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CHAPTER 23 Credit Derivatives

Practice Questions

Problem 23.8.

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.

The table corresponding to Tables 23.2, giving unconditional default probabilities, is Time (years) 1 2 3 4 5 Default Probability 0.0300 0.0291 0.0282 0.0274 0.0266 Survival Probability 0.9700 0.9409 0.9127 0.8853 0.8587 The table corresponding to Table 23.3, giving the present value of the expected regular payments (payment rate is s per year), is Time (years) 1 2 3 4 5 Total Probability of Survival 0.9700 0.9409 0.9127 0.8853 0.8587 Expected Payment 0.9700s 0.9409s 0.9127s 0.8853s 0.8587s Discount Factor 0.9324 0.8694 0.8106 0.7558 0.7047 PV of Expected Payment 0.9044s 0.8180s 0.7398s 0.6691s 0.6051s 3.7364s The table corresponding to Table 23.4, giving the present value of the expected payoffs (notional principal =$1), is Time (years) 0.5 1.5 2.5 3.5 4.5 Total Probability of Default 0.0300 0.0291 0.0282 0.0274 0.0266 Recovery Rate 0.3 0.3 0.3 0.3 0.3 Expected Payoff 0.0210 0.0204 0.0198 0.0192 0.0186 Discount Factor 0.9656 0.9003 0.8395 0.7827 0.7298 PV of Expected Payoff 0.0203 0.0183 0.0166 0.0150 0.0136 0.0838 The table corresponding to Table 23.5, giving the present value of accrual payments, is Time (years) Probability of Default Expected Accrual Payment Discount Factor PV of Expected Accrual Payment 0.5 1.5 2.5 3.5 4.5 Total 0.0300 0.0291 0.0282 0.0274 0.0266 0.0150s 0.0146s 0.0141s 0.0137s 0.0133s 0.9656 0.9003 0.8395 0.7827 0.7298 0.0145s 0.0131s 0.0118s 0.0107s 0.0097s 0.0598s

The credit default swap spread s is given by:

3?7364s?0?0598s?0?0838

It is 0.0221 or 221 basis points.

Problem 23.9.

What is the value of the swap in Problem 23.8 per dollar of notional principal to the protection buyer if the credit default swap spread is 150 basis points?

If the credit default swap spread is 150 basis points, the value of the swap to the buyer of protection is: 0?0838?(3?7364?0?0598)?0?0150?0?0269 per dollar of notional principal.

Problem 23.10.

What is the credit default swap spread in Problem 23.8 if it is a binary CDS?

If the swap is a binary CDS, the present value of expected payoffs per dollar of notional principal is 0.1197 so that

3?7364s?0?0598s?0?1197

The spread, s, is 0.0315 or 315 basis points.

Problem 23.11.

How does a five-year nth-to-default credit default swap work. Consider a basket of 100 reference entities where each reference entity has a probability of defaulting in each year of 1%. As the default correlation between the reference entities increases what would you expect to happen to the value of the swap when a) n?1 and b) n?25. Explain your answer.

A five-year nth to default credit default swap works in the same way as a regular credit default swap except that there is a basket of companies. The payoff occurs when the nth default from the companies in the basket occurs. After the nth default has occurred the swap ceases to exist. When n?1 (so that the swap is a “first to default”) an increase in the default correlation lowers the value of the swap. When the default correlation is zero there are 100 independent events that can lead to a payoff. As the correlation increases the probability of a payoff decreases. In the limit when the correlation is perfect there is in effect only one company and therefore only one event that can lead to a payoff.

When n?25 (so that the swap is a 25th to default) an increase in the default correlation increases the value of the swap. When the default correlation is zero there is virtually no chance that there will be 25 defaults and the value of the swap is very close to zero. As the correlation increases the probability of multiple defaults increases. In the limit when the correlation is perfect there is in effect only one company and the value of a 25th-to-default credit default swap is the same as the value of a first-to-default swap.

Problem 23.12.

How is the recovery rate of a bond usually defined? What is the formula relating the payoff on a CDS to the notional principal and the recovery rate?

The recovery rate of a bond is usually defined as the value of the bond a few days after a default occurs as a percentage of the bond’s face value. The payoff on a CDS is L(1?R) where L is the notional principal and R is the recovery rate.

Problem 23.13.

Show that the spread for a new plain vanilla CDS should be 1?R times the spread for a similar new binary CDS where R is the recovery rate.

The payoff from a plain vanilla CDS is 1?R times the payoff from a binary CDS with the same principal. The payoff always occurs at the same time on the two instruments. It follows that the regular payments on a new plain vanilla CDS must be 1?R times the payments on a new binary CDS. Otherwise there would be an arbitrage opportunity.

Problem 23.14.

Verify that if the CDS spread for the example in Tables 23.2 to 23.5 is 100 basis points and the probability of default in a year (conditional on no earlier default) must be 1.61%. How does the probability of default change when the recovery rate is 20% instead of 40%. Verify that your answer is consistent with the implied probability of default being approximately proportional to 1?(1?R) where R is the recovery rate.

The 1.61% implied default probability can be calculated by setting up a worksheet in Excel and using Solver. The present value of the regular payments becomes 4?1170s, the present value of the expected payoffs becomes 0.0415, and the present value of the expected accrual payments becomes 0?0346s. When s?0?01 the present value of the expected payments equals the present value of the expected payoffs.

When the recovery rate is 20% the implied default probability (calculated using Solver) is 1.21% per year. Note that 1.21/1.61 is approximately equal to (1?0?4)?(1?0?2) showing that the implied default probability is approximately proportional to 1?(1?R). .

Problem 23.15.

A company enters into a total return swap where it receives the return on a corporate bond paying a coupon of 5% and pays LIBOR. Explain the difference between this and a regular swap where 5% is exchanged for LIBOR.

In the case of a total return swap a company receives (pays) the increase (decrease) in the value of the bond. In a regular swap this does not happen.

Problem 23.16.

Explain how forward contracts and options on credit default swaps are structured.

When a company enters into a long (short) forward contract it is obligated to buy (sell) the protection given by a specified credit default swap with a specified spread at a specified future time. When a company buys a call (put) option contract it has the option to buy (sell) the protection given by a specified credit default swap with a specified spread at a specified