On coherency and other properties of MAXVAR

TL;DR

This paper investigates the MAXVAR risk measure on L^2 space, providing a straightforward proof of its coherency and averseness, and deriving an explicit formula for its risk envelope based on its relation to CVaR.

Contribution

It offers the first elementary proof of MAXVAR's properties and explicitly characterizes its risk envelope through its connection to CVaR.

Findings

MAXVAR is coherent and aversive.

An explicit formula for MAXVAR's risk envelope is derived.

MAXVAR can be expressed as a convex combination of CVaR.

Abstract

This paper is concerned with the MAXVAR risk measure on L^2 space. We present an elementary and direct proof of its coherency and averseness. Based on the observation that the MAXVAR measure is a continuous convex combination of the CVaR measure, we provide an explicit formula for the risk envelope of MAXVAR.

Equations

$${\mathrm{MAXVAR}}_n(X):=\mathbb{E}(\max \{X_1,\cdots ,X_n\}),$$

$${\mathrm{MAXVAR}}_n(X+Y)\le {\mathrm{MAXVAR}}_n(X)+{\mathrm{MAXVAR}}_n(Y)\forall X,Y.$$

$${\mathbb{P}}_0\left(X_1+Y_1\le t_1,\cdots ,X_n+Y_n\le t_n\right)$$

$$=$$

$${\mathbb{P}}_0((X_1,Y_1)\in B_1,\cdots ,(X_n,Y_n)\in B_n)$$

$$=$$

$$B_i=\left\{(x,y)\in {\mathbb{R}}^2:x+y\le t_i\right\}$$

$${\mathrm{MAXVAR}}_n(X)$$

$${\mathrm{MAXVAR}}_n(Y)$$

$${\mathrm{MAXVAR}}_n(X+Y)$$

$$\max \{X_1+Y_1,\cdots ,X_n+Y_n\}\le \max \{X_1,\cdots ,X_n\}+\max \{Y_1,\cdots ,Y_n\},$$

$${\mathrm{MAXVAR}}_n(X+Y)$$

$$\le \mathbb{E}(\max \{X_1,\cdots ,X_n\})+\mathbb{E}(\max \{Y_1,\cdots ,Y_n\})$$

$$={\mathrm{MAXVAR}}_n(X)+{\mathrm{MAXVAR}}_n(Y).$$

$${\mathrm{MAXVAR}}_n(X)$$

$$={\int }_{-\infty }^0\left[{\mathbb{P}}_0(\max \{X_1,\cdots ,X_n\}>t)-1\right]dt+{\int }_0^{+\infty }{\mathbb{P}}_0(\max \{X_1,\cdots ,X_n\}>t)dt$$

$$=-{\int }_{-\infty }^0({\mathbb{P}}_0(X\le t){)}^{n}dt+{\int }_0^{+\infty }\left[1-({\mathbb{P}}_0(X\le t){)}^n\right]dt$$

$$\le -{\int }_{-\infty }^0({\mathbb{P}}_0(Y\le t){)}^{n}dt+{\int }_0^{+\infty }\left[1-({\mathbb{P}}_0(Y\le t){)}^n\right]dt$$

$$={\int }_{-\infty }^0\left[{\mathbb{P}}_0(\max \{Y_1,\cdots ,Y_n\}>t)-1\right]dt+{\int }_0^{+\infty }{\mathbb{P}}_0(\max \{Y_1,\cdots ,Y_n\}>t)dt$$

$$={\mathrm{MAXVAR}}_n(Y).$$

$$F(t)={\mathbb{P}}_0(\max \{X_1,\cdots ,X_n\}\le t)$$

$$\mathbb{E}(\max \{X_1,\cdots ,X_n\})$$

$$=$$

$${\mathrm{MAXVAR}}_n(X^k)$$

$${\mathrm{MAXVAR}}_n(X)=\mathbb{E}(\max \{X_1,\cdots ,X_n\})\ge \mathbb{E}(X_1)=\mathbb{E}(X).$$

$${\mathrm{CVaR}}_{\alpha }(X)=\underset{\beta \in \mathbb{R}}{\min }\left\{\beta +\frac{1}{1-\alpha }\mathbb{E}(X-\beta {)}_{+}\right\},$$

$${\mathrm{VaR}}_{\alpha }(X):=\inf \left\{\nu \in \mathbb{R}:{\mathbb{P}}_0(X>\nu )<1-\alpha \right\}.$$

$${\mathrm{MAXVAR}}_n(\cdot )={\int }_0^1{\mathrm{CVaR}}_{\alpha }(\cdot )w_n(\alpha )d\alpha ,$$

