期权期货与其他衍生产品第九版课后习题与答案Chapter.docx

1、期权期货与其他衍生产品第九版课后习题与答案ChapterCHAPTER 24Credit RiskPractice QuestionsProblem 24.1.The spread between the yield on a three-year corporate bond and the yield on a similar risk-free bond is 50 basis points. The recovery rate is 30%. Estimate the average hazard rate per year over the three-year period.Fro

2、m equation (24.2) the average hazard rate over the three years is 0 0050 (1 0 3) 0 0071 or 0.71% per year.Problem 24.2.Suppose that in Problem 24.1 the spread between the yield on a five-year bond issued by the same company and the yield on a similar risk-free bond is 60 basis points. Assume the sam

3、e recovery rate of 30%. Estimate the average hazard rate per year over the five-year period.What do your results indicate about the average hazard rate in years 4 and 5?From equation (24.2) the average hazard rate over the five years is0 0060 (1 0 3) 0 0086 or 0.86% per year. Using the results in th

4、e previous question, the hazard rate is 0.71% per year for the first three years and0 0086 5 0 0071 32or 1.07% per year in years 4 and 5.Problem 24.3.0 0107Should researchers use real-world or risk-neutral default probabilities for a) calculating credit value at risk and b) adjusting the price of a

5、derivative for defaults?Real-world probabilities of default should be used for calculating credit value at risk. Risk-neutral probabilities of default should be used for adjusting the price of a derivative for default.Problem 24.4.How are recovery rates usually defined?he recovery rate for a bond is

6、 the value of the bond immediately after the issuer defaults as a percent of its face value.Problem 24.5.Explain the difference between an unconditional default probability density and a hazard rate.The hazard rate, h(t)at timet is defined so that h(t )tis the probability of defaultbetween times tan

7、dttconditional on no default prior to time t . The unconditionaldefault probability densityq(t ) is defined so that q(t)tis the probability of defaultbetween times tandttas seen at time zero.Problem 24.6.Verify a) that the numbers in the second column of Table 24.3 are consistent with the numbers in

8、 Table 24.1 and b) that the numbers in the fourth column of Table 24.4 are consistent with the numbers in Table 24.3 and a recovery rate of 40%.The first number in the second column of Table 24.3 is calculated as1 ln(1 0 00245) 0 00035047or 0.04% per year when rounded. Other numbers in the column ar

9、e calculated similarly. The numbers in the fourth column of Table 24.4 are the numbers in the second column of Table 24.3 multiplied by one minus the expected recovery rate. In this case the expected recovery rate is 0.4.Problem 24.7.Describe how netting works. A bank already has one transaction wit

10、h a counterparty on its books. Explain why a new transaction by a bank with a counterparty can have the effect ofincreasing or reducing the bank exposurecredit to the counterparty.Suppose company A goes bankrupt when it has a number of outstanding contracts with company B. Netting means that the con

11、tracts with a positive value to A are netted against those with a negative value in order to determine how much, if anything, company A owescompany B. Company A is not allowed to “ cherry pick ” by keeping-valuethe positivecontracts and defaulting on the negative-value contracts.The new transaction

12、will increase the bank s exposure to the counterparty if the contracttends to have a positive value whenever the existing contract has a positive value and a negative value whenever the existing contract has a negative value. However, if the new transaction tends to offset the existing transaction,

13、it is likely to have the incremental effect of reducing credit risk.Problem 24.8.“DVA can improve the bottom line when a bank is experiencing financial difficulty. ” Explain why this statement is true.When a bank is experiencing financial difficulties, its credit spread is likely to increase. This i

14、ncreasesqi* and DVA increases. This is a benefit to the bank: the fact that it is more likely to default means that its derivatives are worth less.Problem 24.9.Explain the difference between the Gaussian copula model for the time to default and CreditMetrics as far as the following are concerned: a)

15、 the definition of a credit loss and b) the way in which default correlation is modeled.(a)In the Gaussian copula model for time to default a credit loss is recognized only when a default occurs. In CreditMetrics it is recognized when there is a credit downgrade as well as when there is a default.(b

16、)In the Gaussian copula model of time to default, the default correlation arises because the value of the factor M . This defines the default environment or average defaultrate in the economy. In CreditMetrics a copula model is applied to credit ratings migration and this determines the joint probab

17、ility of particular changes in the credit ratings of two companies.Problem 24.10.Suppose that the LIBOR/swap curve is flat at 6% with continuous compounding and a five-year bond with a coupon of 5% (paid semiannually) sells for 90.00. How would an asset swap on the bond be structured? What is the as

