C The Diebold-Gunther-Tay method

Diebold, Gunther, and Tay (1998) propose a way to evaluating density forecasts. The basic idea is that since under the null hypothesis the forecasts are equal to the true densities (conditioned on past information), applying the cumulative distribution function (the probability integral transform or PIT) to the series of observations should yield a series of iid uniform-[0, 1] variables. Whether the transformed variables are iid uniform can be checked in various ways. Diebold, Gunther, and Tay (1998) suggest plotting histograms and autocorrelation functions to visualize the quality of the density forecasts.

In order to apply the PIT to our predicted recovery rate densities, we create a vector y† in which we stack all recovery rate observations. For each element in the vector, we can now create a conditional density forecast from our estimated model.

Applying the cumulative distribution function associated with these density forecasts to the vector y† yields a vector of transformed variables. Under the null hypothesis that the density forecasts are correct, the elements of the vector of transformed variables should be an iid uniform series. Serial correlation of the series would indicate that we have not correctly conditioned on the relevant information. A departure from uniformity would indicate that the marginal distributions are inappropriate.

D Supplementary tables


Table 1: Recovery Rate Statistics by Year

This table reports some annual statistics for the data used in the paper. First column figures are issuer-weighted default rates of US bond issuers provided by Moody’s. The other three columns are the number of default events, and the mean and standard deviation for recovery rates in the Altman data.

 Year Default  rate  Number of  observations  Mean  Recovery  Standard  Deviation
 1981 0.17% 1 12.00 -
 1982 1.08% 12 39.51 14.90
 1983 1.02% 5 48.93 23.53
 1984 0.98% 11 48.81 17.38
 1985 1.01% 16 45.41 21.87
 1986 2.07% 24 36.09 18.82
 1987 1.65% 20 53.36 26.94
 1988 1.52% 30 36.57 17.97
 1989 2.43% 41 43.46 28.78
 1990 4.14% 76 25.24 22.28
 1991 3.55% 95 40.05 26.09
 1992 1.85% 35 54.45 23.38
 1993 1.13% 21 37.54 20.11
 1994 0.80% 14 45.54 20.46
 1995 1.25% 25 42.90 25.25
 1996 0.77% 19 41.90 24.68
 1997 0.89% 25 53.46 25.53
 1998 1.60% 34 41.10 24.56
 1999 2.61% 102 28.99 20.40
 2000 3.43% 120 27.51 23.36
 2001 4.98% 157 23.34 17.87
 2002 3.33% 112 30.03 17.18
 2003 2.36% 57 37.33 23.98
 2004 1.28% 39 47.81 24.10
 2005 1.12% 33 58.63 23.46
 


Table 2: Recovery Rates by Seniority and Industry

Panel A: Number of observations and the mean and standard deviation of recovery rates in our sample classified by seniority, for the whole sample (all default events), for default events for which we only observe recovery on a single instrument (with only one seniority), and for default events for which we observe recoveries on at least two different seniorities.
Panel B: Number of observations and the mean and standard deviation of recovery rates in our sample classified by industry.

Panel A: Recovery Rates by Seniority
 Seniority  Number of  observations  Mean  Recovery  Standard  Deviation
All default events
 Senior Secured 203 42.08 25.48
 Senior Unsecured 366 36.88 23.29
 Senior Subordinated 326 32.90 23.77
 Subordinated 154 34.51 23.05
 Discount 75 21.29 18.48
Default events with single recovery
 Senior Secured 145 39.29 23.35
 Senior Unsecured 239 36.45 22.45
 Senior Subordinated 209 34.52 23.30
 Subordinated 87 37.86 20.22
 Discount 29 21.72 19.67
Default events with multiple recoveries
 Senior Secured 58 49.04 29.25
 Senior Unsecured 127 37.68 24.87
 Senior Subordinated 117 30.00 24.42
 Subordinated 67 30.16 25.79
 Discount 46 21.03 17.91
Panel B: Recovery Rates by Industry
 Industry
 Building 10 33.56 36.24
 Consumer 149 35.66 22.21
 Energy 47 36.47 16.66
 Financial 95 35.60 25.54
 Leisure 69 41.43 29.40
 Manufacturing 395 35.08 23.83
 Mining 14 35.52 17.50
 Services 65 34.16 28.09
 Telecom 169 29.43 20.90
 Transportation 66 38.07 23.79
 Utility 23 51.34 27.97
 Others 22 37.94 19.30
 


Table 3: Parameter estimates (2)

Parameter estimates and measures of fit for various models that differ in the combination of variables that influence the hazard rate λ and the two parameters of the beta distribution α and β. Note that with the specification of Shumway (2001), a positive coefficient on a explanatory variable for λ implies that λ falls when the variable rises. sen2, sen3, sen4, sen5 are seniority dummies for Senior Unsecured, Senior Subordinated, Subordinated and Discount respectively, mult is a dummy that is one for observations corresponding to default events for which we observe multiple recoveries, cycle is the unobserved credit cycle, and lagged def. rate and rec. rates are the previous annual default rate and mean recovery rate respectively. indB and indC are dummies corresponding to industry groups B and C. * denotes individual significance at 5%.

