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.

ES captures the magnitude of losses in the tail (not just the threshold), so it is sensitive to the shape and fatness of the tail beyond VaR. This makes ES more informative about extreme outcomes.