Liquidity, Risk Measures, and Concentration of Measure

This paper studies curves of the form (ρ(λX))_λ≥0, called risk profiles, where ρ is a convex risk measure and X a random variable. Financially, this captures the sensitivity of risk to the size of the investment in X, which the original axiomatic foundations of convex risk measures suggest to interpret as liquidity risk. The shape of a risk profile is intimately linked with the tail behavior of X for some notable classes of risk measures, namely shortfall risk measures and optimized certainty equivalents, and shares many useful features with cumulant generating functions. Exploiting this link leads to tractable necessary and sufficient conditions for pointwise bounds on risk profiles, which we call concentration inequalities. These inequalities admit useful dual representations related to transport inequalities, and this leads to efficient uniform bounds for risk profiles for large classes of X. Several interesting mathematical results emerge from this analysis, including a new perspective on nonexponential concentration estimates and some peculiar characterizations of classical transport inequalities. Lastly, the analysis is deepened by means of a surprising connection between time consistency properties of law invariant risk measures and the tensorization of concentration inequalities.