VaR is the measure of the maximum (worst case) trading loss for a given portfolio over a certain holding period and for a given confidence interval. The VaR number is thus essentially determined by two parameters, - (holding) time period and a confidence level, and is a measure of the loss (expressed in say Rupees crore) on the portfolio that will not be exceeded by the end of the time period with the specified confidence level.

The most difficult part in VaR estimations is the derivation of the portfolio returns distribution. Two popular approaches in the literature for the calculation of VaR are variance-covariance analysis and historical simulation. Variance-covariance analysis relies on the assumption that financial returns are normally distributed. This method is easy to implement because the VaR can be computed from a simple linear formula with variances and co-variances of the returns as the only inputs. Its major drawback is the assumption that financial market returns are normally distributed, an assumption that has been shown to be invalid in thousands of empirical studies on asset returns. Financial returns are typically characterized by fat-tails and volatility clustering. Fat-tail property of asset returns implies that losses are much more frequent than predicted by the variance-covariance analysis. The variance-covariance analysis is particularly weak where the demands from a VaR model for regulatory purposes and risk control are strong, i.e. in the prediction of extreme quantiles or large losses.

Another variant of variance-covariance analysis is the Exponential weighting approach. This approach applies exponentially declining weights to past returns to calculate conditional volatilities. This technique is justified by the presence of conditional heteroskedasticity or volatility clustering in the data, meaning that a volatile day is typically followed by volatile days. The exponential approach also has the drawback that a conditional normality assumption needs to be made to calculate the VaR of a portfolio from its conditional volatility, an assumption that, more often than not, is not satisfied by financial data. Although the exponential smoothing approach addresses the issue of non-normality in the (unconditional) distribution of returns, it may not be applicable for regulatory VaR purposes on three grounds. First, while daily returns reflect strong volatility clustering, they can hardly be detected in bi-weekly returns such as the regulatory 10-day holding period. Second, the volatility clustering observed in the data largely emanates from medium and small range volatility periods. Extreme events, such as losses at or beyond 99% confidence interval scatter rather independently over time. Finally, as has been established by the theoretical and empirical literature, interest rate and bond return processes typically display a complicated dependence structure in both mean and variance, thereby making simple exponential weighting schemes - so often used in computing the equity VaR - inapplicable.

The historical simulation (HS) method, by using empirical percentiles from the historical return distribution, gets around the problem of making distributional assumptions. By applying the full empirical market return distribution to all the items in the current trading portfolio, the outcome exactly reflects the historical frequency of the large losses over specific data window. A second advantage of this approach compared to variance-covariance analysis is that it can incorporate non-linear positions, such as derivative positions, in a natural way, a property that is also useful in the context of fixed income portfolios. The problem with the HS method is that it is very sensitive to the particular data window, which the Basle committee has chosen to be at least one year of past returns. In other words, whether returns from a highly volatile or a crash period is included or not makes a huge difference for the value-at-risk predicted. Hence VaR predictions based on HS exhibit high variability.

A hybrid approach proposed by Boudokh, Richardson and Whitelaw (Risk, 1998) draws on the strengths of the exponential and the HS approaches to estimate the percentiles of returns directly using declining weights on past observations. Ordering returns over the historical simulation period, the hybrid approach attributes exponentially diminishing weights to each observation in building the conditional empirical distribution. Although this method combines the strengths of above mentioned methods, it still suffers from the tail-discreteness problem that we discussed earlier and it may not be very effective particularly in predicting the extremely large losses.