Using current market debt prices, KPMG uses a net present value (NPV) ap- proach to credit risk pricing that evaluates the loan’s structure. That is, the impact of revaluations, embedded options, exercise strategies, covenants, and penalties on credit risk pricing is evaluated using a lattice or “tree”
analysis. The loan’s value is computed for all possible transitions through various states, ranging from credit upgrades and prepayments, to restruc- turings, to default.
Figure 5.2 from KPMG shows, in a simplified fashion,14the potential transitions of the credit rating of a B rated borrower over a four-year loan period using a tree diagram. Given transition probabilities, the original grade B borrower can migrate up or down over the loan’s life to different
FIGURE 5.2 The multiperiod loan migrates over many periods.
0 1 2 3
Time
4 D C B B+ Risk
Grade A
nodes (ratings), and may even migrate to Dor default (an absorbing state).
Along with these migrations, you can build in a pricing grid that reflects the bank’s current policy on spread repricing for borrowers of different quality (or, alternatively, a grid that reflects the spreads that the “market” charges on loans of different quality). Potentially, at least, this methodology can tell the bank whether it has a “good” or “bad” repricing grid in an expected net present value (NPV) sense (basically, whether the expected return on the loan equals the risk-free rate). When valuing a loan in this framework, valu- ation takes place recursively (from right to left in Figure 5.2), as it does when valuing bonds under binomial or multinomial models. For example, if the expected NPVof the loan in its final year is too “high,” and given some prepayment fee, the model can allow prepayment of the loan to take place at the end of period 3. Working backward through the tree from right to left, the total expected NPVof the four-year loan can be determined. More- over, the analyst can make different assumptions about spreads (the pricing grid) at different ratings and prepayment fees to determine the loan’s value.
In addition, other aspects of a loan’s structure, such as caps, amortization schedules, and so on can be built in and a Value at Risk (VAR,see Appen- dix 1.1) can also be calculated.15
Inputs to the LAS include the credit spreads for one-year option-free zero-coupon primary bonds for each of the 18 S&P or Moody’s ratings clas- sifications. Each node (reflecting annual revaluations) incorporates the risk- neutral probability of transition from one risk rating to another. The LAS uses an average of Moody’s and S&P transition probabilities.16 The loan value at each node is then revalued using the market-based credit spread for each rating classification.
Using the market data on bond yields from Figure 5.1 we can illustrate the LAS approach to price a $100 two-year zero-coupon loan. Using an in- ternal rating system, the loan is given a B rating upon its origination. As- suming LGD=100 percent (for a zero recovery rate), we have shown earlier in this chapter that the PDfor B rated corporate debt in the first year is 5 percent and, assuming there was no default in the first year, the PDis 5.34 percent in the second year. However, default is not the only possibility that will affect the loan’s value. For simplicity, we consider only two other possibilities: the loan’s rating will remain at its current B rating or it will be upgraded one full letter grade to an A rating.17In our example, a hypothet- ical ratings transition matrix shows that the probability of an upgrade from B to A (in any period) is 1 percent and the PDis 5 percent (assuming that the beginning period rating was B). Moreover, the probability of a down- grade from A to B is 5.66 percent and the probability of migrating from A to default is 0.34 percent.18Finally, the probability of no change in credit rating is assumed to be 94 percent for all ratings classifications.
Figure 5.3 shows the backward recursion process used by the LAS in order to price the loan. Starting from period 2, the value of the loan is $100 as long as there is no default and $0 recovery in the event of default. Mov- ing back one year to period 1, let us first examine the B rated node. If the loan is B rated in period 1, then there is a 94 percent chance that it will re- tain that rating until period 2, a 1 percent chance that it will be upgraded to an A rating, and a 5 percent chance that it will default at the beginning of period 2. The D rated node (default) is an absorbing state with a value of zero. Using equation (5.2) and the risk-free forward rates obtained from the yield curve in Figure 5.1,19risk-neutral evaluation of the B rated node in pe- riod 1 is as follows:
Similarly, the A rated node in period 1 is valued at:
Moving back one more year to period 0, using the one-year risk-free spot rate of 8 percent p.a., the loan can be valued as:
0 94 100
1 1204 0 0566 100
1 1204 0034 0 88 95
. . .
. . ( ) $ .
+
+ =
0 94 100
1 1204 0 01 100
1 1204 0 05 0 84 79
. . .
. . $ .
+
+
( )
=FIGURE 5.3 Risky debt pricing.
Period 1
Period 0 Period 2
$100 A Rating
$100 B Rating
$88.95
$74.62
$84.79
$0 Default 5%
5%
0.34%
94%
94%
5.66% 1%
94%
1%
Using a two period version of equation (5.2), we can also solve for the loan’s credit spread, denoted CS,defined to be a constant risk premium added to the risk-free rate to reflect the loan’s risk exposure:20
Using the one-year risk-free rate of 8 percent p.a. and the one year forward risk-free rate of 12.04 percent p.a., we obtain a credit spread of CS=5.8 percent p.a.21This credit spread evaluates unexpected losses/gains from rat- ing migration over the life of the loan as well as the probability of default.
The credit spread can be further decomposed into expected and unexpected losses. Expected losses are derived using actual or historical default rates observed in ratings transition matrices. Unexpected losses are derived as the remaining portion of the total credit spread that compensates the lender for the (higher) risk-neutral default probability.22
This simplified example, while providing the flavor of the LAS, ab- stracts from many of its features. For example, in our example, we assumed that the transition matrix was fixed over the two-year life of the loan. In re- ality, transition matrices are themselves volatile and may be related to eco- nomic conditions.23 In particular, during economic upturns, default rates tend to be low and ratings upgrades tend to be high relative to downgrades, whereas the opposite holds true during economic downturns. KPMG de- fines a Z-risk index of migrations that measures how good or bad credit conditions are after controlling for ratings. That is, if Z<0 (Z>0) then default rates are higher (lower) than average and there are more down- grades (upgrades) than upgrades (downgrades). Thus, the LAS credit spreads fluctuate with economic conditions, since credit rating migrations are driven by the systematic Z-risk component as well as the company- specific component.
Following Ginzberg et al. (1994), it can be argued that this extended risk-neutral valuation framework is valid as long as a replicating (no-arbi- trage) portfolio of underlying assets is available. However, it is unclear how such a replicating portfolio could be established in reality when most loans are not traded in active markets. Moreover, if bond spreads include a liq- uidity premium, carrying costs, and factors other than credit spreads, then
74 62 100
1 08 1 1204
. = . .
+ +
(
CS) (
+ +CS)
0 94 84 79
1 08 0 01 88 95
1 08 0 05 0 74 62
. .
. . .
. . $ .
+
+
( )
=the LAS will overestimate credit risk exposure. Kamakura’s Risk Manager models some of these “noise” factors explicitly.