固定收益证券的复习计算题

Fixed-income treasuryPpt31、公式：Practice Question 3.1Suppose currently, 1-year spot rate is 1% and marketexpects that 1-year spot rate next year wouldbe 2%and 1-year spot rate in 2 years would be 3%. Compute today ’s 2-year spot rate and 3-year spot rate.(已做答案）2、Current YieldCompute the current yield for a 7% 8-year bond whose price is$94.17. How aboutthe current yield if price is $100, $106,respectively?3Case 3.1Consider a 7% 8-year bond paying coupon semiannually which is sold for $94.17. Thepresent value using various discount rate is:A. What is the YTM for this bond?B. How much is the total dollar return on this bond?C. How much is the total dollar return if you put the same amount of dollars into a depositaccount with the same annual yield?4、 Forward Rates注： 6-month bill spot rate is 3% 是年化利率（ 3%要除以 2）1-year bill spot rate is 3.3% 是年化利率（ 3.3%要除以 2）Ppt41、 Fixed ‐ Coupon BondsPractice Question 4.2A. What is the value of a 4-year 10% coupon bond that pays interest semiannually assuming thatthe annual discount rate is 8%? What is the value of a similar 10% coupon bond with an infinite maturity （无期限） ?B. What is the value of a 5-year zero-coupon bond with a maturity value of $100 discounted at an8% interest rate?C. Compute the value par $100 of par value of a 4-year 10% coupon bond, assuming the paymentsare annual and the discount rate for each year is 6.8%, 7.2%, 7.6% and 8.0%, respectively.Infinite maturityPv=($100*10%/2)/(8%/2)(半年付息）Present Value PropertiesPractice Question 4.4A. Suppose the discount rate for the 4-year 10% coupon bond with a par value of $100 is 8%. Compute its present value.B. One year later, suppose that the discount rate appropriate for a 3-year 10%coupon bond increases from 8% to 9%. Redo your calculation in part A and decompose the price change attributable to moving to maturity and to the increase in the discount rate .（期限与贴现率变化）3、 Pricing a Bond between Coupon PaymentsPractice Question 4.6Suppose that there are five semiannual coupon payments remaining for a 10% coupon bond.Also assume the following:①Annual discount rate is 8%②78 days between the settlement date and the next coupon payment date③182 days in the coupon periodCompute the full price of this coupon bond. What is the clean price of this bond?4、 Valuation ApproachCase 4.1A. Consider a 8% 10-year Treasury coupon bond. What is its fair value if traditional approach is used, given yield for the 10-year on-the-run Treasury issue is 8%?B. What is the fair value of above Treasury coupon bond if arbitrage-free approach is used,given the following annual spot rates?C. Which approach is more accurate （准确） ?C、Arbitrage-Free Approach is more accuratePpt52、 ConvexityConsider a 9% 20-year bond selling at $134.6722 to yield 6%. For a 20 bp changein yield, its price would either increase to $137.5888 or decrease to $131.8439.pute the convexity for this bond.B.What is the convexity adjustment for a change in yield of 200 bps?C. If we know that the duration for this bond is 10.66, what should the total estimated percentage price change be for a 200 bp increase in the yield? How about a 200 bp decrease in the yield?Ppt61、 Measuring Yield Curve RiskCase 6.1: Panel AConsider the following two $100 portfolios composed of2-year , 16-year , and 30-year issues, all of which are zero-coupon bonds:For simplicity, assume there are only three key rates — 2years , 16 years and 30 years . Calculate the portfolio ’ s key rate durations at these three points and its effective duration.Case 6.1: Panel BConsider the following three scenarios:Scenario 1: All spot rates shift down 10 basis points.Scenario 2: The 2-year key rate shifts up 10 basis points anthe 30-year rate shifts down 10 basis points.Scenario 3: The 2-year key rate shifts down 10 basis pointsand the 30-year rate shifts up 10 basis points.How would the portfolio value change in each scenario?Ppt7Consider a 6.5% option-free bond with 4 years remaining to maturity. If the appropriate binomial interest rate tree is shown as below, calculate the fair price ofthis bond.Ppt81、Valuing Callable and Putable BondsCase 8.1 : Valuing a callable bond with singlecall priceConsider a 6.5% callable bond with 4 years remaining to maturity,callable in one year at $100. Assumethe yield volatility is 10%and the appropriate binomial interest rate tree is same as Case 6.4. Calculate the fair price of this callable bond.2、Case 8.2 : Valuing a callable bond with call scheduleConsider a 6.5% callable bond with 4 years remaining tomaturity, callable in one year at a call schedule as below:Assumethe yield volatility is 10%and the appropriate binomial interest rate tree is same as Case 6.4. Calculate the fair price of thiscallable bond.3、Case 8.3 : Valuing a putable bond Consider a 6.5% putable bond with 4years remaining to maturity, putable in one year at $100. Assumethe yield volatilityis 10%and the appropriate binomial interest rate tree is same as Case 6.4. Calculate the fair price of this putable bond.Va pppp lue of aCapppppppConvertible BondsCase 9.1 :Suppose that the straight value of a 5.75% ADCconvertible bond is $981.9 per $1,000 of par value and its market price is $1,065. The market price per share of commonstock is $33 and the conversion ratio is 25.32 sharesper $1,000 of parvalue. Also assume that the common stock dividend is $0.90per share. ption公式：Minimum Value: the greater of its conversion price and its straight value.Conversion Price= Market price of common stock ×Conversion ratioStraight Value/Investment Value: present value of the bond’ scash flowsdiscounted at the required return on a comparable option-free issue.Market Conversion Price/Conversion ParityPrick= Market