Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders

Abstract

This paper proposes some new classes of risk measures, which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, and show that if the physical probability is atomless then a comonotonic subadditive (resp. convex) risk measure respecting stop-loss order is in fact a law-invariant coherent (resp. convex) risk measure.

Introduction

In finance and economics the term “risk” stands for potential losses of a financial position over a period. Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space, where $\Omega$ represents all possible states of nature at the end of a period, $\mathbb{P}$ is the objective or physical probability measure. Random variables on $(\Omega ,\mathcal{F},\mathbb{P})$ are interpreted as potential losses of financial positions over the time period. A negative outcome of a loss variable means a gain. Throughout this paper, we will denote $L^{\infty }(\Omega ,\mathcal{F})$ and $L^{\infty }(\Omega ,\mathcal{F},\mathbb{P})$ by $\mathcal{X}$ and $\mathcal{X}\left(\mathbb{P}\right)$, respectively. The aim of risk measures is to quantify the risk of $X$ by some number $\rho \left(X\right)$, where $X\in \mathcal{X}$. So a risk measure $\rho$ is a mapping from the set $\mathcal{X}$ to the real numbers. The most popular risk measure is $VaR$ (Value at Risk). For example, the banking regulation “Basel Accord II” specifies a risk measure as $VaR$ at level 0.01. For $X\in \mathcal{X}\left(\mathbb{P}\right)$ and $\alpha \in (0,1)$, the value at risk of $X$ at level $\alpha$ is defined as

$$Va{R}_{\alpha }\left(X\right)≔inf\{x:\mathbb{P}(X>x)\le \alpha \}\text{.}$$

It is in fact an $(1-\alpha )$-quantile of $X$, denoted by $q_X(1-\alpha )$. As a function of $\alpha$, $Va{R}_{1-\alpha }\left(X\right)$ (or $q_X\left(\alpha \right)$) is the right-continuous inverse of the distribution function of $X$.

$VaR$ gives only the probability of certain losses to occur, but not the magnitude of potential losses. To overcome this shortcoming one introduced the Average value at risk ($AVaR$) at level $\alpha$:

$$AVa{R}_{\alpha }\left(X\right)=\frac{1}{\alpha }{\int }_0^{\alpha }Va{R}_p\left(X\right)dp\text{,}$$

and the weighted $VaR$ ($WVaR$) as follows:

$$WVa{R}_{\mu }\left(X\right)={\int }_{[0,1]}AVa{R}_{\lambda }\left(X\right)\mu \left(d\lambda \right)\text{,}$$

where $\mu$ is a probability measure on $[0,1]$. However, the main objection to $VaR$ is that it lacks the subadditivity: adding two separate risks together does not increase the overall risk. So people tried to find some new risk measures which satisfy the subadditivity.

As the first great stage in this direction, Artzner et al., 1997, Artzner et al., 1999 proposed an axiomatic approach to risk measures, and introduced the so-called coherent risk measures, which satisfy the following axioms:

(a1) (monotonicity) $X\le Y$ implies $\rho \left(X\right)\le \rho \left(Y\right)$;

(a2) (positive homogeneity) $\rho \left(cX\right)=c\rho \left(X\right)$ if $c\ge 0$;

(a3) (translation invariance) $\rho (X+c)=\rho \left(X\right)+c,c\in \mathbb{R}$;

(a4) (subadditivity) If $X,Y\in \mathcal{X}$, then

$$\rho (X+Y)\le \rho \left(X\right)+\rho \left(Y\right)\text{.}$$

$AVaR$ and $WVaR$ are typical examples of coherent risk measures.

But in general, a measure of risk should not increase in a linear way with the size of risk. This motivated Föllmer and Schied (2002), and independently Frittelli and Rosazza Gianin (2002), to relax the requirements of positive homogeneity and subadditivity to the requirement of convexity:

(a5) (convexity): For $\lambda \in [0,1],\rho (\lambda X+(1-\lambda )Y)\le \lambda \rho \left(X\right)+(1-\lambda )\rho \left(Y\right)$.

A mapping $\rho :\mathcal{X}\to \mathbb{R}$ is called a convex risk measure, if it satisfies Axioms (a1), (a3) and (a5).

Coherent risk measures take subadditivity as the most basic requirement for a good risk measure. However, a merger of two portfolios may create extra risk and subadditivity of a risk measure for all risks may be too restrictive sometimes. This observation motivated us in Song and Yan (2006) to relax the subadditivity or convexity requirement for all risks to that for comonotonic risks as follows:

(A4) (comonotonic subadditivity) If $X,Y$ are comonotonic, then

$$\rho (X+Y)\le \rho \left(X\right)+\rho \left(Y\right)\text{;}$$

(A5) (Comonotonic convexity) If $X,Y$ are comonotonic, then

$$\rho (\lambda X+(1-\lambda )Y)\le \lambda \rho \left(X\right)+(1-\lambda )\rho \left(Y\right)\text{for }\lambda \in (0,1)\text{.}$$

Here, two real functions $X,Y$ are called comonotonic if there is no pair ${\omega }_1,{\omega }_2\in \Omega$ such that $X\left({\omega }_1\right)<X\left({\omega }_2\right)$ and $Y\left({\omega }_1\right)<Y\left({\omega }_2\right)$.

