Value at Risk (VaR)
A loss threshold: “on a normal day, losses should not exceed this amount with X% confidence.”
VaR is a percentile (quantile) of the loss distribution over a fixed horizon, for example the 99th percentile of one-day losses.
It answers: “How bad is a bad day?” but not “How bad can it get beyond that bad day?” — tail severity is left out.
In practice, VaR is estimated from models (normal, t, GARCH), historical data, or Monte Carlo, and results can change a lot when tails are fat.
Engineering analogy: designing a bridge to withstand loads up to a chosen stress percentile, but not explicitly specifying what happens in truly extreme events.
Simulated daily returns under a normal model. The shaded red tail shows the 5% worst days. The solid red line marks VaR as a loss threshold; the dashed red line marks the average loss on those bad days (Expected Shortfall).
The grey bars show a normal return distribution centred on μ with spread σ. The solid red line is VaR: only about 5% of simulated days fall to the left. The dashed red line (ES) is the average of those bad days.
As you increase σ, the histogram widens, and both VaR and ES move further left (bigger losses). The portfolio slider translates those percentages into money: a risk manager will often read “95% 1-day VaR of 2.70% on a 10m book” directly as an amount at risk on a normal day.
Control analogy: VaR is like a design limit for typical disturbances on a system state. ES tells you how severe the excursions are once you actually cross that limit — which is why regulators now often prefer ES to VaR for capital.