Implications on Portfolio Level

Systematic effects in modeling DRT might not only affect the DRT itself, but also the

loss involved. In Section 3.2 (see Figure 3.2), the relation between DRTs and the non-

discounted RRs has been shown.24 _{This indicates that recovery cash flows are lower the}

longer the resolution process takes. Furthermore, the DRT directly enters the calculation

of the LGD by discounting the recovery cash flows. To put it simple, assuming a constant

risk adjusted interest rate of 5% and recovery cash flows being paid at the end of the

resolution process, the LGD of a single loan is derived as

LGD = 1− RRT

(1 +r)T, (3.11)

where RRT denotes the time dependent non-discounted RR. Its value is set to the mean of the related DRT bucket. Table 3.10 summarizes the six DRT buckets and the cor-

responding means. For example, a loan with a DRT of two years is assigned with a

non-discounted RR of 72.72%.

Table 3.10: Non-discounted RR by DRT buckets

DRT bucket non-discounted RR 0<DRT≤1 84.23% 1<DRT≤2 72.72% 2<DRT≤3 62.80% 3<DRT≤4 59.82% 4<DRT≤5 59.09% DRT>5 43.70%

Notes: The table summarizes the mean of the non-discounted RR per bucket of DRTs. The

first row of the table (0 < DRT ≤ 1) includes loans with DRTs up to one year. The second row

(1<DRT≤2 ) includes loans with DRTs longer than one year up to two years and so on. In the last

row (DRT > 5) loans with DRTs longer than five years are summarized. The means meet the ones

illustrated in the upper left panel of Figure 3.2.

We further study the representative portfolio of the randomly sampled 1,000 loans by

considering implications on portfolio level. The exposure weighted portfolio loss distri-

bution is generated via Monte-Carlo simulation. DRTs for the 1,000 loans are randomly

drawn according to Model I, II, and III, respectively.25 _{The corresponding LGDs are}

calculated by Equation (3.11). Finally, the portfolio loss is given by:

LGDP F = 1 n EAD n X i=1 (LGDi EADi), (3.12)

where, EAD indicates the average EAD of the portfolio. The procedure is repeated 100,000 times to generate the portfolio loss distribution.

Table 3.11 summarizes the mean and the 95% quantile of the simulated portfolio

DRTs. As the baseline hazard rates are calibrated on the empirical mean of the DRTs

(see Table 3.1), the average portfolio DRT in Model I corresponds to this value for both

economic scenarios. In Model II, the mean is higher in a recession and lower in an

expansion period. This effect is more pronounced in Model III.

Table 3.11: Inferences of systematic factors on the distribution of portfolio DRTs

Recession Expansion

Model I mean 1.59 1.59

95% quantile 1.69 1.69

Model II mean 2.01 1.29

95% quantile 2.14 1.37

Model III mean 2.42 1.10

95% quantile 3.83 1.75

Notes: The table summarizes the mean and 95% quantile of the portfolio DRT for Model I, II,

and III in an exemplary recession (realizations of macroeconomic variables as of Q1 2009) and expansion (realizations of macroeconomic variables as of Q2 2011) period. For every loan in the representative portfolio a DRTs is drawn according to the underlying model. Afterwards, the mean of the random draws is calculated to generate the average portfolio DRT. The procedure is repeated 100,000 times to generate the distribution of portfolio DRTs.

The portfolio loss distribution is simulated based on the portfolio DRTs and the non-

discounted RRs as of Table 3.10. Figure 3.11 displays the portfolio loss distributions for

25_{In Model III, we initially draw a frailty from the Normal distribution with mean 0 and varianceσ}2_.

This frailty realizationuis constant for all 1,000 loans. Given this realization, the resolution intensity is

constant among the loans in the homogeneous portfolio and we then draw the DRTs from the conditional

Model I, II, and III for the exemplary recession and expansion period. In the left panel

the portfolio loss distribution of a recession is shown. Compared to Model I, the portfolio

loss distribution of Model II is shifted to the right and slightly wider. This indicates that

not only the mean of the portfolio loss but also its variation increases compared to Model

I. This is mainly due to the exponential distribution of the DRT. Since it is fully specified

by one parameter, mean and variance of the DRT are solely driven by this parameter

and, thus, move in parallel. This effect is also reflected in the portfolio loss. However,

the difference to Model III is much more pronounced than the difference between Model

