1 Introduction

It is well-known that a distorted expectation of a random variable (r.v.) can be expressed as a mixture of its corresponding quantiles; see e.g. Wang [8] or Denuit et al. [2]. Although this statement is true, one has to be careful to formulate it in an appropriate and correct way. In this short note, we explore this statement and the conditions under which it holds.

A second goal of this note is to present a complete proof for the additivity property which holds for distorted expectations of a comonotonic sum. The proofs of this theorem that are presented in the literature are often incomplete, in the sense that they only hold for a particular type of distortion functions, such as the class of concave distortion functions. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of quantiles.

2 Distortion risk measures as mixtures of quantiles

In this section, we investigate the representation of a distorted expectation of a r.v. as a weighted average of its quantiles. All r.v.’s that we consider are defined on a common probability space \({( \Upomega,\mathcal{F} ,\mathbb{P})}.\) The cumulative distribution function (cdf) and the decumulative distribution function (ddf) of a r.v. X are denoted by F X and \(\overline{F}_{X}, \) respectively.

2.1 Distorted expectations

For a given r.v. X, we define its càglàd (continue à gauche, limitée à droite) inverse cdf F −1 X , as well as its càdlàg (continue à droite, limitée à gauche) inverse cdf F −1+ X as follows.

Definition 1

(The inverse cdf’s F −1 X and F −1+ X ) For any \(p\in[ 0,1], \) the inverse cdf F −1 X (p) is defined by

$$F_{X}^{-1}(p)=\inf\{ x\mid F_{X}(x)\geq p\}, $$

whereas the inverse cdf F −1+ X (p) is defined by

$$F_{X}^{-1+}(p)=\sup\{ x\mid F_{X}(x)\leq p\}. $$

In these expressions, \(\inf\varnothing=+\infty\) and \(\sup\varnothing=-\infty\) by convention.

We recall the following equivalence relations:

$$p\leq F_{X}(x)\Leftrightarrow F_{X}^{-1}(p)\leq x,\quad x\in {\mathbb{R}} \hbox { and }p\in[ 0,1],$$
(1)

and

$${\mathbb{P}}[X<x] \leq p\Leftrightarrow x\leq F_{X}^{-1+} (p),\quad x\in {\mathbb{R}} \hbox { and }p\in[0,1], $$
(2)

which will be used in the derivations hereafter.

In order to define the distorted expectation of a r.v., we have to introduce the notion of distortion function.

Definition 2

(Distortion function) A distortion function is a non-decreasing function \(g:[ 0,1] \rightarrow[ 0,1]\) such that g(0) = 0 and g(1) = 1.

Any distortion function g can be represented as the following convex combination of distortion functions:

$$g(q)=p_{1}g^{(c)}(q)+p_{2}g^{(d)}(q)+p_{3}g^{(s)}(q),\quad q\in[ 0,1], $$

where p i  ≥ 0 for i = 1, 2, 3 and p 1 + p 2 + p 3 = 1. In this expression, g (c) is absolutely continuous, g (d) is discrete and g (s) is singular continuous.

Wang [8] introduced a class of risk measures in the actuarial literature, the elements of which are known as distortion risk measures.

Definition 3

(Distorted expectation) Consider a distortion function g. The distorted expectation of the r.v. X, notation \(\rho_{g}[ X],\) is defined as

$$\rho_{g}[X] =-\int\limits_{-\infty}^{0}[1-g(\overline {F}_{X}(x))] \hbox {d}x+\int\limits_{0}^{+\infty}g( \overline{F}_{X}(x)) \hbox{d}x,$$
(3)

provided at least one of the two integrals in (3) is finite.

The functional ρ g is called the distortion risk measure with distortion function g. Both integrals in (3) are well-defined and take a value in \([0,+\infty].\) Provided at least one of the two integrals is finite, the distorted expectation \(\rho_{g}[X] \) is well-defined and takes a value in \([-\infty,+\infty].\) Hereafter, when using a distorted expectation \(\rho_{g}[X],\) we silently assume that both integrals in the definition (3) are finite, or equivalently, that \({\rho_{g}[X] \in \mathbb{R}},\) unless explicitly stated otherwise.

