# An Optimal Three-Way Stable and Monotonic Spectrum of Bounds on Quantiles: A Spectrum of Coherent Measures of Financial Risk and Economic Inequality

## Abstract

**:**

## 1. Introduction

A very serious shortcoming of VaR, in addition, is that it provides no handle on the extent of the losses that might be suffered beyond the threshold amount indicated by this measure. It is incapable of distinguishing between situations where losses that are worse may be deemed only a little bit worse, and those where they could well be overwhelming. Indeed, it merely provides a lowest bound for losses in the tail of the loss distribution and has a bias toward optimism instead of the conservatism that ought to prevail in risk management.

- (I)
- The common risk measures $\phantom{\rule{0.166667em}{0ex}}VaR$ and $\phantom{\rule{0.166667em}{0ex}}CVaR$ are in the spectrum: ${Q}_{0}(X;p)={\phantom{\rule{0.166667em}{0ex}}VaR}_{p}(X)$ and ${Q}_{1}(X;p)={\phantom{\rule{0.166667em}{0ex}}CVaR}_{p}(X)$; thus, ${Q}_{\alpha}(X;p)$ interpolates between ${\phantom{\rule{0.166667em}{0ex}}VaR}_{p}(X)$ and ${\phantom{\rule{0.166667em}{0ex}}CVaR}_{p}(X)$ for $\alpha \in (0,1)$ and extrapolates from ${\phantom{\rule{0.166667em}{0ex}}VaR}_{p}(X)$ and ${\phantom{\rule{0.166667em}{0ex}}CVaR}_{p}(X)$ on towards higher degrees of risk sensitivity for $\alpha \in (1,\infty ]$. Details on this can be found in Section 5.1.
- (II)
- The risk measure ${Q}_{\alpha}(\xb7;p)$ is coherent for each $\alpha \in [1,\infty ]$ and each $p\in (0,1)$, but it is not coherent for any $\alpha \in [0,1)$ and any $p\in (0,1)$. Thus, $\alpha =1$ is the smallest value of the sensitivity index for which the risk measure ${Q}_{\alpha}(X;p)$ is coherent. One may also say that for $\alpha \in [1,\infty ]$ the risk measure ${Q}_{\alpha}(\xb7;p)$ inherits the coherence of ${\phantom{\rule{0.166667em}{0ex}}CVaR}_{p}={Q}_{1}(\xb7;p)$, and for $\alpha \in [0,1)$ it inherits the lack of coherence of ${\phantom{\rule{0.166667em}{0ex}}VaR}_{p}={Q}_{0}(\xb7;p)$. For details, see Section 5.3.
- (III)
- ${Q}_{\alpha}(X;p)$ is three-way stable and monotonic: in $\alpha \in (0,\infty ]$, in $p\in (0,1)$, and in X. Moreover, as stated in Theorem 3.4 and Proposition 3.5, ${Q}_{\alpha}(X;p)$ is nondecreasing in X with respect to the stochastic dominance of any order $\gamma \in [1,\alpha +1]$; but, this monotonicity property breaks down for the stochastic dominance of any order $\gamma \in (\alpha +1,\infty ]$. Thus, the sensitivity index α is in a one-to-one correspondence with the highest order of the stochastic dominance respected by ${Q}_{\alpha}(X;p)$.

- *
- In Section 2, the three-way stability and monotonicity, as well as other useful properties, of the spectrum ${\left({P}_{\alpha}(X;x)\right)}_{\alpha \in [0,\infty ]}$ of upper bounds on tail probabilities are established.
- *
- In Section 3, the corresponding properties of the spectrum ${\left({Q}_{\alpha}(X;p)\right)}_{\alpha \in [0,\infty ]}$ of risk measures are presented, as well as other useful properties.
- *
- The matters of effective computation of ${P}_{\alpha}(X;x)$ and ${Q}_{\alpha}(X;p)$, as well as optimization of ${Q}_{\alpha}(X;p)$ with respect to X, are considered in Section 4.
- *
- An extensive discussion of results is presented in Section 5, particularly in relation with existing literature.
- *
- Concluding remarks are collected in Section 6.
- *
- The necessary proofs are given in Appendix A.

