4% v[M3OQ%Q'44e?Q?Q?eŏ17@z): @v@e ~v  <NMM? ]`h  I"h??3`  `  `  `  `  `  ?п3d23 M NM4 3QQ ;Q ;Q3_4E4D$% |M 3O&Q4$% |M 3O&Q4FApX }3O 3 b#M43*??N(#M&! M4%  9ZM3O^&Q $Time to Maturity'4% B0MZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% GjM:3O}&Q .Duration and Maturity'4% +sZM3Og%Q'4%  MsZM3O\%Q'4%  ysZM3O%Q'44e@@@eӼ?5;NQ@0L F% @e ~v  <NMM?z ]`L&  I"L&??3`  `  `  `  ` | ` | ?3d23 M NM4 3QQ ;Q ;Q3_4E4D$% M 3O&Q4$% M 3O&Q4FAk 3O1 3 bo#M43*@@??N(#M&! M4% 0 M3OM&Q  Coupon Rates '4% AMZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% Rk M:3O&Q 6Duration and Coupon Rates'4% C jpM3O~%Q'4% pM3Ox%Q'4% wpM3Og%Q'44eQ? ףp= ?)\(?e8gDio@gs@"uq@e ~v  <NMM?z9 C]`0  I"0??3` | ` | ` | ` | ` | ?3d23 M NM4 3QQ ;;=Q ;;=Q3_4E4D$% M 3O&Q4$% M 3O&Q4FAk 3O1 3 bo#M43*@@??N(#M&! M4% 0 M3OM&Q  Coupon Rates '4% AMZ3O"&Q Years'4523  NM43" 3_ M NM  MM< 444% Rk M:3O&Q 6Duration and Coupon Rates'4% pM3Og%Q'4% pM3Ox%Q'4% C J M3O~Q'44e{Gz??Q?e`TR' @\C@K@e ~v  <NMM?zg q]`3  I"3??3` |  ` | !` | "` | #` | (?3d23 M NM4 3QQ ;ikQQ3_4E4D$% M 3O&"Q4$% M 3O&!Q4FAkN 3O 3 bo#M43*?@??N(#M&#! 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" %# $" # ##  ( %" %#  "  # #  &" !" !   ( # ( $ ( #8 "< !< %#8 %"< !8 #"8 ##8 "8  ,  ,  # , "<  , '# ,! (,!  '" #! `L$Chart15Sheet1Sheet2֪Sheet3zr8 3  @@   lb%Solutions to end-of-chapter problems.TimeCouponMaturity Cash FlowYTMPVIFPV of CFPrice =PV*CF*T Numerator = Duration =a. Coupon BondZero Coupon BondHb. Increasing the yield-to-maturity decreases the duration of the bond.Jd. Changing the YTM does not affect the duration of the zero coupon bond.\e. Increasing the YTM on the coupon bond allows for a higher reinvestment income which more_quickly recovers the initial investment. The zero coupon bond has no cash flow until maturity.Ch. 9. Problem 2.= Numerator/Price Problem 4Five-year Treasury BondiAs the yield to maturity increases, duration decreases because of the reinvestment of interim cash flows at higher rates.Duration.PVIF = 1/(1+YTM/2)^(Time*2) Problem 5Two-year Treasury Bondb. Change in Problem 6 Six-year CDPVIF = 1/(1+YTM)^(Time)oDuration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being greceived more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. Problem 7Semiannual paymentsAnnual payments Consol Bond D = 1 + 1/RlA consol bond is a bond that pays a fixed coupon each year forever. A consol trading at a YTM of 10 percentnhas a duration of 11 years, while a 20-year zero-coupon bond trading at a YTM of 10 percent, or any other YTM,Qhas a duration of 20 years because no cash flows occur before the twentieth year. Problem 8Five-year Bond*Duration decreases as the amount of couponinterest increases. Problem 9 Problem 10No annual paymentsAmortized payment of $57.619.059Two-year loan: Principal and interest at end of year two.(Two-year loan: Amortized over two years. Repayment ProvisionsP&I @ 2I@1Amortize5Duration decreases dramatically when a portion of the;principal is repaid at the end of year one. Duration often:is described as the weighted average maturity of an asset,7so if more weight is given to early payments, then the *effective maturity of the loan is reduced. Problem 13 Two-year BondCoupon = Maturity = Par value =YTM == PVIF = 1/(1+YTM)^(Time) PVIF = 1/(1+YTM/2)^(Time*2)Four-year Treasury Bond Three-year Treasury Bond HTwo-year loan: Interest at end of year one, P & I at end of year two. Problem 15 Problem 17.c(Calculation in millions)Ten-year CD:Bank BTen-year Loan: Bank ATen-year CD: Bank A Problem 19Two-year Capital Note Problem 20.aFive-year Loan Portfolio Problem 20.cTwo-year Core Deposits Problem 22. 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