HullFund8eCh23ProblemSolutions
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
The table corresponding to Table 23.3, giving the present value of the expected regular payments (payment rate is s per year), is
The table corresponding to Table 23.4, giving the present value of the expected payoffs (notional principal =$1), is
The table corresponding to Table 23.5, giving the present value of accrual payments, is
The credit default swap spread s is given by:
373640059800838s s .+.=. 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: 00838(3736400598)0015000269.-.+.?.=. 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
373640059801197s s .+.=. The spread, s , is 0.0315 or 315 basis points.
Problem 23.11.
How does a five-year n th-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) 1n = and b) 25n =. Explain your answer.
A five-year n th 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 n th default from the companies in the basket occurs. After the n th default has occurred the swap ceases to exist. When 1n = (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 25n = (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 (1)L 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 1R - times the spread for a similar new binary CDS where R is the recovery rate.
The payoff from a plain vanilla CDS is 1R - 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 1R - 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 41170s ., the present value of the expected payoffs becomes 0.0415, and the present value of the expected accrual payments becomes 00346s .. When 001s =. 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 (104)(102)-./-. 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
future time. Both contracts are normally structured so that they cease to exist if a default occurs during the life of the contract.
Problem 23.17.
“The position of a buyer of a credit default swap is similar to the position of someone who is long a risk-free bond and short a corporate bond.” Explain this statement.
A credit default swap insures a corporate bond issued by the reference entity against default. Its approximate effect is to convert the corporate bond into a risk-free bond. The buyer of a credit default swap has therefore chosen to exchange a corporate bond for a risk-free bond. This means that the buyer is long a risk-free bond and short a similar corporate bond. Problem 23.18.
Why is there a potential asymmetric information problem in credit default swaps?
Payoffs from credit default swaps depend on whether a particular company defaults. Arguably some market participants have more information about this that other market participants (see Business Snapshot 23.3).
Problem 23.19.
Does valuing a CDS using actuarial default probabilities rather than risk-neutral default probabilities overstate or understate its value? Explain your answer.
Real world default probabilities are less than risk-neutral default probabilities. It follows that the use of real world default probabilities will tend to understate the value of a CDS. Further Questions
Problem 23.20.
Suppose that the risk-free zero curve is flat at 6% per annum with continuous compounding and that defaults can occur at times 0.25 years, 0.75 years, 1.25 years, and 1.75 years in a two-year plain vanilla credit default swap with semiannual payments. Suppose that the recovery rate is 20% and the unconditional probabilities of default (as seen at time zero) are 1% at times 0.25 years and 0.75 years, and 1.5% at times 1.25 years and1.75 years. What is the credit default swap spread? What would the credit default spread be if the instrument were a binary credit default swap?
The table corresponding to Table 23.3, giving the present value of the expected regular payments (payment rate is s per year), is
The table corresponding to Table 23.4, giving the present value of the expected payoffs (notional principal =$1) is
The table corresponding to Table 23.5, giving the present value of accrual payments, is
The credit default swap spread s is given by:
s s
.+.=.
180410011700375
It is 0.0206 or 206 basis points.
If the instrument were a binary credit default swap the expected payoff would be
0.0375/0.8=0.0468 and the spread s would be given by
.+.=.
1804100117004675
s s
It would be 0.0257 or 257 basis points.
Problem 23.21.
Assume that the default probability for a company in a year, conditional on no earlier defaults is λand the recovery rate is R. The risk-free interest rate is 5% per annum. Default always occur half way through a year. The spread for a five-year plain vanilla CDS where payments are made annually is 120 basis points and the spread for a five-year binary CDS where payments are made annually is 160 basis points. Estimate R and λ. Estimating the recovery rate is fairly easy. The spread for the vanilla CDS should be 1R
-
R=.. To find λwe set up a spread times the spread for the binary swap. It follows that 025
sheet containing tables similar to Tables 23.2 to 23.5 in the text and search with Solver for the value of λthat causes the spread to be 120 basis points. The answer is 0.0155 or 1.55% per year.
