Advances in Risk Management

RAFFAELE ZENTI, MASSIMILIANO PALLOTTA AND CLAUDIO MARSALA 221

Table 11.2Proportion of failures

Model 1 Model 2 Model 3

POF 1.8% 1.5% 1.2%

Kupiec statistic 39.24 47.03 56.04

p-value 100.0% 100.0% 100.0%

rt,t+H

i.i.d.
∼t−studentN(0, I, 3dgf), after a single run we see that all the models
are in difficulties, as can be seen in Table 11.2.
After a small number of days, the picture becomes clearer. In order to get
more information, one can use some Bayesian analysis, as outlined below.

11.5.2 Comparative Bayesian analysis of the performance of
the VaR models

In the case of the POF test, we are testing the null hypothesis that the pro-
portion of failuresp=n_failures/Kis equal toBinomial(α,K)=pα,K≈5.8% in
our case (because, for the sake of simplicity, we are assuming absence of
correlation).
Hence, it would be reasonable to assume as a prior, a Beta distribution
with parametersa 0 =K·pα,K+1,b 0 =K·(1−pα,K)+1. Thus, our prior can
be written as:

π(p)∝pa^0 −^1 (1−p)b^0 −^1 (11.9)

Basically, we center our prior distribution on 5.8 percent, that is the
theoretical POF if the model is correct.
After the first run we have some data: we observen_failuresout ofK
results. The likelihood functionl(n_failures,K,p) is given by the binomial
distribution:

l(n_failures,K,p)=

(

K

n_failures

)

pn_failures(1−p)K−n_failures (11.10)

that can be rewritten as:

l(n_failures,K,p)∝pn_failures(1−p)K−n_failures (11.11)

We combine the likelihood function with our prior distribution and we get
the posterior distribution which turns out to be a Beta distribution (as we