Bootstrapping default probabilities from CDS quotes

http://billiontrader.com/post/41[3/1/2014 4:23:16 PM]

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APR,14

06Bootstrapping default probabilities from CDS quotesFixed Income • Leonid Sopotnitskiy

The objective of this article is to give a brief overview of CDS pricing and the methodology of calibrating the default probabilities to the real market. A sample spreadsheet with respective calculations is included.

Credit Default Swaps (CDS) are usually quoted in terms of a basis point spread, which virtually indicates the market’s view on the riskiness of the reference entity. The swap deal represents an exchange of cash flows forming 2 payment legs. One paid by the protection buyer on a regular basis is the premium leg, and the other is the default leg, where payments occur only in case of a default event. A fundamental concept for CDS pricing is that a CDS transaction has a zero NPV at inception for the deal participants. This means that the present value of the premium leg (PL) and the default leg (DL) must be equal. And this is where we analyze the CDS term structure and derive the hazard rates from each tenor, because default probabilities are inferred from them, which basically allows us to calibrate the pricing model to the market. The following system of equations shows the formation of leg prices, whereas the spread represents the relationship of the DL and PL:

where P(T) is the survival probability at time T, DF(0;T) is the respective discount factor calculated form the spot risk-free interest rate curve. The question is how to calculate the PVs since both legs have uncertain cash flows? Premiums are paid up to either the credit event or maturity of the swap, whereas the payments of the default leg (1 – Recovery Rate) will occur only upon a credit event. This implies that we will require a grid of credit event probabilities, payment cash flows and discount factors in our calculations. The discount factors can be derived using the following formula:

The next step is to solve for the survival probabilities and hazard rates (λ) that virtually represent a default probability per period, provided the reference entity has survived all previous periods (P(T_(n-1) )-P(T_n )). Our current example provides a calculated grid of survival probabilities (including historical series) using a coded function CDS_Survival_Probability(), which can be found in the “CDS_Bootstrapping” module, in VBA assuming a constant recovery rate R = 40%, which is usually used by the market. Once this is done default probabilities are easily obtained as 1 – Survival Probability (refer to the sample Excel file for results).

Chart 1 provides the term structure of the default probabilities of 5 underlying entities with the assumption that at T = 0 the reference entity is solvent, hence PD(T) = 0. The hazard rates are derived by the CDS_Hazard_Rate() function coded in VBA, which uses the following equation:

Bootstrapped default intensities can be viewed on chart 2.

Bootstrapping default probabilities from CDS quotes

http://billiontrader.com/post/41[3/1/2014 4:23:16 PM]

In order to obtain survival probabilities we can reorganized the CDS pricing equation into the following form:

The above statement is verified by the constructed chart displaying the dependency of default probabilities on recovery rates and tenors (see chart 3).

Bootstrapping default probabilities from CDS quotes

http://billiontrader.com/post/41[3/1/2014 4:23:16 PM]

MAY,14

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This chart in general tells us that higher levels of recovery rate induce higher default probability for a given premium – this happens primarily due to the fact that the default leg becomes cheaper (i.e. we assume that the reference entity will be solvent enough to cover more losses) at a fixed spread and hence the conditional default probability increases.

The results can be applied in relative value analysis when comparing CDS curves with bond data on a Z-spread basis: in order to do this the bond cash flows will need to be adjusted by the default probabilities; and this plays a major role in adjusting OTC derivative positions to reflect credit risk (CVA/DVA). This topic is further discussed and illustrated in articles on relative value analysis of credit instruments and pricing basket CDS using copulas

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2 Comments

visitor says:

thanks for this invaluable article. wanna see more stuff in regards to the application in busket CDS/CDO pricing using copula from here.

visitor says:

A complete spreadsheet with VBA would be great! the current one looks like extracted partially from a CDS pricing project.

Reply

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Bootstrapping default probabilities from CDS quotes

http://billiontrader.com/post/41[3/1/2014 4:23:16 PM]

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