Risk Measures: Rationality and Diversification

Abstract

When there is uncertainty about interest rates (typically due to either illiquidity or defaultability of zero coupon bonds) the cash-additivity assumption on risk measures becomes problematic. When this assumption is weakened, to cash-subadditivity for example, the equivalence between convexity and the diversification principle no longer holds. In fact, this principle only implies (and it is implied by) quasiconvexity. For this reason, in this paper quasiconvex risk measures are studied. We provide a dual characterization of quasiconvex cash-subadditive risk measures and we establish necessary and sufficient conditions for their law invariance. As a byproduct, we obtain an alternative characterization of the actuarial mean value premium principle.

Representation results for law invariant time consistent functions

“…Building on the Nagumo-Kolmogorov-de Finetti Theorem, Cerreia-Vioglio et al [7] recently proved that a function c 0 : L ∞ → R is of the form (5) if and only if c 0 is normalized on constants, law invariant, monotone, has the Lebesgue property and satisfies…”

Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals

“…The question of model uncertainty was incorporated in the axiomatic approaches starting with the work of Gilboa and Schmeidler [35], later followed by, among others, Maccheroni et al [57] and Cerreia-Vioglio et al [11]; see also [10,13]. Axioms are then placed on preferences over a set of so-called random lotteries, also referred to as horse acts.…”

“…[6,28], such results are yet lacking for the quasiconvex case. 13 Finally, recall that the risk preferences of the investor are modelled via a standard continuous and concave utility function in (2.7). While this is a natural assumption, the value function u(x) satisfies only weaker properties.…”

Entropy Coherent and Entropy Convex Measures of Risk

“…While assuming that a liquidly traded risk-less asset exists is common in the literature on coherent and convex measures of risk, some interesting recent papers depart from this assumption, by replacing translation invariance by cash-subadditivity; see El Karoui and Ravanelli [16] and Cerreia-Vioglio et al [9]. For a setting in which the regulator accepts an array of possibly risky securities instead of restricting to cash, see, e.g., Frittelli and Scandolo [21].…”

“…This induces that X can be viewed as the minimal amount of capital that the economic agent holding X is required to commit and add to the financial position to make the position acceptable. We note that translation invariance in this context is, in fact, an assumption of cash-additivity (El Karoui and Ravanelli [16] and Cerreia-Vioglio et al [9]). It amounts to assuming that there are no frictions on the risk-free asset market and, in particular, that an institution can borrow and lend cash at the same risk-free rate.…”