RAFFAELE ZENTI, MASSIMILIANO PALLOTTA AND CLAUDIO MARSALA 2193Att+H, it is possible to compute the actual returns for theKportfolios,
collected in the vectorRt,t+H=Wt·rt,t+Ht.
4 Armed withVaRt,jandRt,t+H, it is possible to calculate the number (or
alternatively the relative frequency) of failuresn_failuresacross theK
portfolios through the Hit function.
5 We apply one or more statistical tests, in order to assess VaR modelj, eval-
uating ifn_failuresis significantly different from the theoretical frequency,
under the null hypothesis that the model is correct. This can be done using
most of the tests outlined in Section 11.3. The theoretical frequency of fail-
ures depends on the forecasted multivariate distribution of returns. In the
scholastic case of independent returns, the total number of failures fol-
lows a binomial distribution Pr[n_failures=x]=Binomial(α,K), while in
presence of some degree of comovements, it must be computed using the
estimated multivariate distribution. This involves computing a cumula-
tivedistributionfunction, whichcanbedonenumerically, forexample, by
Monte Carlo simulation. Given the degree of computing power currently
available in most cases this is not a major obstacle for most practitioners.
For Gaussian models like RiskMetrics® this computation is rather fast.
6 It is possible to check that, over time, the VaR model under examination
does not display serial correlation. This can be done, for example, by
monitoring the time series of failures{n_failurest}, where the matrix ofK
portfolios is kept fixed over time, that isWt=W. Basically we keep track
of a large number of portfolios’ failures.It is clear that there is a lot of additional information available on the per-
formance of one (or more) VaR models, as we focus on a high number of
portfolios on each assessment date, that is, we keep a cross-section perspec-
tive. After a small number of runs of this procedure, it is possible to judge a
model in a rather precise way. Of course, this can also be done using an his-
torical backtesting method (running the model in the past). However, this
requires a lot of historical data, with the associated problems, for example
some securities have no price in the past, there are corporate actions that
alter the history, and so on.
11.5 Applications
11.5.1 An intuitive example
Let us look at a simple example. We assume that the data generation process
(DGP) is anN-dimensional process such thatrt,t+H
i.i.d.
∼NormalN(0, I), where
I is the identity matrix. We assume thatNis equal to 1,000. We then estimate