Consider a distortion function g which can be expressed as a strictly convex combination of two distortion functions g 1 and g 2, i.e.

$$g=c_{1}g_{1}+c_{2}g_{2} $$

with weights 0 < c i  < 1, i = 1, 2, and c 1 + c 2 = 1. Assuming that \({\rho_{g}[X] \in \mathbb{R}}\) is then equivalent with assuming that \({\rho_{g_{i}}[X] \in \mathbb{R}, i=1,2}.\) Under any of these assumptions, we have that \(\rho_{g}[X]\) is additive with respect to g, in the sense that

$$\rho_{g}[X] =c_{1}\rho_{g_{1}}[X] +c_{2} \rho_{g_{2}}[X].$$
(4)

The proofs of the equivalence of the stated assumptions and of (4) follow from the observation that the additivity property (with respect to g) holds for both integrals in (3). Notice that the statements above remain to hold in case c i  = 0 for i = 1 or i = 2, provided g i is chosen such that \(\rho_{g_{i}}[X]\) is finite.

Hereafter, we will often consider distortion functions that are left continuous (l.c.) on (0, 1] or right continuous (r.c.) on [0, 1).

The inverse F −1 X defined above belongs to the class of distortion risk measures. Indeed, for \(p\in(0,1),\) consider the l.c. distortion function g defined by

$$g(q)={\mathbb I}(q>1-p), \quad 0 \leq q \leq 1,$$
(5)

where we use the notation \({\mathbb {I} (A)}\) to denote the indicator function, which equals 1 when A holds true and 0 otherwise. From definition (3) and equivalence relation (1), we find that the corresponding distorted expectation is equal to the p-quantile of X:

$$\rho_{g}[X] =F_{X}^{-1}(p).$$

2.2 Distorted expectations and r.c. distortion functions

In the following theorem, it is shown that any distorted expectation \(\rho _{g}[X]\) with r.c. distortion function g can be expressed as a weighted average of the quantiles \(F_{X}^{-1+}(q)\) of X.

Theorem 4

When g is a r.c. distortion function, the distorted expectation \(\rho_{g}[X]\) has the following Lebesgue–Stieltjes integral representation:

$$\rho_{g}[X] =\int\limits_{[0,1]}F_{X}^{-1+}(1-q) \hbox {d}g(q).$$
(6)

Proof

Taking into account that F X has at most countably many jumps, we have that \({\overline{F}_{X}(x) =\mathbb{P}[X\geq x]}\) a.e., and we can rewrite expression (3) for \(\rho_{g}[X]\) as follows:

$$\rho_{g}[X] =-\int\limits_{-\infty}^{0}[1-g( {\mathbb{P}}[X\geq x])] \hbox{d}x+\int\limits_{0}^{+\infty}g({\mathbb{P}}[X\geq x]) \hbox{d}x.$$
(7)

As the distortion function g is r.c., we find that \({g(\mathbb{P}[X\geq x])}\) can be expressed as \({\int_{[0,\mathbb{P}[X\geq x]]}}\)dg(q), which has to be understood as a Lebesgue–Stieltjes integral. Applying Fubini’s theorem to change the order of integration and noticing (2), the second integral in (7) can be transformed into

$$\begin{aligned} \int\limits_{0}^{+\infty}g({\mathbb{P}}[X\geq x]) \hbox {d}x & =\int\limits_{[0,{\mathbb{P}}[X\geq0] ]}\hbox{d}g(q) \int\limits_{0}^{F_{X}^{-1+}(1-q)}\hbox {d}x\cr & =\int\limits_{[0,{\mathbb{P}}[X\geq0]]}F_{X} ^{-1+}(1-q) \hbox {d}g(q). \end{aligned}$$
(8)

