Covering your credit risk

Credit derivatives may send shivers down the spines of some, but really they are nothing to be afraid of, as Markit's Tom McNerney and Penny Davenport explain

A credit default swap (CDS) is a contract where one party (the credit protection buyer) pays the other (the credit protection seller) a fixed periodic coupon for the life of the contract.

The party paying the premium is effectively buying insurance against specific credit events on an agreed reference entity. Credit events are usually defined to include default, bankruptcy or debt restructuring for a specified reference asset. If such a credit event occurs, the party receiving the premium makes a payment to the protection buyer, and the swap terminates.

Once thought of as opaque and complex instruments, single-name CDS are now one of the most vanilla, highly liquid products that the derivatives market has to offer.

The CDS market has matured considerably over the past few years and there is general consensus among market participants about how CDS trades should broadly be valued, using a survival curve methodology typified by the JPMorgan model.

a CDS Survival Curve

As with any swap, the valuation of a CDS involves calculating the present value of the two legs of the transaction, which, in the case of CDSs, 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:

nThe credit curve, which is the term structure of the CDS spread curve ranging across the standard maturity points, that is, 6M, 1Y, 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 15Y, 20Y and 30Y.

nThe recovery-rate assumption - that is, if a default occurs, how much of their money bond investors will recover.

nA discount rate for calculating a present value - typically zero-coupon swap curve for the trade currency.

nThe traded spread of the CDS.

For a fair-market trade, the present value (PV) of the two legs should be equal, so PV of all future fee payments should be exactly equal to the PV of the contingent payment at the time of default. Consider the following example CDS trade:

Given the mark-to-market CDS spreads for Ford Motor Co. as at 15 January 2007 are:

Let's now assume:

nA constant annual interest rate for discounting, say 5%;

nA recovery rate of 40% on default;

nIf a default occurs, it takes place in the middle of a coupon period;

nDefault 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 (accrued premium for three months), plus the case where there isn't a default (payment at six months), appropriately discounted:

P1 x 104bps x ¼ / (1+5%)(¼)+(1-P1) x 104bps x ½ /(1+5%)(½)

The value of the contingent leg is the payment on default multiplied by the probability of default and discounted:

(1-40%)*P1/(1+5%)(¼)

Setting these equal and solving for P1, we see it is about 0.85%

The equations for the one-year default 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. This gives:

P1 x 146bps x ¼ / (1+5%)(¼) (default in period 1)

+(1-P1) x P2 x [146bps x 1⁄4 /(1+5%)(¾) +146bps x 1/2 /(1+5%)(½))(default in period 2)

+(1-P1)x[(1-P2) x 146bps x ½/(1+5%)(¼) + 146bps x ½ /(1+5%))(no default)

For the contingent leg, we consider default in both the first two periods, giving:

(1-40%) x P1/(1+5%)(¼)(default in period 1)

+(1-40%) x (1-P1) x P2 / (1+5%)(¼)(default in period 2)

The algebra is a little more complex, but again we can solve for P2 and we obtain a value of about 1.54%.

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 point, this is just:

S(six months) = 1-P1 = 99.15%

and for the one-year point, it is:

S(one year) = (1-P1)(1-P2) = 97.62%

Valuing a Trade

In terms of trade valuation, 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 280bps, we use the same equations for each leg, but replace the market rate of 146bps with the traded rate of 280bps in the fee leg, and use the values of P1 and P2 we deduced above.

For the fee leg:

P1 x 280bps x ¼/(1+5%)(¼) (default in period 1)

+ (1-P1) x P2 x [280bps x ¼ /(1+5%)(¾)+280bps x ½ /(1+5%)(½))(default in period 2)

+(1-P1) x [(1-P2) x 280bps x ½ /(1+5%)(¼) + bps x ½ /(1+5%)) (no default)

For the contingent:

(1-40%) x P1/(1+5%)(¼)(default in period 1)

+(1-40%) x (1-P1) x P2 / (1+5%)(¼)(default in period 2)

which, substituting the values of P1 and P2, gives a present value of approximately 1.3%. Intuitively, this number 'feels' about right - if we traded at 280bps and the market is now 146, we would expect the PV to be approximately equal to the discounted value of 134 basis points, which it is.