Convex risk measure

A convex risk measure keeps two of the four coherence axioms, monotonicity and translation invariance (adding an amount m of cash reduces risk by m), and replaces the other two, subadditivity and positive homogeneity, with a single condition: ρ(λX + (1 - λ)Y) ≤ λρ(X) + (1 - λ)ρ(Y) for every λ between 0 and 1. Hans Föllmer and Alexander Schied introduced it in "Convex measures of risk and trading constraints," Finance and Stochastics, vol. 6, no. 4, 2002, pp. 429-447; Marco Frittelli and Emanuela Rosazza Gianin did so independently, with a slightly different definition, in "Putting order in risk measures," Journal of Banking & Finance, vol. 26, no. 7, the same year. The motive is liquidity. Positive homogeneity says that doubling a position doubles its risk; a trader who has to exit ten times the usual volume knows that is false, since market impact grows faster than size. Convexity keeps the part of diversification that matters, mixing two positions creates no risk, without imposing proportionality. Every coherent measure is convex; the converse does not hold. The representation theorem gives ρ(X) = sup over Q of (E_Q[-X] - α(Q)): one takes the worst of the probabilistic scenarios Q, each penalized by a function α. Coherent measures are the case where α takes only the values 0 and +∞, a set of scenarios admitted without penalty. The textbook example is the entropic measure, ρ(X) = (1/θ) ln E[e^(-θX)], whose penalty is the relative entropy of Q with respect to P, divided by θ; written on a loss, it is the actuaries' exponential premium principle. Convexity has a practical cost. Capital allocation by the Euler rule assumes homogeneity, and it no longer applies as is: the contributions of the lines no longer add up to the total.