Distortion Risk Measures of Increasing Rearrangement

Abstract

Increasing rearrangement is a rewarding instrument in financial risk management. In practice, risks must be managed from different perspectives. A common example is the portfolio risk, which often can be seen from at least two perspectives: market value and book value. Different perspectives with different distributions can be coupled by increasing rearrangement. One distribution is regarded as underlying, and the other distribution can be expressed as an increasing rearrangement of the underlying distribution. Then, the risk measure for the latter can be expressed in terms of the underlying distribution. Our first objective is to introduce increasing rearrangement for application in financial risk management and to apply increasing rearrangement to the class of distortion risk measures. We derive formulae to compute risk measures in terms of the underlying distribution. Afterwards, we apply our results to a series of special distortion risk measures, namely the value at risk, expected shortfall, range value at risk, conditional value at risk, and Wang’s risk measure. Finally, we present the connection of increasing rearrangement with inverse transform sampling, Monte Carlo simulation, and cost-efficient strategies. Butterfly options serve as an illustrative example of the method.

1. Introduction

Increasing rearrangement often appears in finance and risk management in a natural context. For example, consider the market value and book value of a portfolio, where the book value is reported according to some reporting standard. In many cases, the portfolio risk has to be managed based on the market value as well as book value—both points of view may be important, and there may be even more perspectives to be controlled. Mostly, the book value can be regarded as an increasing function of the market value. In this case, risk measures of book and market value are related in a unique manner. Given that the distribution of the market value does not have atoms, distortion risk measures of book value can be written in terms of the corresponding market value. This gives insight and computational advancements.

The example of book and market value can be generalized to any two distributions when one of them does not have atoms, as, in this case, an increasing rearrangement exists that transforms the continuous distribution into the other. Hence, given any distribution, we may find the increasing rearrangement with respect to some continuous distribution of our choice (e.g., a uniform or normal distribution) and compute the risk of the given distribution in terms of the continuous distribution that we choose.

The class of distortion risk measures to which this idea applies is a rather general yet useful class of risk measures. All distortion risk measures as defined by Artzner et al. (1999) are coherent, i.e., they are translation-invariant, monotonous, sub-additive, and positively homogeneous. Distortion risk measures were introduced by Denneberg (1990) and Wang Wang et al. (1997). Their roots lie in the dual utility theory of Yaari (1987), where it is shown that there has to exist a function such that a prospect is valued at its distorted expectation. In place of using the tail probabilities in order to quantify risk, the decision-maker uses the distorted tail probabilities. A general overview of the distortion risk measures and their relations with the ordering of risk and the concept of comonotonicity can be found in Denuit et al. (2005) and Dhaene et al. (2006).

Distortion risk measures allow an asset manager to reflect a client’s attitude towards risk by choosing an appropriate distortion function. For instance, actuaries capture the riskiness of a distribution with distortion risk measures and apply them to price extreme events, to assess reserves, to design risk transfer strategies, and to allocate capital.

This article is organized as follows.

In Section 2, we introduce increasing rearrangement. The following Section 3 brings the application of increasing rearrangement to distortion risk measures. This is applied in Section 4 to a series of special distortion risk measures, namely the value at risk, expected shortfall, range value at risk, conditional value at risk, and Wang’s risk measure.

We apply increasing rearrangement to inverse transform sampling and Monte Carlo simulation in Section 5. An example is the butterfly spread with the R code provided in the Appendix A. Finally, we point out the relation of increasing rearrangement to the theory of cost-efficient strategies.

2. Increasing Rearrangement

We introduce increasing rearrangement in two versions called the “quantile version” and the “transport version”. While the quantile version is especially advantageous in finance—as it has the transformation property needed when applied to risk measures—the transport version is somewhat smoother and, for this reason, is preferred when applied to optimal transport.

Increasing rearrangement is related to the generalized inverse of distribution functions. Therefore, we first introduce this concept. Here and in the following, increasing means

$$f\left(a\right)\le f\left(b\right)\mathrm{for}\mathrm{all}a\le b\text{in the range of definition of}f.$$

(1)

Definition 1.

Let $T:\mathbb{R}\to \mathbb{R}$ be an increasing function. The generalized inverse $T^{\leftarrow }$ of T is defined by

$$T^{\leftarrow }\left(y\right)inf\left\{x\in \mathbb{R}\left|T\right(x)\ge y\right\},$$

(2)

with $inf\varnothing =\infty$.

Let F be any distribution function. For $\alpha \in (0,1)$, we denote the corresponding quantile function with $q_{\alpha }\left(F\right)$. It coincides with the generalized inverse $F^{\leftarrow }\left(\alpha \right)$ and is given by

$$q_{\alpha }\left(F\right)=F^{\leftarrow }\left(\alpha \right)=inf\left\{x\in \mathbb{R}\left|F\right(x)\ge \alpha \right\}.$$

(3)

If X is a random variable with distribution function F, we write $q_{\alpha }\left(X\right)\equiv q_{\alpha }\left(F\right)$ as well. To specify the distribution functions, we write $F_X$ for the distribution function of X. If X and Y have the same distribution function, we write $X\stackrel{d}{=}Y$. If X has distribution $F_X$, we write $X\sim F_X$. If X is the argument of some function g, we write $g\left(X\right)\equiv g\circ X$.

