There are three components necessary to obtain the conditional migration matrix:
1. Explanatory processes. Macroeconomic variables must be identified to simulate future values of macroeconomic states (i.e., systematic risk fac- tors).
2. Speculative default rate processes. The stochastic relationship between the macroeconomic explanatory variables and default rates for specula- tive rating grades must be estimated. Different relationships are esti- mated for each industry segment.
3. Shift factors to transform unconditional into conditional migration ma- trices. Assesses the impact of changes in each segment’s conditional speculative default rate on the one-year unconditional rating migration matrix.
Explanatory Processes
CPV-Macro uses fundamental macroeconomic variables to describe the evo- lution of the macroeconomy. Although varying from country to country, some examples of explanatory variables are: unemployment rates, GDP growth, long-term interest rates, foreign exchange rates, public disburse- ments, and aggregate savings rates. Wilson (1997b) suggests that at least three different macroeconomic factors are required to capture the systematic
variation in speculative default rates for each country.19Since different in- dustry segments have different sensitivities to these macroeconomic factors, CPV-Macro estimates the model for each industry segment in each country individually. Throughout this appendix, it is assumed (and therefore sub- script notation is suppressed for ease of exposition) that we are focused on a particular industry segment, j; in a particular country, c;for a particular initial rating class, Z;at time period, t.The analysis is then replicated for each industry segment, for all possible rating classes, at different time peri- ods across different countries.
The estimation of the explanatory process captures the momentum (cyclical dynamics) of each macroeconomic variable. The momentum is mea- sured by the coefficients on the lagged macroecomic variables and the error terms, ki,pandki,qfrom a set of univariate, auto-regressive, moving average processes, ARMA(p,q), for each macroeconomic variable, i,as follows:20
Xi,t=ki,0+ Σp=1 . . . P(ki,pXi, t−p)+ Σq=1 . . . Q(ki,qεi,t−q) (7.8) whereki,pandki,qare the moving average constants to be estimated for each macroeconomic variable i,andεi,t - qare the moving average error terms, as- sumed to be independent and identically distributed as N(0,σ2i). This esti- mation is performed separately for each macroeconomic variable (e.g., GDP innovations, unemployment rates, interest rates), thereby yielding i sets of cyclical momentum estimates, ki,pandki,q.21
Speculative Default Rate Processes
Macroeconomic conditions have the most impact on the default probabili- ties of bonds rated in the speculative grades. Thus, CPV-Macro estimates a speculative grade default probability, denoted PD,conditional on the cycli- cal macroeconomic risk factors estimated in the previous section. Suppress- ing all subscripts for simplicity, the following estimation is performed for each industry segment jat time period t:
where PDis the default probability for a speculative grade issue from in- dustry segment jat time t,22andyis an explanatory index variable that is es- timated using the N different macroeconomic factors Xi estimated in equation (7.8), constructed for each segment j at time t (subscripts sup- pressed) as follows:
(7.9) PD
e y
=
[
1+1−]
y= β0+ Σi=1 . . . NβiXi+V (7.10) where the βi coefficients are estimated specifically for each industry seg- ment jat each point in time t.Thus, the βicoefficients reflect the impact of the cyclical dynamics estimated in equation (7.8) on the probability of de- fault for speculative grades of debt. The error term Vrepresents the unsys- tematic risk component remaining after the systematic risk component is captured by the (sector-weighted) influence of the macroeconomic variables from equation (7.8). The error term Vcan be interpreted as a segment-spe- cific surprise that is similar in function to the jump process assumed in in- tensity-based models; that is, Vreflects sudden shifts in default probabilities that are not a function of domestic macroeconomic conditions. (However, V could include the effect of an external macroeconomic shock; e.g., an Asian crisis on Germany.)
The functional form of equation (7.9) was chosen because it offered, on average, a better fit to the historical data, as measured by R-squared, and because for any value of the index y,equation (7.9) yields a PDbetween 0 and 1.
Once the relationships in equations (7.8 through 7.10) are estimated, the implementation of the model proceeds as follows:
Step 1. Simulate the future value of each macroeconomic explanatory variables, Xi,using equation (7.8). These will largely reflect the shocks in the Xisince the lagged variables of Xiare predetermined. Indeed, the εitandVwill drive y. Step 2. Use the simulated values of all of the macroeconomic variables Xialong with Vto construct the index y[equation (7.10)] for each in- dustry sector. Step 3. Estimate the conditional default probabilities for speculative grade debt in each industry segment using equation (7.9). The simu- lated PDis normalized so that the mean of the first year’s PDacross all simulation runs is equal to its historical average. This assures that the mean of the first-year conditional migration matrix is equal to its un- conditional expected value; that is, the migration adjustment ratio R (defined in the text of this chapter) is centered around 1. Step 4. Define the ratio of the simulated value of PDfor each industry sector from equation (7.9) divided by the long run average default probability (calculated over the historical data series), denoted ; is used in the next section to construct a left- and right-shift operator.Intuitively, if >1, then the simulated speculative default probability for the segment is greater than the long-run average and the right shift operator increases the probability of being downgraded; alternatively,
PD
PD PD
if <1, then the simulated speculative default probability for the seg- ment is less than the long-run average and downgrades are less likely.
