Monetary Measures of Risk

TL;DR

This survey introduces monetary measures of risk, explaining their mathematical properties, representations, and construction methods, with extensions to broader contexts.

Contribution

It provides a comprehensive overview of monetary risk measures, including primal and dual representations and construction techniques, advancing understanding in the field.

Findings

Discusses primal and dual representation results

Provides examples of monetary risk measures

Outlines methods for constructing risk measures

Abstract

This survey gives an introduction to monetary measures of risk as monotone and cash additive functions on spaces of univariate random variables. Primal and dual representation results as well as several examples are discussed. Principal ways to construct risk measures are given and extensions to more general situations indicated.

Equations

$$\forall X\in L^p,\forall r\in \mathrm{I}\mathrm{R}:\varrho (X+r1\mathrm{I})=\varrho (X)-r.$$

$${\varrho }_A(X)=\inf \left\{s\in \mathrm{I}\mathrm{R}\mid X+s1\mathrm{I}\in A\right\}$$

$$A_{\varrho }=\left\{X\in L^p\mid \varrho (X)\le 0\right\}$$

$$X+\epsilon 1\mathrm{I}=X+s_{\epsilon }1\mathrm{I}+\left(\epsilon -s_{\epsilon }\right)1\mathrm{I}\in A+{\mathrm{I}\mathrm{R}}_{+}1\mathrm{I}\subseteq A.$$

$$q_{\alpha }^{+}(X)=\inf \left\{t\in \mathrm{I}\mathrm{R}\mid P[X\le t]>\alpha \right\}=\sup \left\{t\in \mathrm{I}\mathrm{R}\mid P[X<t]\le \alpha \right\}$$

$$X\mapsto V@R_{\alpha }(X)=-q_{\alpha }^{+}(X)=\inf \left\{t\in \mathrm{I}\mathrm{R}\mid P[X+t1\mathrm{I}<0]\le \alpha \right\}$$

$$X\mapsto AV@R_{\alpha }(X)=\frac{1}{\alpha }{\int }_0^{\alpha }V@R_{\beta }(X)d\beta$$

$$\tau(X)=\left\{\begin{array}[]{ccc}-r&\colon&X=r\mathrm{1\negthickspace I},\;r\in\mathrm{I\negthinspace R}\\ +\infty&\colon&X\;\text{is non-constant}\end{array}\right.$$

$$\left(I_A\text{boxempty}\tau \right)(X)$$

$$=\inf \left\{I_A(X_1)-r\mid X_1+r1\mathrm{I}=X,r\in \mathrm{I}\mathrm{R}\right\}$$

$$=\inf \left\{-r\mid X-r1\mathrm{I}\in A,r\in \mathrm{I}\mathrm{R}\right\}={\varrho }_A(X).$$

$${\varrho }^{\ast }\left(Y\right):=\underset{X\in L^p}{\sup }\left\{E\left[XY\right]-\varrho (X)\right\}\text{and}{\varrho }^{\ast \ast }\left(X\right):=\underset{Y\in L^q}{\sup }\left\{E\left[XY\right]-{\varrho }^{\ast }(Y)\right\}$$

$$\varrho^{*}(Y)=\left\{\begin{array}[]{ccc}\sup_{X\in A_{\varrho}}E\left[-XY\right]&:&E\left[Y\right]=1,\;Y\in L^{\infty}_{+}\\ +\infty&:&\text{otherwise}\end{array}\right.$$

$$Q(A)={\int }_{A}Y(\omega )dP\text{for}A\in \mathcal{F}$$

$$\forall X\in L^p:\varrho (X)=\underset{Q\in {\mathcal{Q}}_{\varrho }}{\sup }\left\{E^Q\left[-X\right]-\gamma \left(Q\right)\right\}.$$