 
M2a
M2b
M5
M6
 Default Rates
λ
λ
λ
λ
 constant
3.36*
3.85*
3.36*
3.41*
 cycle
1.04*
1.05*
1.03*
 lagged def. rate
-1.20
 Recovery Rates
α
β
α
β
α
β
α
β
 constant 0 .40*  1 .00*  0 .52*  1 .48*  0 .34*  1 .15*  0 .38   1 .53* 
 sen2 0 .04   0 .15  -0 .02   0 .10  -0 .06  -0 .09  -0 .07  -0 .07  
 sen3 -0 .18   0 .00  -0 .16  -0 .02  -0 .35* -0 .40* -0 .30  -0 .32  
 sen4 0 .34   0 .38*  0 .54   1 .11*  0 .06   0 .01  -0 .04  -0 .11  
 sen5 -0 .32   0 .47  -0 .23   0 .32  -0 .32   0 .19  -0 .23   0 .45  
 mult -0 .22  -0 .56* -0 .39  -0 .54  -0 .29  -0 .49  -0 .26  -0 .56  
 mult×sen2 -0 .21  -0 .26   0 .00   0 .03  -0 .18  -0 .35  -0 .25  -0 .40  
 mult×sen3 -0 .14  -0 .08  -0 .72  -0 .57  -0 .28  -0 .17  -0 .22  -0 .05  
 lagged rec. rate
0 .37  -0 .40  
 cycle
0 .18  -0 .65*  0 .48* -0 .44   0 .55  -0 .63  
 cycle×sen2
0 .08   0 .15  -0 .15   0 .13  -0 .01   0 .29  
 cycle×sen3
-0 .25  -0 .01  -0 .26   0 .25  -0 .20   0 .26  
 cycle×sen4
-0 .46  -0 .56  -0 .09   0 .41   0 .20   0 .79  
 cycle×sen5
-0 .5  -0 .16  -0 .46  -0 .14  -0 .72  -0 .61  
 cycle×mult
0 .67   0 .31   0 .65   0 .20   0 .73   0 .35  
 cycle×mult×sen2
-0 .49  -0 .42   0 .06   0 .22  -0 .03   0 .10  
 cycle×mult×sen3
0 .76   0 .80   0 .52   0 .42   0 .34   0 .20  
 cycle×lagged rec. rate
-0 .39   0 .44  
 indB
0 .29*  0 .43* 
 indC
0 .09   0 .40* 
 Transition Prob.
 p
0.8487
0.9523
0.8742
0.8232
 q
0.7872
0.7634
0.7432
0.6443
 Measures of Fit
 Log Likelihood
131.726
91.214
202.032
193.003
 AIC
-221.45
-110.43
-322.06
-302.01
 BIC
-0.1344
0.0695
-0.1395
-0.1118
 

E Supplementary Figures

PIC

Figure 1: Q-Q plots of the PIT series of the static and cycle models.

Dashed line is the static model (M1), solid line is a dynamic model (M2). The upper panel is a Q-Q plot for periods which are identified as upturn by the dynamic model, the lower panel is a Q-Q plot for periods which are identified as downturns by the dynamic model.

PIC

Figure 2: Correlograms of the PIT series of the static and cycle model.

Upper panel is the correlogram of the static model (M1), the lower panel is the correlogram of a dynamic model (M2). Horizontal lines are 5% two-sided confidence intervals for a single autocorrelation coefficient.

PIC

Figure 3: Loss densities for Models 1, 2, 2a, and 2b

Loss densities for Models 1, 2, 2a and 2b. The solid black line is the unconditional loss density (i.e. assuming the probability of being in an upturn is equal to the unconditional probability), dotted green line is the upturn loss density (probability of being in upturn of 1), the red dashed line is the downturn loss density (probability of being in upturn of 0).

References

   Diebold, F. X., T. A. Gunther, and A. S. Tay, 1998, “Evaluating Density Forecasts with Applications to Financial Risk Management,” International Economic Review, 39, 863–883.

   Shumway, T., 2001, “Forecasting Bankruptcy More Accurately: A Simple Hazard Model,” Journal of Business, 74, 101–124.