price of convertible security ÷Conversion ratioMarket Conversion Premium Per Share= Market conversion price –Market price of common stockMarket Conversion Premium Ratio= Market conversion premium per share ÷ Market price of common stockPremium over straight value=(Market price of convertible bond/Straight value)–1The higher this ratio, the greater downside risk andthe less attractive the convertible bond.Premium Payback Period= Market conversion premium per share ÷ Favorable income differential pershareFavorable Income Differential Per Share= [Coupon interest –(Conversion ratio Common× stock dividend per share)]÷Conversion ratioA. What is the minimum value of this convertible bond?perB. Calculate its market conversion price , market conversion premiumshare and market conversion premium ratio.C. What is its premium payback period?D. Calculate its premium over straight value. ppMarket price of common stock=$33,conversion ratio = 25.32Straight Value=$981.9，market price of conversible bond = $1,065common stock dividend = $0.90Coupon rate=5.75%A、Conversion Price= Market price of common stock×Conversion ratio=$33*25.32=$835.56the minimum value of this convertible bond=max{$835.56, $981.9}=$981.9B、Market Conversion Price/Conversion ParityPrick= Market price of convertible security ÷Conversion ratio=$1065/25.32=$42.06Market Conversion Premium Per Share=Market conversion price –Market price of common stock=$42.06 -$33=$9.06Market Conversion Premium Ratio=Market conversion premium per share ÷ Market price of common stock=$9.06/$33=27.5%C、Premium Payback Period= Market conversion premium per share ÷ Favorable income differential pershareFavorable Income Differential Per Share= [Coupon interest –(Conversion ratio Common× stock dividend per share)]÷Conversion ratioCoupon interest from bond = 5.75%× $1,000 =$57.50Favorable income differential per share = ($57.50–25.32× $0.90) ÷ 25.32 =$1.37 Premium payback period = $9.06/$1.37 = 6.6 yearsD、Premium over straight value=(Market price of convertible bond/Straight value)–1=$1,065/$981.5 –1 =8.5%Ppt10No-Arbitrage Principle:no riskless profits gained from holding a combination of a forward contract position as well as positions in other assets.FP = Price that would not permit profitable riskless arbitrage in frictionless markets, that is:Case 10.1Consider a 3-month forward contrac t on a zero-coupon bond with a face value of $1,000 that is currently quoted at $500, and assume a risk-free annual interest rate of 6%. Determine the price of the forward contract under the no-arbitrage principle.Solutions.Case 10.2Suppose the forward contract described in case 10.1 is actually trading at $510, which is greater than the noarbitrage price. Demonstrate how an arbitrageur can obtain riskless arbitrage profit from this overpriced forward contrac t and how much the arbitrage profit would be.Case 10.3If the forward contract described in case 10.1 is actually trading at $502, which is smaller thanthe no-arbitrage price. Demonstrate how an arbitrageur can obtain riskless arbitrage profit fromthis underpriced forward contract and how much the arbitrage profit would be.Case 10.4 :Calculate the price of a 250-day forward contract on a 7% U.S.Treasurybond with a spot price of $1,050 (including accrued interest) that hasjust paid a coupon and will make another coupon payment in 182 days.The annual risk-free rate is 6%.Solutions.Remember that T-bonds make semiannual coupon payments, soCase 10.6Solutions.The semiannual coupon on a single, $1,000 face-value7% bond is $35. Abondholder will receive one payment 0.5 years from now (0.7 yearsleft to expiration of futures) and one payment 1 year from now (0.2years until expiration). Thus,Ppt11Payoffs and ProfitsCase 11.1Consider a European bond call option with an exercise price of $900. The call premium for this option is $50. At expiration, if the spot price for the underlying bondis $1,000, what is the call option’s payoff as well as its gain/loss? Is this option in the money, out of money, or at the money? Will you exercise this option? How aboutyour answers if the spot price at expiration is $920, and $880, respectively? Solutions.A. If the spot price at expiration is $1,000, the payoff to the call optionis max{0, $1,000 - $900}=$100. So, the call is in the money and it willbe exercised with a gain of $50.B. If the spot price at expiration is $920, the payoff to the call option ismax{0, $920 - $900}=$20. So, the call is in the money and it will beexercised with a loss of $30. (why?)C. If the spot price is $880 at expiration, the payoff to the call option ismax{0, $880 - $900}=0. So, the call is out of money and it will not beexercise. The loss occurred would be $50.Case 11.2Consider a European bond put option with an exercise price of $950. Theput premium for this option is $50. At expiration, if the spot price for the underlying bond is $1,000, what is the put option’s payoff as well as its gain/loss? Is this option in the money, out of money, or at the money? Will you exercise this option? How about your answers if the spot priceat expiration is $920, and $880, respectively?Solutions.A. If the spot price at expiration is $1,000, the payoff to the put option ismax{0, $950 - $1,000}=0. So, the put is out of money and it will not be exercised. The loss occurred would be $50.B. If the spot price at expiration is $920, the payoff to the put option ismax{0, $950 - $920}=$30. So, the put is in the money and it will be exercised with a loss of $20. (why?)C. If the spot price is $880 at expiration, the payoff to the call option ismax{0, $950 - $880}=$70. So, the put is in the money and it will not be exercise with a gain of $20.。