I and II. Through the frailty effect substantially more uncertainty is introduced into the

model and the portfolio loss distribution is characterized by a higher mean and a much

wider range. This indicates that not only the expected loss but also extreme quantiles

of the portfolio loss distribution rise. In the right panel of Figure 3.11, the portfolio

loss distribution of an expansion is displayed. The portfolio loss distribution of Model II

Figure 3.11: Kernel density estimates of loss on portfolio level

recession portfolio loss density 0.15 0.35 0.55 0 10 20 30 expansion portfolio loss density 0.15 0.35 0.55 0 10 20 30

Model I Model II Model III

Notes: The figure illustrates the kernel density estimates of the exposure weighted portfolio loss dis- tribution based on simulated DRT of Model I, II, and III as of Equation (3.7), (3.8), and (3.9) in an exemplary recession (realizations of macroeconomic variables as of Q1 2009) and expansion (realizations of macroeconomic variables as of Q2 2011) period. For every loan in the representative portfolio a DRTs is drawn according to the underlying model. In Model III, a frailty is drawn from the Normal distribution

with mean 0 and varianceσ2_{for each run. The corresponding loss is calculated. Afterwards, the mean of}

the losses is calculated to generate the average loss. The procedure is repeated 100,000 times to generate the distribution of portfolio losses.

is shifted to the left and is narrower compared to Model I. Comparing Model I and II

with Model III, major differences arise. Although the distribution is shifted to the left

of Model II, its wide range persists. The expected loss of Model III is lower compared to

Model I and II. However, extreme quantiles are still higher.

While the former analysis considered exemplary portfolio loss distributions in a reces-

sion and expansion period, we now extend it to all possible scenarios in the estimation

sample to analyze potential portfolio risk if similar scenarios arise in the future. Fig-

ure 3.12 displays the mean (left panel) and the the VaR(95%) as well as the VaR(99%)26

(right panel) of the portfolio loss distribution for all macroeconomic scenarios in the es-

timation sample for the three models. In Model I, the mean is constant over time as the

resolution intensity λI _{is constant. The mean of Model II lies above the one of Model I} in quarters characterized by adverse economic conditions, e.g., Q1 2009. In favorable

economic surroundings, e.g., Q2 2011, it lies below the one of Model I. Comparing Model

II and III, the mean of Model III seems to be more extreme in the majority of the cases

(e.g., Q1 2009 and Q2 2011). The right panel of Figure 3.12 shows the VaR(95%) and

the VaR(99%) of the portfolio loss distribution. As the resolution intensity and, thus,

the portfolio loss distribution is constant over time regarding Model I, the corresponding

extreme quantiles are time-invariant. The course of the VaR(95%) and the VaR(99%)

in Model II seems strongly related to the course of its mean. In recessions, the extreme

quantiles of Model II lie above the ones of Model I, whereas, they lie below in expan-

sions. This might be due to the rather similar shape of the portfolio loss distributions

of Model I and II. Although, the range of the distribution of Model II slightly increases

(decreases) if it is shifted to the right (left), the deviation seems marginal. A clearer

contrast emerges considering Model III where the extreme quantiles are shifted upwards

throughout. Generally, this shows that the stochastic frailty introduces non diversifiable

systematic risk and co-movement between DRTs. This could have a substantial impact

on losses on portfolio level.

Figure 3.12: Mean and VaR(95%) of loss on portfolio level expected loss time mean Q1 2004 Q3 2008 Q1 2013 0.25 0.35 0.45 VaR time V aR Q1 2004 Q3 2008 Q1 2013 0.25 0.35 0.45

Model I Model II Model III VaR (95%) VaR (99%)

Notes: The figure illustrates mean, VaR(95%) and VaR(99%) of the exposure weighted portfolio loss distribution based on simulated DRTs of Model I, II and III as of Equation (3.7), (3.8), and (3.9) for all quarters in the estimation sample. For every loan in the representative portfolio a DRTs is drawn according to the underlying model. In Model III, a frailty is drawn from the Normal distribution with

mean 0 and varianceσ2_{for each run. The corresponding loss is calculated. Afterwards, the mean of the}

losses is calculated to generate the average loss. The procedure is repeated 100,000 times to generate the distribution of portfolio losses.