Similarly, taking into account that \({1-g(\mathbb{P}[X\geq x])}\) can be expressed as \({\int_{(\mathbb{P}[X\geq x],1]}}\)dg(q), the first integral in (7) can be transformed into

$$\int\limits_{-\infty}^{0}[1-g({\mathbb{P}}[X\geq x])] \hbox {d}x=-\int\limits_{({\mathbb{P}}[X\geq0],1] }F_{X}^{-1+}(1-q)\hbox {d}g(q).$$
(9)

Inserting the expressions (8) and (9) into (7) leads to (6).□

Theorem 4 can be strengthened in the following sense: if either the distorted expectation \(\rho_{g}[X]\) or the Lebesgue–Stieltjes integral \(\int_{[0,1]}^{{}}F_{X}^{-1+}(1-q)\)dg(q) is finite, then also the other quantity is finite and both are equal. Indeed, the case where one starts from a finite ρ g [X] is considered in the proof of the theorem. On the other hand, in case the integral in (6) is finite, it can be written as the sum of the finite integrals \(\int_{[0,\mathbb{P}[X\geq0]]}F_{X}^{-1+}(1-q) d g(q)\) and \(\int_{(\mathbb{P}[X\geq0],1]}F_{X}^{-1+}(1-q) d g(q).\) Applying Fubini’s theorem leads to the relations (8) and (9), which proves that relation (6) holds.

Using integration by parts, Theorem 4 can be considered as a consequence of Corollary 2.1 in Gzyland and Mayoral [7]. The proof presented above is different and is based on Fubini’s theorem.

In order to prove that the càdlàg inverse F −1+ X also belongs to the class of distortion risk measures, let p ∈ (0,1) and consider the r.c. discrete distortion function g defined by

$$g(q)={\mathbb{I}}(q\geq1-p), \quad 0\leq q\leq1.$$

Taking into account expression (6) for \(\rho_{g}[X],\) we find that

$$\rho_{g}[X] =F_{X}^{-1+}(p).$$

The assumption that g is r.c. is essential for (6) to hold. If we assume e.g. that g is l.c., expression (6) for \(\rho_{g}[X]\) above is not valid anymore. This can be illustrated by the l.c. distortion function g that we defined in (5) and for which \(\rho _{g}[X] =F_{X}^{-1}(p).\) Suppose for a moment that expression (6) is valid for l.c. distortion functions. Applying this formula to the distortion function defined in (5), we find that \(\rho_{g}[X] =F_{X}^{-1+}(p).\) As F −1 X (p) and F −1+ X (p) are in general not equal, we can indeed conclude that (6) is in general not valid for a l.c. distortion function. The situation where the distortion function g is left continuous will be considered in Theorem 6.

2.3 Distorted expectations and l.c. distortion functions

In order to present a left continuous version of Theorem 4, we introduce the notion of a dual distortion function. Therefore, consider a distortion function g and define the related function \(\overline{g}:[0,1] \rightarrow[0,1]\) by

$$\overline{g}(q) =1-g(1-q), \quad 0 \leq q \leq1.$$

Obviously, \(\overline{g}\) is also a distortion function, called the dual distortion function of g. The relation between the distorted expectations with distortion functions g and \(\overline{g}, \) respectively, is explored in the following lemma. A proof of this lemma can be found e.g. in Dhaene et al. [5].

Lemma 5

For any r.v. X and distortion function g, we have

$$\rho_{\overline{g}}[X] =-\rho_{g}[-X]$$

and

$$\rho_{g}[X] =-\rho_{\overline{g}}[-X].$$
(10)

The following theorem can be considered as an adapted version of Theorem 4 for l.c. distortion functions. Notice that for a l.c. distortion function g, we have

$$\int\limits_{[0,1]}F_{X}^{-1}(1-q) \hbox {d}g(q) =\int\limits_{[0,1]}F_{X}^{-1}(q) \hbox {d}\overline{g}(q),$$
(11)

by the definition of Lebesgue–Stieltjes integration for l.c. distortion functions.

