Value at Risk (VaR)

Value at risk (VaR) is a statistical estimate of how much a portfolio, trading desk or whole firm could lose over a given time horizon, at a given level of confidence, under normal market conditions. A statement such as "one-day 99% VaR is $5 million" means that, according to the model, losses should exceed $5 million on only about one trading day in a hundred. VaR is used by banks, asset managers, hedge funds, corporate treasuries and regulators to summarize market risk in a single number.

VaR spread widely after the US bank J.P. Morgan published its RiskMetrics methodology in 1994. In 1996 the Basel Committee on Banking Supervision allowed banks to use internal VaR models to set capital for trading risk. Its successor framework, the Fundamental Review of the Trading Book, replaces VaR with expected shortfall for regulatory capital, but VaR remains in wide everyday use for limits, reporting and back-testing.

Imagine a shop owner who looks back at years of daily takings and says: "On 99 days out of 100, I never lose more than 500 on a bad day." That sentence is a value at risk. It gives a sensible line for normal bad days, so the owner knows how much cash to keep in reserve.

What the sentence does not say is how bad the hundredth day can be. It might be a loss of 600, or a flood that wipes out the shop. VaR draws a line under ordinary bad luck but is silent about what lies beyond it, which is both why it is useful and why it has been heavily criticized.

There are three main methods:

• Parametric (variance–covariance). Assumes returns follow a normal distribution and uses the standard deviation and correlations of positions.
• Historical simulation. Revalues today's portfolio using actual price changes from a past window, often the last one or two years, and reads off the loss at the chosen percentile.
• Monte Carlo simulation. Generates thousands of random scenarios from a statistical model, useful for portfolios with options and other non-linear positions.

For a single position under the parametric method, the formula is:

• $z$ is the number of standard deviations for the confidence level: about 1.65 for 95% and 2.33 for 99%.
• $\sigma$ is the daily standard deviation of returns, written as a decimal.
• $V$ is the value of the position.
• $t$ is the horizon in days; the square-root rule assumes independent daily returns.

Worked example: a $10 million portfolio has a daily volatility of 1%. One-day 99% VaR is $2.33 \times 0.01 \times 10{,}000{,}000 = 233{,}000$ dollars. Scaled to ten days, it becomes $233{,}000 \times \sqrt{10} \approx 737{,}000$ dollars.

Under the 1996 Market Risk Amendment to the Basel Accord, banks with approved models held capital based on a 10-day, 99% VaR, multiplied by a factor of at least three, with the multiplier raised if back-testing showed too many days on which losses exceeded VaR. After the 2008 financial crisis showed that VaR models had understated risk, the Basel 2.5 reforms added a "stressed VaR" calibrated to a period of severe market stress.

The Fundamental Review of the Trading Book, finalized by the Basel Committee in January 2019 as part of the Basel III package, replaces VaR in internal models with expected shortfall at a 97.5% confidence level, with horizons that vary by how liquid each risk factor is. Implementation dates differ: the European Union, after two postponements, has set 1 January 2027 (with further relief under discussion in 2026), while the UK's Prudential Regulation Authority applies the new internal-model approach from 1 January 2028. VaR continues to be used for back-testing and internal limits.

Expected shortfall, also called conditional VaR, is the average loss on the days when losses exceed VaR:

Here $L$ is the loss and $\alpha$ the confidence level. If a desk's 97.5% VaR is $1 million and the losses in the worst 2.5% of days average $1.6 million, the expected shortfall is $1.6 million. Because it looks into the tail, it responds to fat tails that VaR ignores.

• Silent about the tail. VaR says nothing about how large losses beyond the threshold can be, so two portfolios with the same VaR may carry very different crash risk.
• Not always additive. Philippe Artzner and co-authors showed in 1999 that VaR can fail "subadditivity": combining two portfolios can produce a VaR higher than the sum of their separate VaRs, which conflicts with the logic of diversification. Expected shortfall does not have this flaw.
• Backward-looking. Models calibrated on calm periods understate risk just before turbulence, and they can miss a black swan entirely.
• Feedback effects. When many firms use similar VaR limits, a rise in volatility can force them all to sell at once, adding to systemic risk and liquidity risk.
• False comfort. Critics, including Nassim Nicholas Taleb, argue that a single precise-looking figure can encourage more leverage. Supervisors therefore pair VaR with stress tests and scenario analysis.

Firms also feed VaR into risk-adjusted return measures when allocating capital, as part of their wider management of financial risk.

• Basel Committee on Banking Supervision: Minimum capital requirements for market risk (January 2019)