Risk Measures in Depth: VaR, CTE, and Coherent Measures

Properties of Risk Measures

A coherent risk measure satisfies four axioms: subadditivity (rho(X+Y) <= rho(X) + rho(Y), so diversification is not penalized), positive homogeneity (rho(lambda*X) = lambda*rho(X)), translation invariance (rho(X+c) = rho(X) + c), and monotonicity (if X <= Y then rho(X) <= rho(Y)). VaR violates subadditivity for non-elliptical distributions, meaning the VaR of a portfolio can exceed the sum of individual VaRs. This is problematic for risk aggregation and capital allocation. CTE (Conditional Tail Expectation) is coherent for continuous distributions.

Advanced Risk Measures

Spectral risk measures are coherent measures of the form rho_phi(X) = integral from 0 to 1 of phi(p)*VaR_p(X)dp, where phi is a non-negative, increasing weight function integrating to 1. CTE is a special case with phi constant above threshold p. Distortion risk measures use g(S(x)) where g is a concave distortion function. Wang's transform uses g(u) = Phi(Phi^{-1}(u) + lambda) where Phi is the standard normal CDF. For Exam MAS-II, understand the axioms of coherent measures, why VaR fails subadditivity, and how CTE and distortion risk measures satisfy the axioms. Capital allocation using the Euler principle is also tested.