1、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 would be 2%and 1-year spot rate in 2 years would be 3%. Compute today s 2-year spot rate and 3-yearspot rate.(已做答案）2、 Current YieldCompute the cu
2、rrent yield for a 7% 8-year bond whose price is$94.17. How about the current yield if price is $100, $106,respectively?3 Case 3.1Consider a 7% 8-year bond paying coupon semiannually which is sold for $94.17. The present value using various discount rate is:A. What is the YTM for this bond?B. How muc
3、h 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 deposit account 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 B
4、ondsPractice Question 4.2A. What is the value of a 4-year 10% coupon bond that pays interest semiannually assuming that the 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 va
5、lue of $100 discounted at an 8% interest rate?C. Compute the value par $100 of par value of a 4-year 10% coupon bond, assuming the payments are 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 PropertiesP
6、ractice 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 t
7、he 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: Annual discount rate
8、 is 8% 78 days between the settlement date and the next coupon payment date 182 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
9、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、 Co
10、nvexityConsider a 9% 20-year bond selling at $134.6722 to yield 6%. For a 20 bp change in yield, its price would either increase to $137.5888 or decrease to $131.8439.A. Compute 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 du
11、ration 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 portfolioscomposed of2-year , 16-year , and 30-year
12、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
13、 rates shift down 10 basis points.Scenario 2: The 2-year key rate shifts up 10 basis points an the 30-year rate shifts down 10 basis points.Scenario 3: The 2-year key rate shifts down 10 basis points and the 30-year rate shifts up 10 basis points.How would the portfolio value change in each scenario
14、?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 of this bond.Ppt81、 Valuing Callable and Putable BondsCase 8.1 : Valuing a callable bond with singlecall priceConsider a 6.5% callable
15、bond with 4 years remaining to maturity, callablein one year at $100. Assumethe yield volatility is 10%and the appropriatebinomial interest rate tree is same as Case 6.4. Calculate the fair priceof this callable bond.2、Case 8.2 : Valuing a callable bond with call schedule Consider a 6.5% callable bo
16、nd with 4 years remaining to maturity, callable in one year at a call schedule as below:Assumethe yield volatility is 10%and the appropriate binomial interestrate tree is same as Case 6.4. Calculate the fair price of this callable bond.3、Case 8.3 : Valuing a putable bond Consider a 6.5% putable bond
17、 with 4years remaining to maturity, putable 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 putable bond.Vapppp lue of aCapppppppConvertible BondsCase 9.1 :Suppose that the straight value of a
18、 5.75% ADCconvertible bond is $981.9per $1,000 of par value and its market price is $1,065 . The market priceper 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 $0.90 per share. ption公式：Minimum Value: the great
19、er 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 flows discounted at the required return on a comparable option-free issue.Market Conversion Price/Conversion ParityPrick=
20、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 bon
21、d/Straight value)1The higher this ratio, the greater downside risk and the 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 di
22、vidend per share) Conversion ratioA. What is the minimum value of this convertible bond ?B. 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 . ppperMarket pr
23、ice 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
24、.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 p
25、er 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 ratioCommon stock dividend per share)Conversion ratioCoupon interest from bon
26、d = 5.75% $1,000 =$57.50Favorable income differential per share = ($57.5025.32 $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 risklessprofitsg
27、ainedfrom holding acombinationof a forwardcontract positionas well aspositions in other assets.FP = Pricethatwould notpermit profitablerisklessarbitrage infrictionless markets, that is:Case 10.1Consider a 3-month forwardcontrac t on a zero-coupon bond with a facevalueof $1,000 that is currentlyquote
28、d at $500, and assume a risk-freeannualinterest rate of 6%. Determinethe price of the forward contractunderthe 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 ar
29、bitrageur canobtain riskless arbitrage profit from this overpriced forward contract 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 than the no-arbitrage price. Demonstrate how an arbitrageur can obtain
30、riskless arbitrage profit from this 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.Treasury bond with a spot price of $1,050 (including accrued interest) that hasjust paid a coupon and will make another
31、 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 years left to expiration of futures) and one payment 1
