Value-at-risk (VaR) is a category of risk metrics that describe probabilistically the market risk of a trading portfolio. Value-at-risk is widely used by banks, securities firms, commodity merchants, energy merchants, and other trading organizations. Such firms could track their portfolios' market risk by using historical volatility as a risk metric. They might do so by calculating the historical volatility of their portfolio's market value over a rolling 100 trading days. The problem with doing this is that it would provide a retrospective indication of risk. The historical volatility would illustrate how risky the portfolio had been over the previous 100 days. It would say nothing about how much market risk the portfolio was taking today. For institutions 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 gives institutions the ability to do so. Unlike retrospective risk metrics, such as historical volatility, value-at-risk is prospective. It quantifies market risk while it is being taken. Measure time in trading days. Let 0 be the current time.
We know a portfolio's current market value
Value-at-risk can be measured in other ways. For example, bank regulations require that value-at-risk be calculated as a 99%-quantile of loss over a two-week horizon. Still other metrics are possible. We could measure value-at-risk as the standard deviation of portfolio value or the standard deviation of portfolio return. Essentially, any parameter of the distribution of a portfolio's future value can be used to measure value-at-risk. Let's formalize this. value-at-risk is applicable to any
liquid
portfolio—that is, any portfolio that can reasonably be marked to market
on a regular basis. Value-at-risk is not applicable to
illiquid assets, such as real estate or fine art. Value-at-risk considers a portfolio's performance over a specific horizon—a
trading day, two weeks, a month, etc. We call this the VaR horizon.
Value-at-risk is measured in a particular currency, USD in the
example above, but any
currency can be used. This is called the base currency.
Finally, the portfolio's market risk is summarized with a single
number. Informally, we called this a parameter of the distribution of
portfolio value. More formally, it is any function of both the portfolio's current value
We distinguish between a VaR measure and a VaR metric. A VaR measure is the procedure by which we arrive at a VaR measurement. It is some computational algorithm, which is typically coded on a computer. A VaR metric is our interpretation of the VaR measurement. In our examples, the VaR metric was one-day 90% USD VaR. Other example of VaR metrics are:
Value-at-risk became popular with trading organizations during the 1990s. It was during this period that the name "value-at-risk" entered the financial lexicon. However, VaR measures had been used long before this. An early user was Harry Markowitz. In his groundbreaking (1952) paper "Portfolio Selection", he adopted a VaR metric of single period variance of return and used this to develop techniques of portfolio optimization. In the early 1980s, the United States Securities and Exchange Commission (SEC) adopted a crude VaR measure for use in assessing the capital adequacy of broker-dealers' trading non-exempt securities. A few years later, Bankers Trust implemented a VaR measure for use with its RAROC capital allocation system. During the late 1980s and early 1990s, a number of institutions implemented VaR measures to support capital allocation or market risk limits. In the early 1990s, three events popularized value-at-risk as a practical tool for use on trading floors:
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 financial firms, corporate treasuries, commodities merchants, and energy merchants. For a detailed discussion of how value-at-risk is calculated, see the article measuring value-at-risk.
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. Its
market value
in one trading day is unknown. It is a
random variable. We may ascribe it a probability distribution. With
value-at-risk,
we summarize a portfolio's market risk by reporting some parameter of this
distribution. For 
two-week 99% 