We assume all random variables to be elements of some domain $\mathcal{X}\subseteq L^0$ of random variables with values in $\mathbb{R}\cup \{\infty \}$ and $\mathcal{X}$ to be a convex cone, i.e., $\alpha X+\beta Y\in \mathcal{X}$ for all $X,Y\in \mathcal{X}$ and $\alpha ,\beta >0$.

Given any increasing function $T:\mathbb{R}\to \mathbb{R}$, the generalized inverse $T^{\leftarrow }$ of T is increasing and left-continuous (cf. (McNeil et al. 2015, p. 641 f.)).

Proposition 1.

If X is a random variable with distribution function $F_X$ and $g:\mathbb{R}\to \mathbb{R}$ is increasing and left-continuous, then

$$F_{g\circ X}^{\leftarrow }\left(\alpha \right)=g\circ F_X^{\leftarrow }\left(\alpha \right)for all \alpha \in (0,1).$$

(4)

In terms of the quantile function, this means

$$q_{\alpha }(g\circ X)=g\circ q_{\alpha }\left(X\right)for all \alpha \in (0,1).$$

(5)

Proof. 

See (McNeil et al. 2015, p. 642).    □

Definition 2.

Let $T:\mathbb{R}\to \mathbb{R}$ be an increasing function. The right-continuous generalized inverse $T^{\to }$ of T is defined by

$$T^{\to }\left(y\right)inf\left\{x\in \mathbb{R}\left|T\right(x)>y\right\},$$

(6)

with $inf\varnothing =\infty$.

Definition 3.

Let X and Y be random variables with distribution functions $F_X$ and $F_Y$. The increasing rearrangement (quantile version) of Y with respect to X is given by

$$g^{\leftarrow }{F}_Y^{\leftarrow }\circ F_X.$$

(7)

The increasing rearrangement (transport version) of Y with respect to X is given by

$$g^{\to }{F}_Y^{\to }\circ F_X.$$

(8)

Lemma 1.

Let X and Y be random variables. Let $g^{\to }$ be the increasing rearrangement (transport version) of Y with respect to X. Then, the following holds:

1. 

$g^{\to }$ is increasing;

2. 

$g^{\to }$ is right-continuous;

3. 

If X does not have atoms, then $Y\stackrel{d}{=}{g}^{\to }\left(X\right).$

Proof. 

See (Villani 2008, p. 19 f.).    □

Lemma 2.

Let X and Y be random variables. Let $g^{\leftarrow }$ be the increasing rearrangement (quantile version) of Y with respect to X. Then, the following holds:

1. 

$g^{\leftarrow }$ is increasing;

2. 

If X does not have atoms, then $g^{\leftarrow }$ is left-continuous;

3. 

If X does not have atoms, then $Y\stackrel{d}{=}{g}^{\leftarrow }\left(X\right)$.

Proof. 

(i) The functions $F_X$ and $F_Y^{\leftarrow }$ are increasing, and so is $g^{\leftarrow }$.

(ii) X does not have atoms if $F_X$ is continuous. Moreover, $F_Y^{\leftarrow }$ is left-continuous. This implies (2).

(iii) The continuity of $F_X$ implies $F_X\left(X\right)\sim U[0,1]$ (uniformly distributed on $[0,1]$). Hence, $F_Y^{\leftarrow }\left(F_X\left(X\right)\right)\stackrel{d}{=}Y$ (cf. (Embrechts and Hofert 2013, p. 429)).    □

While the quantile version of increasing rearrangement seems to be a natural choice in terms of the generalized inverse and favorable in view of Proposition 1, the transport version of increasing rearrangement is right-continuous without further assumptions on X. Villani praises the transport version of increasing rearrangement: “This rearrangement is quite simple, explicit, as smooth as can be, and enjoys good geometric properties” (Villani 2008, p. 20).

3. Risk Measures of Increasing Rearrangement

Distortion risk measures are law-invariant risk measures and include the value at risk and expected shortfall. They are defined in the following way.

Definition 4

(Tsukahara (2009)). A distortion function D is an increasing, right-continuous function on $[0,1]$ satisfying $D\left(0\right)=0$ and $D\left(1\right)=1$. The distortion risk measure associated with a distortion function D is defined by

$${\varrho }_D\left(X\right){\int }_0^1{q}_u\left(X\right)dD\left(u\right).$$

(9)

Note that, in our notation, we regard X as a loss function, i.e., positive values of X stand for losses, and negative values of X stand for profits. If one uses a P&L function with the opposite interpretation, the formulas here and in the following must be altered accordingly.

In McNeil et al. (2015), the definition of distortion risk measures is given for the smaller class of convex, absolutely continuous distortion functions. With respect to these distortion functions, the corresponding distortion risk measures are coherent. However, this excludes the value at risk from the class, which is why we prefer the more general Definition 4 of Tsukahara.

Let D be an absolutely continuous distortion function and $\phi$ be the right derivative of D. Then, $\phi$ is a non-negative function, and

$$D\left(u\right)={\int }_0^u\phi \left(s\right)ds.$$

(10)

One defines the following.

Definition 5.

Let φ be the right derivative of an absolutely continuous distortion function. The spectral risk measure with respect to φ is

$${\varrho }_{\phi }\left(X\right){\int }_0^1{q}_u\left(X\right)\phi \left(u\right)du.$$

(11)

One calls φ the spectrum of ${\varrho }_{\phi }$.

If $\phi$ is the right derivative of an absolutely continuous distortion function D, then ${\varrho }_D={\varrho }_{\phi }$ holds by construction.

