CONTRACT DEFINITION
"Worst of" default swaps are default swaps on a basket of issuers in which only the first default is covered. Losses due to further defaults do not lead to payments by the writer of the default swap. A "worst of " default swap provides relatively cheap partial insurance against defaults in a bond portfolio compared to buying the basket of default swaps, in particular for correlated defaults. This article will analyze two cases, zero default correlation and positive default correlation.
ZERO DEFAULT CORRELATION
First we assume that defaults in a basket are uncorrelated, i.e. default of one issuer does not increase the probability that any other issuer will default. This is only true for a well-diversified basket. The payment to the buyer of a default swap if the underlying bond defaults is 1-*, where * is the recovery price of the bond. The buyer pays an annual premium to the seller until default occurs.
Let P(A), P(B), P(AUB) and P(AB) be respectively the risk neutral probability that issuer A, issuer B, either A or B and both issuers will default during the life of the contract. P(A) and P(B) are estimated from the market prices of default swaps using a stripping algorithm.
The "worst of" contract insures the holder against the event
P(AUB) = P(A) + P(B) - P(AB)
Since the defaults are uncorrelated,
P(AB) = P(A)P(B)
If the default probabilities are very low, the probability of a joint default, not covered by a "worst of" default swap is very small. The price of a "worst of" default swap is approximately equal to the sum of the two individual default swaps. This will be true for short-dated default swaps on good credits. The "worst of" provides almost complete insurance against default.
In the following analysis it is assumed that the issuers in the basket have identical credit curves and the same recovery rate. From Figure 1 it can be deduced that if the default probabilities are substantial, the possibility of a joint default cannot be ignored and we have to take into account a substantial convexity term. This is particularly true for long maturities and low credit quality issuers.
POSITIVE DEFAULT CORRELATION
In a typical basket, defaults are positively correlated, i.e. default of an issuer will trigger defaults of other issuers. The correlation will depend on the degree of concentration (geographical, industrial,...) in the portfolio. The probability of joint default is given by,
P(AB) = P(A)P(B/A),
where P(B/A) is the conditional probability that B will default if A has defaulted. This quantity can be significant and the price will be lower than the sum of the individual default swaps. To analyze the impact of the default correlation we need to impose a certain structure for the evolution of the credit spreads. Modeling the credit spreads is equivalent to modeling the instantaneous default probabilities. The two quantities are proportional with proportionality constant (1 - *), where * is the recovery rate. We assume that credit spreads follow correlated Brownian motions with drift.
Figure 2 shows the incremental effect of the correlation for five-year default swaps. It is clear that ignoring correlation can result in substantial overpricing of the "worst of".
Negative Correlation: For completeness of the analysis I will look at the negative default correlation scenario, even though it is very unlikely to be observed in real markets. A counter cyclical company would potentially be negatively correlated with the rest of the market. Under this scenario P(AB) will be smaller than P(A)P(B) and the price of the "worst of" will be higher than in the zero correlation case. The intuition is that if issuer A defaults then the probability that B will default becomes smaller. In the extreme case the two defaults are mutually exclusive, i.e. it is impossible for both issuers to default during the life of the contract. Then the price equals the sum of the prices of the default swaps.
LARGE NUMBER OF ASSETS
The properties demonstrated in the two cases become even more pronounced as we increase the number of assets. In Figure 3 the credit spread correlation has been assumed to be 50%.
Discounting effect: The expected payment time on the "worst of" is shorter than the average payment time on the basket of the individual default swaps. This increases the price of the "worst of" relative to the basket. The effect is more pronounced as the number of underlying bonds in the basket increases.
This week's Learning Curve was written by Angelo Arvanitis, head of quantitative credit & risk research at Paribas in London.