In Song and Yan (2006), we gave a representation for functionals on $L^{\infty }(\Omega ,\mathcal{F})$ satisfying (a1)–(a3) and (A4) in terms of Choquet integrals. One referee reported that Laeven (2005) had obtained a similar result for the case of finite state space $\Omega$.

Independently, Heyde et al. (2006) proposed a so-called natural risk statistic which is data based and satisfies also the comonotonic subadditivity. As is pointed out in Heyde et al. (2006), the comonotonic subadditivity is consistent with the prospect theory, and Ellsberg’s paradox illustrates that risk associated with non-comonotonic random variables may violate subadditivity. In fact, the prospect theory postulates that people evaluate uncertain prospects using “decision weights” that may be viewed as distorted probabilities of outcomes. Yaari (1987) and Schmeidler (1989) further introduced “Choquet expected utility” and “rank-dependent models”, which impose preference on comonotonic random variables rather than on arbitrary random variables. Schmeidler (1989) also indicated that risk preference for comonotonic random variables are easier to justified than the risk preference for arbitrary random variables.

In insurance, risk measures (or premium principles) are interpreted as prices of insurance risks. Requiring a risk measure to be law-invariant seems very natural. Here by law-invariance we mean: If $X,Y$ are identically distributed, denoted by $X{=}^{d}Y$, then $\rho \left(X\right)=\rho \left(Y\right)$. For example, $VaR$, $AVaR$ and $WVaR$ are all law-invariant. Law-invariant risk measures were first discussed systematically by Kusuoka (2001) for coherent case, and then extended to the convex case by Dana (2005), and Föllmer and Schied (2004).

On the other hand, stochastic orders play an important role in insurance mathematics. For two real random variables $X,Y$, $X$ is said to precede $Y$ in (first order) stochastic dominance (denoted by $X{\le }_{st}Y$) if $F_X\left(x\right)\ge F_Y\left(x\right)$ for all $x\in \mathbb{R}$; $X$ is said to precede $Y$ in the stop-loss order sense (denoted by $X{\le }_{sl}Y$), if $\mathbb{E}\left[{(X-d)}_{+}\right]\le \mathbb{E}\left[{(Y-d)}_{+}\right]$, for all $d\in \mathbb{R}$. By respecting stochastic dominance order (resp. stop-loss order) we mean: $X{\le }_{st}Y$ (resp. $X{\le }_{st}Y$) implies $\rho \left(X\right)\le \rho \left(Y\right)$. It is obvious that the property of respecting the stochastic dominance implies the properties of monotonicity and law-invariance. Conversely, if $\mathbb{P}$ is atomless, then the converse statement is true. In fact, since $\mathbb{P}$ is atomless, there exists a random variable $U\in \mathcal{X}\left(\mathbb{P}\right)$ on $(\Omega ,\mathcal{F},\mathbb{P})$, which is uniformly distributed on $(0,1)$. Now assume that $\rho :\mathcal{X}\left(\mathbb{P}\right)\to R$ satisfies monotonicity and law-invariance. If $X{\le }_{st}Y$, then $X{=}^d{q}_X\left(U\right)$, $Y{=}^d{q}_Y\left(U\right)$, and for each $t\in (0,1)$, $q_X\left(t\right)\le q_Y\left(t\right)$, so we get

$$\rho \left(X\right)=\rho \left(q_X\left(U\right)\right)\le \rho \left(q_Y\left(U\right)\right)=\rho \left(Y\right)\text{,}$$

which means that $\rho$ respects the stochastic dominance.

In addition, as pointed out in Dhaene et al., 2006a, Dhaene et al., 2006b, the stop-loss order has a clear economic meaning: it represents the common preferences of all risk averse decision makers, because $X{\le }_{sl}Y$ iff $\mathbb{E}\left[u(w-X)\right]\ge \mathbb{E}\left[u(w-Y)\right]$ for all concave utility functions $u$ and real number $w$. So requiring a risk measure to respect the stop-loss order for risk averse decision makers is reasonable. In fact, if a premium principle can be represented as a Choquet integral with respect to a distorted probability (resp. concave distorted probability), then it respects the stochastic dominance order (resp. stop-loss order) automatically (see Dhaene et al., 2006a, Dhaene et al., 2006b). Such kinds of premium principle are very popular in insurance, and called distortion premium principles (see Hurlimann, 1998).

The above observations suggested us to investigate those risk measures which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. This is the objective of the present paper. We will name the property of respecting the (first) stochastic dominance (resp. stop-loss order) as Axiom ST (resp. Axiom SL).

It is worth mentioning that for risk measures that respect stop-loss order, the comonotonic subadditivity property implies subadditivity. This follows immediately from Lemma 2.1 further in the paper.

The rest of the paper is organized as follows. In Section 2 we give some preliminary results about Choquet integrals, stochastic orders and distortions, some of which are new. Our main results are presented in Section 3. We prove that a risk measure satisfying Axioms (a2), (a3), (A4) and Axiom ST can be represented in terms of Choquet integrals w.r.t. distortions, and that if the physical probability is atomless, a risk measure satisfying Axioms (a2), (a3), (A4) and Axiom SL can be represented in terms of Choquet integrals w.r.t. concave distortions. In addition, we show that if the physical probability is atomless, then Axioms (a2), (a3), (A4) and Axiom SL are equivalent to Axioms (a2), (a3), (a4) and Axiom ST. In this section we also give representation results for risk measures satisfies Axioms (a3), (A5), and Axiom ST or SL. We end this paper with some concluding remarks in Section 4.