$${\mathrm{MAXVAR}}_n(X)={\int }_0^1{\mathrm{CVaR}}_{\alpha }(X)w_n(\alpha )d\alpha ,$$

$$w_n(\alpha ):=n(n-1)(1-\alpha ){\alpha }^{n-2},\alpha \in [0,1]$$

$${\int }_0^1{w}_n(\alpha )d\alpha =n(n-1){\int }_0^1({\alpha }^{n-2}-{\alpha }^{n-1})d\alpha =n(n-1)\left[\frac{1}{n-1}-\frac{1}{n}\right]=1.$$

$$f_{\alpha }(A):$$

$$=g_{\alpha }[{\mathbb{P}}_0(A)],$$

$$g_{\alpha}(x):=\left\{\begin{array}[]{ll}\frac{1}{1-\alpha}x~{}~{}~{}~{}~{}~{}~{}\text{if}~{}x\in[0,1-\alpha),\\ ~{}~{}~{}1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if}~{}x\in[1-\alpha,1].\end{array}\right.$$

$${\mathrm{CVaR}}_{\alpha }(X)={\int }_{-\infty }^0[f_{\alpha }(X>t)-1]dt+{\int }_0^{+\infty }{f}_{\alpha }(X>t)dt.$$

$${\int }_{-\infty }^0[f_{\alpha }(X>t)-1]dt+{\int }_0^{+\infty }{f}_{\alpha }(X>t)dt$$

$$=$$

Full text

2010 MR subject classification: 49N15, 91G40

Key words: coherent risk measure, risk averse, risk envelope.

1 Introduction

In Cherny and Madan [2] and Cherny and Orlov [3], a new kind of risk measure –“MAXVAR” – is proposed, which is useful in the analysis of large portfolios. Given a probability space $(\Omega,\Sigma,\mathbb{P}_{0})$ and a random variable $X\in\mathscr{L}^{2}(\Omega,\Sigma,\mathbb{P}_{0})$, where $\mathscr{L}^{2}(\Omega,\Sigma,\mathbb{P}_{0})$ is the square integrable Lebesgue space ($\mathscr{L}^{2}$ for short), the MAXVAR is defined as

[TABLE]

where $X_{1},\cdots,X_{n}$ are i.i.d. copies of $X$. We call $\mathrm{MAXVAR}_{n}(\cdot)$ the “MAXVAR risk measure”.

Note that $\mathrm{MAXVAR}_{n}(\cdot)$ is always finite on $\mathscr{L}^{2}$ since $|\mathrm{MAXVAR}_{n}(X)|\leq n\mathbb{E}(|X|)<+\infty$ for any $X\in\mathscr{L}^{2}$.

In [2, 3], the name of “MINVAR risk measure” was used. Since we treat risk measures as a nondecreasing function, we use “MAXVAR risk measure” instead. Obviously, we have $\mathrm{MAXVAR}_{n}(X)=-\mathrm{MINVAR}_{n}(-X)$. Different from papers [2, 3], which considered coherency of MINVAR in $\mathscr{L}^{\infty}$ space, this paper deals with the $\mathscr{L}^{2}$ space. Our proof of the coherency of MAXVAR risk measure is direct and independent of [2, 3]. Moreover, we show risk averseness of MAXVAR and give an explicit formula for its risk envelope.

In Section 2, we present a simple proof for the coherency of MAXVAR. We show its aversity in Section 3. Section 4 is devoted to the discussion of a continuous representation of MAXVAR and Section 5 provides an explicit formula for its risk envelope.

2 Coherency of MAXVAR

In this section, we show that MAXVAR is a coherent risk measure in basic sense of Rockafellar.

Definition 1

(Rockafellar [5]) A functional $\mathcal{R}:\mathscr{L}^{2}\rightarrow(-\infty,+\infty]$ is a coherent risk measure in basic sense if it satisfies

(A1)

$\mathcal{R}(C)=C$* for all constant $C$;*

(A2)

(“convexity”) $\mathcal{R}(\lambda X+(1-\lambda)Y)\leq\lambda\cdot\mathcal{R}(X)+(1-\lambda)\cdot\mathcal{R}(Y)$ for any $X,Y\in\mathscr{L}^{2}$ and any fixed $0\leq\lambda\leq 1$;

(A3)

(“monotonicity”) $\mathcal{R}(X)\leq\mathcal{R}(Y)$ for any $X,Y\in\mathscr{L}^{2}$ satisfying $X\leq Y$;

(A4)

(“closedness”) If $\|X^{k}-X\|_{2}\rightarrow 0$ and $\mathcal{R}(X^{k})\leq 0$ for all $k\in\mathbb{N}$, then $\mathcal{R}(X)\leq 0$;

(A5)

(“positive homogeneity”) $\mathcal{R}(\lambda X)=\lambda\mathcal{R}(X)$ for any $\lambda>0$ and $X\in\mathscr{L}^{2}$.

