The VaR answers the question: what the maximum loss with a specified confidence level over a given time horizon could be. To put it another way, the VaR is a measure of the maximum potential change in the value of a portfolio of financial instruments with a given probability over a defined horizon.

For example, if we divide the portfolio return distribution into two areas of 95 percent and 5 percent, as shown in Figure below then an investor may anticipate losing less than the VaR over the given time horizon, for instance, one day. The VaR can be specified over various time frames (commonly, between one day and one month) and probability of loss levels (most often between 1 percent and 10 percent). The VaR can be expressed as a percentage or in absolute profit-loss context (for example, US$). Obviously, to calculate the VaR over one day horizon, we should analyze the daily distribution of returns, for one month horizon - monthly, and so on.

The explained idea derives the formal definition of the VaR as follows:

where

*V* = the portfolio value*V _{T}* = the portfolio value at the end of time horizon

*T*

- relative change in the portfolio value

*F(V)*= cumulative distribution function of

*V*(cdf)

*c*= the confidence level.

The VaR may reflect either underperformance of the portfolio relative to its mean or absolute losses depending on the benchmark used. In the first case, it is calculated as a loss relative to a zero return on a portfolio:

whereas a drawdown relative the mean of distribution is calculated as

Quant KB » Risk Management» Value-at-Risk