$${\varrho }_{max}(X)=\underset{Q\in {\mathcal{M}}_1(P)}{\sup }{E}^Q[-X],$$

$$AV@R_{\alpha }(X)=\sup \left\{E^Q[-X]\mid Q\in {\mathcal{M}}_1(P),\frac{dQ}{dP}\le \frac{1}{\alpha }\right\}.$$

$${\varrho }_{ent}(X)=\underset{Q\in {\mathcal{M}}_1(P)}{\sup }\left\{E^Q[-X]-\frac{1}{\beta }H(Q\mid P)\right\}$$

$$\forall X\in L^{\infty }:\varrho (X)=\underset{m\in {\mathcal{M}}_1((0,1])}{\sup }\left\{{\int }_0^{1}AV@R_{\alpha }(X)dm(\alpha )-\pi (m)\right\}.$$

$${\varrho }_{\varphi }(X)=\inf \left\{\varphi (X_1)+\tau \left(X_2\right)\mid X_1+X_2=X\right\}=\inf \left\{\varphi \left(X-r1\mathrm{I}\right)-r\mid r\in \mathrm{I}\mathrm{R}\right\}$$

$${\varrho }_{\ell }(X)={\varrho }_{A_{\ell }}(X)=\inf \left\{s\in \mathrm{I}\mathrm{R}\mid E\left[\ell (-X-s1\mathrm{I})\right]\le r_0\right\}$$

$$\forall X\in L^{\infty }:{\varrho }_{\ell }(X)=\underset{Q\in {\mathcal{M}}_1(P)}{\max }\left[E^Q[-X]-\underset{\lambda >0}{\inf }\frac{1}{\lambda }\left(r_0+E\left[{\ell }^{\ast }\left(\lambda \frac{dQ}{dP}\right)\right]\right)\right]$$

$${\varrho }_{\varphi }(X)=-{\int }_0^1\varphi (s)q_X^{-}(s)ds$$

$$X{⪯}_{SSD}Y\iff \forall \alpha \in (0,1]:AV@R_{\alpha }(X)\ge AV@R_{\alpha }(Y).$$

Full text

Monetary Measures of Risk111This paper was written in 2015 as a contribution to Wiley Encyclopedia of Operations Research and Management Science. It is accepted and will appear in the second edition of EORMS.

Andreas H. Hamel222Free University of Bozen-Bolzano, Faculty of Economics and Management, \hrefmailto:andreas.hamel@unibz.itandreas.hamel@unibz.it

A monetary risk measure is a mathematical tool for quantifying the risk of a random future gain (or loss) which is denoted in (discounted) units of a reference instrument (a currency, for example). As such, it is a real-valued function, and it is convenient to allow for the value $+\infty$. The greater the value of the risk measure, the higher the risk.

Two elementary mathematical properties turned out to be crucial. Both have a straightforward and convincing economic interpretation.

The first one is a monotonicity property: if gain $X$ is not less than gain $Y$ no matter what happens in the world, then the risk of $X$ should not be greater than the risk of $Y$.

The second one is additivity with respect to a riskless reference instrument: if one adds $s$ units of the reference instrument (e.g., cash) to the (discounted) random gain (e.g., as a deposit), then the risk (i.e., the value of the risk measure) decreases by $s$. Because of the immediate interpretation of the value of such risk measures as capital requirements, they are also called monetary measures of risk [27].

This second property, called cash-additivity, has remarkable mathematical consequences. Its economic interpretation, ‘linearity in payments’ has been pointed out already very clearly in [54, p. 101] by Yaari, and it became popular through the work [4] by Artzner et al. Cash-additivity goes by many names, for example “translation invariance,” (already in [53], also [4]), “translation equivariance” ([51]), or just “additivity” ([5, p. 1455]).