Theorem 1

$\mathrm{MAXVAR}_{n}(\cdot)$* is a coherent risk measure in basic sense.*

Proof. (A1) is obvious by definition. (A5) is also easy to check since if $X_{1},\cdots,X_{n}$ are i.i.d. copies of $X$ and $\lambda>0$, then $\lambda X_{1},\cdots,\lambda X_{n}$ are i.i.d. copies of $\lambda X$.

Proof of (A2). We only need to show the following subadditive property of $\mathrm{MAXVAR}$

[TABLE]

Then (1) and (A5) imply (A2). For any $X,Y\in\mathscr{L}^{2}$, take $(X_{1},Y_{1}),\cdots,(X_{n},Y_{n})$ as i.i.d. copies of the two dimensional random vector $(X,Y)$. That is, the random vectors $(X_{1},Y_{1}),\cdots,(X_{n},Y_{n})$ are independent and have the same joint distribution as the random vector $(X,Y)$. Then $X_{1},\cdots,X_{n}$ are i.i.d. copies of $X$ and $Y_{1},\cdots,Y_{n}$ are i.i.d. copies of $Y$. We next show that $X_{1}+Y_{1},\cdots,X_{n}+Y_{n}$ are i.i.d. copies of $X+Y$.

Since $(X_{i},Y_{i})$ has the same joint distribution as $(X,Y)$, $i=1,...,n$, it follows that $X_{i}+Y_{i}$ has the same distribution as $X+Y$. In order to prove that $X_{1}+Y_{1},\cdots,X_{n}+Y_{n}$ are independent, we only need to prove that for any $t_{1},\cdots,t_{n}\in\mathbb{R}$,

[TABLE]

In fact, since the random vectors $(X_{1},Y_{1}),\cdots,(X_{n},Y_{n})$ are independent, we have

[TABLE]

for any Borel sets $B_{1},\cdots,B_{n}\subseteq\mathbb{R}^{2}$. In particular, if we take

[TABLE]

for any $1\leq i\leq n$ in (2), we can get (2). Therefore, $X_{1}+Y_{1},\cdots,X_{n}+Y_{n}$ are independent. Moreover, they are i.i.d. copies of $X+Y$.

Since the definition of MAXVAR does not depend on the choice of the i.i.d. copies, we have

[TABLE]

Furthermore, since

[TABLE]

we get

[TABLE]

*Proof of *(A3). For any $X,Y\in\mathscr{L}^{2}$ satisfying $X\leq Y$, suppose $X_{1},\cdots,X_{n}$ are i.i.d. copies of $X$ and $Y_{1},\cdots,Y_{n}$ are i.i.d. copies of $Y$. We can see that $\mathbb{P}_{0}(X\leq t)\geq\mathbb{P}_{0}(Y\leq t)$ for any $t\in\mathbb{R}$ since $X\leq Y$. Then we have

[TABLE]

The detail of the first equality is as follows. Denote by

[TABLE]

the cumulative distribution function of $\max\{X_{1},\cdots,X_{n}\}$. Then

[TABLE]

And the second equality comes from the fact that $F(t)=(\mathbb{P}_{0}(X\leq t))^{n}$.

Proof of (A4). Suppose $X^{k}~{}(k=1,2,\cdots),X\in\mathscr{L}^{2}$ and $\|X^{k}-X\|_{2}\rightarrow 0$ as $k$ tends to infinity. Then $X^{k}\rightarrow X$ in distribution. Denote by $F_{k}(t)$ the distribution function of $X^{k}~{}(k=1,2,\cdots)$ and by $F(t)$ the distribution of $X$. Then $\lim\limits_{k\rightarrow\infty}F_{k}(t)=F(t)$ for all continuous points of $F(\cdot)$. It implies that $\lim\limits_{k\rightarrow\infty}[F_{k}(t)]^{n}=[F(t)]^{n}$ for all continuous points of $[F(\cdot)]^{n}$. Note that $[F_{k}(t)]^{n}$ is the distribution function of $\max\{X^{k}_{1},\cdots,X^{k}_{n}\}$ and $[F(t)]^{n}$ is the distribution function of $\max\{X_{1},\cdots,X_{n}\}$, where $X^{k}_{1},\cdots,X^{k}_{n}$ are i.i.d. copies of $X^{k}~{}(k=1,2,\cdots)$ and $X_{1},\cdots,X_{n}$ are i.i.d. copies of $X$. Therefore, we have $\max\{X^{k}_{1},\cdots,X^{k}_{n}\}\rightarrow\max\{X_{1},\cdots,X_{n}\}$ in distribution, and

