Value at risk

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.^[1]

For example, if a portfolio of stocks has a one-day 5% 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".^[2]

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).^[3]

VaR has four main uses in finance: risk management, financial control, financial reporting and computing regulatory capital. VaR is sometimes used in non-financial applications as well.^[4] However, it is a controversial risk management tool.

Important related ideas are economic capital, backtesting, stress testing, expected shortfall, and tail conditional expectation.^[5]

^ Jorion, Philippe (2006). Value at Risk: The New Benchmark for Managing Financial Risk (3rd ed.). McGraw-Hill. ISBN 978-0-07-146495-6.
^ Holton, Glyn A. (2014). Value-at-Risk: Theory and Practice second edition, e-book.
^ Cite error: The named reference Einhorn II was invoked but never defined (see the help page).
^ McNeil, Alexander; Frey, Rüdiger; Embrechts, Paul (2005). Quantitative Risk Management: Concepts Techniques and Tools. Princeton University Press. ISBN 978-0-691-12255-7.
^ Dowd, Kevin (2005). Measuring Market Risk. John Wiley & Sons. ISBN 978-0-470-01303-8.

[Jorion-1] Jorion, Philippe (2006). Value at Risk: The New Benchmark for Managing Financial Risk (3rd ed.). McGraw-Hill. ISBN 978-0-07-146495-6.

[Holton-2] Holton, Glyn A. (2014). Value-at-Risk: Theory and Practice second edition, e-book.

[Einhorn_II-3] Cite error: The named reference Einhorn II was invoked but never defined (see the help page).

[McNeil-4] McNeil, Alexander; Frey, Rüdiger; Embrechts, Paul (2005). Quantitative Risk Management: Concepts Techniques and Tools. Princeton University Press. ISBN 978-0-691-12255-7.

[Dowd-5] Dowd, Kevin (2005). Measuring Market Risk. John Wiley & Sons. ISBN 978-0-470-01303-8.

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