Following the famous [38], the variance of a random variable was used as a risk measure in portfolio selection problems (see also 3.2.2.2333Such labels refer to other articles in EORMS. and 3.2.2.4). However, it is neither monotone, nor cash-additive. Moreover, it weighs (random) gains and losses in a symmetrical way which is not a desirable property of a (financial) risk measure and, on an even deeper level, the variance is not consistent with important stochastic dominance orders as already discussed, e.g., in [41]. On the other hand, in the financial practice the (cash-additive) so-called Value-at-Risk was (and still is) widely used as a risk measure. Its drawback turned out to be the missing convexity: diversification is not generally supported by the Value-at-Risk. Therefore, a new class of (monotone and cash-additive) risk measures, called coherent, was introduced in [4] (see also 4.6.3.3).

1 Risk measures and acceptance sets

Let $\left(\Omega,\mathcal{F},P\right)$ be a probability space, $p\in[0,\infty]$ and $L^{p}:=L^{p}(\Omega,\mathcal{F},P)$ be the linear space of all (equivalence classes of) univariate, to the $p$th power (absolutely) integrable random variables where $X_{1}\colon\Omega\to\mathrm{I\negthinspace R}$ and $X_{2}\colon\Omega\to\mathrm{I\negthinspace R}$ generate the same element of $L^{p}$ whenever $P\left(\left\{\omega\in\Omega\mid X_{1}(\omega)=X_{2}(\omega)\right\}\right)=1$. If $p=0$, $L^{0}$ is the space of all random variables. If $p=\infty$, $L^{\infty}:=L^{\infty}(\Omega,\mathcal{F},P)$ is the space of essentially bounded random variables. An inequality like $X_{1}\leq X_{2}$ for two elements of $X_{1},X_{2}\in L^{p}$ is understood $P$-almost surely, i.e. $P\left(\left\{\omega\in\Omega\mid X_{1}(\omega)\leq X_{2}(\omega)\right\}\right)=1$. The element $\mathrm{1\negthickspace I}\in L^{p}$ denotes the function whose value is 1 $P$-almost surely, and $L^{p}_{+}=\left\{X\in L^{p}\mid 0\leq X\right\}$.

A function $\varrho\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ is called monotone if $X,Y\in L^{p}$, $X\leq Y$ imply $\varrho(Y)\leq\varrho(X)$, and it is called cash-additive if

[TABLE]

Definition 1

A risk measure is a function $\varrho\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ which is monotone, cash-additive and satisfies $\varrho(0)\in\mathrm{I\negthinspace R}$.

Risk measures can also be defined on other linear spaces of random variables, see, for example, [12]. It is remarkable, though mathematically not difficult, that risk measures are basically in one-to-one correspondence with their acceptance sets (see also 3.2.4.1). This fact depends almost entirely on property (1). Here are the necessary concepts. A set $A\subseteq L^{p}$ is called monotone if $A+L^{p}_{+}\subseteq A$, and it is called directionally closed if $X\in A$, $\left\{r_{n}\right\}_{n=0,1,\ldots}\subseteq\mathrm{I\negthinspace R}_{+}$, $\lim_{n\to\infty}r_{n}=0$ and $X+r_{n}\mathrm{1\negthickspace I}\in A$ for all $n=0,1,\ldots$ imply $X\in A$.

Definition 2

An acceptance set is a set $A\subseteq L^{p}$ which is monotone, directionally closed and satisfies $A\cap\mathrm{I\negthinspace R}\mathrm{1\negthickspace I}\neq\emptyset$ as well as $(L^{p}\backslash A)-\mathrm{I\negthinspace R}\mathrm{1\negthickspace I}=L^{p}$.

The correspondence between acceptance sets and risk measures is established in the following result.

Proposition 3

If $A\subseteq L^{p}$ is an acceptance set, then the function $\varrho_{A}$ on $L^{p}$ defined by

[TABLE]

is a risk measure. If $\varrho\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ is a risk measure, then the set

[TABLE]

is an acceptance set. Moreover, it holds $A=A_{\varrho_{A}}$ and $\varrho=\varrho_{A_{\varrho}}$.

