3.9 Discussion of Simulation Results

In Figure 3.7 the P-P plots for the historical simulation with the full yield curve ( INAAA ) as risk factor are displayed for the different variants of the simulation. From the P-P plots it is apparent that mean adjustment significantly improves the predictive power in particular for intermediate confidence levels (i.e., for small risk factor changes).

Figure: P-P Plots variants of the simulation. 8722 XFGpp.xpl
\includegraphics[width=1.5\defpicwidth]{PP1.ps}

Figure 3.8 displays the P-P plots for the same data set and the basic historical simulation with different choices of risk factors. A striking feature is the poor predictive power for a model with the spread as risk factor. Moreover, the over-estimation of the risk in the conservative approach is clearly reflected by a sine-shaped function, which is superposed on the ideal diagonal function.

Figure: P-P Plots choice of risk factors. 8726 XFGpp.xpl
\includegraphics[width=1.5\defpicwidth]{PP2.ps}

In Figs. 3.9 and 3.10 we show the Q-Q plots for basic historic simulation and volatility updating using the INAAA data set and the full yield curve as risk factors. A striking feature of all Q-Q plots is the deviation from linearity (and, thus, normality) for extreme quantiles. This observation corresponds to the leptokurtic distributions of time series of market data changes (e.g. spread changes as discussed in section 3.3.2).

Figure 3.9: Q-Q Plot for basic historical simulation.
\includegraphics[width=1.5\defpicwidth]{QQ1.ps}

Figure 3.10: Q-Q plot for volatility updating.
\includegraphics[width=1.5\defpicwidth]{QQ2.ps}


3.9.1 Risk Factor: Full Yield

The results in Table 3.12 indicate a small under-estimation of the actually observed losses. While volatility updating leads to a reduction of violations, this effect is not clearly recognizable for the mean adjustment. The positive results for volatility updating are also reflected in the corresponding mean squared deviations in Table 3.15. Compared with the basic simulation, the model quality can be improved. There is also a positive effect of the mean adjustment.


Table: MSD P-P Plot for the full yield and the spread curve( $ \times10\,000$)
full yield spread curve
Curve V1 V2 V3 V4 V1 V2 V3 V4
INAAA 0,87 0,28 0,50 0,14 8,13 22,19 8,14 16,15
INAA2 0,45 0,36 0,32 0,16 6,96 21,41 7,25 15,62
INAA3 0,54 0,41 0,43 0,23 7,91 21,98 7,97 15,89
INA1 0,71 0,27 0,41 0,13 7,90 15,32 8,10 8,39
INA2 0,50 0,39 0,42 0,17 9,16 15,15 9,51 6,19
INA3 0,81 0,24 0,58 0,24 9,53 12,96 9,61 7,09
INBBB1 0,71 0,29 0,54 0,13 9,59 15,71 9,65 11,13
INBBB2 0,33 0,34 0,26 0,12 11,82 14,58 11,59 10,72
INBBB3 0,35 0,59 0,40 0,34 7,52 11,49 7,78 6,32
INBB1 0,31 0,95 0,26 0,28 4,14 4,57 3,90 1,61
INBB2 0,52 0,49 0,36 0,19 6,03 3,63 5,89 2,12
INBB3 0,53 0,41 0,36 0,17 3,11 3,65 3,09 1,67
INB1 0,51 0,29 0,38 0,15 3,59 1,92 2,85 1,16
INB2 0,51 0,48 0,31 0,22 4,29 2,31 3,41 1,42
INB3 0,72 0,38 0,32 0,16 3,70 2,10 2,99 3,02
BNAAA 0,59 0,19 0,48 0,56 10,13 17,64 9,74 11,10
BNAA1/2 0,54 0,21 0,45 0,46 5,43 13,40 5,73 7,50
BNA1 0,31 0,12 0,29 0,25 8,65 17,19 8,09 8,21
BNA2 0,65 0,19 0,57 0,59 6,52 12,52 6,95 6,45
BNA3 0,31 0,19 0,32 0,29 6,62 9,62 6,59 3,80
Average 0,54 0,35 0,40 0,25 7,04 11,97 6,94 7,28



3.9.2 Risk Factor: Benchmark

The results for the number of violations in Table 3.13 and the mean squared deviations in Table 3.16 are comparable to the analysis, where risk factors are changes of the full yield. Since the same relative changes are applied for all yield curves, the results are the same for all yield curves. Again, the application of volatility updating improves the predictive power and mean adjustment also has a positive effect.


