The models used to create a time-series of survival probability rates for a default swap can be quite complex, but we can simplify the main points in the process:
* A basic starting assumption is that market observable credit spreads capture the market view of the riskiness of an obligor's debt (although they may well reflect other structural factors). This risk, as implied by credit spreads, depends on the probability of default as well as the severity of loss following default. Hence for a given credit spread and a given recovery rate assumption, we can approximate the probability of default.
* Given the close linkages between asset swap spreads and default swap premiums, this information is also contained in default swap premiums, although default swap premiums also reflect other issues such as counterparty risk. The market observable default swap curve is used, or a flat default swap curve is assumed if there are relatively few traded points along the curve.
* We assume a recovery value for a deliverable obligation in the default swap contract. As mentioned last week, this will be a function of the seniority of the obligation.
* Under certain assumptions about the default process, we can use the market observable information together with a recovery rate assumption to interpolate a time-series of default probabilities. These are used to calculate a cumulative default probability function:
Ft = ... P(Dj), j = 0 .... t
where P(Dt) is the probability of default during time t and Ft is the cumulative default probability at time t
* Survival probability is defined as:
St = 1-Ft
where St is the survival probability at time t. This provides us with a time series of survival probabilities.
An Example Of A Survival Probability Time-Series
The table opposite shows the results of this process in constructing a time-series of survival probabilities for a default swap. We assume five-year senior protection is quoted at 380 basis points and a 40% recovery rate on the senior debt of the reference entity. Using the relationship between survival probability and cumulative probability of default, we can infer the marginal probabilities of default. Marginal default probability is a strictly decreasing function of time.
Default != Credit Event
For the purposes of this series of articles, we have interchangeably used "default" and "credit event". In reality, "default" as captured by rating agency statistics may sometimes be a more severe test than certain credit events, which ultimately trigger default swaps. Moody's Investors Service, for example, notes three categories of default for the purposes of its ratings and historical default statistics:
* Missed or delayed interest or principal payments;
* Bankruptcy or receivership; and
* Distressed exchange either leaving investors with a diminished financial obligation or an exchange for the apparent reason of avoiding default.
In a default swap contract, restructuring can sometimes be considered a "soft" credit event. Additionally post-recovery statistics reflect such "hard default" whereas the expected recovery following for example a "soft" restructuring credit event would likely be significantly higher than for a liquidation. Against this, however, protection sellers assume cheapest-to-deliver risk following a credit event. In general, the possibility of "soft" credit events other than default will not be taken into account in the construction of survival probability rates.
Sensitivity Of Survival Probability
To Recovery Rate Assumption
Assumptions about recovery rates will be a factor determining the shape of the survival probability curve. This relationship can be summarized as follows:
For a given credit spread, a high recovery assumption implies a higher probability of default, relative to a low recovery assumption, and hence a lower survival probability.
Similarly, for a given credit spread, a low recovery assumption implies a lower probability of default, relative to a high recovery assumption, and hence a higher survival probability.
The chart below shows survival probability is a decreasing function of recovery rate and time.
How Survival Probability Varies With Time For Different
Recovery Assumptions (150bps Default Premium)
MTM Differences: Bonds and Default Swaps
The resultant mark to market will typically be different from a comparable cash-market unwind. In general, a long or short default swap position will have a smaller positive or negative change in value for a given spread change than a comparable asset swap. In other words, the differing valuation methodology of the two instruments leads to the default swap having a lower "risky duration".
Take the following simple example. An investor purchases USD10 million of five-year bonds at par, which asset swaps to LIBOR plus 100bps. The investor also undertakes a similar risk position in the default swap market by selling USD10 million of default protection to the same maturity generating a premium of 100bps. If both spreads immediately widen by 20bps, then the loss on the default swap would be lower than the loss on the asset swap. Conversely, however, a simultaneous tightening of spreads would yield a greater profit on the bonds than the gain on the default swap.
Difference In P&L Between the Two Transactions (Default Less Cash) for Simultaneous Spread Changes
Assuming CDS unwind of 150bps and 45% recovery assumption
Sensitivities Of The Mark-To-Market Amount
A main driver of the mark-to-market value is the recovery rate assumption but the value also has other sensitivities: Time Sensitivity, or 'Theta'.
Over time, the mark-to-market declines toward zero with its shortening maturity as less risky cash flows in the annuity remain. The mark to market becomes more sensitive to changes in the recovery value assumption the longer the default swap has to maturity.
Incremental Mark-to-Market
For a given recovery rate assumption, survival probability rates are a decreasing function of default premiums. In other words, for a given recovery rate assumption, wider default premiums reflect greater probability of default and hence a lower survival probability.
Below, we show the mark-to-market value increase on a long protection position as a result of an increase in premiums. The influence of survival probability at wider premiums can be seen from the declining slope of the mark-to-market curve. The incremental mark to market from a long protection position declines as premiums move wider.
Incremental Mark-To-Market Declines As Default
Premiums Increase
This week's Learning Curve was written by Barnaby Martin (left) and Chris Francis in credit derivatives research at Merrill Lynch in London.