The condition $A\cap\mathrm{I\negthinspace R}\mathrm{1\negthickspace I}\neq\emptyset$ means that there is an amount of cash which the financial agent accepts, and $(L^{p}\backslash A)-\mathrm{I\negthinspace R}\mathrm{1\negthickspace I}=L^{p}$ says that there is a limit to cash withdrawals starting from whatever position $X$. Both conditions together make sure that $\varrho_{A}$ never attains $-\infty$ as a value and that $\varrho_{A}(0)\in\mathrm{I\negthinspace R}$. Everything else being straightforward, one can verify $A_{\varrho_{A}}\subseteq A$ as follows: If $X\in A_{\varrho_{A}}$, then $\varrho_{A}\left(X\right)=\inf\left\{s\in\mathrm{I\negthinspace R}\mid X+s\mathrm{1\negthickspace I}\in A\right\}\leq 0$, and the very definition of the infimum implies the following: For each $\varepsilon>0$ there is $s_{\varepsilon}\leq\varepsilon$ such that $X+s_{\varepsilon}\mathrm{1\negthickspace I}\in A$. Using the fact that $A$ is monotone one obtains

[TABLE]

Since $A$ is directionally closed, $X\in A$, hence $A_{\varrho_{A}}\subseteq A$.

Directional closedness of $A$ implies the closedness of the set $\left\{s\in\mathrm{I\negthinspace R}\mid X+s\mathrm{1\negthickspace I}\in A\right\}$ which means that the infimum in the definition of $\varrho_{A}(X)$ is attained if it is not $+\infty$: there is $s_{0}\in\mathrm{I\negthinspace R}$ such that $\varrho_{A}(X)=s_{0}$ and $X+s_{0}\mathrm{1\negthickspace I}\in A$. This implies $X+\varrho_{A}(X)\mathrm{1\negthickspace I}\in A$, i.e., $X$ can be made acceptable by depositing $\varrho_{A}(X)$ units of cash (or more).

In most cases, risk measures and acceptance sets have to satisfy further requirements. Again, property (1) provokes one-to-one correspondences between properties of risk measures and acceptance sets:

(a) A risk measure $\varrho$ is convex if, and only if, the “induced” acceptance set $A_{\varrho}$ is convex. This implies that a risk measure is convex if, and only if, it is quasiconvex.

(b) $\varrho$ is positively homogeneous (sublinear) if, and only if, $A_{\varrho}$ is a cone (a convex cone).

(c) $\varrho$ has (only) real values if, and only if, $A_{\varrho}-\mathrm{I\negthinspace R}\mathrm{1\negthickspace I}=L^{p}$.

Corresponding statements are obtained for acceptance sets $A$ and the “induced” risk measures $\varrho_{A}$.

Following [4], it became custom in the math finance community to call sublinear risk measures coherent. However, the authors of [4] probably intended to use the word “coherent” in a more literal sense: for example in [33] convex, but not necessarily sublinear risk measures are also called (weakly) coherent. Moreover, “coherent” is also used in the sense which is usually associated with “arbitrage-free”–as in [52].

A few elementary examples for risk measures are the following. The function $X\mapsto E\left[-X\right]$ is a linear risk measure on $L^{1}$, the function $X\mapsto-\operatorname*{essinf}X$ is a sublinear risk measure on $L^{\infty}$. A large class of risk measures is based on quantiles: For $\alpha\in(0,1]$, the number

[TABLE]

is called the upper $\alpha$-quantile of $X$; the function

[TABLE]

is called the Value-at-Risk of $X$ at level $\alpha$; the function

[TABLE]

is called the Average-Value-at-Risk of $X$ at level $\alpha$. Whereas $V@R$ is a positively homogeneous, but in general non-convex risk measure on $L^{0}$, the $AV@R$ is a sublinear risk measure on $L^{1}$, i.e., it is coherent.