Lemma 3.

Let D be a distortion function and ${\varrho }_D$ the risk measure associated with D. Then, ${\varrho }_D$ can be written in the form

$${\varrho }_D\left(X\right)={\int }_{-\infty }^{\infty }xdD\circ F_X\left(x\right).$$

(12)

Proof. 

The proof given in (McNeil et al. 2015, p. 287) for convex and absolutely continuous distortion function D applies literally.    □

The chain rule, applied to (12), implies the following.

Lemma 4.

Let D be an absolutely continuous distortion function, φ its spectrum and ${\varrho }_{\phi }$ the risk measure associated with φ. Let X be a random variable with an absolutely continuous distribution function $F_X$ and let $f_X$ be the right derivative of $F_X$ (density of X). Then, ${\varrho }_{\phi }$ can be written in the form

$${\varrho }_{\phi }\left(X\right)={\int }_{-\infty }^{\infty }x\phi \circ F_X\left(x\right)f_X\left(x\right)dx.$$

(13)

Theorem 1.

Let X and Y be random variables and X be atomless. Let $g^{\leftarrow }$ be the quantile version of the increasing rearrangement of Y with respect to X. Let D be a distortion function and ${\varrho }_D$ the distortion risk measure associated with D. Then, the following holds:

$$\begin{array}{c}{\varrho }_D\left(Y\right)={\int }_0^1{q}_u\left(Y\right)dD\left(u\right)={\int }_0^1{g}^{\leftarrow }\circ q_u\left(X\right)dD\left(u\right),\end{array}$$

(14)

$$\begin{array}{c}{\varrho }_D\left(Y\right)={\int }_{-\infty }^{\infty }xdD\circ F_Y\left(x\right)={\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right)dD\circ F_X\left(x\right).\end{array}$$

(15)

If D is absolutely continuous and φ is the corresponding spectrum, we have

$${\varrho }_{\phi }\left(Y\right)={\int }_0^1{q}_u\left(Y\right)\phi \left(u\right)du={\int }_0^1{g}^{\leftarrow }\circ q_u\left(X\right)\phi \left(u\right)du.$$

(16)

Moreover, if X is absolutely continuous with density $f_X$,

$${\varrho }_{\phi }\left(Y\right)={\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right)\phi \circ F_X\left(x\right)f_X\left(x\right)dx.$$

(17)

Note that representation (17) cannot be written in terms of $F_Y$, because $F_Y$ is not supposed to be absolutely continuous (not even continuous).

Proof. 

By Lemma 7, $g^{\leftarrow }$ is an increasing, left-continuous function, and $Y\stackrel{d}{=}{g}^{\leftarrow }\left(X\right)$. Thus, Proposition 1 applies and yields (14) and (16).

To show (15), we write $G\left(x\right)=D\circ F_X\left(x\right)$ and observe $G^{\leftarrow }\left(x\right)=F_X^{\leftarrow }\circ D^{\leftarrow }\left(x\right)$ (Cf. (McNeil et al. 2015, p. 287)). Now,

$$\begin{array}{cc}\hfill {\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right)dG\left(x\right)& ={\int }_0^1{g}^{\leftarrow }\left(G^{\leftarrow }\left(u\right)\right)du={\int }_0^1{g}^{\leftarrow }\circ F_X^{\leftarrow }\circ D^{\leftarrow }\left(u\right)du\hfill \\ & =\mathbb{E}\left[g^{\leftarrow }\circ F_X^{\leftarrow }\circ D^{\leftarrow }\left(U\right)\right],U\sim U[0,1].\hfill \end{array}$$

(18)

We introduce the random variable $V=D^{\leftarrow }\left(U\right)$ and obtain with (14)

$${\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right)dD\circ F_X\left(x\right)=\mathbb{E}\left[g^{\leftarrow }\circ F_X^{\leftarrow }\left(V\right)\right]={\int }_0^1{g}^{\leftarrow }\circ F_X^{\leftarrow }\left(v\right)dD\left(v\right)={\varrho }_D\left(Y\right).$$

(19)

This shows (15). Finally, applying the chain rule to (15) implies (17) and this finishes the proof.    □

4. Application to Special Risk Measures

4.1. Application to Value at Risk

The value at risk of a loss distribution is defined to be the quantile at some confidence level $\alpha \in (0,1)$, and we write

$${VaR}_{\alpha }\left(Y\right)q_{\alpha }\left(Y\right).$$

(20)

The value at risk is a distortion risk measure with distortion function

$$D\left(u\right)={\mathbb{1}}_{[\alpha ,1]}\left(u\right).$$

(21)

(See Tsukahara (2009) for more details.) Hence, Theorem 1 applied to VaR yields the following well-known lemma.

Lemma 5.

Let X and Y be random variables, and X does not have atoms. Let $g^{\leftarrow }$ be the quantile version of the increasing rearrangement of Y with respect to X. Then,

$${VaR}_{\alpha }\left(Y\right)=g^{\leftarrow }\circ {VaR}_{\alpha }\left(X\right).$$

(22)

Proof. 

