Valuation of CDS

The protection buyer periodically pays the CDS spread, while the protection seller assumes the obligation to compensate for losses if the organization defaults.

However, this raises the main question:

How much should this protection cost?

If the CDS spread is too low, the protection seller will not receive sufficient compensation for assuming the risk. If the spread is too high, purchasing the protection will become unattractive to the other party.

Therefore, to determine the fair price of a CDS, the contract is divided into two parts:

• Premium Leg — what the protection buyer pays;
• Protection Leg — what the protection seller pays in the event of default.

When a CDS contract is entered into, the present, or discounted, values of these two sides must be equal:

Premium Leg

The protection buyer periodically pays the CDS spread until either:

• the contract reaches maturity; or
• the reference company defaults.

If the company defaults before the maturity date, the remaining regular payments stop.

Therefore, when valuing CDS premiums, it is not sufficient simply to multiply the annual payment by the number of years in the contract.

For each future premium payment, we must consider:

• the probability that the company will not default before the payment date;
• the present value of that future payment.

Survival Probability means the probability that the company will not default by the end of the relevant period.

For example, if the annual conditional default probability is 2%:

The probability of surviving until the end of the first year is: 100%-2%=98%

The probability of surviving until the end of the second year is: 0.98^2=96.04%

The probability of surviving until the end of the third year is: 0.98^3=94.12%

The expected present value of each regular premium payment is calculated as follows:

In other words, the premium is paid only if the company survives until the relevant payment date.

Accrued Premium — What Happens If Default Occurs Between Two Payment Dates?

CDS premiums are often paid quarterly or semiannually.

If the company defaults between two payment dates, the protection buyer must still pay the portion of the premium that accrued up to the date of default.

For example, if the premium is paid annually and default occurs halfway through the year, the protection buyer will pay the premium accrued for half of the year.

Therefore, the Premium Leg consists of two components:

In Hull’s simplified example, default is assumed to occur halfway through each year. Therefore, the accrual fraction is: 0.5

Accordingly, the accrued premium factor for each year is calculated as follows:

Ultimately, we obtain the following table for the Premium Leg.

Protection Leg

The Protection Leg represents the amount that the protection seller must pay in the event of default.

If the CDS notional is (L), and the recovery rate in the event of default is (R), the payment upon default will be: L*(1-R)

For example, suppose:

• the CDS notional is USD 100 million;
• the recovery rate is 40%.

In that case, the unrecovered portion in the event of default is USD 60 million.

However, this USD 60 million will be paid only if default actually occurs.

Therefore, to calculate the Protection Leg for each year, we must take the annual default probabilities into account.

Ultimately, we obtain the following table.

In simple terms, on one side we have the expected discounted value of the premium payments, and on the other side we have the expected discounted value of the loss in the event of default.

For a new CDS, these two values must be equal:

The spread is the rate (s) that makes the two sides equal:

Therefore:

Risk-Neutral Default Probability

The Premium Leg and the Protection Leg of a CDS are discounted using the risk-free rate.

Therefore, the default and survival probabilities must be consistent with the same valuation framework.

It would be incorrect to combine real-world default probabilities with a risk-neutral valuation model, because the compensation required by the market for bearing credit risk would then be omitted from the valuation.

Real-world probabilities can also be used, but this would require an appropriate risk-adjusted discount rate or stochastic discount factor.

Estimating this adjustment is difficult. Therefore, the first approach is generally used for market valuation of CDS contracts.

From the CDS Spread to the Default Probability

Sometimes we do not know the default probability, but we do know the market CDS spread.

In that case, we can perform the calculation in reverse.

A simple approximation between the CDS spread and the hazard rate is:

where:

• (s) is the CDS spread;
• (\lambda_Q) is the risk-neutral hazard rate;
• (R) is the recovery rate.

Using this formula, if we know (s) and (R), we can calculate (\lambda_Q), or the implied conditional default probability.

This means:

We determine the approximate default probability required in a risk-neutral world to explain the CDS spread observed in the market.

This approach is similar to calculating implied volatility in the options market:

• the option price is used to derive implied volatility;
• the CDS spread is used to derive the implied default probability.

The Excel file also contains a mechanism for performing this calculation.

The Value of a CDS Also Changes After the Contract Is Entered Into

Like other swaps, a CDS is valued continuously, and its mark-to-market value can become either positive or negative.

Suppose the spread on an existing CDS contract is: 1.50%

However, the current fair spread has declined to: 1.24%

The protection seller under the existing contract continues to receive 1.50%, while selling equivalent new protection would generate only 1.24%.

Therefore, the existing contract has value to the protection seller.

For a notional amount of USD 100 million, this creates a positive mark-to-market value for the seller.

Naturally, this also creates opportunities for speculation.

Excel File — CDS Valuation

Source:

John C. Hull, Options, Futures, and Other Derivatives, Chapter 24 — Credit Derivatives.