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Value at risk (VaR) is a measure of the risk of loss of investment/Capital. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses. For a given portfolio, time horizon, and probability p, the p VaR can be defined informally as the maximum possible loss during that time after excluding all worse outcomes whose combined probability is at most p. This assumes mark-to-market pricing, and no trading in the portfolio. For example, if a portfolio of stocks has a one-day 95% VaR of

1 million, that means that there is a 0.05 probability that the portfolio will fall in value by more than

1 million over a one-day period if there is no trading. Informally, a loss of

1 million or more on this portfolio is expected on 1 day out of 20 days (because of 5% probability). More formally, p VaR is defined such that the probability of a loss greater than VaR is (at most) (1-p) while the probability of a loss less than VaR is (at least) p. A loss which exceeds the VaR threshold is termed a "VaR breach". It is important to note that, for a fixed p, the p VaR does not assess the magnitude of loss when a VaR breach occurs and therefore is considered by some to be a questionable metric for risk management. For instance, assume someone makes a bet that flipping a coin seven times will not give seven heads. The terms are that they win

100 if this does not happen (with probability 127/128) and lose

12,700 if it does (with probability 1/128). That is, the possible loss amounts are

0 or

12,700. The 1% VaR is then

0, because the probability of any loss at all is 1/128 which is less than 1%. They are, however, exposed to a possible loss of $12,700 which can be expressed as the p VaR for any p ≤ 0.78125% (1/128). VaR has four main uses in finance: risk management, financial control, financial reporting and computing regulatory capital.

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