The assertion follows from Equation (14) of Theorem 1. Moreover, it is a consequence of Lemma 7 in combination with Proposition 1, namely

$${VaR}_{\alpha }\left(Y\right)=F_Y^{\leftarrow }\left(\alpha \right)=F_{g^{\leftarrow }\circ X}^{\leftarrow }\left(\alpha \right)=g^{\leftarrow }\circ F_X^{\leftarrow }\left(\alpha \right)=g^{\leftarrow }\circ {VaR}_{\alpha }\left(X\right).$$

(23)

□

4.2. Application to Expected Shortfall

The expected shortfall of a random variable Y with $\mathbb{E}\left(Y^{+}\right)<\infty$ and confidence level $\alpha \in (0,1)$, e.g., $\alpha =0.995$, is the spectral risk measure defined as follows:

$${ES}_{\alpha }\left(Y\right){\int }_0^1{q}_u\left(Y\right){\phi }_{ES}\left(u\right)du$$

(24)

with specific spectrum

$${\phi }_{ES}\left(u\right){\displaystyle \frac{1}{1-\alpha }}{\mathbb{1}}_{[\alpha ,1)}\left(u\right)\mathrm{for}u\in [0,1].$$

(25)

Alternatively, the expected shortfall can be represented in terms of the distortion function of the form

$$D_{ES}\left(u\right)={\int }_{\alpha }^u{\phi }_{ES}\left(s\right)ds=max\left\{{\displaystyle \frac{u-\alpha }{1-\alpha }},0\right\},u\in [0,1],$$

(26)

(This follows from Definition 5 of distortion risk measures.) For continuous random variables X, the expected shortfall has an intuitive interpretation as conditional expectation, namely

$${ES}_{\alpha }\left(X\right)=\mathbb{E}\left[X|X\ge q_{\alpha }\left(X\right)\right].$$

(27)

For discontinous random variables, though, one has to be careful, and the proper definition in this general case amounts to (McNeil et al. 2015, p. 283)

$${ES}_{\alpha }\left(X\right)={\displaystyle \frac{1}{1-\alpha }}\left\{\mathbb{E}\left[X{\mathbb{1}}_{(q_{\alpha }\left(X\right),\infty )}\left(X\right)\right]+q_{\alpha }\left(X\right)\left(1-\alpha -P(X>q_{\alpha }\left(X\right))\right)\right\}.$$

(28)

Nevertheless, the theorem on increasing rearrangement and distortion risk measures allows us to find a version of Formula (27) for discontinuous random variables as follows.

Theorem 2.

Let X and Y be random variables and X be absolutely continuous. Let $g^{\leftarrow }$ be the quantile version of the increasing rearrangement of Y with respect to X. Let $\mathbb{E}\left(Y^{+}\right)<\infty$. Then,

$${ES}_{\alpha }\left(Y\right)=\mathbb{E}\left[g^{\leftarrow }\left(X\right)|X\ge q_{\alpha }\left(X\right)\right].$$

(29)

Proof. 

As X is absolutely continuous, there exists a density function $f_X$. Consequently, the expected shortfall can be written in the form of Equation (17) with spectrum ${\phi }_{ES}$. We derive

$$\begin{array}{cc}\hfill {ES}_{\alpha }\left(Y\right)& ={\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right){\phi }_{ES}\circ F_X\left(x\right)f_X\left(x\right)dx\hfill \\ & ={\displaystyle \frac{1}{1-\alpha }}{\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right){\mathbb{1}}_{[\alpha ,1)}\left(F_X\left(x\right)\right)f_X\left(x\right)dx\hfill \\ & ={\displaystyle \frac{1}{1-\alpha }}{\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right){\mathbb{1}}_{[q_{\alpha }\left(X\right),\infty )}\left(x\right)f_X\left(x\right)dx\hfill \\ & ={\displaystyle \frac{1}{1-\alpha }}\mathbb{E}\left[g^{\leftarrow }\left(X\right){\mathbb{1}}_{[q_{\alpha }\left(X\right),\infty )}\left(X\right)\right]=\mathbb{E}\left[g^{\leftarrow }\left(X\right)|X\ge q_{\alpha }\left(X\right)\right]\hfill \end{array}$$

(30)

as was to be shown.    □

In case $Y=g^{\leftarrow }\left(X\right)$, X-a.s., Theorem 2 takes the striking form

$${ES}_{\alpha }\left(Y\right)=\mathbb{E}\left[Y|X\ge q_{\alpha }\left(X\right)\right].$$

(31)

The following theorem shows that this holds for any (not necessarily left-continuous) increasing transformation g with $Y=g\circ X$, X-a.s. as well.

Theorem 3.

Let X be an absolutely continuous random variable, and $Y=g\circ X$, X-a.s., where g is increasing. Let $E\left(Y^{+}\right)<\infty$. Then,

$${ES}_{\alpha }\left(Y\right)=\mathbb{E}\left(Y\right|X\ge q_{\alpha }\left(X\right)).$$

(32)

Proof. 