18、set swap spread that would be calculated in this situation?Suppose that the principal is $100. The asset swap is structured so that the $10 is paid initially. After that $2.50 is paid every six months. In return LIBOR plus a spread is received on the principal of $100. The present value of the fixed

19、 payments is10 2 5e 0 06 0 5 2 5e 0 06 1 卐 2 5 0 06 5 100e 0 06 5 105 3579The spread over LIBOR must therefore have a present value of 5.3579. The present value of $1 received every six months for five years is 8.5105. The spread received every six months must therefore be5 3579 8 5105 $ 0 6296. The

20、 asset swap spread is therefore2 0 6296 1 2592% per annum.Problem 24.11.Show that the value of a coupon-bearing corporate bond is the sum of the values of its constituent zero-coupon bonds when the amount claimed in the event of default is the no-default value of the bond, but that this is not so wh

21、en the claim amount is the face value of the bond plus accrued interest.When the claim amount is the no-default value, the loss for a corporate bond arising from a default at time t is?v(t)(1 R) Bwhere v(t ) is the discount factor for time t and B is the no-default value of the bond at time t . Supp

22、ose that the zero-coupon bonds comprising the corporate bond have no-defaultvalues at time t of Z1 , Z2 , Zn , respectively. The loss from the i th zero-coupon bondarising from a default at timet is?v(t)(1R)ZiThe total loss from all the zero-coupon bonds isn垐v(t )(1R) Bv(t)(1 R) ZiiThis shows that t

23、he loss arising from a default at time tis the same for the corporate bondas for the portfolio of its constituent zero-coupon bonds. It follows that the value of the corporate bond is the same as the value of its constituent zero-coupon bonds.When the claim amount is the face value plus accrued inte

24、rest, the loss for a corporate bond arising from a default at time t is?v(t) B v(t )R L a(t)where L is the face value and a(t) is the accrued interest at time t . In general this is not the same as the loss from the sum of the losses on the constituent zero-coupon bonds.Problem 24.12.A four-year cor

25、porate bond provides a coupon of 4% per year payable semiannually and has a yield of 5% expressed with continuous compounding. The risk-free yield curve is flat at 3% with continuous compounding. Assume that defaults can take place at the end of each year (immediately before a coupon or principal pa

26、yment and the recovery rate is 30%. Estimate the risk-neutral default probability on the assumption that it is the same each year.Define Q as the risk-free rate. The calculations are as followsTimeDef.RecoveryRisk-freeLoss GivenDiscountPV of Expected(yrs)Prob.Amount ($)Value ($)Default ($)FactorLoss

27、 ($)1.0Q30104.7874.780.970472 57Q2.0Q30103.8873.880.941869 58Q3.0Q30102.9672.960.913966 68Q4.0Q30102.0072.000.886963 86QTotal272 69QThe bond pays a coupon of 2 every six months and has a continuously compounded yield of 5% per year. Its market price is 96.19. The risk-free value of the bond is obtai

28、ned by discounting the promised cash flows at 3%. It is 103.66. The total loss from defaults shouldtherefore be equated to 103 6696 197 46. The value of Q implied by the bond price istherefore given by 272 69Q7 46 . orQ 0 0274 . The implied probability of default is2.74% per year.Problem 24.13.A com

29、pany has issued 3- and 5-year bonds with a coupon of 4% per annum payable annually. The yields on the bonds (expressed with continuous compounding) are 4.5% and 4.75%, respectively. Risk-free rates are 3.5% with continuous compounding for all maturities. The recovery rate is 40%. Defaults can take p

30、lace half way through each year. The risk-neutral default rates per year are Q1 for years 1 to 3 and Q2 for years 4 and 5.Estimate Q1 andQ2 .The table for the first bond isTimeDef.RecoveryRisk-freeLoss GivenDiscountPV of(yrs)Prob.Amount ($)Value ($)Default ($)FactorExpectedLoss ($)0.5Q140103.0163.01

31、0.982761 92Q11.5Q140102.6162.610.94895941Q12.5Q140102.2062.200.91625698Q1Total178 31Q1The market price of the bond is 98.35 and the risk-free value is 101.23. It follows thatQ1 is given by178 31Q1 101 23 98 35so that Q1 0 0161 .The table for the second bond isTimeDef.RecoveryRisk-freeLossDiscountPV of Expected(yrs)Prob.AmountValue ($)GivenFactorLoss ($)($)Default