218 EVALUATING VALUE-AT-RISK ESTIMATES: A CROSS-SECTION APPROACHof any process aimed to assess VaR model there is an evaluation of the
forecasted multivariate distribution of assets returns. The problem is that all
the methods outlined in section 11.3 lose the multivariate framework as they
focus on portfolio’s VaR and portfolio’s actual returns that are conceptually
univariate objects. We wish to make this point clear.
Considering an investment universe made ofNassets, the actual returns
rt,t+Hon given period [t,t+H]areanN-dimensional process, portfolio
weights are collected in aN-dimensional column vector, but the portfo-
lio return, for exampleRt,t+H=r′t,t+H·wt, is a univariate process. Hence a
‘portfolio perspective’ implies a reduction of dimensionality, fromNto 1.
Inasimilarway, whenestimatingVaRonestarts(atleastinprinciple)with
a forecast of the multivariate probability density function of returns, a very
granular piece of information. However, in order to estimate a portfolio’s
VaR, this granular information is combined with the assets’ weight, and
because of this mapping, at the end of the process there is a univariate
object. Again, a portfolio perspective involves a decrease of the number
of dimensions we are dealing with. This has a big impact. Think about
a portfolio whose weights are unequally distributed: a small number of
assets have a large weight, while the others exhibit small weights. Note
that this is a fairly common situation in the asset management industry, as
many portfolios have approximately the same structure of indexes made of
hundreds of securities. Passive mutual funds are a typical example, as well
as low tracking-error portfolios, very common in the industry (just to name
two real-world situations). Now, if a large number of securities has a small
weight, a portfolio’s VaR estimate would be determined mainly by a subset
of the information content of multivariate distribution: the subset that relates
to larger positions. This tends to obscure VaR model capability, as the model
could be overall poor but, by chance, could be good at estimating the risk of
the dominating assets. This could lead to an erroneous positive assessment,
and subject to more sampling error. One could argue that small positions are
not important by definition. The key point is that a portfolio’s composition
changesovertime: asmallpositiontodaycouldbealargepositiontomorrow.
Therefore the idea is to recover more information from the multivari-
ate estimated distribution, which allows us to measure more correctly the
forecasting capability of the model under consideration.
The methodology is simple and is based on the following steps:
1 At timet, considering an investment universe made ofNassets, we
randomly generateKportfolios (whereKis large number, say some hun-
dreds), whose weights are collected in theK×NmatrixWt(each column
is a portfolio).
2 We estimate VaR for each portfolio using modeljaccording to (1), so that
we have a vector of VaRs,VaRt,j(α,Wt,H).