**Value-at-risk** (VaR) is a probabilistic metric of market risk (PMMR) used by banks and other organizations to monitor risk in their trading portfolios. For a given probability and a given time horizon, value-at-risk indicates an amount of money such that there is *that* probability of the portfolio not losing more than *that* amount of money over *that* horizon. For example, if a portfolio has a one-day 90% value-at-risk of USD 3.2 million, such a portfolio would be expected to not lose more than USD 3.2 million, nine days out of ten.

Different choices for the probability and time horizon correspond to different **value-at-risk metrics**. Actually, a value-at-risk metric is specified with three items:

- a time horizon—one trading day in our example;
- a probability—90% in our example;
- a currency—USD in our example.

We name a value-at-risk metric by listing those three items followed by “VaR”, so the value-at-risk metric of our example is called one-day 90% USD VaR. Whenever the horizon is expressed in days, those are understood to be trading days.

As another example, under the Basel II Accord, Eurozone banks were required to monitor market risk with two-week 99% EUR VaR. A portfolio with two-week 99% EUR VaR of 21.3 million would be expected to not lose more than EUR 21.3 million over 99 two-week periods out of 100.

Value-at-risk has two advantages over many other metrics of market risk:

- Value-at-risk is prospective: For example, a firm might monitor market risk by tracking daily fluctuations in the value of a trading portfolio and reporting the 100-day rolling standard deviation of those values. Doing so would be easier than calculating value-at-risk, but it would be retrospective. The rolling standard deviation would show how risky a portfolio had been over the previous 100 days. It would say nothing about how risky the portfolio is today. For organizations to manage risk, they must know about risks while they are being taken. If a trader mis-hedges a portfolio, his employer needs to find out before a loss is incurred. Value-at-risk quantifies market risk while it is being taken, based on the current composition of the portfolio and current market conditions.
- Value-at-risk is broadly applicable: Many metrics of market risk are narrow. For example, if a portfolio’s delta for IBM stock is USD 800,000, that says nothing about its gamma for IBM stock. It says nothing about the portfolio’s delta for Microsoft stock. It says nothing about the portfolio’s overall market risk. With a single number, value-at-risk summarizes the market risk of an entire portfolio, taking into account—at least in theory—all holdings and all components of market risk.[2] As an analogy, risk metrics such as delta show us the trees. With value-at-risk, we see the forrest.

These advantages are not unique to value-at-risk. They are shared by other probabilistic metrics of market risk (PMMRs), such as volatility or semivariance. The reason we use value-at-risk and not some other PMMR is largely historical.

Value-at-risk emerged on trading floors during the 1980s, possibly in response to the SEC’s Uniform Net Capital Rule (UNCR), which required broker-dealers trading non-exempt securities to calculate capital charges for market risk.[1] In the early 1990s, three events dramatically expanded use of value-at-risk:

- The Group of 30 (1993) published a groundbreaking report on derivatives practices. It was influential and helped shape the emerging field of financial risk management. It promoted the use of value-at-risk by derivatives dealers and appears to be the first publication to use the phrase “value-at-risk.”
- JP Morgan (1994) released the first detailed description of value-at-risk as part of its free RiskMetrics service. This was intended to promote the use of value-at-risk among the firm’s institutional clients. The service comprised a technical document describing how to implement a VaR measure and a covariance matrix for several hundred key factors updated daily on the internet.
- In 1995, the Basel Committee on Banking Supervision implemented market risk capital requirements for banks. These were based upon a crude value-at-risk measure, but the committee also approved, as an alternative, the use of banks’ own proprietary VaR measures in certain circumstances.

These three initiatives came during a period of heightened concern about systemic risks due to the emerging—and largely unregulated—OTC derivatives market. It was also a period when a number of organizations—including Orange County, Barings Bank, and Metallgesellschaft—suffered staggering losses due to speculative trading, failed hedging programs or derivatives. Financial risk management was a priority for firms, and value-at-risk was rapidly embraced as the tool of choice for quantifying market risk. It was implemented by banks, investment firms, corporate treasuries, commodities merchants and energy merchants.

For a deeper discussion of value-at-risk, or for worked examples of actual value-at-risk measures, see my book *Value-at-Risk: Theory and Practice*. I distribute the latest edition free online at http://value-at-risk.net.

## Notes

- [1] See Section 1.9.1 of Holton (2014).
- [2] Value-at-risk is only applicable to market risk, though. Assets such as real estate or fine art cannot be marked to market on a day-to-day basis, rendering the notions of market risk and value-at-risk meaningless for them. Efforts to extend value-at-risk to credit risk or operational risk have generally been ill advised. See Section 0.2 of Holton (2013).

## References

- Group of 30 (1993).
*Derivatives: Practices and Principles*, Washington: Group of 30. - Holton, Glyn A. (2014).
*Value-at-Risk: Theory and Practice*, 2^{nd}Edition, e-book. - Morgan Guaranty (1994).
*RiskMetrics Technical Document*, 2^{nd}Edition, New York: Morgan Guaranty.

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