Advances in Risk Management

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

propose a methodology based on cross-section analysis of randomly gener-
ated portfolios. This exploits in a better way the information content of the
multivariate distribution of returns used to estimate VaR.
The plan of this chapter is as follows. The next section (11.2) formalizes
the notion of VaR and section 11.3 reviews existing backtesting methods. In
section 11.4 we present our methodology and in section 11.5 we give some
examples that show how this approach can be successfully applied. Section
11.6 concludes.

11.2 Value-at-risk

In general, VaR models attempt to forecast the time-varying distributions
of portfolio returns, and different models provide different estimates. In
addition, VaR estimates vary over time as market conditions and portfolio
composition change. From a formal point of view, throughout the rest of
this chapter, we will refer to a portfolio’s value at risk as theαquantile of
portfolio’s return distribution:

VaRt,j(α,wt,H)=F−t,j^1 (α|It,wt) (11.1)

wherewtis the vector of weights at timetfor a given portfolio,His the time
horizon (say 10 days),F−t,j^1 (·) is the inverse of the cumulative probability

distribution of portfolio’s returns, estimated at timetusing modelj, and It
is the information set on market conditions available at timetto estimate
VaR. For a survey on VaR models and current risk management practice, see
Jorion (2001).

11.3 Review of existing methods for backtesting

An assortment of tests has been proposed so far to check the accuracy of VaR
models. We try to identify a taxonomy of the most popular testing methods.

11.3.1 Tests based on the hit function

These are by far the most popular. Consider the event thatRt,t+H(wt), the
portfolio return on given period [t,t+H], is less thanVaRt,j(α,wt,H), for
example, its reported VaR at timet. This event is a VaR failure.
The Hit function for modeljcounts failures as follows:

Hitt+H,j(α,wt)=

{

1 if Rt,t+H≤VaRt,j(α,wt,H)

0 if Rt,t+H>VaRt,j(α,wt,H)

(11.2)

Thus, the Hit function over time is a binary time series,{Hitt+H,j}, that
registers the history of VaR failures. As pointed out by Christoffersen (1998),
the sequence{Hitt+H,j}should satisfies two fundamental properties.