Credit Risk

Credit risk is the likelihood that the borrower will default. Rating agencies like Standard & Poor's, Fitch, and Moody's all rate bonds, and these ratings (AAA, AA, BBB, and so on) signal the underlying credit risk of the issuer. Credit risks can be quantified by looking at spreads over Treasuries (which are considered to be risk-free). So, for example, if the investment grade corporate bond yield is 7 percent and the 10-year Treasury yield is 3 percent, then the spread of 4 percent can be thought of as the expected loss on the corporate bond. A simple model of default is therefore the credit spread over the risk-free yield on Treasuries.

The holder of the corporate bond is compensated for the expected loss and the 400 basis points is the risk premium. If we consider, too, that with default comes less than full recovery, then a more precise estimate of default likelihood is the spread adjusted for the expected recovery. For example, suppose the recovery rate is 50 percent where, upon default, the bondholder expects to recover half of the bond's value when the assets of the firm are liquidated. Then the expected credit risk is equal to the spread divided by (1 – R), where R is the recovery rate. In this case, the expected default rate is 8 percent per year.

We can estimate more accurate default probabilities. Using the spreadsheet developed in Chapter 2, suppose we have a five-year corporate bond paying a 6 percent coupon with a yield to maturity of 7 percent and that the yield on the five-year Treasury (also paying a 6 percent coupon) is 5 percent. The corporate bond is therefore priced at 95.84, while the Treasury is priced at 104.38. The difference is $8.54, and this must be the expected loss on the corporate bond. With continuous compounding, this value would be slightly higher than $8.75. The expected probability of default would appear to be 8.54/95.84 or about 8.9 percent (9.1 percent in the continuously compounded case).

This is a useful approximation to expected default loss but it doesn't provide a guide to default probabilities over the life of the bond. To get at default probabilities, we need to estimate the present value of expected losses over the life of the bond, and using this number, solve for the probability of default. Let's look at this by way of an example provided in Hull's text (2008). Suppose we are analyzing the same five-year corporate bond with the expected loss to default equal to $8.75. On the credit risk spreadsheet for this chapter, I show the cash flows for this bond discounted forward at a 5 percent risk-free rate. The risk-free value is therefore the value of the bond paying the corporate coupon discounted using the risk-free yield to maturity. We assume that losses occur on the half-year directly preceding coupon payment (these are the green shaded cells in Table 11.1).

Table 11.1 Estimating Default Probability.

The risk-free value at any given time is the present value of the current coupon of $3 plus the future cash flows on that bond until maturity. We assume a recovery rate of 40 percent in the table and sum the PV of expected losses to $288.48 over the life of the bond. If the expected default loss is $8.75, then this must equal the probability weighted value of the total expected loss. For example, $8.75 = ρ $288.48 where ρ, the default probability, is solved to be 3.03 percent. This is the annual default hazard for this bond. (See Chapter 13 for a deeper discussion of hazard rates.)

This can be improved upon further. Consider three corporate bonds—a three-, five-, and seven-year bond and their Treasury equivalents. We use the three-year bond to derive a default probability for the first three years. Then we use the five-year bond to estimate the default probability for years four and five and the seven-year bond for default probabilities in years six and seven. Doing this gives us a more realistic and complete picture of default risk over time. In principle, given available bond data, we could bootstrap out the term structure of default risks.