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 diversi…cation principle no longer holds. In fact, this principle only implies (and it is imp...

We study a framework for constructing coherent and convex measures of risk that is inspired by infimal convolution operator, and which is shown to constitute a new general representation of these classes of risk functions. We then discuss how this scheme may be effectively applied to obtain a class of certainty equivalent measures of risk that can directly incorporate preferences of a rational ...

Regulation and risk management in banks depend on underlying risk measures. In general this is the only purpose that is seen for risk measures. In this paper we suggest that the reporting of risk measures can be used to determine the loss distribution function for a financial entity. We demonstrate that a lack of sufficient information can lead to ambiguous risk situations. We give examples, sh...

In the context of complete nancial markets, we study dynamic measures for the risk associated with a given liability C at time t = T , of the form (x; C) := sup 2D inf ()2A(x) S 0 (T) + : Here x is the initial capital available at time t = 0, A(x) the class of admissible portfolio strategies, S 0 () the price of the risk-free instrument in the market, P = fP g 2D a suitable family of probabilit...

The paper considers modeling of risk-averse preferences in stochastic programming problems using risk measures. We utilize the axiomatic foundation of coherent risk measures and deviation measures in order to develop simple representations that express risk measures via specially constructed stochastic programming problems. Using the developed representations, we introduce a new family of highe...

We define (d, n)−coherent risk measures as set-valued maps from Ld into IR satisfying some axioms. We show that this definition is a convenient extension of the real-valued risk measures introduced by Artzner, Delbaen, Eber and Heath (1998). We then discuss the aggregation issue, i.e. the passage from IR−valued random portfolio to IR−valued measure of risk. Necessary and sufficient conditions o...

Can the usage of a risky numeraire with a greater than risk free expected return reduce the capital requirements in a solvency test? I will show that this is not the case. In fact, under a reasonable technical condition, there exists no optimal numeraire which yields smaller capital requirements than any other numeraire. 1 Statement and Proof of the Result Can the usage of a risky numeraire wit...

Coherent, convex and monetary risk measures were introduced in a setup where uncertain outcomes are modelled by bounded random variables. In this paper, we study such risk measures on Orlicz hearts. This includes coherent, convex and monetary risk measures on L-spaces for 1 ≤ p < ∞ and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. We prove that ...

The paper provides an axiomatic characterization of dynamic risk measures for multi-period financial positions. For the special case of a terminal cash flow, we require that risk depends on its conditional distribution only. We prove a representation theorem for dynamic risk measures and investigate their relation to static risk measures. Two notions of dynamic consistency are proposed. A key i...

Measures of risk appear in two categories: Risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures (safety measures). Pure risk measures (risk deviation measures) are natural generalizations of the standard deviation. While pur...