固定收益证券的复习计算题

1、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 todays 2-year spot rate and 3-year spot rate.(已做答案）2、Current YieldCompute the current yield f

2、or a 7% 8-year bond whose price is$94.17. How about the current yield if price is $100, $106,respectively?3、uCase 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 much is the tot

3、al 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、FixedCoupon Bondsu Practice Questi

4、on 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 value of $100 discounte

5、d 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 PropertiesuPractice Question 4.4

6、A. 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 att

7、ributable 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 PaymentsuPractice Question 4.6Suppose that there are five semiannual coupon payments remaining for a 10% coupon bond. Also assume the following: Annual discount rate is 8% 78 days betwe

8、en 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 ApproachuCase 4.1A. Consider a 8% 10-year Treasury coupon bond. What is its fair value if traditional approach isused, gi

9、ven 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-

10、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 duration for this bond is

11、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 RiskuCase 6.1: Panel AConsider the following two $100 portfolios composed of 2-year, 16-year, and 30-year issues, all of which ar

12、e zero-coupon bonds:For simplicity, assume there are only three key rates2years, 16 years and 30 years. Calculate the portfolios 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

13、points.Scenario 2: The 2-year key rate shifts up 10 basis points an the30-year rate shifts down 10 basis points.Scenario 3: The 2-year key rate shifts down 10 basis points andthe 30-year rate shifts up 10 basis points.How would the portfolio value change in each scenario?Ppt7Consider a 6.5% option-f

14、ree 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 BondsuCase 8.1: Valuing a callable bond with singlecall priceConsider a 6.5% callable bond with 4 years remaining to

15、 maturity, callable in one year at $100. Assume the 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 t

16、omaturity, callable in one year at a call schedule as below:Assume the 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.3、Case 8.3: Valuing a putable bond Consider a 6.5% putable bond with 4 years remaining to

17、 maturity, putable in one year at $100. Assume the yieldvolatility is 10% and the appropriate binomial interest rate tree is same as Case 6.4. Calculate the fair price of this putable bond.Vapppplue of aCapppppppConvertible BondsuCase 9.1:Suppose that the straight value of a 5.75% ADC convertible bo

18、nd is $981.9 per $1,000 of par value and its market price is $1,065. The market price per share of common stock is $33 and the conversion ratio is 25.32 shares per $1,000 of parvalue. Also assume that the common stock dividend is $0.90 per share.ption公式：Minimum Value: the greater of its conversion p

19、rice and its straight value.Conversion Price = Market price of common stock ×Conversion ratioStraight Value/Investment Value: present value of the bonds cash flows discounted at the required return on a comparable option-free issue.Market Conversion Price/Conversion ParityPrick = Market price o

20、f 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 bo

21、nd/Straight value) 1The higher this ratio, the greater downside risk and theless attractive the convertible bond.Premium Payback Period= Market conversion premium per share ÷ Favorable income differential per shareFavorable Income Differential Per Share = Coupon interest (Conversion ratio ×

22、; Common stock dividend per share) ÷ Conversion ratioA. What is the minimum value of this convertible bond?B. Calculate its market conversion price, market conversion premium per share and market conversion premium ratio.C. What is its premium payback period?D. Calculate its premium over straig

23、ht 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 co

24、nvertible 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 P

25、remium 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 per shareFavorable Income Differential Per Share = Coupon interest (Conversion ratio × Common

26、 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 = $1.37Premium payback period = $9.06/$1.37 = 6.6 yearsD、Premium over straight value = (Market price of convertible

27、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 10.1

28、Consider a 3-month forward contract 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 underthe no-arbitrage principle.Solutions.Case 10.2Suppose the forward contract descr

29、ibed 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 contract and how much the arbitrage profit would be.Case 10.3If the forward contract described in case 10.1 is

30、actually trading at $502, which is smaller than the no-arbitrage price. Demonstrate how an arbitrageur can obtain 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.Treasu

31、ry bond with a spot price of $1,050 (including accrued interest) that has just 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,

32、000 face-value7% bond is $35. A bondholder will receive one payment 0.5 years from now (0.7 years left to expiration of futures) and one payment 1 year from now (0.2 yearsuntil expiration). Thus,Ppt11Payoffs and ProfitsCase 11.1Consider a European bond call option with an exercise price of $900. The

33、 call premium for this option is $50. At expiration, if the spot price for the underlying bond is $1,000, what is the call options 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 price at e

34、xpiration is $920, and $880, respectively?Solutions.A. If the spot price at expiration is $1,000, the payoff to the call option is max0, $1,000 - $900=$100. So, the call is in the money and it will be exercised with a gain of $50.B. If the spot price at expiration is $920, the payoff to the call option is max0, $920 - $900=$20. So, the call is in the money and it will be exercised with a loss of $30. (why?)C. If the spo