## 2. An Optimal Three-Way Stable and Three-Way Monotonic Spectrum of Upper Bounds on Tail Probabilities

**Proposition 2.1.**

- (i)
- ${P}_{\alpha}(X;x)$ is nonincreasing in $x\in \mathbb{R}$.
- (ii)
- If $\alpha \in (0,\infty )$ and $\mathsf{E}{X}_{+}^{\alpha}=\infty $, then ${P}_{\alpha}(X;x)=\infty $ for all $x\in \mathbb{R}$.
- (iii)
- If $\alpha =\infty $ and $\mathsf{E}{e}^{\lambda X}=\infty $ for all real $\lambda >0$, then ${P}_{\infty}(X;x)=\infty $ for all $x\in \mathbb{R}$.
- (iv)
- If $\alpha \in (0,\infty )$ and $\mathsf{E}{X}_{+}^{\alpha}<\infty $, then ${P}_{\alpha}(X;x)\to 1$ as $x\to -\infty $ and ${P}_{\alpha}(X;x)\to 0$ as $x\to \infty $, so that $0\u2a7d{P}_{\alpha}(X;x)\u2a7d1$ for all $x\in \mathbb{R}$.
- (v)
- If $\alpha =\infty $ and $\mathsf{E}{e}^{{\lambda}_{0}X}<\infty $ for some real ${\lambda}_{0}>0$, then ${P}_{\alpha}(X;x)\to 1$ as $x\to -\infty $ and ${P}_{\alpha}(X;x)\to 0$ as $x\to \infty $, so that $0\u2a7d{P}_{\alpha}(X;x)\u2a7d1$ for all $x\in \mathbb{R}$.

**Proposition 2.2.**

- (i)
- For all $x\in [{x}_{*},\infty )$, one has ${P}_{\alpha}(X;x)={P}_{0}(X;x)=\mathsf{P}(X\u2a7ex)=\mathsf{P}(X=x)={p}_{*}\phantom{\rule{0.166667em}{0ex}}\mathsf{I}\{x={x}_{*}\}.$
- (ii)
- For all $x\in (-\infty ,{x}_{*})$, one has ${P}_{\alpha}(X;x)>0$.
- (iii)
- The function $(-\infty ,{x}_{*}]\cap \mathbb{R}\ni x\mapsto {P}_{\alpha}{(X;x)}^{-1/\alpha}$ is continuous and convex if $\alpha \in (0,\infty )$; we use the conventions ${0}^{-a}:=\infty $ and ${\infty}^{-a}:=0$ for all real $a>0$; concerning the continuity of functions with values in the set $[0,\infty ]$, we use the natural topology on this set. Also, the function $(-\infty ,{x}_{*}]\cap \mathbb{R}\ni x\mapsto -ln{P}_{\infty}(X;x)$ is continuous and convex, with the convention $ln0:=-\infty $.
- (iv)
- If $\alpha \in (0,\infty ]$, then the function $(-\infty ,{x}_{*}]\cap \mathbb{R}\ni x\mapsto {P}_{\alpha}(X;x)$ is continuous.
- (v)
- The function $\mathbb{R}\ni x\mapsto {P}_{\alpha}(X;x)$ is left-continuous.
- (vi)
- ${x}_{\alpha}$ is nondecreasing in $\alpha \in [0,\infty ]$, and ${x}_{\alpha}<\infty $ for all $\alpha \in [0,\infty ]$.
- (vii)
- If $\alpha \in [1,\infty ]$, then ${x}_{\alpha}=\mathsf{E}X$; even for $X\in {\mathcal{X}}_{\alpha}$, it is of course possible that $\mathsf{E}X=-\infty $, in which case ${P}_{\alpha}(X;x)<1$ for all real x.
- (viii)
- ${x}_{\alpha}\u2a7d{x}_{*}$, and ${x}_{\alpha}={x}_{*}$ if and only if ${p}_{*}=1$.
- (ix)
- ${E}_{\alpha}(1)=({x}_{\alpha},\infty )\ne \mathsf{\xd8}$.
- (x)
- ${P}_{\alpha}(X;x)=1$ for all $x\in (-\infty ,{x}_{\alpha}]$.
- (xi)
- If $\alpha \in (0,\infty ]$, then ${P}_{\alpha}(X;x)$ is strictly decreasing in $x\in [{x}_{\alpha},{x}_{*}]\cap \mathbb{R}$.

**Proposition 2.3.**${P}_{\alpha}(X;x)$ is continuous in $\alpha \in [0,\infty ]$ in the following sense: Suppose that $({\alpha}_{n})$ is any sequence in $[0,\infty )$ converging to $\alpha \in [0,\infty ]$, with $\beta :={sup}_{n}{\alpha}_{n}$ and $X\in {\mathcal{X}}_{\beta}$; then ${P}_{{\alpha}_{n}}(X;x)\to {P}_{\alpha}(X;x)$.

**Proposition 2.4.**Suppose that $\alpha \in (0,\infty ]$. Then ${P}_{\alpha}(X;x)$ is continuous in X in the following sense. Take any sequence ${({X}_{n})}_{n\in \mathbb{N}}$ of real-valued r.v.’s such that ${X}_{n}\underset{n\to \infty}{\stackrel{\mathrm{D}}{\u27f6}}X$ and the uniform integrability condition (2.20)- (2.21) is satisfied. Then one has the following.

