The essential idea is represented in the transition matrix for a given country, shown in Figure 7.1. Note especially the cell of the matrix in the bottom right-hand corner (pCD). Each cell in the transition matrix shows the proba- bility that a particular counterparty, rated at a given grade at the beginning of the period, will move to another rating by the end of the period. In Fig- ure 7.1, pCD shows the estimated probability that a C-rated borrower (a speculative-grade borrower) will default over the next year, that is, it will move from a C rating to a D (default) rating. The unconditional one-year transition matrix shown in Figure 7.1 is derived as follows: the historic fre- quency of transitions from each initial rating to each other rating divided by the total number of issuers that began the year in the initial rating classifi- cation; that is, pCDis the observed number of issues, averaged over the entire sample period, that started out the year with a C rating and ended up with a D rating one year later divided by the total number of C ratings at the
FIGURE 7.1 Historical (unconditional) transition matrix.
AAA AAA
C D AA
C Beginning
of Period
AA
End of Period
pCD
start of each year. This approach assumes that each rating transition (e.g., pCD) is a constant parameter.
In general, we would expect this probability to move significantly during the business cycle and to be higher in recessions than in expansions.6 Be- cause the probabilities in each row of the transition matrix must sum to 1, an increase in pCDmust be compensated for by a decrease in other probabilities, for example, those involving upgrades of initially C-rated debt, where pCB andpCArepresent the probabilities of the C-rated borrower’s moving to, re- spectively, a B grade and an A grade during the next year. The density, or mass, of the probabilities in the transition matrix moves increasingly in a southeast direction as a recession proceeds.7
With this in mind, let pCDvary at time talong with a set of macro factors indexed by variable y. For convenience, the subscripts (C and D) will be dropped. However, we are implicitly modeling the probability that a C-rated borrower will default over the next period (say, one year). In general terms:8
pt=f(yt) (7.1)
wheref′ <0; that is, there is an inverse link between the state of the econ- omy and the probability of default. The macro indicator variable ytcan be viewed as being driven by a set of i(systematic) macroeconomic variables at time t(Xit) as well as (unsystematic) random shocks or innovations to the economic system (Vt). In general:9
yt=g(Xit, Vt) (7.2)
where i =1,. . . , n andVt∼N(0,σ2).
In turn, macroeconomic variables (Xit) such as gross domestic product (GDP) growth, unemployment, and so on, can themselves be viewed as being determined by their past histories (e.g., lagged GDP growth) as well as being sensitive to shocks themselves (εit).10Thus:
Xit=h(Xit−1, Xit−2, . . . , εit) (7.3)
where i =1,. . . , nandεit∼N(0,σε2).
Different macro model specifications can be used in the context of equa- tions (7.2) and (7.3) to improve model fit, and different models can be used
to explain transitions for different countries and industries. This is dis- cussed in greater detail in Appendix 7.1.
Substituting equation (7.3) into equation (7.2), and equation (7.2) into equation (7.1), the probability of a speculative (grade C) loan moving to grade D during the next year will be determined by:
pt=f(Xit−j;Vt,εit) (7.4) Essentially, equation (7.4) models the determinants of this transition probabil- ity as a function of lagged macro variables, a general economic shock factor or
“innovation” (Vt), and shock factors or innovations for each of the imacro variables (εit). Because the Xit - jare predetermined, the key variables driving pt will be the innovations or shocks Vtand εit.Using a structured Monte Carlo simulation approach, values for Vtandεitcan be generated for periods in the future that occur with the same probability as that observed from history.11 We can use the simulated V’s and ε’s, along with the fitted macro model, to simulate scenario values for pCDin periods t, t+1,t+2,. . . , t+n,and on into the future.
Suppose that, based on current macroeconomic conditions, the simu- lated value for pCD,labeled pt*, is 0.174, and the number in the historic (un- conditional) transition matrix is 0.15 (where * indicates the simulated value of the transition probability). Because the (unconditional) transition value, of 0.15 is less than the value estimated conditional on the macro economic state (0.174), we are likely to underestimate the VARof loans and a loan portfolio—especially at the low-quality end.
Define the migration adjustment ratio (Rt):12
Based on the simulated macro model, the probability of a C-rated borrower’s defaulting over the next year is 16 percent higher than the average (uncondi- tional) historical transition relationship implies. We can also calculate this ratio for periods t+1,t+2, and so on. For example, suppose, based on sim- ulated innovations and macro-factor relationships, the simulation predicts p*t+1to be 0.21. The migration adjustment ratio relevant for the next year (Rt+1) is then:
(7.6)
R p
t p
t t
+ +
+
= = =
1
1 1
21 15 1 4
* .
. .
(7.5) R p
t p
t t
= * =. =
. .
174 15 1 16
Again, the unconditional transition matrix will underestimate the risk of default on low-grade loans in this period. These calculated ratios can be used to adjust the elements in the projected t, t+1,. . . , t+ntransition ma- trices. In CPV–Macro, the unconditional value of pCD is adjusted by the ratio of the conditional value of pCDto its unconditional value. Consider the transition matrix for period t;then R×0.15=0.174 (which is the same as pt*). Thus, we replace 0.15 with 0.174 in the transition matrix (Mt), as shown in Figure 7.2. This also means that we need to adjust all the other el- ements in the transition matrix (e.g., pCA, pCB,and so on). A number of pro- cedures can be used to do this, including linear and nonlinear regressions of each element or cell in the transition matrix on the ratio Rtand the use of a diffusion parameter, λ[see Wilson (1997a, b) and Appendix 7.1; remember that the rows of the transition matrix must sum to one13]. For the next pe- riod (t+1), the transition matrix would have to be similarly adjusted by multiplying the unconditional value of pby Rt+1, or .15 ×1.4=.21.
Thus, there would be different transition matrices for each year into the future (t, t+1, . . . , t+n), reflecting the simulated effect of the macroeco- nomic shocks on transition probabilities. We could use this type of ap- proach, along with CreditMetrics, to calculate a cyclically sensitive VARfor one year, two years, . . . nyears.14Specifically, the simulated transition ma- trix Mt,would replace the historically based unconditional (stable Markov) transition matrix, and, given any current rating for the loan (say, C), the dis- tribution of loan values based on the macro-adjusted transition probabili- ties in the C row of the matrix Mt could be used to calculate VAR at the
FIGURE 7.2 Conditional transition matrix (Mt).
AAA AAA
C D
AA
C
M t =
AA
.174 . . .
. . .
one-year horizon, in a fashion similar to that used under CreditMetrics in Chapter 6.
We could also calculate VARestimates using longer horizons. Suppose we are interested in transitions over the next two years (tandt+1). Multi- plying the two matrixes,
Mt, t+1=Mt×Mt+1 (7.7) produces a new matrix, Mt,t+1. The final column of this new matrix will give the simulated (cumulative) probabilities of default on loans of all rat- ings over the next two years.
We have considered just one simulation of values for pt* from one set of shocks (Vt,εit). Repeating the exercise over and over again (e.g., taking 10,000 random draws) would produce 10,000 values of pt*and 10,000 pos- sible transition matrices.
Consider the current year (t). We can plot hypothetical simulated val- ues for pt*,as shown in Figure 7.3. The mean simulated value of pt* is .174, but the extreme value (99th percentile, or worst-case value) is .45. When
FIGURE 7.3 Probability distribution of simulated values of pt* in year t.
.174 .45 pt*
Probability
%
Expected Value
99th Percentile (Maximum) Value
calculating capital requirements—that is, when considering unexpected de- clines in loan values—the latter figure for pt*,and the transition matrix as- sociated with this value, might be considered most relevant.