Question

Calculate the leveraged adjusted duration gap of this position. Is the bank exposed to interest rate increases or decreases and why? [50 marks] Consider the following problem: 3. Suppose that the probability of an entity defaulting during a year conditional on no earlier default is 2%. Assume that defaults always happen halfway through a year. The risk-free rate is 5% per annum with continuous compounding and the recovery rate is 40%. Assume that payments on a 3-year credit default swap (CDS) are made once a year, at the end of each year. The credit default spread is equal to s basis points. That is, payments are made at the rate of s per year and the notional principal is £1. The accrual payment of 0.5s made in the event of a default. (a) Caculate the survival probabilities and unconditional default prosabilities for each of the three years. [20 marks) (b) Calculate the present value of the expected payments made on the CDS. [20 marks] (c) Calculate the present value of the expected payoff. [20 marks] (d) Calculate the present value of the expected accrual payments. [20 marks] (e) From the answer of (a)- (d), what is s equal to? Suppose the CDS had been negotiated some time ago for a spread of 200 basis points. Calculate the value of swap to the seller/buyer.

Answer 1

Suppose that the probability of an entity defaulting during a year conditional on no earlier default is 2%. Assume that defaults always happen halfway through a year. The risk-free rate is 5% per annum with continuous compounding and the recovery rate is 40%. Assume that payments on a 3-year credit default swap (CDS) are made once a year, at the end of each year. The credit default spread is equal to s basis points. That is, payments are made at the rate of s per year and the notional principal is £1. The accrual payment of 0.5s made in the event of a default.

(a) Calculation of the survival probabilities and unconditional default probabilities for each of the three years.

Time (years)	Default Probability	Survival Probability
1	0.0200 (2%)	0.9800
2	0.0196	0.9604
3	0.0192	0.9412

1-.02 = 0.98 ; 0.98*0.98 = 0.9604 ; 0.98*0.9604 = 0.9412

(b) Calculation of the present value of the expected payments made on the CDS

Time (years)	Survival Probability	Expected Payment	Discount Factor	PV of expected payment
1	0.9800	0.9800s	0.9704	0.9510s
2	0.9604	0.9604s	0.9418	0.9045s
3	0.9412	0.9412s	0.9139	0.8602s
Total				2.7157s

(c) Calculation of the present value of the expected payoff

Time (years)	Default Probability	Recovery rate	Expected Payoff	Discount Factor	PV of expected payoff
0.5	0.0200	0.4	0.0120	0.9753	0.0117
1.5	0.0196	0.4	0.0118	0.9277	0.0109
2.5	0.0192	0.4	0.0115	0.8825	0.0102
Total					0.0328

(d) Calculation the present value of the expected accrual payments

Time (years)	Default Probability	Expected Accrual Payment	Discount Factor	PV of Accrual payment
0.5	0.0200	0.0100s	0.9753	0.0097s
1.5	0.0196	0.0098s	0.9277	0.0091s
2.5	0.0192	0.0096s	0.8825	0.0085s
Total				0.0273s

(e) From the total table values in (b) and (d) the present value of expected payments is

2.7157s + 0.0273s = 2.743s

The present value of the expected payoff is 0.0328. Equating the two, the CDS spread for a new CDS is given by,

2.743s = 0.0328

Therefore, s = 0.01195

The mid-market spread should be 0.01195 times the principal or 119 basis points per year.

Suppose the CDS had been negotiated some time ago for a spread of 200 basis points.

The value of a swap negotiated some time ago with a CDS spread of 200 basis points would be = 2.743 × 0.0200 − 0.0328 = 0.02206 per pound of the principal.