As with any swap, valuing credit default swaps (CDS) involves calculating the present value of the two legs of the transaction. In the case of CDS, these are the premium leg (the regular fee payments) and the contingent leg (the payment at the time of default). The basic inputs to the model are: the credit curve, which is the term structure of the CDS spread curve ranging across the standard maturity points, i.e. 6M, 1Y, 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 15Y, 20Y and 30Y; the recovery rate assumption, meaning if a default occurs, how much of their money bond investors will recover; a discount rate for calculating a present value, typically the zero-coupon swap curve for the trade currency; and the traded spread of the CDS.
For a fair market trade, the present value of the two legs should be equal, so the present value of all future fee payments should be exactly equal to the present value of the contingent payment at the time of default. Consider the following example CDS trade:
The mark-to-market CDS spreads for Ford as at September 13 were 100bp for 6M and 150bp for 1Y. Let us assume: a 5% constant annual interest rate for discounting; a recovery rate of 40% on default; that if a default occurs, it takes place in the middle of a coupon period; and default probabilities of P1 and P2 for the two periods of the CDS.
For the six-month CDS, value of the premium leg is made up of two parts: the case where there is a default (at three months), plus the case where there isn't a default (payment at six months), appropriately discounted:
The value of the contingent leg is the payment on default multiplied by the probability of default and discounted:
Setting these equal and solving for P1, we see it is about 0.82%.
The equations for the 1Y swap are a little more complicated. On the premium leg we consider the cases of default in period 1, default in period 2 and no default in either period. For the contingent leg we consider default in both the first two periods. The algebra is a little more complex, but solving for P2, we obtain a value of about 1.64%.
The common way of quoting these probabilities is as a survival function S(t), which is the probability of no default. For the six-month period this is:
And for one year:
Once the survival function has been determined from current market spreads, both the fee and contingent legs may be valued using similar principles. For example, if we are valuing a one-year trade on Ford where we traded at 200bp, we use the same equations for each leg but replace the market rate of 150bp with the traded rate of 200bp in the fee leg and use the values of P1 and P2 we deduced above.
Substituting the values of P1 and P2, this gives a present value of approximately 0.47%. Intuitively this number 'feels' about right: if we traded at 200bp and the market is now 150, we would expect the present value to be approximately equal to the discounted value of 50bp, which it is.
Ford Motor Co.
Start date: September 1, 2005 Trade maturity: September 1, 2006
Premium coupon: 200 basis points Premium frequency: six months
CDS: a contract where one party (credit protection buyer) pays the other (credit protection seller) a fixed periodic coupon for the life of the contract on a specified reference asset. The party paying the premium is effectively buying insurance against specific credit events, such as default, bankruptcy or debt restructuring. If such a credit event occurs, the party receiving the premium makes a payment to the protection buyer, and the swap then terminates.
Zero-coupon curve: a set of interest rates for various maturities for deposits paying all interest in a single payment at maturity.