RAFFAELE ZENTI, MASSIMILIANO PALLOTTA AND CLAUDIO MARSALA 217The crucial point about this methods is that they need the entire cumulative
probability distribution of portfolio’s returns estimated at timetwith model
j, for example,Ft,j(·). This is not a trivial requirement, as not every VaR model
has the goal to predict the entire distribution, or at least it does not pretend
to predict well the entire distribution. The reason is that VaR models focus
on the left tail of returns’ distribution. Typical examples are models based on
Extreme Value Theory or Quantile Regression. They could possibly perform
very well on lower quantiles, say 1 percent or 5 percent quantiles, but they
cannot provide useful information on less extreme quantiles.
Tests based on a given Loss function
Alternative methods are based on loss functions that assign numerical scores
to VaR estimates according to some metric that measures the impact of
failures. Formally the loss function is any function of the general form:
Lt+H,j(VaRt,j(α,wt,H),Rt,t+H,α,!) (11.8)that depends on VaR estimates and realized returns (typically according to a
distance measure, because realized returns far below VaR have to be penal-
ized), on the probability levelα(as differentαcan be evaluated differently),
and on some parameter!that reflects the specific concerns that this func-
tion has to take into consideration. The Hit function (11.2) is a very simple
example of (11.8). Different VaR models can be evaluated based on the scores
arising from (11.8).
Lopez (1999), who pioneered this approach, suggests this methodology
as a flexible alternative to statistical hypothesis testing. Once a regulatory
loss function is specified, he argues, VaR estimates could be compared across
time and across financial institutions. Anyway, it might be a difficult task
to specify a proper loss function. Another drawback comes from the fact
that, in order to calibrate the assessment procedure, it is necessary to make
assumptions about the distribution of portfolio’s returns.
11.4 An extension: the cross-section approach
So far we have seen that there is an intrinsic difficulty in testing the accuracy
of a VaR model: intuitively, this is becauseαis usually small, thus failures are
rare events, and one needs large data-sets to test rare events. Data limitation
is the issue. So it is crucial to increase the amount of information at our
disposal when assessing a VaR model.
The starting point is that a portfolio’s VaR depends on its assets’ weights
and a forecast of the multivariate distribution of the assets’ returns. As
portfolio weights are exogenous, for example are given, at the very roots