Question

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.

What is the value of the swap in Problem 4.1 per dollar of notional principal to the protection buyer if the credit default swap spread is 150 basis points?

What is the credit default swap spread in Problem 4.1 if it is a binary CDS?

Answer 1

Below are the steps to calculate the solution along with the formulas

First we need to prepare a table 5 years giving each default probability and survival probability for each year

	Years	Hazard Rate	Default Probability	Survival probability
	0			1
	1	0.03	0.03*1	1-(0.03*1)
	2	0.03	0.03*Survival probability of 1st year	(Previous year survival probability - current year default probability)
	3	0.03	0.03*Survival probability of 2nd year	(Previous year survival probability - current year default probability)
	4	0.03	0.03*Survival probability of 3rd year	(Previous year survival probability - current year default probability)
	5	0.03	0.03*Survival probability of 4th year	(Previous year survival probability - current year default probability)
	Now we would calculate the present value of expected regular payment we would assume the expected payment rate is s
		A	B	C	D
	Years	Probaility of survival (to be taken as calculated above)	Expected payment(Probability of survival*s)	Discount factor	PV of expected payment
	1		ColumnA*s	e^-(0.07*1)	Column B* Column C
	2		ColumnA*s	e^-(0.07*2)	Column B* Column C
	3		ColumnA*s	e^-(0.07*3)	Column B* Column C
	4		ColumnA*s	e^-(0.07*4)	Column B* Column C
	5		ColumnA*s	e^-(0.07*5)	Column B* Column C
	Total
	The next step would be to calculate the present value of the expected payoff
	Assuming the notional principal is $ 1
	Years	Probability of default( to be taken from above)	Recovery rate	Expected payoff	Discount factor
	0.5		0.3	Notinal principal*(1-recovery rate)*probability of default	e^-(0.035*0.50)
	1.5		0.3	Notinal principal*(1-recovery rate)*probability of default	e^-(0.035*1.50)
	2.5		0.3	Notinal principal*(1-recovery rate)*probability of default	e^-(0.035*2.50)
	3.5		0.3	Notinal principal*(1-recovery rate)*probability of default	e^-(0.035*3.50)
	4.5		0.3	Notinal principal*(1-recovery rate)*probability of default	e^-(0.035*4.50)
	Total
	Now we would calculate the present value of expected accrual payments
	Years	Probability of default( to be taken from above)	Expected accrual payment	Discount factor	PV of expected Accrual payment(Expected accrual payment*discount factor)
	0.5		Probability of default*0.50*s	e^-(0.035*0.50)
	1.5		Probability of default*0.50*s	e^-(0.035*1.50)
	2.5		Probability of default*0.50*s	e^-(0.035*2.50)
	3.5		Probability of default*0.50*s	e^-(0.035*3.50)
	4.5		Probability of default*0.50*s	e^-(0.035*4.50)
	Total
	The credit default swap spread s would be given by
	Present value of the expected payment + PV of expected accrual payment = Present value of expected payoff
	When we put the total of the value received from the above table we would get the value of s which is the credit default swap spread
2)	If the credit default swap is 150 basis point, the value of the swap to buyer would be
	Present value of expected payoff - (Present value of the expected payment+present value of the expected accrual payment)*credit default swap
3)	If the swap is a binary CDS, the present value of the expected payoffs would be
		Present value of the expected payment + PV of expected accrual payment = Present value of expected payoff of binary cds