[TABLE]

as $k$ tends to infinity. Thus, if $\mathrm{MAXVAR}_{n}(X^{k})\leq 0$ for all $k=1,2,\cdots$ then $\mathrm{MAXVAR}_{n}(X)\leq 0$. The proof of the theorem is completed. $\Box$

3 Risk-averseness of MAXVAR

Suppose $\mathcal{R}$ is a functional from $\mathscr{L}^{2}$ to $(-\infty,+\infty]$. Recall that an averse risk measure is defined by axioms (A1), (A2), (A4), (A5) and

(A6)

$\mathcal{R}(X)>\mathbb{E}(X)$ for all non-constant $X$.

We then have the next theorem.

Theorem 2

*If $n\geq 2$, then $\mathrm{MAXVAR}_{n}(\cdot)$ is averse.

Föllmer and Schied [4] proved that if $\mathcal{R}$ is a coherent, law-invariant risk measure in $\mathscr{L}^{\infty}$ (not $\mathscr{L}^{2}$) other than $\mathbb{E}(\cdot)$, then $\mathcal{R}$ is averse, where “law-invariant” stands for that $\mathcal{R}(X)=\mathcal{R}(Y)$ whenever $X$ and $Y$ have the same distribution under $\mathbb{P}_{0}$. Since we are now considering the $\mathscr{L}^{2}$ case, we cannot use the result in Föllmer and Schied [4] directly. We next give a separate proof.

Proof of Theorem 2 On one hand, for any $X\in\mathscr{L}^{2}$, let $X_{1},\cdots,X_{n}$ be i.i.d. copies of $X$. Then we have

[TABLE]

On the other hand, if $\mathrm{MAXVAR}_{n}(X)=\mathbb{E}(X)=\mathbb{E}(X_{1})~{}(n\geq 2)$, then

$\max\{X_{1},\cdots,X_{n}\}=X_{1}$ almost surely. Similarly, $\max\{X_{1},\cdots,X_{n}\}=X_{2}$ almost surely. Therefore, $X_{1}=X_{2}$ almost surely. Since $X_{1}$ and $X_{2}$ are independent, we must have $X_{1}$ equals to a constant almost surely, which is equivalent to say $X$ equals to a constant almost surely. Therefore, $\mathrm{MAXVAR}_{n}(X)>\mathbb{E}(X)$ for nonconstant $X$, which implies that $\mathrm{MAXVAR}_{n}(\cdot)$ is averse when $n\geq 2$.

Remark. In fact, Theorems 1 and 2 can be obtained as corollaries of Theorem 3 in the next section. See the remark after the proof of Theorem 3 for details. However, we think it is of interest to provide an elementary proof only based on definition of MAXVAR.

4 MAXVAR as a continuous convex combination of CVaR

An important coherent risk measure in basic sense is the conditional value at risk (CVaR) popularized by Rockafellar and Uryasev [6]. Among several equivalent definitions of CVaR, the most familiar one is probably the following.

[TABLE]

where $(t)_{+}=\max(t,0)$ and $\alpha\in[0,1)$. The minimum is attained at $\beta^{*}=\mathrm{VaR}_{\alpha}(X)$, and the VaR ( “Value-at-Risk”) is defined as

[TABLE]

In this section, we show that $\mathrm{MAXVAR}_{n}(\cdot)$ is certain “continuous convex combination” of the CVaR measure in the sense that

[TABLE]

where $w_{n}(\alpha)~{}(\alpha\in[0,1])$ is the “weight function” which satisfies $w_{n}(\alpha)\geq 0$ on $[0,1]$ and $\int_{0}^{1}w_{n}(\alpha)d\alpha=1$. Specifically, we have the next theorem.

Theorem 3

For any $X\in\mathscr{L}^{2}$, we have

[TABLE]

where

[TABLE]

is the weight function.

Remark. It can be easily checked that $w_{n}(\alpha)\geq 0$ on $[0,1]$, and

[TABLE]

Therefore, $w_{n}(\alpha)$ is indeed a weight function.

