Advances in Risk Management

RAFFAELE ZENTI, MASSIMILIANO PALLOTTA AND CLAUDIO MARSALA 215

Unconditional coverage property

A given modeljsatisfies this property if:

PrHitt+H,j(α,wt)= 1 =α (11.3)

then on average the model is correct.
There are a variety of tests able to verify this property. Kupiec (1995)
models{Hitt+H,j}as a sequence of independent draws from a binomial dis-
tribution with probability of occurrence equal toα. Campbell (2005) suggests
a test performed directly on the sample average of failures. It is well-known,
see among others Kupiec (1995) and Lopez (1999), that a key issue with these
tests is their statistical power: they exhibit low power in relatively small sam-
ples (for example, 250 days, as in the regulatory framework). This implies
that the chance of misclassifying an erroneous VaR model as accurate is high.

Independence property

A given modeljsatisfies this property if the sequence{Hitt+H,j}is inde-
pendently and identically distributed (i.i.d.), that is, failures do not exhibit
serial correlation. A model that exhibits the correct unconditional coverage
property but that violates the independence property might display clusters
of failures over time. This is undesirable, as the consequence could be an
exposure to financial losses for quite a few periods in a row. The two proper-
ties, for example, unconditional coverage and independence, can be jointly
expressed as:

Hitt+H,j(α,wt)

i.i.d.

∼ Binomial(α) (11.4)

where Binomial(α) is a binomial variate with probabilityα. In words,
the sequence of failures must be a series of independent and identically
distributed binomial events. The main contribution in this area is by
Christoffersen (1998) that depicts a joint test of these two properties, as they
are both crucial. If the two properties are tested in a standalone fashion,
there is the risk of not detecting a model which is poor in one or the other
property. Conversely, joint tests lack of the ability to identify models that
are deficient only in one of the two properties.
These methods are quite data-intensive, since they only make use of
whether or not a failure occurred. They need a lot of information. The reason
is simple: withαequal to 1 percent as in the banking regulatory framework,
or equal to 5 percent (more common among asset managers that use VaR for
portfolio management purposes), one has to collect a lot of history in order
to see some failures, if the model adopted is decent. The power of these tests
is rather scarce.