Theorem 6

When g is a l.c. distortion function, the distorted expectation \(\rho_{g}[X]\) has the following Lebesgue–Stieltjes integral representation:

$$\rho_{g}[X] =\int\limits_{[0,1]}F_{X}^{-1}(1-q) \hbox {d}g(q).$$
(12)

Proof

Let g be a l.c. distortion function. The dual distortion function \(\overline{g}\) of g is r.c. Applying (6) and (10) leads to

$$\rho_{g}[X] =-\rho_{\overline{g}}[-X] =-\int\limits_{[0,1]}F_{-X}^{-1+}(1-q)\hbox {d}\overline{g}(q).$$

Taking into account the expression

$$F_{-X}^{-1+}(1-q)=-F_{X}^{-1}(q), $$

as well as the equality (11), we find (12).□

An alternate proof of Theorem 6 follows from first rewriting \({g(\mathbb{P}[X\geq x])}\) as \({\int_{[0,\mathbb{P}[X\geq x])}}\)dg(q) and \(1-g(\mathbb{P}[X\geq x])\) as \(\int_{[\mathbb{P}[X\geq x],1]}\)dg(q), respectively, and then proceeding as in the proof of Theorem 4.

Theorem 6 can be strengthened in the following sense: if either the distorted expectation ρ g [X] or the Lebesgue–Stieltjes integral \(\int_{[0,1]}F_{X}^{-1}(1-q)\) d g(q) is finite, then also the other quantity is finite and both are equal.

The distortion function g defined in (5) is an example of a l.c. discrete distortion function. Its dual distortion function \(\overline{g}\) is given by

$$\overline{g}(q)={\mathbb{I}}(q\geq p), \quad 0 \leq q \leq 1.$$

From Theorem 6 it follows that \(\rho_{g}[X]\) is given by F −1 X (p), as we found before.

There are at most countably many values of \(q\in[0,1]\) where the inverses \(F_{X}^{-1}(q) \) and \(F_{X}^{-1+}(q) \) differ. This implies that in case g is continuous on \([0,1],\) we can replace F −1+ X by F −1 X in (6) without changing the value of the integral. This observation leads to the following implication:

$$g\hbox { is continuous }\Longrightarrow\rho_{g}[X] =\int\limits_{[0,1]}F_{X}^{-1}(1-q) \hbox {d}g(q).$$
(13)

Notice that this implication follows also directly from (12). Furthermore, when g is absolutely continuous, we can replace d\(g(q)\) by \(g^{\prime}(q)\)dq in (13), and we find that

$$g\hbox { is absolutely continuous }\Longrightarrow\rho_{g}[X] ={\mathbb{E}}[F_{X}^{-1}(1-U) g^{\prime}(U)],$$
(14)

where U is a r.v. uniformly distributed on the unit interval [0, 1].

In the literature, much attention is paid to the class of concave (resp. convex) distortion functions. A concave distortion function is continuous on \((0,1]\) and can only jump at 0, while a convex distortion function is continuous on \([0,1)\) and can only jump at 1. Concave (resp. convex) distortion functions without jumps in the endpoints of the unit interval are absolutely continuous, which implies that the expressions for \(\rho_{g}[X]\) in (13) and (14) hold in particular for these functions.

Consider a concave distortion function g without a jump at 0. Taking into account (14), one can rewrite the corresponding distorted expectation \(\rho_{g}[X]\) as

$$\rho_{g}[X] =-\int\limits_{0}^{1}F_{X}^{-1}(q) \phi(q) \hbox {d}q,$$
(15)

with

$$\phi(q) =-g^{\prime}(1-q).$$

Notice that \(\phi(q)\) may not exist on a set of Lebesgue measure 0, but this observation does not hurt the validity of (15). A risk measure of the form (15) is called a spectral risk measure with risk spectrum \(\phi(q);\) see e.g. Gzyland and Mayoral [7].

