Advances in Risk Management

220 EVALUATING VALUE-AT-RISK ESTIMATES: A CROSS-SECTION APPROACH

Table 11.1 Proportion of failures

Model 1 Model 2 Model 3

Run 1 POF 5.4% 4.7% 3.6%
Kupiec statistic 0.27 2.28 10.00
p-value 39.6% 86.9% 99.8%
Run 2 POF 5.1% 4.6% 3.7%
Kupiec statistic 0.88 2.73 9.04
p-value 65.2% 90.2% 99.7%

Run 3 POF 4.9% 3.8% 3.5%
Kupiec statistic 1.49 8.13 11.02
p-value 77.8% 99.6% 99.9%
Run 4 POF 6.5% 5.0% 4.2%
Kupiec statistic 0.92 1.16 50.3
p-value 66.3% 71.9% 97.5%
Run 5 POF 5.2% 4.0% 3.1%
Kupiec statistic 0.64 6.48 15.72
p-value 57.5% 98.9% 100.0%

VaR, withαequal to 5 percent, according to several models that differ for
the hypothesis made about the DGP:

1 model 1 assumesrt,t+H

i.i.d.

∼NormalN(0, I);

2 model 2 assumesrt,t+H

i.i.d.

∼NormalN(0, 1. 1 ·I), that is, variance is 10%

greater than reality; and

3 model 3 assumesrt,t+H

i.i.d.

∼NormalN(0, 1. 2 ·I), that is, variance is 20%

greater than reality;

We apply our procedure, generatingKrandom portfolios withKequal
to 1,000. We also create a market scenario, generating a vector of returns
rt,t+Husing the chosen DGP. We then test the unconditional coverage prop-
erty using the binomial test proposed by Kupiec (1995), probably the most
popular method among practitioners. We calculate the proportion of fail-
ures (POF), the Kupiec statistic and the associatedp-value. We apply the
procedure for five consecutive periods (5 days). Results are reported in
Table 11.1.
It is apparent that VaR estimates obtained using Model 1 (corresponding
to the DGP) are better than those obtained from the other models: the
number of failures and the other statistics say that the performance of
this model is closer to what we expect in theory. If we assume that the
DGP is a completely different process, for example a multivariatet-student
with 3 degrees of freedom a with correlation equal to zero, such that