VaR, and VaR-like concepts, likely emerged from the market toil of the late 1980s and the increasing use of so-called quants to supply a degree of mathematical rigour to the task of estimating the likelihood of near-term potential losses of investments. Like many financial-mathematical concepts, its a rephrasing of the much older statistical concepts of confidence intervals and critical points (essentially). As is well-known, JP-Morgan (and later as RiskMetrics) standardised the concepts in the mid 1990s in the publication Risk Management: A Practical Guide.

Computation of VaR can be loosely grouped into two camps: the parametric and non-parametric. Parametric methods attempt to describe the underlying data points (here collective loses of some investments) by statistical distributions that are controlled by parameters, such as the mean and standard deviation for the normal distribution, for instance. Non-parametric methods side-step this parameterisation, and instead work with empirical data or apply a different class of transformations.

Much of the criticism directed at VaR methods, most notably post the events of 2007, is actually a criticism of the choice of parametric distribution underlying the most ubiquitous of analytical approaches - namely the use of the normal distribution to describe the distribution of potential losses. This led to a revived interest in methods such as H-VaR which do not make such assumptions.

H-VaR takes a conceptually simple approach to estimating VaR by using real past returns to estimate the likelihood of losses over different intervals. For example, suppose we calculate the 1-day 95% VaR for an equity using 100 days of (daily) data. The 95th percentile corresponds to the least worst of the worst 5% of returns. As we are using 100 days of data, the H-VaR corresponds to the 5th worst day.

Although the concept of H-VaR is simple, when used in a factor model portfolio risk framework, the implementation is not so straightforward. In our single stock example, it was easy enough to rank the realised daily P&Ls to compute H-VaR. In a factor model, for a stock the P&L becomes parametric P&L, parameterised by the factors in the factor model and the VaR is computed on this P&L series.

H-VaR is a conceptually simple method to compute VaR, which has several advantages over methods making distributional assumptions. When used within a factor model portfolio risk framework, it becomes implicitly a parametric method - depending upon the risk factors and risk model. Care must be taken in how results are interpreted, and in particular how idiosyncratic features are taken account of in such a model - as this would naturally be embedded in the realised P&L time-series which is not used in a factor model context.

Portfolio risk: historical VaR