As an example of a concave distortion function, for \(p\in[0,1),\) consider

$$g(q)=\min\left(\frac{q}{1-p},1\right), \quad 0 \leq q \leq 1.$$

The corresponding distorted expectation \(\rho_{g}[X]\) is denoted by TVaR\(_{p}[X].\) From (14) we find that TVaR\(_{p}[X]\) is given by

$$\hbox{TVaR}_{p}[X] =\frac{1}{1-p}\int\limits_{p}^{1}F_{X} ^{-1}(q)\hbox{d}q.$$

2.4 Distorted expectations and general distortion functions

In Theorems 4 and 6, we derived expressions for distortion risk measures \(\rho_{g}[X]\) related to r.c. and l.c. distortion functions g, in terms of the quantile functions F −1+ X and F −1 X , respectively. In general, distortion functions may be neither r.c. nor l.c. However, as will be proven in the following theorem, a general distortion function can always be represented by a convex combination of a r.c. and a l.c. distortion function.

Theorem 7

Any distortion function g can be represented by a convex combination

$$g=c_{r}g_{r}+c_{l}g_{l},$$
(16)

where g r and g l are a r.c. and a l.c. distortion function, respectively, and the non-negative weights c r and c l sum to 1.

When \(c_{r}\in(0,1),\) the distorted expectation \(\rho_{g}[X]\) can be expressed as

$$\rho_{g}[X] =c_{r}\rho_{g_{r}}[X] +c_{l} \rho_{g_{l}}[X].$$
(17)

Proof

Consider a general distortion function g. For any \(p\in(0,1],\) we define

$$D(p)=\sum_{q\in[0,p)}[g(q+) -g(q)],$$

where the sum is taken over the finite or countable set of all values of q in \([0,p)\) where the distortion function is right discontinuous. Furthermore, we set D(0) = 0.

In case \(D(1) =0,\) we have that g is r.c., while in case D(1) = 1, we find that g is l.c., and in both cases (16) and (17) are obvious.

Let us now assume that \(0<D(1) <1.\) Define

$$g_{l}(p)=\frac{D(p)}{D(1)}, \quad 0 \leq p \leq1,$$

and

$$g_{r}(p)=\frac{g(p)-D(1) g_{l}(p)}{1-D(1)}, \quad 0 \leq p \leq 1.$$

It is easy to check that g l and g r are a l.c. and a r.c. distortion function, respectively. Moreover,

$$g=(1-D(1))g_{r}+D(1)g_{l},$$

so that (16) holds. From (4) and the discussion of that result, we can conclude that under the implicit assumption that \({\rho _{g}[X] \in \mathbb{R}},\) or equivalently, that \(\rho_{g_{r}}[X]\) and \(\rho_{g_{l}}[X]\) are real-valued, relation (17) holds.□

Expression (17) remains to hold in case c r  = 0, provided g r is chosen such that \(\rho_{g_{r}}[X]\) is finite, while it also holds in case c r  = 1, provided g l is chosen such that \(\rho_{g_{l}}[X]\) is finite. Notice that it is always possible to choose such a distortion function, and hereafter, we will make this appropriate choice when c r  = 0 or c r  = 1.

The intuitive idea behind the proof of the theorem above is that we form a piecewise constant l.c. distortion function g l by successively adding all jumps corresponding to right-side discontinuities of g. The rescaled difference \((g-D(1)g_{l})/(1-D(1))\) is a distortion function that is obtained from g by pulling down its graph at its right-side discontinuities, making it a r.c. distortion function. The reader is referred to Dudley and Norvaiša [6] for related discussions on Young type integrals where the integrand and the integrator may have any kind of discontinuities.

As an illustration of Theorem 7, consider the distortion function g defined by

$$g(q) =\frac{1}{2}{\mathbb{I}}\left( \frac{1}{3}<q<\frac{2} {3}\right) +{\mathbb{I}}\left( \frac{2}{3}\leq q\leq1\right), \quad 0 \leq q \leq 1.$$

