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 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.
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).
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.
|
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.
|
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.
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.
|
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.