EXAMPLE 4.24 Credit Ratings
Corporate bonds are assigned a credit rating that provides investors with a guide of the general creditworthiness of a corporation as a whole. The most well-known credit rating agencies are Moody's, Standard & Poor's, and Fitch. These rating agencies assign a letter grade to the bond issuer. For example, Fitch uses the letter classifications of AAA, AA, A, BBB, BB, B, CCC, and D. Over time, the credit ratings of the corporation can change. Credit rating specialists use the terms of “credit migration” or “transition rate” to indicate the probability of a corporation going from letter grade to letter grade over some particular span of time. For example, based on a large amount of data from 1990 to 2013, Fitch estimates that the five-year transition rates to be graded AA in the fifth year based on each of the current (“first year”) grades to be:20
| Current rating | AA (in 5th year) |
|---|---|
| AAA | 0.2283 |
| AA | 0.6241 |
| A | 0.0740 |
| BBB | 0.0071 |
| BB | 0.0012 |
| B | 0.0000 |
| CCC | 0.0000 |
| D | 0.0000 |
Recognize that these values represent conditional probabilities. For example, . In the financial institution sector, the distribution of grades for year 2013 are
| Rating | AAA | AA | A | BBB | BB | B | CCC | D |
| Proportion | 0.010 | 0.066 | 0.328 | 0.358 | 0.127 | 0.106 | 0.004 | 0.001 |
The transition rates give us probabilities rating changes moving forward. An interesting question is where might a corporation have come from looking back retrospectively. Imagine yourself now in year 2018, and you randomly pick a financial institution that has a AA rating. What is the probability that institution had a AA rating in year 2013? A knee jerk reaction might be to answer 0.6241; however, that would be incorrect. Define these events:
We are seeking while the transition table gives us . From the distribution of grades for 2013, we have . Because grades are disjoint and their probabilities add to 1, we can employ Bayes's rule. It will be convenient to present the calculations of the terms in Bayes's rule as a table.
| 2013 grade | |||
|---|---|---|---|
| AAA | 0.010 | 0.2283 | |
| AA | 0.066 | 0.6241 | |
| A | 0.328 | 0.0740 | |
| BBB | 0.358 | 0.0071 | |
| BB | 0.127 | 0.0012 | |
| B | 0.106 | 0.0000 | |
| CCC | 0.004 | 0.0000 | |
| D | 0.001 | 0.0000 |
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Here is the computation of the desired probability using Bayes's rule along with the preceding computed values:
The probability is 0.5848, not 0.6241, that a corporation rated AA in 2018 was rated AA five years earlier in 2013. This example demonstrates the important general caution that we must not confuse with .