The function $\tau\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ defined by

[TABLE]

is sublinear and satisfies the requirements of Definition 1 except for monotonicity. It could be seen as an extreme way to evaluate risk: non-constant payoffs are considered as intolerable risks, and acceptable are only the non-negative constant ones. It turns out that every risk measure has a representation in terms of $\tau$. Indeed, defining the indicator function (in the sense of convex analysis) $I_{A}\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ of an acceptance set $A\subseteq L^{p}$ by $I_{A}(X)=0$ whenever $X\in A$ and $I_{A}(X)=+\infty$ otherwise, formula (2) can be written as

[TABLE]

Thus, the position $X$ is split into a constant and a remaining position $X_{1}$ which should be acceptable. If this is possible, one looks for the minimal risk of the constant evaluated by $\tau$. If such a split is not possible, $I_{A}(X_{1})=+\infty$ always holds and $\left(I_{A}\boxempty\tau\right)(X)=+\infty$. Mathematically, $\varrho_{A}$ is the infimal convolution of $I_{A}$ and $\tau$. Every risk measure that satisfies the assumptions in Proposition 3 has such a representation: $\varrho(X)=\left(I_{A_{\varrho}}\boxempty\tau\right)(X)$ for all $X\in L^{p}$. This is very convenient, in particular for duality purposes, since the two functions $I_{A}$ and $\tau$ are easy to handle.

2 Closedness and dual representation

The space $L^{0}$ is a complete metric, linear space for any Lévy-metric. If $p\geq 1$, $L^{p}$ is a Banach space with the norm $\left\|X\right\|_{p}=\left(\int_{\Omega}\left|X\right|^{p}dP\right)^{\frac{1}{p}}$ for $1\leq p<\infty$ and $\left\|X\right\|_{\infty}=\operatorname*{esssup}\left|X\right|$ for $p=\infty$. In the following, $p\in\left\{0\right\}\cup[1,\infty]$ is assumed.

Proposition 4

The following statements are equivalent for a risk measure $\varrho\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$:

(a) At each $X\in L^{p}$, $\varrho$ is lower semicontinuous, i.e., $\varrho(X)\leq\liminf_{n\to\infty}\varrho(X_{n})$ whenever $\lim_{n\to\infty}X_{n}=X$ in $L^{p}$.

(b) $\left\{X\in L^{p}\mid\varrho(X)\leq r\right\}$ is closed for each $r\in\mathrm{I\negthinspace R}$;

(c) $A_{\varrho}=\left\{X\in L^{p}\mid\varrho(X)\leq 0\right\}\subseteq L^{p}$ is closed.

A parallel statement holds for $\varrho_{A}$ replaced by a risk measure $\varrho$ and $A$ by $A_{\varrho}$.

The equivalence of (a) and (b) is standard in variational analysis, while (c) enters the picture because of the cash-additivity (1). A risk measure that satisfies one (and hence) all of the conditions in Proposition 4 is called closed.

For $p\in[1,\infty)$, the topological dual of $L^{p}$ is the Banach space $L^{q}$ for $\frac{1}{p}+\frac{1}{q}=1$ with $q=\infty$ whenever $p=1$. If $p=\infty$ then $L^{\infty}$ is supplied with the weak topology generated by the dual pair $(L^{1},L^{\infty})$ (of locally convex spaces, see [3, Section 5.14]), and this ensures that $L^{\infty}$ and $L^{1}$ become topological duals of each other. Note that the topology on $L^{\infty}$ influences the closedness of $\varrho$: There are functions on $L^{\infty}$ which are closed with respect to the norm topology, but not closed with respect to the (weak) topology generated by $L^{1}$. A condition that ensures the latter turns out to be equivalent to the so-called Fatou property, see [27, Section 4.3].

Let $\varrho\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ be a closed, convex risk measure. According to Definition 1, it never attains the value $-\infty$, and it has at least one real value $\varrho(0)$. This means that $\varrho$ is proper in the sense of convex analysis (see [55, p. 39]), and it satisfies all the assumptions of the Fenchel-Moreau theorem [55, Theorem 2.3.3]: it coincides with its Legendre-Fenchel biconjugate $\varrho^{**}$ which is given by the two formulas

[TABLE]

where the Legendre-Fenchel conjugate $\varrho^{*}\colon L^{\infty}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ of $\varrho$ is defined on the topological dual space $L^{q}$ of $L^{p}$.