This distortion function is neither r.c. nor l.c., but it can be represented as follows:

$$g(q) =\frac{1}{2}(g_{r}(q) +g_{l}(q)) ,\quad 0 \leq q \leq 1,$$

with

$$g_{r}(q) ={\mathbb{I}}\left( \frac{2}{3}\leq q\leq1\right) \hbox { and }g_{l}(q) ={\mathbb{I}}\left( \frac{1}{3} <q\leq1\right),$$

where g r (q) and g l (q) are a r.c. and a l.c. distortion function, respectively. Taking into account (17), we find that

$$\rho_{g}[X] =\frac{1}{2}(\rho_{g_{r}}[X] +\rho_{g_{l}}[X]).$$

3 Distortion risk measures and comonotonic sums

A random vector \(\underline {X}=(X_{1},\ldots,X_{n})\) is said to be comonotonic if

$$\underline {X}\overset{\hbox {d}}=( F_{X_{1}}^{-1}(U),\ldots,F_{X_{n}}^{-1}(U)),$$
(18)

where U is a uniform \((0,1)\) r.v. and \(\overset{\hbox {d}}{=}\) stands for equality in distribution.

For a general random vector \(\underline {X}=(X_{1},\ldots,X_{n}),\) we call \((F_{X_{1}}^{-1}(U),\ldots,F_{X_{n}}^{-1}(U))\) the comonotonic modification of \(\underline {X},\) corresponding to the uniform r.v. U. Furthermore, the sum of the components of the comonotonic modification is denoted by S c:

$$S^{c}=F_{X_{1}}^{-1}(U) +F_{X_{2}}^{-1}(U) +\cdots+F_{X_{n}}^{-1}(U).$$

For an overview of the theory of comonotonicity and its applications in actuarial science and finance, we refer to Dhaene et al. [3]. Financial and actuarial applications are described in Dhaene et al. [4]. An updated overview of applications of comonotonicity can be found in Deelstra et al. [1].

The following theorem states that distorted expectations related to general distortion functions are additive for comonotonic sums.

Theorem 8

(Additivity of ρ g for comonotonic r.v.’s) Consider a random vector \(\underline {X}=(X_{1},\ldots,X_{n}),\) a distortion function g and the distorted expectations \(\rho_{g}[X_{i}],\) \(i=1,2,\ldots,n.\) The distorted expectation of the comonotonic sum S c is then given by

$$\rho_{g}[S^{c}] = \sum\limits_{i=1}^{n} \rho_{g}[X_{i}].$$
(19)

Proof

Applying the decomposition (17) in the first and the last steps of the following derivation, while taking into account Theorems 4 and 6 in the second and the fourth steps and, finally, applying the additivity property of the càglàd and càdlàg inverses F −1 and F −1+ for comonotonic r.v.’s in the third step, we find that

$$\begin{aligned} \sum_{i=1}^{n}\rho_{g}[X_{i}] & =\sum_{i=1}^{n}(c_{r}\rho_{g_{r} }[X_{i}]+c_{l}\rho_{g_{l}}[X_{i}])\\ & =c_{r}\int\limits_{[0,1]}\sum_{i=1}^{n}F_{X_{i}}^{-1+}(1-q)\hbox {d}g_{r} (q)+c_{l}\int\limits_{[0,1]}\sum_{i=1}^{n}F_{X_{i}}^{-1}(1-q)\hbox {d}g_{l}(q)\\ & =c_{r}\int\limits_{[0,1]}F_{S^{c}}^{-1+}(1-q)\hbox {d}g_{r}(q)+c_{l}\int\limits _{[0,1]}F_{S^{c}}^{-1}(1-q)\hbox {d}g_{l}(q)\\ & =c_{r}\rho_{g_{r}}[S^{c}]+c_{l}\rho_{g_{l}}[S^{c}]\\ &=\rho_{g}[S^{c}]\\ \end{aligned}$$

Given that \(\rho_{g}[X_{i}], i=1,2,\ldots,n,\) is finite by assumption, we have that ρ g r [X i ] and ρ g l [X i ] are finite too, so that all steps in the derivation above are allowed. We can conclude that \(\rho_{g}[S^{c}]\) is finite and given by (19).□

The additivity property of distorted expectations for comonotonic sums presented in Theorem 8 is well-known. However, most proofs that appear in the literature only consider the case where the distortion function is continuous or concave. The proof that we presented here is simple and considers the general case.