In this paper, we give the axiomatic characterization of risk measures and discuss the treads of developments in this area. The main recently proposed risk measures are presented, and their properties and relations are discussed. The corresponding versions of dynamic risk measure are also briefly introduced.

In this paper we introduce convex imprecise previsions as a special class of imprecise previsions, showing that they retain or generalise most of the relevant properties of coherent imprecise previsions but are not necessarily positively homogeneous. The broader class of weakly convex imprecise previsions is also studied and its fundamental properties are demonstrated. The notions of weak conve...

We extend the definition of a convex risk measure to a conditional framework where additional information is available. We characterize these risk measures through the associated acceptance sets and prove a representation result in terms of conditional expectations. As an example we consider the class of conditional entropic risk measures. A new regularity property of conditional risk measures ...

Because of their simplicity, risk measures are often employed in financial risk evaluations and related decisions. In fact, the risk measure ρ(X) of a random variable X is a real number customarily determining the amount of money needed to face the potential losses X might cause. At a sort of second-order level, the adequacy of ρ(X) may be investigated considering the part of the losses it does...

This note extends the concept of a coherent risk measure to make it more consistent with a firm’s capital budgeting perspective. A coherent risk measure defines the risk of a portfolio to be that amount of cash that must be added to the portfolio such that it becomes acceptable to a regulator. As such, a coherent risk measure implicitly assumes that the firm has already made its capital budgeti...

Tail risk refers to the risk associated with extreme values and is often affected by extremal dependence among multivariate extremes. Multivariate tail risk, as measured by a coherent risk measure of tail conditional expectation, is analyzed for multivariate regularly varying distributions. Asymptotic expressions for tail risk are established in terms of the intensity measure that characterizes...

In this paper, we argue that a distinction exists between risk measures and decision principles. Though both are functionals assigning a real number to a random variable, we think there is a hierarchy between the two concepts. Risk measures operate on the first “level”, quantifying the risk in the situation under consideration, while decision principles operate on the second “level”, often bein...

Although financial risk measurement is a largely investigated research area, its relationship with imprecise probabilities has been mostly overlooked. However, risk measures can be viewed as instances of upper (or lower) previsions, thus letting us apply the theory of imprecise previsions to them. After a presentation of some well known risk measures, including Value-at-Risk or VaR, coherent an...

It is well known that a random vector with given marginals is comonotonic if and only if it has the largest convex sum, and that a random vector with given marginals (under an additional condition) is mutually exclusive if and only if it has the minimal convex sum. This paper provides an alternative proof of these two results using the theories of distortion risk measure and expected utility.

We study the problem of diversifying a given initial capital over a finite number of investment funds that follow different trading strategies. The investment funds operate in a market where a finite number of underlying assets may be traded over finite discrete time. We present a numerical procedure for finding a diversification that is optimal in the sense of a given convex risk measure. The ...