We consider the sets

$$\mathcal{Y}\equiv \left\{x\in \mathbb{R}|g\left(x\right)\ge q_{\alpha }\left(Y\right)\right\} \mathrm{and} \mathcal{X}\equiv \left\{x\in \mathbb{R}|x\ge q_{\alpha }\left(X\right)\right\}.$$

(33)

In the first place, $\mathcal{X}\subseteq \mathcal{Y}$, or, equivalently,

$${\mathbb{1}}_{\mathcal{Y}}-{\mathbb{1}}_{\mathcal{X}}\ge 0.$$

(34)

Indeed, for an arbitrary $x\in \mathcal{X}$, we obtain

$$\alpha \le P(X\le x)\le P\left(g\right(X)\le g(x\left)\right)=P(Y\le g(x\left)\right),$$

(35)

which means $x\in \mathcal{Y}$. This implies (34). Moreover, we find that

$${\mathbb{1}}_{\mathcal{Y}}\left(x\right)-{\mathbb{1}}_{\mathcal{X}}\left(x\right)>0 \mathrm{implies} g\left(x\right)=q_{\alpha }\left(Y\right).$$

(36)

This can be shown by contradiction. Indeed, let $x\in \mathcal{Y}\setminus \mathcal{X}$ be arbitrary and assume $g\left(x\right)>q_{\alpha }\left(Y\right)$. Then, we obtain by the monotony of g and the absolute continuity of X

$$\begin{array}{cc}\hfill P(X\le x)& =P(X<x)\ge P\left(g\right(X)<g(x\left)\right)=P(Y<g(x\left)\right)\hfill \\ & \ge P\left(Y\le q_{\alpha }\left(Y\right)+{\displaystyle \frac{g\left(x\right)-q_{\alpha }\left(Y\right)}{2}}\right)\ge \alpha ,\hfill \end{array}$$

(37)

hence $x\ge q_{\alpha }\left(X\right)$ in contradiction to $x\notin \mathcal{X}$. Thus, statement (36) holds.

Now, we prove the theorem by showing

$$(1-\alpha )\left({ES}_{\alpha }\left(Y\right)-\mathbb{E}(g\circ X|X\ge q_{\alpha }\left(X\right))\right)=0.$$

(38)

Representation (28) of the expected shortfall is equivalent to

$${ES}_{\alpha }\left(Y\right)={\displaystyle \frac{1}{1-\alpha }}\left\{\mathbb{E}\left(Y{\mathbb{1}}_{Y\ge q_{\alpha }\left(Y\right)}\right)+q_{\alpha }\left(Y\right)\left(1-\alpha -P(Y\ge q_{\alpha }\left(Y\right)\right)\right\},$$

(39)

as follows from

$$\mathbb{E}\left(Y{\mathbb{1}}_{Y>q_{\alpha }\left(Y\right)}\right)=\mathbb{E}\left(Y{\mathbb{1}}_{Y\ge q_{\alpha }\left(Y\right)}\right)-q_{\alpha }\left(Y\right)P(Y=q_{\alpha }\left(Y\right)).$$

(40)

The random variable X is assumed to be absolutely continuous and hence possesses a probability density function $f^X$. We calculate with (39)

$$\begin{array}{c}{\displaystyle (1-\alpha )\left({ES}_{\alpha }\left(Y\right)-\mathbb{E}(g\circ X|X\ge q_{\alpha }\left(X\right))\right)}\hfill \\ =\mathbb{E}\left(Y|Y\ge q_{\alpha }\left(Y\right)\right)+q_{\alpha }\left(Y\right)\left(1-\alpha -P(Y\ge q_{\alpha }\left(Y\right)\right)\hfill \\ -(1-\alpha )\mathbb{E}(g\circ X|X\ge q_{\alpha }\left(X\right))\hfill \\ =\int g\left(x\right)1_{\mathcal{Y}}\left(x\right)f^X\left(x\right)dx+q_{\alpha }\left(Y\right)\int 1_{\mathcal{X}}\left(x\right)f^X\left(x\right)dx\hfill \\ -q_{\alpha }\left(Y\right)\int 1_{\mathcal{Y}}\left(x\right)f^X\left(x\right)dx-\int g\left(x\right)1_{\mathcal{X}}\left(x\right)f^X\left(x\right)dx\hfill \\ =\int \left[g\left(x\right)-q_{\alpha }\left(Y\right)\right]\left[1_{\mathcal{Y}}\left(x\right)-1_{\mathcal{X}}\left(x\right)\right]f^X\left(x\right)dx.\hfill \end{array}$$

(41)

By (34) and (36), the integrand vanishes. This shows (38) and thus the theorem is proven.    □

4.3. Application to Range Value at Risk

Next, we highlight the range value at risk introduced by Cont et al. (2010). It is a modified version of the expected shortfall such that an average of the VaR levels is calculated across a pre-defined range of loss probabilities. Let $0<\alpha <1$, $0<\beta \le 1-\alpha$, then

$${RVaR}_{\alpha ,\beta }\left(X\right){\displaystyle \frac{1}{\beta }}{\int }_{\alpha }^{\alpha +\beta }{VaR}_u\left(X\right)du.$$

(42)

The range value at risk has good statistical properties (“$\mathcal{C}$-robustness”), which carry over to the expected shortfall in a certain sense, as is discussed in Cont et al. (2010). In applying our main Theorem 1, we derive a formula for the range value at risk of Y in terms of X.

Theorem 4.

Let X and Y be random variables and X be absolutely continuous. Let $g^{\leftarrow }$ be the quantile version of the increasing rearrangement of Y with respect to X. Let $E\left(Y^{+}\right)<\infty$ and $\beta >0$, $\alpha +\beta \le 1$. Then,

$${RVaR}_{\alpha ,\beta }\left(Y\right)=\mathbb{E}\left[g^{\leftarrow }\left(X\right)|q_{\alpha +\beta }\left(X\right)\ge X\ge q_{\alpha }\left(X\right)\right].$$

(43)

Proof. 

