Encyclopedia of Quantitative Finance, IV Volume Set

Value-at-Risk

The Value at Risk (VaR) is the maximum likely loss that a portfolio experiences over a specified horizon period, where the key term is “likely” to be interpreted in terms of a specified probability, known as a confidence level. For example, if the daily VaR at the 95% confidence level is $1m, then we would expect a maximum loss of $1m on the best prospective 95 days out of 100. More formally, the VaR at the confidence level p can be defined as the 100 × p% quantile of the loss distribution [8, 9, 16]. Note, therefore, that the VaR is measured in units of loss: a positive VaR means that the likely worst outcome is a monetary loss, whereas a negative VaR implies that the likely worst outcome is a profit. Also note that although we have defined the VaR here in terms of a quantile of the loss distribution, the VaR can equivalently be defined in terms of a quantile of the profit/loss (P/L) distribution (i.e., the negative of our loss distribution) or in terms of a quantile of the distribution of financial returns (see Market Risk).

The values of the parameters on which the VaR is defined, the horizon period and the confidence level, would be chosen by the user and depend on the context and the use to which the VaR is put. The horizon might be anything from a fraction of a day to decades ahead, but is typically somewhere between 1 day and a month. Confidence levels are ...