Question 14.6: A firm has a bond in issue that has a Standard and Poor’s cr...
A firm has a bond in issue that has a Standard and Poor’s credit rating of BBB. The projected values of the bond, allowing for changes in gross redemption yield and, when relevant, default and recovery, are given below for each credit rating. Using a credit migration approach based on Standard and Poor’s migration rates from 1981 to 2009, what are the expected value and variance of the value of the bond?
| Year-end rating | Value given rating |
| AAA | 104.27 |
| AA | 103.18 |
| A | 102.10 |
| BBB | 100.00 |
| BB | 94.98 |
| B | 90.29 |
| CCC-C | 81.78 |
| Default | 61.97 |
| Unrated | – |
Using the process outline above, an additional column is needed giving the probability of migration to the various credit ratings. A second column is then added giving the probabilities adjusted for rating withdrawals. This column is then multiplied by the value given the credit rating to arrive at a probability weighted value. The sum of these values gives the mean, which can then be deducted from the values given the credit ratings to arrive at the difference from the mean. Each of these values is then squared and multiplied by the probability of occurrence. The sum of these results gives the variance of returns.
| Year-end rating | Value given rating | Pr of rating (%) | Adjusted Pr of rating (%) | Pr-weighted value | Value less mean | Pr-weighted squared value |
| AAA | 104.27 | 0.01 | 0.01 | 0.01 | 4.61 | 0.0021 |
| AA | 103.18 | 0.14 | 0.15 | 0.15 | 3.52 | 0.0173 |
| A | 102.10 | 3.76 | 4.03 | 4.11 | 2.45 | 0.2250 |
| BBB | 100.00 | 84.16 | 90.18 | 90.18 | 0.34 | 0.0985 |
| BB | 94.98 | 4.13 | 4.43 | 4.2 | -4.68 | 0.9033 |
| B | 90.29 | 0.70 | 0.75 | 0.68 | -9.37 | 0.6146 |
| CCC-C | 81.78 | 0.16 | 1.17 | 0.14 | -17.87 | 0.5111 |
| Default | 61.97 | 0.26 | 0.28 | 0.17 | -37.69 | 3.6931 |
| Unrated | – | 6.68 | – | – | – | – |
| Total | 99.66 | 6.0651 |
Therefore the expected value of the bond is 99.66 with a variance of 6.0651.