The assumption of the absolute continuity of X gives the possibility to write the range value at risk in terms of the density function $f_X$ by (17), as follows:

$$\begin{array}{cc}\hfill {RVaR}_{\alpha ,\beta }\left(Y\right))& ={\displaystyle \frac{1}{\beta }}{\int }_{\alpha }^{\beta +\alpha }{VaR}_u\left(Y\right)du={\displaystyle \frac{1}{\beta }}{\int }_0^1{\mathbb{1}}_{(\alpha ,\alpha +\beta )}\left(u\right)q_u\left(Y\right)du\hfill \\ \hfill & ={\displaystyle \frac{1}{\beta }}{\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right){\mathbb{1}}_{(\alpha ,\alpha +\beta )}\left(F_X\left(x\right)\right)f_X\left(x\right)dx\hfill \\ \hfill & ={\displaystyle \frac{1}{\beta }}{\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right){\mathbb{1}}_{\left(q_{\alpha }\left(X\right),q_{\alpha +\beta }\left(X\right)\right)}\left(x\right)f_X\left(x\right)dx\hfill \\ \hfill & =\mathbb{E}\left[g^{\leftarrow }\left(X\right)|q_{\alpha +\beta }\left(X\right)\ge X\ge q_{\alpha }\left(X\right)\right],\hfill \end{array}$$

(44)

as was to be shown.    □

4.4. Application to Conditional Value at Risk

The distributional transform is defined as follows; see Burgert and Rüschendorf (2006); Rüschendorf (2009). Let X be a real random variable with distribution function F and let $V\sim U(0,1)$ be uniformly distributed on $(0,1)$ and independent of X. The modified distribution function $F(x,\lambda )$ is defined by

$$F(x,\lambda )P(X<x)+\lambda P(X=x).$$

(45)

The (generalized) distributional transform of X is then defined by

$$UF(X,V).$$

(46)

An equivalent representation of the distributional transform is

$$U=F(X-)+V\left(F\right(X)-F(X-\left)\right).$$

(47)

An early source for this transform is the statistics book of Ferguson (1967). See (Rüschendorf 2013, pp. 21–28) for a range of applications of the distributional transform. By means of the distributional transform, one can give a proper definition of the conditional value at risk, namely (see Burgert and Rüschendorf (2006); Rüschendorf (2009))

$${CVaR}_{\alpha }\left(X\right)=\mathbb{E}\left(X\right|U\ge \alpha ).$$

(48)

The main theorem on the conditional value at risk is that it actually coincides with the expected shortfall. We prove this result of Burgert and Rüschendorf by applying Theorem 2.

Corollary 1

(Burgert and Rüschendorf (2006)). Let Y be a random variable and U its distributional transform. Let $E\left(Y^{+}\right)<\infty$. Then,

$${ES}_{\alpha }\left(Y\right)=\mathbb{E}\left(Y\right|U\ge \alpha ).$$

(49)

Proof. 

For the distributional transform U of Y, it holds that $U\sim U(0,1)$ and $Y=F^{\leftarrow }\left(U\right)$ a.s.; see Rüschendorf (2009).

On $[0,1]$, $F_U$ is the identity. Hence, the increasing rearrangement $g^{\leftarrow }$ of Y with respect to U equals $F^{\leftarrow }$ on $[0,1]$ by (7). This implies $Y=g^{\leftarrow }\left(U\right)$ a.s.

U possesses a density (namely, the density of the uniform distribution on $(0,1)$). This means that U is absolutely continuous and Theorem 2 applies. We obtain

$${ES}_{\alpha }\left(Y\right)=\mathbb{E}\left(g^{\leftarrow }\left(U\right)\right|U\ge q_{\alpha }\left(U\right)).$$

(50)

Using $Y=g^{\leftarrow }\left(U\right)$ a.s., this implies

$${ES}_{\alpha }\left(Y\right)=\mathbb{E}\left(Y\right|U\ge q_{\alpha }\left(U\right))={CVaR}_{\alpha }\left(Y\right),$$

(51)

as was to be shown.    □

4.5. Application to Wang’s Risk Measure

For the pricing of financial and insurance risks, Wang (2000) has introduced a specific distortion function that is based on the standard normal distribution $\Phi$ in combination with a shift parameter $\lambda \in \mathbb{R}$:

$$D_{\mathrm{Wang},\lambda }\left(u\right)=\Phi \left({\Phi }^{-1}\left(x\right)-\lambda \right).$$

(52)

Formula (15) of Theorem 1 yields the following.

Lemma 6.

Let X and Y be random variables and X be normally distributed with $\mu \in \mathbb{R}$ and ${\sigma }^2>0$. Let $g^{\leftarrow }$ be the quantile version of the increasing rearrangement of Y with respect to X. Let $E\left(Y^{+}\right)<\infty$. Then,

$${\varrho }_{\mathit{Wang},\lambda }\left(Y\right)=\mathbb{E}\left[g^{\leftarrow }\circ Z\right]$$

(53)

holds for $Z\sim N(\mu +\lambda \sigma ,{\sigma }^2)$.

Likewise, let X be log-normally distributed with $\mu \in \mathbb{R}$ and ${\sigma }^2>0$ and $g^{\leftarrow }$ be the quantile version of the increasing rearrangement of Y with respect to X, $E\left(Y^{+}\right)<\infty$. Then, (53) holds for $log\left(Z\right)\sim N(\mu +\lambda \sigma ,{\sigma }^2)$.

Proof. 

