Value at risk: a one-sided confidence interval

In short

• A confidence interval for a mean is only meaningful when the data have a stable mean and a finite, stable variance. Trending price levels fail that test, so analysts work with returns instead.
• Value at risk (VaR) is the lower bound of a one-sided confidence interval: a 99% one-month VaR of EUR 2 million means a 1% chance of losing more than EUR 2 million in a month.
• VaR can be stated as a percentage of value or as a money amount.
• Parametric (variance–covariance) VaR with a zero mean return: $\text{VaR} = z \times \sigma \times P$, with one-sided z of 1.645 (95%) or 2.33 (99%).
• Other methods: historical simulation and Monte Carlo simulation.
• Limits of parametric VaR: assumes normal returns, the same distribution at every horizon, that history is a guide, and linear exposures (a poor fit for options).