###### Implementation Shortfall

The implementation shortfall approach involves taking the difference between the prevailing price and... **Read More**

Credit rating agencies come up with transition matrixes of credit ratings based on the historical experience of issuers. A * transition matrix* captures the probability that a certain obligor will transition (migrate) from one credit state (rating) to another over a given time period, usually a year.

The table below presents an example of a rating transition matrix according to S&P’s rating categories:

$$ \textbf{One-year transition matrix}$$

$$ \small{\begin{array}{l|cccccccc} \textbf{Initial}& {} & \textbf{Rating} & \textbf{at} & \textbf{Year} & \textbf{End} & {} & {} & {} \\ \textbf{Rating} & \textbf{AAA} & \textbf{AA} & \textbf{A} & \textbf{BBB} & \textbf{BB} & \textbf{B} & \textbf{CCC} & \textbf{Default}\\ \hline \text{AAA} & {90.81\%} & {8.33\%} & {0.68\%} & {0.06\%} & {0.12\%} & {0.00\%} & {0.00\%} & {0.00\%} \\ \text{AA} & {0.70\%} & {90.65\%} & {7.79\%} & {0.64\%} & {0.06\%} & {0.14\%} & {0.02\%} & {0.00\%} \\ \text{A} & {0.09\%} & {2.27\%} & {91.05\%} & {5.52\%} & {0.74\%} & {0.26\%} & {0.01\%} & {0.06\%} \\ \text{BBB} & {0.02\%} & {0.33\%} & {5.95\%} & {86.93\%} & {5.30\%} & {1.17\%} & {0.12\%} & {0.18\%} \\ \text{BB} & {0.03\%} & {0.14\%} & {0.67\%} & {7.73\%} & {80.53\%} & {8.84\%} & {1.00\%} & {1.06\%} \\ \text{B} & {0.00\%} & {0.11\%} & {0.24\%} & {0.43\%} & {6.48\%} & {83.46\%} & {4.07\%} & {5.20\%} \\ \text{CCC} & {0.22\%} & {0.00\%} & {0.22\%} & {1.30\%} & {2.38\%} & {11.24\%} & {64.86\%} & {19.79\%} \\ \end{array}}$$

The expected percentage price change due to credit migration can be calculated as:

$$ {\Delta}\%P =\ – (\text{Modified duration of the bond}) \times (\Delta \text{Credit spread}) $$

The expected return will then be calculated as:

$$ \text{Expected return}=\text{Yield to maturity (YTM)}+{\Delta}\%P $$

An A-rated corporate bond with a yield to maturity of 4% will have a modified duration of 6.5 at the end of the year. You have been given the following partial corporate transition matrix:

$$ \begin{align*} & {\textbf{ One-Year Transition Matrix for}} \\ & \textbf{A-Rated Bonds and Credit Spreads} \end{align*} \\ \begin{array}{c|c|c|c|c|c|c|c} & \bf{AAA} & \bf{AA} & \bf A & \bf{BBB} & \bf{BB} & \bf B & \bf{CCC,} \\ & & & & & & & \bf{CC,C} \\ \hline {\text{Probability}} & 0.018 & 0.263 & 75.010 & 16.704 & 6.081 & 1.531 & 0.394 \\ {(\%)} & & & & & & \\ \hline \text{Credit} & 0.60\% & 0.90\% & 1.10\% & 1.50\% & 3.40\% & 6.50\% & 9.50\% \\ \text{Spread} & & & & & & \end{array} $$

The expected return of the bond over the next year is *closest to*:

**Solution**

The bond’s expected return over the next year is calculated as its yield to maturity, plus the expected percentage price change in the bond over the same year.

$$ \begin{align*} {\text{Expected }} & {\text{percentage price change} } {\left({\Delta}\%P\right)} \\ & =\ –(\text{Modified duration of the bond} \times {\Delta \text{Credit spread}} \end{align*} $$

$$ \begin{array}{c|c|c|c} & \bf{\text{Expected} \%} & \textbf{Probability} & \bf{\text {Expected } \%} \\ & \textbf{Price} & & \bf{\text{Price Change} \times} \\ & \textbf{Change} & & \textbf{Probability}\\ \hline \text{From A to AAA} & 3.25\% & 0.018 & 0.0006\% \\ \hline \text{From A to AA} & 1.30\% & 0.263 & 0.0034\% \\ \hline \text{From A to A} & 0.00\% & 75.010 & 0.0000\% \\ \hline \text{From A to BBB} & -2.60\% & 16.704 & -0.4343\% \\ \hline \text{From A to BB} & -14.95\% & 6.081 & -0.9091\% \\ \hline \text{From A to B} & -35.10\% & 1.531 & -0.5375\% \\ \hline \text{From A to CCC,CC,C} & -54.60\% & 0.394 & -0.2150\% \end{array} $$

$$ \text{Total expected percentage price change} =-2.0919\% $$

$$ \begin{align*} \text{Expected return} & = \text{Yield to maturity (YTM)}+ {\Delta}\%P \\ & = 4.0\% – 2.0919\% = 1.91\% \end{align*} $$

## Question

The following is a partial one-year corporate bonds transition matrix:

$$ \begin{array}{c|c|c|c} \text{From/To} & \bf{AAA} & \bf{AA} & \bf A \\ \hline AAA & 90.00 & 8.00 & 2.00 \\ \hline AA & 2.00 & 87.00 & 14.00 \\ \hline A & 0.60 & 2.40 & 83.00 \\ \hline \text{Credit Spread} & 1.00\% & 1.30\% & 1.50\% \end{array} $$

Consider an AAA-rated corporate bond with 5% coupon payments that matures in six years. The bond has a modified duration of 5. If this bond is downgraded to an A rating over the next year, its expected price change will be

closest to:

- -0.05%.
- -1.30%.
- -2.50%.

Solution

The correct answer is C.The expected percentage price change is the product of the negative of the modified duration and the difference between the credit spread in the new rating and the old rating:

$$ {\Delta}\%P = –5 \times (0.0150 – 0.01) = –0.0250, \text{ or } –2.50\% $$

Reading 31: Credit Analysis Models

*LOS 31 (c) Calculate the expected return on a bond given transition in its credit rating.*