Value at Risk (VaR)

Value at risk (VaR) is one of the most widely used risk measures in finance. It answers the question: over a given period, what is the loss we should not exceed with a certain level of confidence? A one day 99% VaR of $1 million means that on 99 days out of 100, losses should be smaller than $1 million. Banks, funds and trading desks use VaR to set limits, report risk and, for banks, historically to calculate regulatory capital. Its simplicity is its strength and its weakness: it says nothing about how bad losses can be on the remaining days.

The three ingredients#

Three calculation methods#

A parametric example#

VaR = z × Daily volatility × Portfolio value

The z score is about 1.645 for 95% and 2.326 for 99% confidence, from the normal distribution. See Normal Distribution.

Historical VaR in practice#

Take the last 500 daily returns of the current portfolio's holdings, compute what each day would mean for today's positions, sort the results and pick the 5th percentile loss for 95% VaR (the 25th worst of 500). It captures real fat tails and correlations from that window, but if the window contains no crisis, VaR will look deceptively low.

Limitations#

VaR in the 2008 crisis#

Many banks' VaR models showed moderate risk before 2008, because they relied on calm recent history and assumed liquid markets. Losses then exceeded VaR far more often than the models implied. The experience led regulators to add stressed VaR and, later, to move toward expected shortfall for bank market risk capital. See The 2008 Financial Crisis.

Backtesting VaR#

Compare actual daily losses with VaR. At 99% confidence, losses should exceed VaR on about 1% of days, about 2 or 3 days a year. Many more exceptions mean the model underestimates risk. Banks are required to backtest their models.

Using VaR well#

1. Use it with stress tests and expected shortfall. See Stress Testing and Scenario Analysis.
2. Set limits in VaR terms for desks and strategies. See Risk, Position, Loss and Drawdown Limits.
3. Decompose VaR by position to see risk drivers. See Risk Contribution and Risk Decomposition.
4. Never treat it as the worst case.

Frequently asked questions#

What is value at risk?#

An estimate of the loss a portfolio should not exceed over a set period with a given confidence level, such as 99% over one day.

How is VaR calculated?#

With parametric formulas assuming normal returns, historical simulation using past returns, or Monte Carlo simulation from a model.

What is the main weakness of VaR?#

It says nothing about the size of losses beyond the threshold, and it often underestimates risk when returns have fat tails or markets turn illiquid.

Next, learn the measure that looks beyond VaR in Expected Shortfall (CVaR).