Advances in Risk Management

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

are using a conjugate prior):

f(p|n_failures,K)∝π(p)·l(n_failures,K,p)

=Beta(a 0 +n_failures,b 0 +K−n_failures) (11.12)

=Beta(a 1 ,b 1 )

so we have a Bayesian updating scheme that enables us to understand how
a given VaR model performs over time. Aftertsteps, the distribution of
pis:

Betat(at,bt)=Beta

(

a 0 +

∑t

i= 1

n_failuresi,b 0 +

∑t

i= 1

(K−n_failuresi)

)

(11.13)

For example, Figure 11.1 shows the prior distribution and the posterior
distribution, calculated using (11.13), for Model 1 and Model 2 of the pre-
vious example. The prior is common to both models, as the null is that any
given model is correct, but the posterior is different: it is rather evident that
Model 1 is closer to the prior distribution which corresponds to the null. As
time passes, the distribution of Model 1 will eventually converge to the prior
(true by definition). Thus this methodology can help in ranking different risk
models.
Depending on the perspective of the analysis, one could instead use a
non-informative prior, for example a uniform distribution over the unit
interval.
For global risk models (for example, models that can be applied across
the whole spectrum of asset classes – many specialized software products
claim to be global risk analysers), this procedure can be applied in a parallel
fashion to several investment universes. This enables regulators and finan-
cial institutions to assess the model on several areas, for instance European
equities, US equities, Far East equities, emerging markets equities, inter-
national government bonds, international corporate bonds, and so on. It is
possible to understand how and where a model fails. For example, if among
the randomly generatedKportfolios, portfoliosexhibits a large failure, it is
possible to analyse the sources of this loss, as suggested below.

11.5.3 Failures analysis

First, consider the shortfall of modeljon a given period [t,t+H] for portfolio
s, defined by the vector of weightswst:

D

j

t,t+H(α,w

s

t,H)=Rt,t+H(w

s

t)−VaRt,j(α,w

s

t,H). (11.14)