The representation $\varrho=\varrho^{**}$ is useful only if one can determine $\varrho^{*}$. It turns out that

[TABLE]

This follows from the representation $\varrho=I_{A_{\varrho}}\boxempty\tau$ and the fact that the conjugate of an infimal convolution is the sum of the conjugates ([55, Theorem 2.3.1 (ix)]): one has to compute $(I_{A_{\varrho}})^{*}$ and $\tau^{*}$. While the former is known to be the support function of $A_{\varrho}$ (an easy consequence of the definition of the conjugate), the latter is $I_{\left\{Y\in L^{q}\mid E[Y]=-1\right\}}$. Observing that the support function of $A_{\varrho}$ attains the value $+\infty$ whenever $Y\not\in-L^{q}_{+}$ (this follows from monotonicity of $A_{\varrho}$) and then replacing $Y$ by $-Y$ one obtains (4).

The two conditions for $Y$ in (4) admit a striking interpretation of the dual representation formula $\varrho=\varrho^{**}$. To $Y\in L^{q}_{+}$ satisfying $E\left[Y\right]=1$ one can assign a probability measure $Q$ by

[TABLE]

which is absolutely continuous with respect to $P$, i.e., $\frac{dQ}{dP}=Y$. Moreover, the relationship between $Q$ and $Y$ is one-to-one. If one denotes the set of such probability measures by $\mathcal{M}_{1}(P)$, then the dual representation result for risk measures on $L^{p}$ reads as follows.

Theorem 5

The function $\varrho\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ is a closed, convex risk measure if, and only if, there exists a non-empty set $\mathcal{Q}_{\varrho}\subseteq\mathcal{M}_{1}(P)$ and a function $\gamma\colon\mathcal{Q}_{\varrho}\to\mathrm{I\negthinspace R}$ such that

[TABLE]

Moreover, $\gamma(Q)=\sup_{X\in A_{\varrho}}E\left[-XY\right]=\sup_{X\in A_{\varrho}}E^{Q}\left[-X\right]$ whenever $Q$ is the probability measure generated by $Y\in L^{q}_{+}$ which satisfies $E\left[Y\right]=1$ and $\varrho^{*}(Y)<+\infty$.

If $\varrho$ is additionally positive homogeneous (hence sublinear), then $\gamma(Q)=0$ for $Q\in\mathcal{Q}_{\varrho}$.

The worst case risk measure $\varrho_{max}\colon L^{\infty}\to\mathrm{I\negthinspace R}$ defined by $\varrho_{max}(X)=-\operatorname*{essinf}X=\inf\left\{t\in\mathrm{I\negthinspace R}\mid X+t\mathrm{1\negthickspace I}\geq 0\right\}$ has the dual representation

[TABLE]

whereas the Average-Value-at-Risk on $L^{1}$ can be represented as

[TABLE]

Both are sublinear (coherent) risk measures. Note that $\frac{dQ}{dP}\in L^{\infty}$ for $Q\in\mathcal{Q}_{AV@R}$. A verification of this formula can be given via the representation $AV@R_{\alpha}(X)=(\varphi\boxempty\tau)(X)$ with $\varphi(X)=\frac{1}{\alpha}\max\left\{-X,0\right\}$ which is due to [44], [45].

By $\varrho_{ent}(X)=\frac{1}{\beta}\log E[exp(-\beta X)]$ for $\beta>0$, a risk measure $\varrho_{ent}\colon L^{\infty}\to\mathrm{I\negthinspace R}$ is defined; it is convex, but not positively homogeneous. Its dual representation is

[TABLE]

where $H(Q\mid P)=E^{Q}[\log\frac{dQ}{dP}]$ is the relative entropy of $Q$ with respect to $P$.

