Valuation & Risk Models

VaR is the maximum loss not exceeded over a given horizon at a stated confidence level; equivalently, the loss that will be exceeded only with probability (1 - c).

A confidence level (c), a time horizon (holding period), and a base currency/measurement unit. VaR is meaningless without the horizon and confidence level.

Absolute VaR is loss relative to zero (initial value), VaR = -(μ - zασ)·V. Relative VaR is loss relative to the expected value, VaR = zα·σ·V (ignores the mean μ). For short horizons μ ≈ 0 so the two nearly coincide.

VaR only marks the threshold of the tail; it says nothing about how large losses beyond that threshold can be. With fat tails the losses past VaR can be far worse than a normal model implies, which VaR ignores.

ES is the expected (average) loss conditional on the loss exceeding VaR: ES = E[L | L > VaR]. It averages the losses in the tail beyond the VaR quantile, so ES ≥ VaR.