Question

Suppose that the risk-free zero curve is flat at 3% per annum with continuous compounding and that defaults can occur at times 0.25, 0.75, 1.25, and 1.75 years in a two-year plain vanilla credit default swap with semiannual payments.  Suppose, further, that the recovery rate is 25% and the unconditional probabilities of default (as seen at time zero) are 1.5% at times 0.25 years and 0.75 years, and 2.0% at times 1.25 years and 1.75 years.  

What is the credit default swap spread?

What would the credit default spread be if the instrument were a binary credit default swap with a payoff of $1?

Answer 1

Let the payment rate be s.

Calculation of PV of expected regular payments:

1	2	3=2*0.5	4	5=3*4
Time (yrs)	Prob of survival	Expected Payment	Discount Factor	PV of Expected payments
0.5	0.985	0.4925	0.9852	0.485211
1	0.97	0.485	0.9707	0.4707895
1.5	0.95	0.475	0.9563	0.4542425
2	0.93	0.465	0.9422	0.438123
Total				1.8483s

Let notional value of principal be $1. Then, PV of expected payoffs is given as:

Time (yrs)	Prob of Default	Recovery Rate	Expected Payoff	Discount Factor	PV of Expected Payoff
0.25	0.015	0.25	0.015	0.9926	0.014889
0.75	0.015	0.25	0.015	0.978	0.01467
1.25	0.02	0.25	0.02	0.9639	0.019278
1.75	0.02	0.25	0.02	0.9501	0.019002
Total					0.067839

Now PV of accrual payments is calculated as follows:

Time (yrs)	Prob of Default	Expected Accrual Payment	Discount factor	PV of Expected Accrual Payment
0.25	0.015	0.0025	0.9926	0.0024815
0.75	0.015	0.0025	0.978	0.002445
1.25	0.02	0.00375	0.9639	0.003614625
1.75	0.02	0.00375	0.9501	0.003562875
Total				0.012104s

The credit default swap spread is given by:

1.8483s+0.012104s= 0.067839

s= 0.0365 or 365 basis points.