3 Law invariance and Kusuoka representation

A risk measure $\varrho\colon L^{p}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ is called law invariant if $\varrho(X)=\varrho(Y)$ whenever $X$ and $Y$ have the same distribution under $P$. Standard examples of law invariant risk measures are the quantile based $V@R$ and $AV@R$. For risk measures on $L^{\infty}$, law invariance has strong implications. A typical result reads as follows.

Theorem 6

Let $(\Omega,\mathcal{F},P)$ be an atomless probability space such that $L^{2}$ is separable. Then, $\varrho\colon L^{\infty}\to\mathrm{I\negthinspace R}$ is a law invariant convex risk measure if, and only if, there exists a convex function $\pi\colon\mathcal{M}_{1}((0,1])\to[0,\infty]$ such that

[TABLE]

where $\mathcal{M}_{1}((0,1])$ is the set of (Borel) probability measures on $(0,1]$.

The characterization in Theorem 6 is due to Kusuoka [37] for the sublinear case (in terms of integrated quantile functions) and due to Jouini, Schachermayer and Touzi [36] in the general convex case. It shows the importance of the Average-Value-at-Risk.

4 Constructing risk measures

Translative envelopes. Let $\phi\colon L^{1}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ be a monotone function. Then, the function $\varrho_{\phi}\colon L^{1}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ defined by

[TABLE]

is a risk measure whenever $\varrho_{\phi}(0)\in\mathrm{I\negthinspace R}$. Note that $\varrho_{\phi}$ is nothing else than the infimal convolution of the two functions $\phi$ and $\tau$ ([55, Theorem 2.1.3 (ix)]). Moreover, it can be shown that $\varrho_{\phi}$ is the pointwise greatest cash-additive function which is pointwise not greater than $\phi$, thus it may be called the (lower) cash-additive envelope of $\phi$. This construction has been introduced in [17] in a different context, and for risk measures in [23]. Moreover, the so-called “optimized certainty equivalent” introduced in [5], [6] has the same form in which $\phi(X)=E[\ell(X)]$ for a monotone (non-increasing) function $\ell\colon\mathrm{I\negthinspace R}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$. As shown above, every risk measure is the cash-additive envelope of the indicator function of its “induced” acceptance set: $I_{A_{\varrho}}$ is monotone since $A_{\varrho}$ is.

Risk measures associated with loss/utility functions. Let $\ell\colon\mathrm{I\negthinspace R}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ be a proper, increasing and not identically constant function and $r_{0}\in{\rm int\,}\ell(\mathrm{I\negthinspace R})$. Define the set $A_{\ell}=\left\{X\in L^{1}\mid E\left[\ell(-X)\right]\leq r_{0}\right\}$. The risk measure $\varrho_{\ell}$ defined by

[TABLE]

is called loss-based shortfall risk measure. It is convex if $\ell$ is convex. If $\ell$ is real-valued and $\varrho_{\ell}$ is considered as a function on $L^{\infty}$, then it is weakly closed with dual representation

[TABLE]

where $\ell^{*}$ is the Fenchel conjugate of $\ell\colon\mathrm{I\negthinspace R}\to\mathrm{I\negthinspace R}$. Shortfall risk measures are law invariant and in some sense dual to divergence risk measures (discussed in [27, Section 4.9], the latter have a primal representation depending on $\ell^{*}$) which in turn also coincide with the “optimized certainty equivalent” introduced by Ben-Tal and Teboulle [5], [6].

