In this section we explain, how a VaR can be calculated and a
backtesting can be implemented with the help of XploRe routines. We
present numerical results for the different yield curves.
The VaR estimation is carried out with the help of the
VaRest
command.
The
VaRest
command calculates a VaR for historical simulation, if one specifies
the method parameter as "EDF" (empirical distribution function). However, one has to be
careful when specifying the sequence of asset returns which are used as input for the
estimation procedure. If one calculates zero-bond returns from relative risk factor changes
(interest rates or spreads) the complete empirical distribution
of the profits and losses must be estimated anew for each day
from the
relative risk factor changes, because the profit/loss
observations are not identical with the risk factor changes.
For each day the profit/loss observations generated with one of
the methods described in subsections 3.4.1 to 3.4.5 are
stored to a new row in an array
PL
. The actual profit and
loss data from a mark-to-model
calculation for holding period
are stored to a one-column-vector
MMPL
.
It is not possible to use a continuous sequence of
profit/loss data with overlapping time windows for the VaR estimation. Instead the
VaRest
command must be called separately for each day. The consequence is that
the data the
VaRest
command operates on consists of a row of
numbers:
profit/loss values contained in the vector (PL[t,])', which has one column and
rows followed by the actual mark-to-model profit or loss MMPL[t,1] within
holding period
in the last row.
The procedure is implemented in the quantlet XFGpl which can be downloaded from
quantlet download page of this book.
The result is displayed for the INAAA curve in Figures. 3.5 (basic historical simulation) and 3.6 (historical simulation with volatility updating). The time plots allow for a quick detection of violations of the VaR prediction. A striking feature in the basic historical simulation with the full yield curve as risk factor is the platform-shaped VaR prediction, while with volatility updating the VaR prediction decays exponentially after the occurrence of peak events in the market data. This is a consequence of the exponentially weighted historical volatility in the scenarios. The peak VaR values are much larger for volatility updating than for the basic historical simulation.
In order to find out, which framework for VaR estimation has the best
predictive power, we count the number of violations of the VaR prediction
and divide it by the
number of actually observed losses. We use the 99% quantile, for which we
would expect an violation rate of 1% for an optimal VaR estimator.
The history used for the drawings of the scenarios consists of
days, and the holding period is
day. For the volatility
updating we use a decay factor of
, J.P. Morgan (1996).
For the simulation we assume that the synthetic zero-bond has a
remaining time to maturity of 10 years at the beginning of the simulations.
For the calculation of the
first scenario of a basic historical simulation
observations are
required. A historical simulation with volatility updating requires
observations preceding the trading day the first scenario
refers to. In order to allow for a comparison between different methods
for the VaR calculation, the beginning of the simulations is
. With these simulation parameters
we obtain 1646 observations for a zero-bond in the industry sector
and 1454 observations for a zero-bond in the banking sector.
In Tables 3.12 to 3.14 we list the percentage of violations for all yield curves and the four variants of historical simulation V1 to V4 (V1 = Basic Historical Simulation; V2 = Basic Historical Simulation with Mean Adjustment; V3 = Historical Simulation with Mean Adjustment; V4 = Historical Simulation with Volatility Updating and Mean Adjustment). In the last row we display the average of the violations of all curves. Table 3.12 contains the results for the simulation with relative changes of the full yield curves and of the yield spreads over the benchmark curve as risk factors. In Table 3.13 the risk factors are changes of the benchmark curves. The violations in the conservative approach and in the simultaneous simulation of relative spread and benchmark changes are listed in Table 3.14.
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