., the present With this default probability the present value of regular payments is 41244s
., and the present value of payoffs is 0.0499.
value of the accrual payments is 00333s
Problem 23.22.
Explain how you would expect the returns offered on the various tranches in a synthetic CDO to change when the correlation between the bonds in the portfolio increases.
When the default correlation is zero the junior tranches (i.e., those that get impacted early by defaults) are much more risky than senior tranches and should receive much higher yields. As the default correlation increases the junior tranches become less risky and the senior tranches become more risky. As a result we should expect the yield offered to junior tranches to go down and that offered to senior tranches go up. In the limit when the correlation is 1.0 there is in effect only one bond. All tranches are then equally risky and should be promised the same yield.
Problem 23.23.
Suppose that
a.The yield on a five-year risk-free bond is 7%
b.The yield on a five-year corporate bond issued by company X is 9.5%
c. A five-year credit default swap providing insurance against company X defaulting costs
150 basis points per year.
What arbitrage opportunity is there in this situation? What arbitrage opportunity would there be if the credit default spread were 300 basis points instead of 150 basis points? Give two reasons why arbitrage opportunities such as those you identify are less than perfect.
A possible trading strategy is to buy a corporate bond with principal $Q, buy a credit default swap with notional principal of $Q, and short a Treasury bond with principal $Q. If there is no default, the corporate bond when combined with the credit default swap creates an
.-.=.. The short Treasury position costs instrument that provides a return of about 951580%
about 7% and so the strategy should provide a return of about 1%. If company X defaults, the short position in the Treasury bond is closed out. There is then a final inflow of $Q from the credit default swap and the corporate bond and a final outflow equal to the value of the Treasury bond. Providing interest rates are not very low at the time of default (so that the Treasury bond is worth significantly more than $Q), the trading strategy should be profitable in this case as well.
The trading strategy shows that we should expect the cost of a credit default swap to be close to the spread between the yield on the corporate bond and the yield on the Treasury bond. In this case we should expect it to be around 250 basis points rather than 150 basis points. We could argue that it should be exactly 250 basis points if (a) the Treasury bond is a par yield bond (b) the corporate bond is a par yield bond (c) term structures are flat and interest rates are constant, and (d) the credit default swap gives the holder the right to sell the bond for its face value plus accrual interest in the event of a default. (In practice credit default swaps give the holder the right to sell the bond for its face value in the event of a default.)
Problem 23.24 (Excel file)
The 1-, 2-, 3-, 4-, and 5-year CDS spreads are 100, 120, 135, 145, and 152 basis points, respectively. The risk-free rate is 3% for all maturities, the recovery rate is 35%, and payments are quarterly. Use DerivaGem to calculate the continuously compounded hazard rate each year. What is the probability of default in year 1? What is the probability of default in year 2? (Use the result in footnote 7.)
The CDS worksheet shows that hazard rates in years 1, 2, 3, 4, 5 are 1.533%, 2.161%,
2.566%, 2.735%, and 2.824%. The probabil ity of default during year 1 is 1? e-0.0153×1 =
1.521%. The probability of default during the second year conditional on no default in the first year is 1? e-0.0216×1 =
2.138%. The unconditional default probability is 0.98479×0.02138
= 2.105%
Problem 23.25 (Excel file)
Table 23.7 shows the five-year iTraxx index was 77 basis points on January 31, 2008. Assume that the risk-free rate is 5% for all maturities, the recovery rate is 40%, and payments are quarterly. Assume also that the spread of 77 basis points applies to all maturities. Use the DerivaGem CDS worksheet to calculate a hazard rate consistent with the spread. Use this in the CDO worksheet with 10 integration points to imply base correlations for each tranche from the quotes for January 31, 2008.
The CDS worksheet shows that the hazard rate is constant at 1.275%. The CDO worksheet shows that the base correlations for the 0-3%, 3-6%, 6-9%, 9-12%, and 12-22% tranches are 0.40, 0.52, 0.58, 0.60, and 0.73.