Traditionally collateralized debt obligations (CDOs) involve a transfer of collateral assets. The CDO liabilities then reference the cash flows (principal and interest) of the collateral assets. But CDOs are increasingly issued in synthetic form, where there is no physical transfer of collateral. In these structures the CDO references default losses, rather than the cash flows of the referenced collateral.
Credit Default Swaps
In their simplest form synthetic CDOs are akin to single name credit default swaps. A credit default swap is a contract where one party ,the protection buyer, makes periodic payments in exchange for payments contingent on a credit event. In a single name credit default swap, the credit event is the default of a single issuer. Thus, the price of the credit default swaps reflects the market's view of the default likelihood of the reference name.
In unfunded form a synthetic CDO is essentially a basket default swap. In this case, in return for a premium, the protection seller makes contingent payments based on the default losses incurred by a portfolio or basket of assets. The protection may be for first loss or equity, in which case the seller pays for the initial default losses, up to some threshold; second loss or mezzanine, where the seller pays losses beyond the first loss threshold up to some higher limit; or senior, where the seller pays all losses beyond a threshold.
Synthetic CDOs may also be structured in funded form. The simplest form of this structure is a combination of a risk free bond and a basket default swap. The initial investment is used to purchase the risk free bond and at the outset the investor receives the risk free rate of interest from the bond as well as premium payments from the basket default swap. As losses occur on the reference portfolio the principal of the bond is reduced to pay the required contingent payments. At the maturity of the structure the investor receives the remaining principal of the risk free position. In both funded and unfunded forms the pricing of synthetic CDOs is primarily a matter of pricing the basket default swaps themselves.
Correlation Products
It is often said that synthetic CDOs are correlation products because their value depends not only on the probability of default for each individual name in the reference portfolio, but also on the correlation between those names. Consider a simple case where the reference portfolio consists of two names, each with a 5% probability of default. Assume two levels of protection, one paying on the first default, and one paying on the second. If the names are independent, then the probability that the first loss protection is triggered is 1 - (0.95)2 = 9.75%, that is, 100% less the chance that there are zero defaults. The probability that the second loss protection is triggered is (0.05)2 = 0.25%, that is, only if there are two defaults. On the other hand, if the defaults are perfectly correlated, then there are only two possibilities: either both or neither of the names default. Thus, the probability that either protection is triggered is the probability that both names default, or 5%. Note that for the first loss protection the increased correlation has reduced the likelihood that the protection is invoked, and therefore reduced the fair price of this protection. For the second loss piece, the increased correlation has increased the trigger likelihood and fair price.
The relationship between correlation and pricing holds generally for the first loss and most senior levels of protection in basket default swaps. For the first loss piece, since losses are capped, the pricing is driven primarily by the likelihood that no losses occur, which increases with correlation. For the senior piece, the pricing is driven by the likelihood that an extreme loss occurs, which is greater in cases of higher correlation.
Pricing
Based on the discussion above most pricing models require individual default probabilities and a correlation across the names in the pool. Since individual default probabilities can often be obtained from single name credit-default swaps correlation is the main pricing parameter for synthetic CDOs.
Consider an example structure with USD36 million of collateral, where first loss protects against the first USD5 million of losses, second loss protects against the next USD10 million of losses, and the remaining losses are protected by the senior tranche. We can model spreads on each of these tranches at different correlation levels. For example, using a correlation of 45%, we obtain model spreads of two basis points, 250bps and 1,642bps for the senior, second loss, and first loss protection respectively. We can repeat this exercise at different correlation levels to obtain the sensitivity of spreads to changes in correlation. Figure 1 shows the normalized cost of protection for each tranche at different correlation levels. We can see that the cost of protection for the senior tranche increases with increasing correlation, that is the cost increases as senior debt gets riskier, while the first loss protection cost decreases with increasing correlation, that is the cost decreases as equity gets less risky. Note that, in this particular example the cost of second loss protection behaves like senior as an increasing function of correlation.
Implied Correlation
We can back out an implied correlation for each tranche from CDO prices in the same way we would obtain implied volatility from option prices and the Black-Scholes pricing model. An implied correlation extracted in this fashion depends on the specific CDO pricing model, but the observations above hold generally. Suppose the market spreads for our example structure are 2bps, 150bps and 2,000bps for the senior, second loss, and first loss protection respectively. We can compare these observed spreads with our model spreads to calibrate a correlation parameter for each tranche. For example the model spread is equal to the observed spread for the senior tranche, 2bps, when we price it using a 45% correlation. In other words, the implied correlation for the senior tranche is 45%. The implied correlation for each tranche can be inferred from Figure 1 as the correlation level with which our pricing model recovers the observed market spread. We can see that for the first and second loss tranches, the implied correlation is roughly 20%, while for the senior tranche the implied correlation is 45%.
A first application of the implied correlation is to mark to market a synthetic tranche on an ongoing basis. As we observe changes in the spreads for the individual names in the pool, we can reprice the tranches above by assuming the correlation level is constant, but updating the spreads. Additionally, implied correlations can be used for relative value analysis. For example, senior and second loss tranche protection increase in value as correlation increases. Since both tranches depend on the same basket of names, we would expect them to have a similar implied correlation. As mentioned above, the implied correlation of the senior tranche (45%) is significantly higher than that of the second loss tranche (20%). This means that senior protection is expensive relative to second loss protection. We can also perform relative value analysis across deals if we believe that the pools in two different CDOs have similar characteristics, in terms of credit quality, industry, and geographical distribution. For example, if two CDOs have similar pools and structure we would expect to observe comparable implied correlations for their senior tranches. All else being equal, a discrepancy in implied correlation can serve as an indicator of a discrepancy in price between the two structures.
Figure 1: Cost of protection as a function of default correlation
This week's Learning Curve was written by Christopher Finger, head of credit products, and Jorge Mina, senior researcher, at the RiskMetrics Groupin New York.
