Question

"Suppose that the risk-free zero curve is flat at 7% per annum with continuous compounding and that defaults can occur halfway through each year in a new five-year credit default swap. Suppose that the recovery rate is 30% and the hazard rate is 3%.

a. Estimate the credit default swap spread. Assume payments are made annually.

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

c. What is the credit default swap spread in Problem 25.8 if it is a binary CDS? "

Answer 1

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

Time (years)	Probability of surviving to year end	Default Probability during year
1	0.9704	0.0296
2	0.9418	0.0287
3	0.9139	0.0278
4	0.8869	0.0270
5	0.8607	0.0262

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

Time (yrs)	Probability of survival	Expected Payment	Discount Factor	PV of Expected Payment
1	0.9704	0.9704s	0.9324	0.9048s
2	0.9418	0.9418s	0.8694	0.8187s
3	0.9139	0.9139s	0.8106	0.7408s
4	0.8869	0.8869s	0.7558	0.6703s
5	0.8607	0.8607s	0.7047	0.6065s
Total				3.7412s

The table corresponding to Table 23.4, giving the present value of the expected payoffs (notional principal =$1), is

Time (yrs)	Probability of default	Recovery Rate	Expected Payoff	Discount Factor	PV of Expected Payment
0.5	0.0296	0.3	0.0207	0.9656	0.0200
1.5	0.0287	0.3	0.0201	0.9003	0.0181
2.5	0.0278	0.3	0.0195	0.8395	0.0164
3.5	0.0270	0.3	0.0189	0.7827	0.0148
4.5	0.0262	0.3	0.0183	0.7298	0.0134
Total					0.0826

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

Time (yrs)	Probability of default	Expected Accrual Payment	Discount Factor	PV of Expected Accrual Payment
0.5	0.0296	0.0148s	0.9656	0.0143s
1.5	0.0287	0.0143s	0.9003	0.0129s
2.5	0.0278	0.0139s	0.8395	0.0117s
3.5	0.0270	0.0135s	0.7827	0.0106s
4.5	0.0262	0.0131s	0.7298	0.0096s
Total				0.0590s

The credit default swap spread is given by:

3.7412s+0.0590s = 0.0826