What is Value at Risk (VaR)?

Value at Risk (VaR) estimates how much a portfolio could lose over a given period, at a given level of confidence. A statement such as "a one-day 95% VaR of €1,500" means: on 95 days out of 100 the portfolio is expected to lose less than €1,500; on the other 5 it may lose more. Banks and regulators use VaR widely, and it turns an abstract volatility figure into an amount in euros.

It always has three ingredients: a time horizon (one day, ten days, a year), a confidence level (usually 95% or 99%), and a loss amount.

The formula

Assuming returns are normally distributed, the parametric VaR is:

\text{VaR}_{\alpha} = V \times (z_{\alpha} \, \sigma - \mu)

where $V$ is the portfolio's value, $\sigma$ and $\mu$ the volatility and expected return over the horizon, and $z_{\alpha}$ the quantile of the standard normal distribution: 1.645 for 95% and 2.326 for 99%. Over a single day $\mu$ is tiny and is often left out.

The historical VaR avoids the normal assumption: it sorts the actual past returns and reads off the loss at the 5th percentile (or the 1st, for 99%).

A simple illustration

A portfolio worth €100,000 has a daily volatility of 1%:

\text{VaR}_{95\%} = 100{,}000 \times 1.645 \times 1\% \approx 1{,}645

On roughly one trading day in twenty — about a dozen days a year — the loss is expected to exceed €1,645. At 99% confidence the threshold rises to about €2,326, exceeded on two or three days a year.

How to read it

It is a threshold, not a maximum. VaR says how often a loss should be exceeded, not by how much. The losses beyond it can be several times larger.
It grows with the horizon, roughly with the square root of time. A one-day VaR of €1,645 corresponds to about €26,000 over a year of 252 trading days, before expected returns are taken into account.
Compare like with like. A 95% one-day VaR and a 99% ten-day VaR of the same portfolio are very different numbers.

Limits

Fat tails. Real market returns have more extreme days than the normal distribution predicts, so parametric VaR tends to understate the risk of crashes.
Calm periods flatter it. A historical VaR estimated from quiet years says little about the next crisis.
It is blind beyond the threshold. Two portfolios with the same VaR can have very different worst cases. Conditional Value at Risk (CVaR) measures exactly that part.

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