Pricing credit default swaps means, above all, trying to attribute a value to the various components of the underlying asset. In this case, the asset is the credit risk of a certain reference entity, or to be more precise, the reference entity's risk of triggering a credit event.

In a plain-vanilla credit default swap structure, in exchange for the credit default swap premium paid by the buyer for protection against the reference entity's credit risk, there are potential cash inflows if a credit event takes place. The structure of the credit default swap's cash flows is explained in the chart below. The structure considered here provides protection against default risk for a period of four years.

In the event of default by the reference entity, the seller of the protection is required to pay an amount equal to the difference between the initial price of the credit exposure and the recovery value relating to the liability on which the payoff provided for in the contract is calculated.

For the purposes of evaluating the expected loss, the main step is to consider the probability of default by the reference entity during a particular time period and the recovery rate relating to the type of issue being analyzed.

The equation that expresses the valuation of the expected loss

is as follows:

where:

EL = Expected Loss

n = Number of periods to the due date

and:

where:

E (DCFi )= Cash flows expected in the case of default

(1-pi-1)= Risk-neutral marginal probability of the Reference Entity surviving at the time i-1

(1-pi) = Risk-neutral marginal probability of the Reference Entity surviving at the time i

RR = Recovery Rate

In this example, the severity of the loss is represented by the recovery rate's complement of one, that is to say, 1-RR. The loss in the case of default by the seller of protection, or more generally the lender of funds in the normal business of granting credit, is given by the product of the severity times the face value of the exposure. This is also known as the "Loss Given Default."

A credit default swap is priced so that the seller of the protection manages to recover the potential outflows in the event that the reference entity defaults. The premium calculated in this way ought to constitute the "threshold" price that the buyer of the credit default swap has to pay for protection against the risk of default.

To this end it is useful to express the expected loss in percentage terms, relating the value of the expected loss to the face value of the credit exposure. This percentage represents the minimum level of the premium for the credit default swap.

In addition to the expected loss, the seller of the protection has to take into consideration the cost of the capital that needs to be put aside to cover the volatility of the expected loss. The pricing of the credit default swap should therefore take into account this aspect as well.

Given a certain percentile level of the loss distribution (for example, 99%), it is possible to obtain a value for the worst-case credit loss and therefore the level of the unexpected loss, that being the difference between the worst-case credit loss and the expected loss. The cost of the amount of capital equal to the unexpected loss has to be included in the pricing of the credit default swap (see chart below).

The cost of capital is shown in the following equation:

where:

ULC = Unexpected Loss Cost

Kei = Cost of capital for period i

ECAi = Amount of capital for period i: this amount is calculated on the basis of the unexpected loss (the difference between the worst case credit loss and the expected loss).

The cost of the capital to be held against the volatility of the expected loss (expressed as a percentage of the face value of the exposure) should therefore be added to the cost of the expected loss, to reach a correct valuation of the credit default swap.

The cost of capital could be estimated by calculating the securities market line equation for the CAPM model. In this case the variables that determine the cost of capital consist of the yield on the risk-free reference security, the beta of the equity relating to the reference entity, and the risk premium for the equity market, according to the following expression:

*Ke _{i} = Rf +*

*ß*

_{i}(E(Rm)-Rf )where:

*Rf* = Risk-free yield

*Rm* = Yield of the equity index of reference

*(E(Rm) -Rf)* = Premium for market risk

*ß** _{i}* = Beta of the security considered

Taking into account the equations given previously, the premium for a credit default swap is shown in the following formula:

where:

*CDP* = Credit default premium

**References**

Das, S.R., 1998, Valuation and Pricing of Credit Derivatives, in "Credit Derivatives," John Wiley & Sons;

E. Elton, M. J. Gruber, 1981, *Modern Portfolio Theory and Investment Analysis*, John Wiley & Sons.

*This week's Learning Curve was written by Andrea Fabbri, deputy head of the credit derivatives group at* *Banca Commerciale Italiana**and visiting professor at* *Bocconi University**in Milan.*