Question

Suppose that the risk-free zero curve is flat at 6% per annum with continuous compounding and that defaults can occur at times 0.25 years, 0.75 years, 1.25 years, and 1.75 years in a two-year plain vanilla credit default swap with semiannual payments. Suppose that the recovery rate is 20% and the unconditional probabilities of default (as seen at time zero) are 1% at times 0.25 years and 0.75 years, and 1.5% at times 1.25 years and1.75 years. What is the credit default swap spread?
What would the credit default spread be if the instrument were a binary credit default swap?

Answer 1

The present value of the expected regular payments (payment rate is “s” per year), is

The present value of the expected regular payments

Time(yrs.)	Probability of survival	Expected Payment	Discount Factor	PV of Expected Payment
0.5	0.99	0.495 s	0.9704	0.4804 s
1	0.98	0.49 s	0.9418	0.4615 s
1.5	0.965	0.4825 s	0.9139	0.441 s
2	0.95	0.475 s	0.8869	0.4213 s
Total				1.8041 s

The present value of the expected payoffs (notional principal =$1), is

present value of the expected payoffs

Time(yrs.)	Probability of default	Recovery Rate	Expected Payoff	Discount Factor	PV of Expected Payment
	0.01	0.2	0.008	0.9851	0.0079
	0.01	0.2	0.008	0.956	0.0076
	0.015	0.2	0.012	0.9277	0.0111
	0.015	0.2	0.012	0.9003	0.0108
Total					0.0375

The present value of accrual payments, is

present value of accrual payments

Time(yrs.)	Probability of default	Expected Accrual Payment	Discount Factor	PV of Expected Payment
0.25	0.01	0.0025s	0.9851	0.0025 s
0.75	0.01	0.0025 s	0.956	0.0024 s
1.25	0.015	0.00375 s	0.9277	0.0035 s
1.75	0.015	0.00375 s	0.9003	0.0034 s
Total				0.0117 s

The credit default swap spread sis given by:

1.804 s + 0.0117 s = 0.0375

It is 0.0206 or 206 basis points. For a binary credit default swap, we set the recovery rate equal to zero in the second table to get the present value of expected payoffs equal to 0.0468 so that

1.804 s + 0.0117 s = 0.0468

and the spread is 0.0258 or 258 basis points.