3.6 VaR Estimation and Backtesting with XploRe

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 8407 VaRest command. The 8410 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 $ N$ relative risk factor changes, because the profit/loss observations are not identical with the risk factor changes.

For each day the $ N$ 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 $ h$ 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 8417 VaRest command must be called separately for each day. The consequence is that the data the 8420 VaRest command operates on consists of a row of $ N+1$ numbers: $ N$ profit/loss values contained in the vector (PL[t,])', which has one column and $ N$ rows followed by the actual mark-to-model profit or loss MMPL[t,1] within holding period $ h$ in the last row. The procedure is implemented in the quantlet XFGpl which can be downloaded from quantlet download page of this book.

Figure: VaR time plot basic historical simulation. 8424 XFGtimeseries.xpl
\includegraphics[width=1.3\defpicwidth]{TPB1.ps}

Figure: VaR time plot historical simulation with volatility updating. 8428 XFGtimeseries2.xpl
\includegraphics[width=1.3\defpicwidth]{TPV1.ps}

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 $ N=250$ days, and the holding period is $ h=1$ day. For the volatility updating we use a decay factor of $ \gamma = 0.94$, 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 $ N+h-1$ observations are required. A historical simulation with volatility updating requires $ 2(N+h-1)$ 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 $ t_0=[2(N+h-1)/N]$. 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.


8434 XFGexc.xpl


Table 3.12: Violations full yield and spread curve (in %)
Full yield Spread curve
Curve V1 V2 V3 V4 V1 V2 V3 V4
INAAA 1,34 1,34 1,09 1,28 1,34 1,34 1,34 1,34
INAA2 1,34 1,22 1,22 1,22 1,46 1,52 1,22 1,22
INAA3 1,15 1,22 1,15 1,15 1,09 1,09 0,85 0,91
INA1 1,09 1,09 1,46 1,52 1,40 1,46 1,03 1,09
INA2 1,28 1,28 1,28 1,28 1,15 1,15 0,91 0,91
INA3 1,22 1,22 1,15 1,22 1,15 1,22 1,09 1,15
INBBB1 1,28 1,22 1,09 1,15 1,46 1,46 1,40 1,40
INBBB2 1,09 1,15 0,91 0,91 1,28 1,28 0,91 0,91
INBBB3 1,15 1,15 1,09 1,09 1,34 1,34 1,46 1,52
INBB1 1,34 1,28 1,03 1,03 1,28 1,28 0,97 0,97
INBB2 1,22 1,22 1,22 1,34 1,22 1,22 1,09 1,09
INBB3 1,34 1,28 1,28 1,22 1,09 1,28 1,09 1,09
INB1 1,40 1,40 1,34 1,34 1,52 1,46 1,09 1,03
INB2 1,52 1,46 1,28 1,28 1,34 1,40 1,15 1,15
INB3 1,40 1,40 1,15 1,15 1,46 1,34 1,09 1,15
BNAAA 1,24 1,38 1,10 1,10 0,89 0,89 1,03 1,31
BNAA1/2 1,38 1,24 1,31 1,31 1,03 1,10 1,38 1,38
BNA1 1,03 1,03 1,10 1,17 1,03 1,10 1,24 1,24
BNA2 1,24 1,31 1,24 1,17 0,76 0,83 1,03 1,03
BNA3 1,31 1,24 1,17 1,10 1,03 1,10 1,24 1,17
Average 1,27 1,25 1,18 1,20 1,22 1,24 1,13 1,15



Table 3.13: Violations benchmark curve (in %)
Curve V1 V2 V3 V4
INAAA, INAA2, INAA3, INA1, INA2, INA3, INBBB1, INBBB2, INBBB3, INBB1, INBB2, INBB3, INB1, INB2, INB3 1,52 1,28 1,22 1,15
BNAAA, BNAA1/2, BNA1, BNA2, BNA3 1,72 1,44 1,17 1,10
Average 1,57 1,32 1,20 1,14



Table 3.14: Violations in the conservative approach and simultaneous simulation(in %)
conservative approach simultaneous simulation
Curve V1 V2 V3 V4 V1 V2 V3 V4
INAAA 0,24 0,24 0,30 0,30 1,22 1,28 0,97 1,03
INAA2 0,24 0,30 0,36 0,30 1,22 1,28 1,03 1,15
INAA3 0,43 0,36 0,30 0,30 1,22 1,15 1,09 1,09
INA1 0,36 0,43 0,55 0,55 1,03 1,03 1,03 1,09
INA2 0,49 0,43 0,49 0,49 1,34 1,28 0,97 0,97
INA3 0,30 0,36 0,30 0,30 1,22 1,15 1,09 1,09
INBBB1 0,43 0,49 0,36 0,36 1,09 1,09 1,03 1,03
INBBB2 0,49 0,49 0,30 0,30 1,03 1,03 0,85 0,79
INBBB3 0,30 0,30 0,36 0,36 1,15 1,22 1,03 1,03
INBB1 0,36 0,30 0,43 0,43 1,34 1,34 1,03 0,97
INBB2 0,43 0,36 0,43 0,43 1,40 1,34 1,15 1,09
INBB3 0,30 0,30 0,36 0,36 1,15 1,15 0,91 0,91
INB1 0,43 0,43 0,43 0,43 1,34 1,34 0,91 0,97
INB2 0,30 0,30 0,30 0,30 1,34 1,34 0,97 1,03
INB3 0,30 0,30 0,36 0,30 1,46 1,40 1,22 1,22
BNAAA 0,62 0,62 0,48 0,48 1,31 1,31 1,10 1,03
BNAA1/2 0,55 0,55 0,55 0,48 1,24 1,31 1,10 1,17
BNA1 0,62 0,62 0,55 0,55 0,96 1,03 1,10 1,17
BNA2 0,55 0,62 0,69 0,69 0,89 1,96 1,03 1,03
BNA3 0,55 0,55 0,28 0,28 1,38 1,31 1,03 1,10
Average 0,41 0,42 0,41 0,40 1,22 1,22 1,03 1,05