期权期货与其他衍生产品第九版课后习题与答案Chapter

CHAPTER 24 Credit Risk

Practice Questions

Problem 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.

From 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 same 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 is 0?0060?(1?0?3)?0?0086 or 0.86% per year. Using the results in the previous question, the hazard rate is 0.71% per year for the first three years and

0?0086?5?0?0071?3 ?0?0107

2or 1.07% per year in years 4 and 5.

Problem 24.3.

Should researchers use real-world or risk-neutral default probabilities for a) calculating credit value at risk and b) adjusting the price of a 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 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 time t is defined so that h(t)?t is the probability of default between times t and t??t conditional on no default prior to time t. The unconditional default probability density q(t) is defined so that q(t)?t is the probability of default between times t and t??t as 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 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 as

1 ?ln(1?0?00245)?0?00035047

or 0.04% per year when rounded. Other numbers in the column are 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 with a counterparty on its books. Explain why a new transaction by a bank with a counterparty can have the effect of increasing or reducing the bank’s credit exposure to the counterparty.

Suppose company A goes bankrupt when it has a number of outstanding contracts with company B. Netting means that the contracts with a positive value to A are netted against those with a negative value in order to determine how much, if anything, company A owes company B. Company A is not allowed to “cherry pick” by keeping the positive-value contracts and defaulting on the negative-value contracts.

The new transaction will increase the bank’s exposure to the counterparty if the contract tends 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, 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 increases qi* 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) 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) 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 default

rate in the economy. In CreditMetrics a copula model is applied to credit ratings migration and this determines the joint probability 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 asset 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 payments is

10?2?5e?0?06?0?5?2?5e?0?06?1?卐?2?5?0?06?5?100e?0?06?5?105?3579

The 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 asset swap spread is therefore 2?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 when 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

?)B? v(t)(1?R

where v(t) is the discount factor for time t and B? is the no-default value of the bond at time t. Suppose that the zero-coupon bonds comprising the corporate bond have no-default values at time t of Z1, Z2, …, Zn, respectively. The loss from the ith zero-coupon bond arising from a default at time t is

?)Zv(t)(1?R i

The total loss from all the zero-coupon bonds is

垐v(t)(1?R)?Zi?v(t)(1?R)B?in

This shows that the loss arising from a default at time t is the same for the corporate bond as 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 interest, the loss for a corporate bond arising from a default at time t is

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 corporate 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 payment 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 follows Time (yrs) Def. Prob. Recovery Amount ($) Risk-free Value ($) Loss Given Default ($) Discount Factor PV of Expected Loss ($) ?[L?a(t)]v(t)B??v(t)R

1.0 30 104.78 74.78 0.9704 72?57Q Q 2.0 30 103.88 73.88 0.9418 69?58Q Q 3.0 30 102.96 72.96 0.9139 66?68Q Q 4.0 30 102.00 72.00 0.8869 63?86Q Q Total 272?69Q The 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 obtained by

discounting the promised cash flows at 3%. It is 103.66. The total loss from defaults should therefore be equated to 103?66?96?19?7?46. The value of Q implied by the bond price is therefore given by 272?69Q?7?46. or Q?0?0274. The implied probability of default is 2.74% per year.

Problem 24.13.

A company 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 place 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 and Q2.

The table for the first bond is

Time (yrs) Def. Prob. Recovery Amount ($) Risk-free Value ($) Loss Given Default ($) Discount Factor PV of Expected Loss ($) 61?92Q1 0.5 1.5 2.5 Total Q1 Q1 Q1 40 40 40 103.01 102.61 102.20 63.01 62.61 62.20 0.9827 0.9489 0.9162 59?41Q1 56?98Q1 178?31Q1 The market price of the bond is 98.35 and the risk-free value is 101.23. It follows that Q1 is given by 178?31Q1?101?23?98?35

so that Q1?0?0161.

The table for the second bond is Time (yrs) Def. Prob. Recovery Amount ($) Risk-free Value ($) Loss Given Default ($) Discount Factor PV of Expected Loss ($) 0.5 1.5 2.5 3.5 4.5 Total Q1 Q1 Q1 Q2 Q2 40 40 40 40 40 103.77 103.40 103.01 102.61 102.20 63.77 63.40 63.01 62.61 62.20 0.9827 0.9489 0.9162 0.8847 0.8543 62?67Q1 60?16Q1 57?73Q1 55?39Q2 53?13Q2 180?56Q1?108?53Q2 The market price of the bond is 96.24 is and the risk-free value is 101.97. It follows that

180?56Q1?108?53Q2?101?97?96?24

From which we get Q2?0?0260 The bond prices therefore imply a probability of default of 1.61% per year for the first three years and 2.60% for the next two years.