We start with the definition of Wang’s risk measure and (15):

$${\varrho }_{\mathrm{Wang},\lambda }\left(Y\right)={\int }_{-\infty }^{\infty }xd{D}_{\mathrm{Wang},\lambda }\circ F_Y\left(x\right)={\int }_{-\infty }^{\infty }{g}^{\leftarrow }\left(x\right)d{D}_{\mathrm{Wang},\lambda }\circ F_X\left(x\right).$$

(54)

Let us assume that $X\sim \mathit{N}(\mu ,\sigma )$. We can equivalently write X in terms of the standard normal distribution, namely $F_X\left(x\right)=\Phi \left({\displaystyle \frac{x-\mu }{\sigma }}\right)$, and obtain for the integrator

$$D_{\mathrm{Wang},\lambda }\circ F_X\left(x\right)=\Phi \left({\Phi }^{-1}\left[\Phi \left({\displaystyle \frac{x-\mu }{\sigma }}\right)\right]-\lambda \right)=\Phi \left({\displaystyle \frac{x-\mu -\sigma \lambda }{\sigma }}\right).$$

(55)

This implies (53) with $Z\sim \mathit{N}\left(\mu +\lambda \sigma ,{\sigma }^2\right)$.

An analogous argument holds for $log\left(X\right)\sim \mathit{N}(\mu ,\sigma )$ and the log-normal distribution function $F_X\left(x\right)=\Phi \left({\displaystyle \frac{log\left(x\right)-\mu }{\sigma }}\right)$:

$$D_{\mathrm{Wang},\lambda }\circ F_X\left(x\right)=\Phi \left({\Phi }^{-1}\left[\Phi \left({\displaystyle \frac{log\left(x\right)-\mu }{\sigma }}\right)\right]-\lambda \right)=\Phi \left({\displaystyle \frac{log\left(x\right)-\mu -\sigma \lambda }{\sigma }}\right).$$

(56)

Thus, Equation (53) holds in both cases and the lemma is proven.    □

One may write the assertion of the lemma in terms of X rather than Z. Namely, in the notation of the lemma and in case $X\sim N(\mu ,{\sigma }^2)$, Equation (53) amounts to

$${\varrho }_{\mathrm{Wang},\lambda }\left(Y\right)=\mathbb{E}\left[g^{\leftarrow }\circ (X+\lambda \sigma )\right].$$

(57)

In case $log\left(X\right)\sim N(\mu ,{\sigma }^2)$, (53) amounts to

$${\varrho }_{\mathrm{Wang},\lambda }\left(Y\right)=\mathbb{E}\left[g^{\leftarrow }\circ \left(e^{\sigma \lambda }X\right)\right].$$

(58)

5. Application to Numerical Simulation and Cost-Efficient Portfolios

5.1. Inverse Transform Sampling

Inverse transform sampling is related to increasing rearrangement. The well-known inversion principle (or inversion method) for random number generation reads

Theorem 5.

Let F be a cumulative distribution function and X be uniformly distributed on $(0,1)$. Then, $Y{F}^{\leftarrow }\left(X\right)$ has cdf F.

Proof. 

See, e.g., (Kolonko 2008, p. 87).    □

In this setting, the increasing rearrangement of Y with respect to X is

$$g^{\leftarrow }\left(X\right)=F_Y^{\leftarrow }\circ F_X\left(X\right)=F_Y^{\leftarrow }\left(X\right)\mathrm{a}.\mathrm{s}.$$

(59)

In other words, $F^{\leftarrow }\left(X\right)$ as given in the theorem is the increasing rearrangement applied to X in special case $X\sim U(0,1)$. Hence, all assertions of the preceding section apply to inverse transform sampling.

5.2. Monte Carlo Simulation

In general, with Monte Carlo simulation, simulated prices are not an increasing rearrangement of random numbers or underlying prices, respectively. Consider, for example, a butterfly spread (Hull 2003, p. 190 ff.) with payoff as displayed in

Table 1. Payoff of butterfly spread.

Stock Price	Payoff
$S_T<K_1$	0
$K_1\le S_T<K_2$	$S_T-K_1$
$K_2\le S_T<K_3$	$K_3-S_T$
$K_3\le S_T$	0

. Notation:

• $S_T$ price of underlying at maturity T;
• $K_1,K_2,K_3$ strike prices, $K_1<K_3$, $K_2=(K_1+K_3)/2$.

The payoff of the butterfly spread reads as follows.

The payoff is depicted in Figure 1 (blue line). Given the model for stock prices, the distribution of the butterfly payoff can be obtained by increasing rearrangement. Let us denote the butterfly payoff with $Y=Y\left(S_T\right)$ and the increasing rearrangement of Y with respect to $S_T$ by $g^{\leftarrow }$. Then, by Lemma 7, we have

$$Y\stackrel{d}{=}{g}^{\leftarrow }\left(S_T\right).$$

(60)

This is easy to compute in the Monte Carlo simulation. One simply has to sort Y. The R code is provided in Appendix A.

Assuming a log-normal distribution for $S_T$ with $\mu =0$, $\sigma =20\%$ and $T=1/52$, we obtain Figure 1. The payoff Y is denoted with the blue line and $g^{\leftarrow }\left(S_T\right)$ with the red line. Note that $g^{\leftarrow }\left(S_T\right)$ is increasing while Y is not. However, both have the same distribution.