Shift Factors to Transform Unconditional into Conditional Migration Matrices
To transform the unconditional migration matrix (obtained from historical migration probabilities) into a migration matrix that is conditional on macro- economic conditions, the model estimates both a systematic risk sensitivity parameter, denoted λ, and an unsystematic risk sensitivity parameter θ. The Systematic Risk Sensitivity Parameter, λ To understand how systematic risk is diffused through the conditional migration matrix, let us elaborate on the example in the text. We consider an unconditional default probabil- ity for C rated debt, pCD,equal to 0.15. Let us fill in the last line of the un- conditional transition matrix as shown in Table 7.1.23Beginning with pCD, we derive the migration adjustment ratio, R,as a function of simulated de- fault probabilities for speculative grade debt. In Step 4 above we defined to be the ratio of the simulated default probability PD, estimated from equation (7.9), to the historic average speculative default probability. CPV- Macro posits a relationship between downgrade/upgrade probabilities and such that if >1, the probability of downgrades increases and if <1 then more of the mass of the transition matrix is shifted into up- grades. This can be estimated as:
whereDt(Ut) is the actual, historical single rating downgrade (upgrade) per- centages at time period tand, ( ) are the average downgrade (upgrade) rates over the entire period. Using historic data to consecutively re-estimate the empirical relationships in equation (7.11), we can obtain measures for one rating classification downgrade (upgrade) α1(β1), and two rating classi- fications downgrade (upgrade) α2(β2), and so on. However, because historic data includes downgrades to the absorbing state of default, the historic relationship introduces an upward drift in expected defaults for all rating classes. Therefore, CPV-Macro constrains each rating downgrade factor to be equal to the upgrade factor for the equivalent number of ratings
U D
(7.11) D
D
PD U
U
PD
t
t
= +
( )
+= +
( )
+1 1
α α
β β PD
PD PD
PD PD
classifications, such that αi= βiand sets them equal to λi,which is defined as the systematic risk sensitivity parameter for ishifts in ratings classifications.
To illustrate the transformation of the unconditional matrix into a con- ditional transition matrix, we use the example shown in Table 7.1. Suppose that estimation of equation (7.11) yields the following estimates of the sys- tematic risk sensitivity parameter λ0=1.18, λ1=0.4, λ2=4.24 Beginning with the bottom right entry in the unconditional transition matrix, pCD= 0.15, we define the (discrete) transition ratio for a one rating shift from C to D to be:
R=1+ λ1τ (7.12)
whereλ1is the risk sensitivity parameter estimated from equation (7.11) for a one-rating classification shift and τis defined to be −1 for >1 and−( −1) for <1; therefore, τ ≥0. Suppose that the estimation of equation (7.9) yields =1.4, then τ =0.4. Suppose further that we es- timated the systematic risk sensitivity parameter for one rating transition from equation (7.11) to be λ1=0.4. Therefore, solving equation (7.12), R= 1.16. Thus, the conditional value of p*CD=rpCD=1.16(.15)=0.174. This is shown in the bottom right hand entry of the conditional transition matrix shown in Table 7.2.
Note that the ∆pCD=0.174−0.15=.024. To see how the diffusion term (the shift operator) is obtained, note that the shift in transition proba- bilities must be diffused throughout the row so that the sum of all probabil- ities still equals one. We use the systematic risk sensitivity parameter λ in
PD PD PD
PD PD
TABLE 7.1 Unconditional Transition Matrix
A B C D
A . . . . . . . . . . . .
B . . . . . . . . . . . .
C 0.01 0.04 0.80 0.15
TABLE 7.2 Conditional Transition Matrix
A B C D
A B
C 0.0124 0.034 0.7796 0.174
order to define the diffusion term for the last row of the conditional transi- tion matrix shown in Table 7.2 as follows:
That is, the shift operator is defined to be the risk sensitivity parameter times the difference between the change in the transition probability in the next higher class minus the change in the transition probability in the next lower class. Equation (7.13) is a system of three equations with three un- knowns which, in our example, can be solved for: ∆pCC= −0.0204,∆pCB=
−0.006, and ∆pCA=0.0024 to obtain the last row in the conditional transi- tion matrix shown in Table 7.2.25This is repeated for each row of the un- conditional transition matrix.
The Unsystematic Risk Sensitivity Parameter, θ Transition probabilities for low grade and speculative grade debt closely follow the cyclical dynamics esti- mated in the previous section. However, high credit quality debt tends to be less sensitive to cyclical movements. Thus, reliance on the systematic risk sensitivity parameter alone will underestimate the default probabilities for highly rated debt classifications.26 CPV-Macro allows users to input a
“spontaneous combustion” unsystematic risk parameter for highly rated obligors. Rather than a gradual shift in the probability mass across the en- tire row, the unsystematic risk sensitivity parameter, θ, is applied directly to the probability of default entry. Thus, the probability of default for invest- ment grade ratings would be increased by a discrete value. This would be netted out in the diffusion of the conditional transition matrix following the procedure outlined in the previous section.
The conditional transition matrix in our example shown in Table 7.2 is obtained for one simulated value of which corresponds to one set of macroeconomic conditions, Xi.To obtain the loss distribution, this process must be simulated using Monte Carlo simulation techniques for many dif- ferent possible future states of the world. CPV-Macro recommends the use of between 250 and 5,000 macroeconomic scenarios in order to ensure that the results are robust.
PD
(7.13)
∆ ∆ ∆
∆ ∆ ∆
∆ ∆
p p p
p p p
p p
CC CB CD
CB CA CC
CA CB
= −
= −
=
λ λ
λ λ
λ
1 1
2 0
1
121