Theorem 3 was mentioned in Cherny and Orlov [3] without details. We now give a detailed proof by using the so called “Choquet integral”. First, we need a lemma. For any $\alpha\in[0,1)$, define $f_{\alpha}(\cdot):~{}\Sigma\rightarrow[0,1]$ in the following way,

[TABLE]

where

[TABLE]

We then have the following lemma, which implies that the CVaR measure can be written as the “Choquet integral” with respect to $f_{\alpha}(\cdot)$.

Lemma 1

For any $X\in\mathscr{L}^{2}$ and $\alpha\in[0,1)$, we have

[TABLE]

Proof. If $\mathrm{VaR}_{\alpha}(X)\leq 0$, then

[TABLE]

The last step above is due to (5) and (6).

If $\mathrm{VaR}_{\alpha}(X)>0$, then

[TABLE]

which completes the proof. $\Box$

Proof of Theorem 3 Define

[TABLE]

It is not difficult to check that

[TABLE]

where $g_{\alpha}(x)$ is as defined in (7). By (4), for any $X\in\mathscr{L}^{2}$ we have

[TABLE]

So by (8), (9) and Lemma 1, together with Fubini’s theorem and the fact that $\int_{0}^{1}w_{n}(\alpha)d\alpha=1$, we get

[TABLE]

for any $X\in\mathscr{L}^{2}$, as desired.

Remark. Theorem 3 says that $\mathrm{MAXVAR}_{n}(\cdot)$ is a continuous convex combination of the CVaR measure, its coherency in basic sense follows from Proposition 2.1 of Ang et al [1], and its averseness follows from the averseness of the CVaR (Proposition 4.4 of Ang et al [1]) together with the basic property of integral. Therefore, Theorem 3 can actually provide an alternative proof of the coherency and averseness of $\mathrm{MAXVAR}_{n}(\cdot)$.

5 The risk envelope of MAXVAR

Since

[TABLE]

is a coherent risk measure on $\mathscr{L}^{2}$, by the dual representation theorem (Rockafellar [5]), there exists a unique, nonempty, convex and closed set $\mathcal{Q}_{n}\subseteq\mathscr{L}^{2}$, called “the risk envelope of $\mathrm{MAXVAR}_{n}(\cdot)$” such that

[TABLE]

for any $X\in\mathscr{L}^{2}$.

In this section we aim at characterizing the risk envelope of $\mathrm{MAXVAR}_{n}(\cdot)$”. First recall the following well-known result for the discrete convex combination of the CVaR measure, which can be found in Rockafellar [5] and whose proof can be found in Ang et al [1].

Proposition 1

Let $\mathcal{R}(\cdot)=\sum\limits_{i=1}^{n}\lambda_{i}\mathrm{CVaR}_{\alpha_{i}}(\cdot)$ with positive weights $\lambda_{i}$ adding up to $1$. Then $\mathcal{R}$ is a coherent risk measure in the basic sense and its risk envelope is

[TABLE]

A continuous version of Proposition 1 gives the risk envelope of MAXVAR as follows.

Theorem 4

The risk envelope of MAXVAR is

[TABLE]

where

[TABLE]

is the weight function ($0^{0}$ is defined to be 1), and “$\mathrm{cl}$” stands for the closure in $\mathscr{L}^{2}$.

Proof. Note that the integration “$\int_{0}^{1}Q_{\alpha}w_{n}(\alpha)d\alpha$” in (10) is defined pointwise. That is, $Y=\int_{0}^{1}Q_{\alpha}w_{n}(\alpha)d\alpha$ means $Y(\omega)=\int_{0}^{1}Q_{\alpha}(\omega)w_{n}(\alpha)d\alpha$ for any $\omega\in\Omega$. Since $0\leq Q_{\alpha}\leq\dfrac{1}{1-\alpha}$ for any $\alpha\in[0,1)$, we have

[TABLE]

for any $\omega\in\Omega$. Therefore, $\mathcal{Q}_{n}\subseteq\mathscr{L}^{\infty}\subseteq\mathscr{L}^{2}$. In addition, we can check that

[TABLE]

for any $X\in\mathscr{L}^{2}$. Furthermore, it is easy to check the convexity of $\mathcal{Q}_{n}$. Since $\mathcal{Q}_{n}$ is closed in $\mathscr{L}^{2}$, it follows from the dual representation theorem that Formula (11) implies that (10) is the risk envelope of $\mathrm{MAXVAR}_{n}(\cdot)$. $\Box$

Acknowledgment. We would like to thank the anonymous referees for their useful suggestions, which are of great help for improving the manuscript. The research of Jie Sun is partially supported by Australian Research Council under Grant DP160102819. The research of Qiang Yao is partially supported by grants from National Science Foundation of China (No.11201150 and No.11126236) and the 111 Project (No.B14019).