Risk Measures and Capital Requirements for Processes

Abstract

In this paper we propose a generalization of the concepts of convex and coherent risk measures to a multiperiod setting, in which payoffs are spread over different dates. To this end, a careful examination of the axiom of translation invariance and the related concept of capital requirement in the one-period model is performed. These two issues are then suitably extended to the multiperiod case, in a way that makes their operative financial meaning clear. A characterization in terms of expected values is derived for this class of risk measures and some examples are presented.

Cash Sub-additive Risk Measures and Interest Rate Ambiguity

“…✷ Next corollary shows a representation of R where the penalty functional depends on constants c ∈ [0, 1] and probability measures. Frittelli and Scandolo (2006) provide examples of general capital requirement with similar representations. It is interesting to observe that, among these, the only capital requirement that satisfies the property of cash sub-additivity is the one reflecting the agent's temporal risk aversion, which is related to the uncertainty in the numéraire.…”

“…✷ Unfortunately, the procedure of computing current reserve amounts discounting forward risk measures (given by q 0 in equation (2.9)) is feasible only when the zero coupon bonds for the relevant maturities are available on the market. In this case the functional q 0 in equation (2.9) is an example of the general capital requirement defined in Frittelli and Scandolo (2006).…”

Conditional and dynamic convex risk measures

“…An interesting issue is the comparison between the initial preference structure, X Y ⇔ E P u(X) ≥ E P u(Y ), and the derived one, X Y ⇔ ρ u (X) ≤ ρ u (Y ), where ρ u is the risk measure induced by u. In [10] the entropic risk measures have been characterized as the only ones -apart from those induced by linear utilities -for which and coincide. ♦…”

Stable solutions for optimal reinsurance problems involving risk measures

“…Among others, Goovaerts et al (2004) intro duced the consistent risk measures, also studied in Burgert and Rüschendorf (2006), Frittelli and Scandolo (2005) analyzed risk measures for stochastic processes, and Rockafellar et al (2006) de fined the deviations and the expectation bounded risk measures.…”