Risk Measures: VaR, TVaR, and Expected Shortfall for STAM

Value at Risk (VaR)

VaR at level p is the p-th percentile of the loss distribution: VaR_p(X) = F^{-1}(p). It answers "what loss level will not be exceeded with probability p?" For example, VaR at 99% is the loss exceeded only 1% of the time. VaR is simple to interpret but has limitations: it provides no information about the severity of losses beyond the threshold, and it is not subadditive, meaning the VaR of a portfolio can exceed the sum of individual VaRs.

Tail Value at Risk (TVaR)

TVaR at level p (also called Conditional Tail Expectation or CTE) is the expected loss given that the loss exceeds VaR_p: TVaR_p(X) = E[X | X > VaR_p(X)]. For continuous distributions, TVaR_p = (1/(1-p)) times the integral from VaR_p to infinity of x*f(x)dx. TVaR is a coherent risk measure (satisfying subadditivity, positive homogeneity, translation invariance, and monotonicity). Exam STAM tests computation of VaR and TVaR from parametric distributions like exponential, Pareto, and lognormal, as well as from empirical data.