CHAPTER 23
Credit Derivatives

Practice Questions

Problem 23.8.
Suppose that the risk-free zero curve is flat at 7% per annum with continuous compounding and that defaults can occur half way through each year in a new five-year credit default swap. Suppose that the recovery rate is 30% and the default probabilities each year conditional on no earlier default is 3% Estimate the credit default swap spread? Assume payments are made annually.

The table corresponding to Tables 23.2, giving unconditional default probabilities, is

Time (years)
Default Probability
Survival Probability
1
0.0300
0.9700
2
0.0291
0.9409
3
0.0282
0.9127
4
0.0274
0.8853
5
0.0266
0.8587

The table corresponding to Table 23.3, giving the present value of the expected regular payments (payment rate is per year), is

Time (years)
Probability of Survival
Expected Payment
Discount Factor
PV of Expected Payment
1
0.9700
0.9700s
0.9324
0.9044s
2
0.9409
0.9409s
0.8694
0.8180s
3
0.9127
0.9127s
0.8106
0.7398s
4
0.8853
0.8853s
0.7558
0.6691s
5
0.8587
0.8587s
0.7047
0.6051s
Total

3.7364s

The table corresponding to Table 23.4, giving the present value of the expected payoffs (notional principal =$1), is Time (years)
Probability of Default
Recovery Rate
Expected Payoff
Discount Factor
PV of Expected Payoff
0.5
0.0300
0.3
0.0210
0.9656
0.0203
1.5
0.0291
0.3
0.0204
0.9003
0.0183
2.5
0.0282
0.3
0.0198
0.8395
0.0166
3.5
0.0274
0.3
0.0192
0.7827
0.0150
4.5
0.0266
0.3
0.0186
0.7298
0.0136
Total

0.0838

The table corresponding to Table 23.5, giving the present value of accrual payments, is

Time (years)
Probability of Default
Expected Accrual Payment
Discount Factor
PV of Expected Accrual Payment
0.5
0.0300
0.0150s
0.9656
0.0145s
1.5
0.0291
0.0146s
0.9003
0.0131s
2.5
0.0282
0.0141s
0.8395
0.0118s
3.5
0.0274
0.0137s
0.7827
0.0107s
4.5
0.0266
0.0133s
0.7298
0.0097s
Total

0.0598s

The credit default swap spread is given by: It is 0.0221 or 221 basis points.

Problem 23.9.
What is the value of the swap in Problem 23.8 per dollar of notional principal to the protection buyer if the credit default swap spread is 150 basis points?

If the credit default swap spread is 150 basis points, the value of the swap to the buyer of protection is: per dollar of notional principal.

Problem 23.10.
What is the credit default swap spread in Problem 23.8 if it is a binary CDS?

If the swap is a binary CDS, the present value of expected payoffs per dollar of notional principal is 0.1197 so that The spread, s, is 0.0315 or 315 basis points.

Problem 23.11.
How does a five-year th-to-default credit default swap work. Consider a basket of 100 reference entities where each reference entity has a probability of defaulting in each year of 1%. As the default correlation between the reference entities increases what would you expect to happen to the value of the swap when a) and b) . Explain your answer.

A five-year th to default credit default swap works in the same way as a regular credit default swap except that there is a basket of companies. The payoff occurs when the th default from the companies in the basket occurs. After the th default has occurred the swap ceases to exist. When (so that the swap is a “first to default”) an increase in the default correlation lowers the value of the swap. When the default correlation is zero there are 100 independent events that can lead to a payoff. As the correlation increases the probability of a payoff decreases. In the limit when the correlation is perfect there is in effect only one company and therefore only one event that can lead to a payoff.
When (so that the swap is a 25th to default) an increase in the default correlation increases the value of the swap. When the default correlation is zero there is virtually no chance that there will be 25 defaults and the value of the swap is very close to zero. As the correlation increases the probability of multiple defaults increases. In the limit when the correlation