To judge the effectiveness of a Value-at-Risk model, it is common to use
backtesting. A simple approach is to compare the predicted and empirical
number of outliers, where the actual loss exceeds the VaR. We implement this
test in a two risk factor model using real life time series, the FX rates
USD/EUR and GBP/EUR, respectively their DEM counterparts before the
introduction of the Euro. Our backtesting investigation is based on a time
series ranging from
2 Jan. 1991 until 9 Mar. 2000 and simple
linear portfolios
:
The Value-at-Risk is computed with confidence level
(
,
, and
) based on a time series for the statistical
estimators of length
business days. The actual next day value change
of the portfolio is compared to the VaR estimate. If the portfolio loss
exceeds the VaR estimate, an outlier has occurred. This procedure is repeated
for each day in the time series.
The prediction error as the absolute difference of the relative number of
outliers
to the predicted number
is averaged over
different portfolios and confidence levels. The average over the portfolios
(
) uses equal weights,
while the average over the confidence levels
emphasizes the tails by a
weighting scheme
(
,
,
). Based on the result, an
overall error and a relative ranking of the different methods is obtained (see
Table 2.2).
As benchmark methods for Value-at-Risk we use the variance-covariance (vcv) method and historical simulation (his), for details see Deutsch and Eller (1999). The variance covariance method is an analytical method which uses a multivariate normal distribution. The historical simulation method not only includes the empirical copula, but also empirical marginal distributions. For the copula VaR methods, the margins are assumed to be normal, the only difference between the copula VaR's is due to different dependence structures (see Table 2.1). Mainly as a consequence of non-normal margins, the historical simulation has the best backtest results. However, even assuming normal margins, certain copulas (5, 12-14) give better backtest results than the traditional variance-covariance method.
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