Advances in Risk Management

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

11.3.2 Tests based on multiple VaR levels or the entire probability
density function

Especially with small samples, a more accurate use of the information at
disposal is vital: models can be more precisely tested examining more quan-
tiles, for example, VaRs for differentαii=1, 2, 3,...,plevels, extending the
procedure outlined in the previous section. In a more general fashion and
broadening the idea, evaluating the entire probability density function of
portfolio’s return. The evaluation of the entire density forecast extracts a
greater amount of information from available data, as it uses the full range
of forecasted outcomes.
Among others, Chatfield (1993), Crnkovic and Drachman (1997), Diebold,
Gunther and Tay (1998), Christoffersen (1998) as well as Berkowitz (2001)
have proposed methods for evaluating VaR according to multiple VaR lev-
els or the entire probability density function. The idea behind this class
of testing methods is to transform ex-post portfolio’s returns into a new
variable, which is defined in (0,1). The transformation is made using the fore-
casted(forexample, ex-ante)cumulativeprobabilitydistributionofportfolio
returns. Formally:

z

j

t,t+H=Ft,j(Rt,t+H|It,wt) (11.5)

Now, for any variatex∈and any probability density functionφ(x), the
cumulative probability distribution:

(x)=

∫x

−∞

φ(s)ds (11.6)

is a uniform variate on the unit interval, as(x) is just a probability measure,
that is uniform by definition. Hence, the testing procedures rely on the time

series{z
j
t,t+H}, that should simultaneously exhibit two properties:

1 it should be uniformly distributed on the unit interval, as implied by

(11.6); and

2 it should not exhibit serial correlation.

The two properties above can be jointly expressed as:

zjt,t+H

i.i.d.

∼Uniform(0, 1) (11.7)

and can be tested, individually or jointly.
One limit of these approaches is that they can require large data sets in
order to check the accuracy of VaR models. This is not always true, for
example, the test proposed by Berkowitz (2001) is relatively parsimonious.