Question

Risk-free zero rate is 4% per annum with continuous compounding for all maturities and defaultsonly occur at times 0.25 years, 0.75 years, 1.25 years, and 1.75 years in a two-year credit default swap with semiannual payments. The probabilities of default are 0.8%, 1%, 1.2% and 1.5% in the first, second, third and forth six months of the CDS’s lifetime. The recovery rate is 60%. Calculate the CDS spread. Do not need in binary CDS

Answer 1

The following table gives unconditional default probabilities

Year	Default probabilities	Survival Probability
0.25	0.8%	99.2%
0.75	1.0%	99.0%
1.25	1.2%	98.8%
1.75	1.5%	98.5%

The following table gives the present value of the expected regular payments (payment rate is s per year)

Time (years)	Probaility of survival	Expected payment	Discount factor	PV of Expected Payment
0.5	99.2%	0.9920s	0.9806	0.9727s
1	99.0%	0.9900s	0.9615	0.9519s
1.5	98.8%	0.9880s	0.9429	0.9316s
2	98.5%	0.9850s	0.9246	0.9107s
Total				3.7669s

The following table gives the present value of the expected payoffs (notional principal =$1),

Time (Years)	Probability of default	Recovery rate	Expected Payoff	Discount Factor	PV of Expected Payoff
0.25	0.80%	60%	0.0032	0.9902	0.0032
0.75	1.00%	60%	0.0040	0.9710	0.0039
1.25	1.20%	60%	0.0048	0.9522	0.0046
1.75	1.50%	60%	0.0060	0.9337	0.0056
Total					0.0172

The following table givesthe present value of accrual payments, is

Time (Years)	Probability of Default	Expected Accrual Payment	Discount Factor	PV of expected Accrual Payment
0.25	0.80%	0.0040s	0.9902	0.0040s
0.75	1.00%	0.0050s	0.9710	0.0049s
1.25	1.20%	0.0060s	0.9522	0.0057s
1.75	1.50%	0.0075s	0.9337	0.0070s
Total				0.0215s

The credit default swap spread s is given by:

s = (3.7669+0.0215)/0.0172 = 0.0045 or 45 basis points