23.21.    Suppose that in a one-factor Gaussian copula model the 5-year probability of default for each of 125 names is 3% and the pairwise copula correlation is 0.2. Calculate, for factor values of -2, -1, 0, 1, and 2: (a) the default probability conditional on the factor value and (b) the probability of more than 10 defaults conditional on the factor value.

Answer:

We use the following equation for the probability of default conditional on a factor value of (M):

For M = -2, - 1, 0, +1, and +2, the default probabilities are 13.5%, 5.4%, 1.77%, 0.46%, and 0.10%, respectively.

The probability of more than 10 defaults for these value of (M) can be calculated using Excel’s BINOMDIST function;
e.g., P( defaults > 10 | -1.0) = 1 – BINOMDIST(10 defaults, 125 names, 5.45% conditional PD, TRUE) = 7.98%

23.20.    What is the difference between a total return swap and an asset swap?

Answer:

In an asset swap the bond’s promised payments are swapped for LIBOR plus a spread. In a total return sap (a.k.a.,TROR), the bond’s actual payments are swapped for LIBOR plus a spread.

23.19.    Does valuing a credit default swap (CDS) using real-world default probabilities rather than risk-neutral default probabilities overstate or understate its value? Explain your answer.

Answer:

Real world default probabilities are less than risk-neutral default probabilities. It follows that the use of actuarial default probabilities will tend to understate the value of a CDS.

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.

[my adds]
23.17b. Hull simplifies. What are the difference(s) between a long CDS and a long bond position, if any?
23.17c. Hull’s simplified equivalence implies that the difference between the CDS spread and the cash bond spread (a.k.a., the CDS basis) should be near to zero. Gary Gorton (FRM assignment: Information, Liquidity, and the Ongoing Panic of 2007) adds another factor into CDS basis. What is Gorton’s equivalence?

Answer:

23.17 Says Hull, “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.”

23.17b. The key difference between long CDS and long bond is FUNDING. The bond buyer must invest principal but the long CDS is UNFUNDED (synthetic) exposure. The bond investor, therefore incurs additional funding cost (funding risk). Further, the long bond investor is exposed to interest rate risk (market risk) and, possibly currency risk. The bond investor is exposed to a bundle of risks (up to four) whereas the CDS protection buyer (if not M2M) potentially has “pure” exposure to default risk only.

23.17c. Gorton’s equation improves on the “naïve” equivalence in Hull by incorporating the FUNDING cost:
CDS basis = CDS spread – cash bond spread;
and if Rcf = cost of funding = repo rate:
CDS spread + Rcf = cash bond spread, or
CDS spread + repo rate = cash spread, or
(-) CDS basis =  repo rate

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

Answer:

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.

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.

Answer:

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

Question:

23.14.    Assume that the risk-free zero curve is flat at 5% per annum with continuous compounding and that defaults can occur halfway through each year in a new 5-year credit default swap (CDS). Assume the recovery rate is 40%. Verify that if the CDS spread is 100 basis points, then 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.

Answer:

If we assume 40% recovery, then the implied conditional probability of default is 1.61%:

If we assume 20% recovery, then the implied conditional probability of default is 1.21%:

The ratio of implied default probabilities is approximately proportional to 1/(1-R):
1.21/1.61 ~ (1-40%)/(1-20%)

Says Hull, “In passing we note that if the CDS spread is used to imply [infer] an unconditional default probability (assumed to be constant each year) then this implied unconditional default probability is exactly proportional to 1/(1-R). When we use the CDS spread to imply [infer] a conditional default probability (assumed to be constant each year) it is only approximately proportional to 1/(1-R).”

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.

Answer:

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 will be an arbitrage opportunity.

23.12. What is the formula relating the payoff on a CDS to the notional principal and the recovery rate?

Answer:

The recovery rate of a 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. Further, the recovery rate = (1 - loss given default).

23.11. How does a 5-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.

Answer:

A five-year nth to default CDS works in the same way as a regular CDS except there is a basket of companies. The payoff occurs when the nth default among the companies occurs. After the nth default has occurred, the swap ceases to exist. When n = 1 (i.e., a first to default CDS), an increase in the default correlation lowers the value of the swap. When there are 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 (1.0), there is in effect only one company and therefore only one even that can lead to a payoff.
When n=23 (i.e., a 25th to default), an increase in the default 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 near to zero. As the correlation increases, the probability of multiple defaults increases. In the limit, when the correlation is perfect (1.0) , there is in effect only and company and the value of the 25th-to-default CDS is the same as the value of a first-to-default CDS.