CREDIT SPREADS AND THE IMPLIED PROBABILITY OF DEFAULT
Statements about corporate bond yields inevitably include the “assuming no default” caveat. That's because the yield to maturity indicates the highest rate of return the buy-and-hold investor can expect to obtain. When there is a risk that the issuer might default, a prudent investor should expect to realize a return lower than the yield to maturity. If we were to draw a probability distribution for outcomes on a 10-year corporate zero-coupon bond, it would look something like Figure 2.3.
Obviously, the best outcome is that there is no default. The issuer pays the bond holder the full par value at the maturity date, and the investor's realized rate of return is 5.174% (s.a.), given the assumed purchase price of
FIGURE 2.3 Probability Distribution for Rates of Return on a Corporate Bond to a Buy-and-Hold Investor
60 (percent of par value). The probability of realizing a rate of return higher than 5.174% is zero. By far the most likely outcome is no default. Fortunately for those who actually buy bonds, events of default are not all that common. But when that unfortunate event does occur, a bondholder's rate of return depends on when the default occurs and on any remaining value to the bond. Given all the possible outcomes for the realized rates of return, the mean of the probability distribution – that is, the expected value – will be less than 5.174%.
The key feature of Figure 2.3 is that the probability distribution is highly skewed. That's because the yield to maturity is the maximum rate the hold-to-maturity investor will ever experience. As we've seen, the horizon yield could be higher or lower if the bond is sold prior to maturity, but now we're assuming the intent is to the hold the bond for its full lifetime. Contrast this skewed distribution to a 10-year investment in equity. For it, we would likely draw a normal-looking probability distribution (i.e., a classic bell-shaped curve), perhaps adding in “fatter tails.” Most of the outcomes would lie within a couple of standard deviations of the expected value. There usually is no reason to assume that higher equity returns are more or less probable than lower returns. That is, the distribution of outcomes for rates of return typically is symmetric and not highly skewed, as it is for bonds.
A nice application of bond math is to infer the probability of default given the prices on the risky zero-coupon corporate bond and an otherwise comparable risk-free security. As you might expect, some arbitrary assumptions are needed. Let's assume that a 10-year risk-free zero-coupon government bond is priced at 64 (percent of par value) to yield 4.513% (s.a.).
The credit spread on the 10-year corporate zero priced to yield 5.174% (s.a.) is 66.1 basis points: 5.174% – 4.513% = 0.661%.
The probability-of-default calculation is carried out in Table 2.1. Essentially, we build a table showing the loss if the bond were to default in any given year. We assume the probability that the bond defaults at the end of the year is Q. The third column shows the value of the corporate bond if it were risk free. This column is just the constant-yield price trajectory on the zero- coupon bond assuming the yield is 4.513% (s.a.). The price rises smoothly from 64 at day zero to 100 at maturity. For example, at the end of year 7, when three years remain, the risk-free value of the bond would be 87.469.
TABLE 2.1 Calculating the Probability of Default
Year |
Probability of Default |
Risk-Free Value |
Recovery |
Default Loss |
Present Value of Default Loss |
Expected Loss |
1 |
Q |
66.921 |
26.768 |
40.153 |
38.400 |
38.400*Q |
2 |
Q |
69.975 |
27.990 |
41.985 |
38.400 |
38.400*Q |
3 |
Q |
73.169 |
29.268 |
43.901 |
38.400 |
38.400*Q |
4 |
Q |
76.508 |
30.603 |
45.905 |
38.400 |
38.400*Q |
5 |
Q |
80.000 |
32.000 |
48.000 |
38.400 |
38.400*Q |
6 |
Q |
83.651 |
33.460 |
50.191 |
38.400 |
38.400*Q |
7 |
Q |
87.469 |
34.988 |
52.481 |
38.400 |
38.400*Q |
8 |
Q |
91.461 |
36.584 |
54.877 |
38.400 |
38.400*Q |
9 |
Q |
95.635 |
38.254 |
57.381 |
38.400 |
38.400*Q |
10 |
Q |
100.000 |
40.000 |
60.000 |
38.400 |
38.400*Q |
384.000*Q |
A particularly important and sensitive assumption is the recovery rate, which here is assumed arbitrarily to be 40% of the risk-free value. When events of default do occur, it's rare that bondholders are completely wiped out. Depending on the industry and the rank of the bond in the debt structure (i.e., senior versus junior), recovery rates can range widely, but assumptions of 30% to 50% are common starting places. A related term is “loss severity.” If the recovery is $35 for every $100 in par value, the severity of loss is $65.
The fifth column is the loss at that time if default occurs – it is the riskfree value minus recovery. The present value of the default loss is shown in the sixth column. The risk-free rate is used to discount the losses. For example, if the bond were to default at the end of year 5, the loss would be 48.000. The present value of that loss is 38.400.
Here the present values turn out to be the same for each year – that's because the bond is zero-coupon and the recovery rate is assumed to be a constant share of the loss. When we revisit this calculation in Chapter 3 for coupon bonds, this is not the case.
The final column in Table 2.1 shows the present value of the expected loss for each year – the present value of the default loss times the probability of that loss. For the 10 years, the sum is 384.000 * Q. The investor is compensated for the risk of those default losses by the lower price paid for the corporate bond, 60, compared to the price for a comparable risk-free bond, 64. The implied probability of default comes from equating the risk to the compensation: 384.000 * Q = 64 – 60 = 4, so Q = 0.0104. Therefore, the market is pricing in an annual default probability of 1.04% for this corporate zero. Technically, this is the unconditional probability of default. An alternative approach (which is harder to illustrate but more theoretically correct) is to estimate the probability of default for each year conditional on no prior default.
An important choice in this calculation is the risk-free rate. Academics often use the terms risk-free and Treasury interchangeably, but for many purposes in practical bond math, a government bond yield is probably too low. That is, the 10-year risk-free bond priced at 64 is not necessarily the 10-year Treasury STRIPS. Treasuries usually are more liquid than corporate bonds and have the benefit of being exempt from state and local income taxes. Ideally, the risk-free rate in this analysis is the yield on a bond having the same liquidity and taxation as the corporate but default risk that approaches zero.
An approximation for the implied default probability directly uses the credit spread.
(2.5)
Here the credit spread is 66.1 basis points and the recovery rate is assumed to be 40%. This approximation would be an annual default probability of 1.10%, [(0.00661/(1 – 0.40) = 0.0110], Although the two results are close, the advantage of the tabular method shown in Table 2.1 is its flexibility and explicit use of the time value of money. You can easily vary the recovery rate across the years, for example, if it is assumed that the current fixed assets of the issuer depreciate over time, or if there is an impending legal change that could affect creditors' rights in bankruptcy court. Also, you could introduce a term structure of risk-free rates instead of the flat yield curve assumed in the example.
CONCLUSION
In many ways a zero-coupon bond is the ultimate building block for the study of bond math because there are just two cash flows. Its yield to maturity is calculated with intuitive time-value-of-money bond math. Unlike the money market, there are no arcane conventions such as discount rates or 360-day years. Moreover, we can assume arbitrarily any compounding frequency for the annual yield, from continuous to just once a year. The price and yield calculations are straightforward and easy compared to what is coming in Chapter 3 for coupon bonds.