Table: MSD P-P-Plot benchmark curve ( $ \times10\,000$)
Curve V1 V2 V3 V4
INAAA, INAA2, INAA3 0,49 0,23 0,26 0,12
INA1 0,48 0,23 0,26 0,12
INA2, INA3, INBBB1, INBBB2, INBBB3, INBB1, INBB2 0,49 0,23 0,26 0,12
INBB3 0,47 0,23 0,25 0,12
INB1 0,49 0,23 0,26 0,12
INB2 0,47 0,23 0,25 0,12
INB3 0,48 0,23 0,26 0,12
BNAAA, BNAA1/2 0,42 0,18 0,25 0,33
BNA1 0,41 0,18 0,23 0,33
BNA2 0,42 0,18 0,25 0,33
BNA3 0,41 0,18 0,24 0,33
Average 0,47 0,22 0,25 0,17



3.9.3 Risk Factor: Spread over Benchmark Yield

The number of violations (see Table 3.12) is comparable to the latter two variants. Volatility updating leads to better results, while the effect of mean adjustment is only marginal. However, the mean squared deviations (see Table 3.15) in the P-P plots are significantly larger than in the case, where the risk factors are contained in the benchmark curve. This can be traced back to a partly poor predictive power for intermediate confidence levels (see Figure 3.8). Mean adjustment leads to larger errors in the P-P plots.


3.9.4 Conservative Approach

From Table 3.14 the conclusion can be drawn, that the conservative approach significantly over-estimates the risk for all credit qualities. Table 3.17 indicates the poor predictive power of the conservative approach over the full range of confidence levels. The mean squared deviations are the worst of all approaches. Volatility updating and/or mean adjustment does not lead to any significant improvements.


Table: MSD P-P Plot for the conservative approach and the simultaneous simulation( $ \times10\,000$)
conservative approach simultaneous simulation
Curve V1 V2 V3 V4 V1 V2 V3 V4
INAAA 14,94 14,56 14,00 13,88 1,52 0,64 0,75 0,40
INAA2 13,65 13,51 14,29 14,31 0,79 0,38 0,40 0,23
INAA3 14,34 13,99 13,66 13,44 0,79 0,32 0,49 0,27
INA1 15,39 15,60 15,60 15,60 0,95 0,40 0,52 0,29
INA2 13,95 14,20 14,32 14,10 0,71 0,55 0,50 0,39
INA3 14,73 14,95 14,45 14,53 0,94 0,30 0,59 0,35
INBBB1 13,94 14,59 14,05 14,10 1,00 0,33 0,43 0,17
INBBB2 13,74 13,91 13,67 13,73 0,64 0,52 0,45 0,29
INBBB3 13,68 14,24 14,10 14,09 0,36 0,78 0,31 0,31
INBB1 19,19 20,68 18,93 19,40 0,73 1,37 0,52 0,70
INBB2 13,21 14,17 14,79 15,15 0,30 0,82 0,35 0,51
INBB3 15,19 16,47 15,40 15,67 0,55 0,65 0,15 0,21
INB1 15,47 15,64 15,29 15,51 0,53 0,44 0,19 0,26
INB2 14,47 14,93 15,46 15,77 0,24 0,55 0,24 0,24
INB3 14,78 14,67 16,77 17,03 0,38 0,44 0,27 0,22
BNAAA 14,80 15,30 16,30 16,64 1,13 0,33 0,99 0,96
BNAA1/2 13,06 13,45 14,97 15,43 0,73 0,16 0,57 0,50
BNA1 11,95 11,83 12,84 13,08 0,52 0,26 0,44 0,41
BNA2 13,04 12,58 14,31 14,56 0,78 0,13 0,51 0,58
BNA3 12,99 12,70 15,19 15,42 0,34 0,18 0,58 0,70
Average 14,33 14,60 14,92 15,07 0,70 0,48 0,46 0,40



3.9.5 Simultaneous Simulation

From Tables 3.14 and 3.17 it is apparent that simultaneous simulation leads to much better results than the model with risk factors from the full yield curve, when volatility updating is included. Again, the effect of mean adjustment does not in general lead to a significant improvement. These results lead to the conclusion that general market risk and spread risk should be modeled independently, i.e., that the yield curve of an instrument exposed to credit risk should be modeled with two risk factors: benchmark changes and spread changes.