Spectral risk measures. The crucial observation is that a convex combination of two risk measures again is a risk measure, and this can even be generalized to mixtures via probability measures on $[0,1]$, see [1, Proposition 2.2]. Acerbi [1] introduced the following concept. Let $\phi\colon[0,1]\to\mathrm{I\negthinspace R}$ be a function satisfying (a) $\phi(\alpha)\geq 0$ for all $\alpha\in[0,1]$, (b) $\int_{0}^{1}\phi(\alpha)d\alpha=1$, (c) $0\leq\alpha_{1}\leq\alpha_{2}\leq 1$ implies $\phi(\alpha_{1})\geq\phi(\alpha_{2})$. Then, the function $\varrho_{\phi}\colon L^{\infty}\to\mathrm{I\negthinspace R}\cup\left\{+\infty\right\}$ defined by

[TABLE]

is a coherent, law invariant risk measure, and the function $\phi$ is called a risk spectrum which can chosen by the decision maker. Here, $q^{-}_{X}(\alpha)=\inf\left\{t\in\mathrm{I\negthinspace R}\mid F_{X}(t)\geq\alpha\right\}$ is the lower $\alpha$-quantile of $X$. $V@R$ and $AV@R$ turn out to be special spectral risk measures. Compare [12] for further properties, dual representation results and relationships to stochastic dominance orders. Note that already the results of Kusuoka [37, Theorem 7] imply that the class of spectral risk measures on $L^{\infty}$ over an atomless probability space coincides with the class of all weakly closed, coherent, law invariant and comonotonic risk measures (compare Remark 4.4 in [1]).

5 Relationships to other concepts in risk evaluation

Stochastic dominance orders. Stochastic dominance orders for probability distributions are important tools for risk evaluation. Therefore, a crucial property of a risk measure is monotonicity with respect to these orders. The Average Value at Risk does even characterize the second order stochastic dominance $\preceq_{SSD}$: If $X,Y\in L^{1}$, then

[TABLE]

This observation goes back to [42], see also [27, Remark 4.49]. In a similar way, the Value-at-Risk characterize first order stochastic dominance.

Other translative functions. Remarkably, many other functions share property (1). In particular, the sub- and superhedging price of a financial position in an incomplete market are versions of a cash-additive function [27, Section 1.3] and also the so-called good deal bounds [34]. Outside finance, Dempster’s belief functions [14, formula (3.9), p. 363], Choquet integrals [15], imprecise lower/upper expectations [52], insurance premiums as discussed, e.g., in [53], exact functionals and games [39], [40] as well as maxmin expected utility functions [29], among many others, share property (1).

Extensions. (a) The famous Markowitz model for portfolio selection [38] involves the variance as a risk evaluating tool - which is neither monotone, nor cash-additive. On the contrary, it is constant on the linear subspace of $L^{2}$ formed by the constant functions. This property is shared by deviation measures introduced by Rockafellar, Uryasev and Zabarankin [47], [48] which are basically the difference of a risk measure and the expected value. They may replace the variance in procedures like regression analysis [49] or portfolio selection [48]. See [46] for an overview. (b) Since a cash-additive risk measure is quasiconvex if, and only if, it is convex, weaker versions of (1) were introduced, see [19] and [9]. In [8], [18], a concise motivation, further results on quasiconvex risk measures (called performance or assessment indices) and many examples can be found. (c) Under market conditions, one may want to make available a dynamic risk assessment procedure. The main issue is time-consistency, i.e., a position which is acceptable at some point in time should already be acceptable at earlier times. Extensions of the above concepts to the dynamic case were initiated in [16], [10], [11], [43]. More recently, the $L^{0}$-module framework was developed mainly motivated by time-dependent, conditional risk measures, see [24] for an overview and references. (d) In markets with transaction costs and illiquidity, the risk of multi-variate positions needs to be evaluated (see also 3.1.5.7). Several approaches have been pursued: scalar risk measures for multivariate payoffs [7], [20], for example, and vector- and set-valued risk measures [35], [30], [31], [32]. (e) Condition (1) requires the existence of a “non-defaultable” (discountable) numéraire which serves as reference instrument. In the light of recent financial and economic crises, this assumption is questionable. Even more reasons for leaving the framework of “constant numéraires” and alternatives can be found in [21], [22].