5.3. Cost-Efficient Strategies

A strategy (or a payoff) is said to be cost-efficient if any other strategy that generates the same distribution costs at least as much. Starting with Cox and Leland (2000); Dybvig (1988) pointed out that there are indeed inefficient strategies in the market. Examples like the so-called constant proportion portfolio insurance (CPPI) or the butterfly spread discussed in Section 5.2 show, as Dybvig states, “how to throw away a million dollars in the stock market”.

It turns out that a payoff is cost-efficient if and only if it is non-increasing in the state price almost surely; see Bernard et al. (2014) for an introduction to the topic, with proofs and references. We will make this statement more explicit in Theorem 6 below.

In the words of Bernard et al., “The cheapest way to achieve a lottery assigns outcomes of the lottery to the states in reverse order of the state-price density”; see Bernard et al. (2014). Thus, we define decreasing rearrangement as follows.

Definition 6.

Let X and Y be random variables with distribution functions $F_X$ and $F_Y$. The decreasing rearrangement (quantile version) of Y with respect to X is given by

$$\tilde{g}{F}_Y^{\leftarrow }\circ (1-F_X).$$

(61)

Lemma 7.

Let X and Y be random variables. Let $\tilde{g}$ be the decreasing rearrangement (quantile version) of Y with respect to X. Then, the following holds:

1. 

$\tilde{g}$ is decreasing;

2. 

If X does not have atoms, then $\tilde{g}$ is left-continuous;

3. 

If X does not have atoms, then $Y\stackrel{d}{=}\tilde{g}\left(X\right)$.

Proof. 

(i) The function $(1-F_X)$ is decreasing, while $F_Y^{\leftarrow }$ is increasing, so $\tilde{g}$ is a decreasing function.

(ii) X does not have atoms if $F_X$ is continuous. Moreover, $F_Y^{\leftarrow }$ is left-continuous. This implies (2).

(iii) The continuity of $F_X$ implies $F_X\left(X\right)\sim U[0,1]$ (uniformly distributed on $[0,1]$), so $1-F_X\left(X\right)\sim U[0,1]$ as well. Hence, $F_Y^{\leftarrow }\left(1-F_X\left(X\right)\right)\stackrel{d}{=}Y$ (cf. (Embrechts and Hofert 2013, p. 429)).    □

We consider an arbitrage-free and complete market with a unique state price density ${\xi }_t$, $t\ge 0$ and assume that ${\xi }_t$ is atomless for all $t>0$. In such a market, the price (or cost) of a strategy (or of a financial investment contract) with terminal payoff $X_T$ at $T>0$ is given by

$$c\left(X_T\right)=\mathbb{E}\left[{\xi }_T{X}_T\right],$$

(62)

where the expectation is taken under the physical measure. The theorem on cost-efficient strategies reads as follows.

Theorem 6

(Bernard et al. (2014)). Let $X_T$ be a given payoff, i.e., any random variable. Let ${\xi }_t$, $t\ge 0$ be the state price density and ${\tilde{g}}_T$ the decreasing rearrangement (quantile version) of $X_T$ with respect to ${\xi }_T$. Then, ${\tilde{g}}_T\stackrel{d}{=}{X}_T$ and ${\tilde{g}}_T$ is the cost-efficient strategy to replicate $X_T$: given any other random variable $Y_T$ with $Y_T\stackrel{d}{=}{X}_T$, it holds that

$$c\left({\tilde{g}}_T\right)\le c\left(Y_T\right).$$

(63)

In the Black–Scholes model with positive expected return $\mu >0$, the stock prices and state prices at $T>0$ are non-increasing functions of each other. Hence, Theorem 6 amounts to the payoff being increasing in the stock price, i.e., the increasing rearrangement of the payoff in terms of the price of the underlying stock yields the cost-efficient strategy that replicates the distribution of the payoff under consideration. An example is given by the butterfly spread discussed in the previous section. While the given payoff is not cost-efficient, its increasing rearrangement is.

Note that the assertion relies (amongst other assumptions) on the assumption that $\mu >0$. Investors who do not believe that the drift of a stock is positive will consequently not conclude that the payoff of the increasing rearrangement of the butterfly spread is cost-efficient.

6. Conclusions

We consider distortion risk measures of increasing rearrangement. The “pull-through property” of increasing rearrangement with respect to quantiles implies formulas for the risk measure of Y in terms of X, given that Y is the increasing rearrangement of X.

The special case of the value at risk illustrates this principle by the well-known formula

$${VaR}_{\alpha }\left(Y\right)=g^{\leftarrow }\circ {VaR}_{\alpha }\left(X\right),$$

where $Y=g^{\leftarrow }\circ X$ and $g^{\leftarrow }$ is the increasing rearrangement.

Moreover, applied to the expected shortfall, this principle allows us to generalize the interpretation of the expected shortfall as a conditional expectation. Namely, let Y be an arbitrary random variable (“book value”), X be an arbitrary absolutely continuous random variable (“market value”) and $g^{\leftarrow }$ be the increasing rearrangement of Y with respect to X. Then, the expected shortfall of Y can be written as a conditional expectation on $g^{\leftarrow }\circ X$ (which has the same distribution as Y) in terms of X. This applies to the range value at risk as well.

Next, the principle implies a theorem of Burgert and Rüschendorf stating that the expected shortfall equals the conditional value at risk. An application to Wang’s risk measure yields a representation of Wang’s risk measure as some expected value.

Finally, we highlight the connection of the principle to inverse transform sampling, Monte Carlo simulation, and cost-efficient strategies.