The data used in this section is a bond portfolio of a German bank
from 1994 to 1995. The portfolio is not adjusted so that the
exposure vector is time dependent. We assume that
(15.7) and (15.8) hold. The VaR forecast is based on
both prediction rules introduced in Section 15.1 that
are used to estimate the parameters
of the forecast
distribution in RMA and EMA given
. In light of the
bond crisis in 1994 it is interesting how both techniques respond
to this stress factor.
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The significance level under consideration is
for
large losses and
for large profits. To investigate
we include plots of time series from the realized P/L (i.e.,
profit-loss) data
as compared to the respective VaR
estimator
calculated with (15.12). If the
model and the estimation of the parameter
based on
forecast distribution are adequate, then approximately
of
the data should lie below the
and above the
VaR
Estimators. In addition in Figure 15.1 the crossings for
the case where VaR is estimated with EMA are marked. We recognize
that in 1994 (1995) there were a total of 10 (9) crossings
determined for the EMA method. This strongly contrasts the 17 (3)
observed values for the RMA Method. It is clear that the RMA
technique leads to, above all during the bond crisis in 1994, too
many crossings for the
VaR estimator, which means that the
probability of larger losses is underestimated. This tendency to
underestimate the risk is produced from the observation width of
days, when the market is moving towards a more volatile
phase. The opposite is true when moving in the other direction;
RMA overestimates risk. The EMA adapts more quickly to market
phases since data in the past has less of influence on the
estimator due to the exponentially deteriorating weights. With
SFEVaRtimeplot.xpl
we have calculated the estimated VaRs
for another bank using the EMA and RMA respectively.
The poor forecast quality of the RMA, in particular for the left
side of the distribution, can also be seen in that for a
particular day the VaR was exceeded by . If the model
(15.7) - (15.8) is correct, then the variable
(15.19) must have a standard deviation of about
. īThe
empirical standard deviation calculated from the data is about
0.62. According to the volatility scale of the RMA the risk is
underestimated on average by
. The EMA plot in Figure 15.1 shows a better
calibration. The empirical standard deviation of (15.19) is in
this case around
, which corresponds to an underestimation
of risk by approximately
.
All other diagnostic measurements are entered into the QQ plot of the variable
A comparison of the graphs in Figure 15.2 and Figure
15.3 show that the EMA method is calibrated better than
the RMA method. The RMA method clearly shows outliers at both
ends. The interval boundaries of are in both cases
clearly exceeded. This indicates a possible inadequacy of an
assumed normal distribution. QQ plots for the year 1995 are not
shown, which also clearly show the dominance of EMA over RMA.
Another important assumption of our model is the independence of
the re-scaled random variable .
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(16.20) |
The exploratory analysis clearly shows the differences between RMA
and EMA. As a supplement we now compare both estimation techniques
with an appropriate test within the framework of the model
(15.7) - (15.8). We again consider the sample
residuals
from (15.16) and set the threshold
value in (15.14) to
, i.e., to the
quantile
of the distribution of
. From
this we obtain
according to (15.14). Due to
the asymptotic distribution (15.18) we can check the
significance of the hypothesis
From Table 15.2 and Table 15.3 it is obvious
that the observed outliers for EMA are calibrated better than for
the RMA method. For a random sample of values we expect
outliers (standard deviation
). For EMA we observe
standard deviation
outliers and for RMA
standard deviation
. Naturally the outliers influence the test
considerably. Therefore, we repeat the analysis excluding the
outliers and obtain (15.4) and (15.5).