- (i)
- The convergence$${P}_{\alpha}({X}_{n};x)\underset{n\to \infty}{\u27f6}{P}_{\alpha}(X;x)$$
- (ii)

**Theorem 2.5.**The following properties of the tail probability bounds ${P}_{\alpha}(X;x)$ are valid.

**Model-independence:**- ${P}_{\alpha}(X;x)$ depends on the r.v. X only through the distribution of X.
**Monotonicity in**X:- ${P}_{\alpha}(\xb7\phantom{\rule{0.166667em}{0ex}};x)$ is nondecreasing with respect to the stochastic dominance of order $\alpha +1$: for any r.v. Y such that $X\stackrel{\alpha +1}{\u2a7d}Y$, one has ${P}_{\alpha}(X;x)\u2a7d{P}_{\alpha}(Y;x)$. Therefore, ${P}_{\alpha}(\xb7\phantom{\rule{0.166667em}{0ex}};x)$ is nondecreasing with respect to the stochastic dominance of any order $\gamma \in [1,\alpha +1]$; in particular, for any r.v. Y such that $X\u2a7dY$, one has ${P}_{\alpha}(X;x)\u2a7d{P}_{\alpha}(Y;x)$.
**Monotonicity in**α:- ${P}_{\alpha}(X;x)$ is nondecreasing in $\alpha \in [0,\infty ]$.
**Monotonicity in**x:- ${P}_{\alpha}(X;x)$ is nonincreasing in $x\in \mathbb{R}$.
**Values:**- ${P}_{\alpha}(X;x)$ takes only values in the interval $[0,1]$.
- α
**-concavity in**x: - ${P}_{\alpha}{(X;x)}^{-1/\alpha}$ is convex in x if $\alpha \in (0,\infty )$, and $ln{P}_{\alpha}(X;x)$ is concave in x if $\alpha =\infty $.
**Stability in**x:- ${P}_{\alpha}(X;x)$ is continuous in x at any point $x\in \mathbb{R}$ – except the point $x={x}_{*}$ when ${p}_{*}>0$.
**Stability in**α:- Suppose that a sequence $({\alpha}_{n})$ is as in Proposition 2.3. Then ${P}_{{\alpha}_{n}}(X;x)\to {P}_{\alpha}(X;x)$.
**Stability in**X:- Suppose that $\alpha \in (0,\infty ]$ and a sequence $({X}_{n})$ is as in Proposition 2.4. Then ${P}_{\alpha}({X}_{n};x)\to {P}_{\alpha}(X;x)$.
**Translation invariance:**- ${P}_{\alpha}(X+c;x+c)={P}_{\alpha}(X;x)$ for all real c.
**Consistency:**- ${P}_{\alpha}(c;x)={P}_{0}(c;x)=\phantom{\rule{0.166667em}{0ex}}\mathsf{I}\{c\u2a7ex\}$ for all real c; that is, if the r.v. X is the constant c, then all the tail probability bounds ${P}_{\alpha}(X;x)$ precisely equal the true tail probability $\mathsf{P}(X\u2a7ex)$.
**Positive homogeneity:**- ${P}_{\alpha}(\kappa X;\kappa x)={P}_{\alpha}(X;x)$ for all real $\kappa >0$.

## 3. An Optimal Three-Way Stable and Three-Way Monotonic Spectrum of Upper Bounds on Quantiles

**Proposition 3.1.**Recall the definitions of ${x}_{*}$ and ${x}_{\alpha}$ in (2.17) and (2.18). The following statements are true.

- (i)
- ${Q}_{\alpha}(X;p)\in \mathbb{R}$.
- (ii)
- If $p\in (0,{p}_{*}]\cap (0,1)$ then ${Q}_{\alpha}(X;p)={x}_{*}$.
- (iii)
- ${Q}_{\alpha}(X;p)\u2a7d{x}_{*}$.
- (iv)
- ${Q}_{\alpha}(X;p)\underset{p\downarrow 0}{\u27f6}{x}_{*}$.

- (v)
- If $\alpha \in (0,\infty ]$, then the function$$({p}_{*},1)\ni p\mapsto {Q}_{\alpha}(X;p)\in ({x}_{\alpha},{x}_{*})$$$$({x}_{\alpha},{x}_{*})\ni x\mapsto {P}_{\alpha}(X;x)\in ({p}_{*},1).$$
- (vi)
- If $\alpha \in (0,\infty ]$, then for any $y\in \left(-\infty ,{Q}_{\alpha}(X;p)\right)$, one has ${P}_{\alpha}(X;y)>p$.
- (vii)
- If $\alpha \in [1,\infty ]$, then ${Q}_{\alpha}(X;p)>\mathsf{E}X$.

**Example 3.2.**Some parts of Proposition 3.1 are illustrated in Figure 1, with graphs $\{\left(p,{Q}_{\alpha}(X;p)\right):0<p<1\}$ in the important case when the r.v. X takes only two values. Then, by the translation invariance property stated below in Theorem 2.5, without loss of generality (w.l.o.g.) $\mathsf{E}X=0$. Thus, $X={X}_{a,b}$, where a and b are positive real numbers and ${X}_{a,b}$ is a r.v. with the uniquely determined zero-mean distribution on the set $\{-a,b\}$. Let us take $a=1$ and $b=3$, with the values of α equal 0 (black), $\frac{1}{2}$ (blue), 1 (green), 2 (orange), and ∞ (red). One may compare this picture with the one for ${P}_{\alpha}(X;x)$ in Example 1.3 in [8] (where the same values of a, b, and α were used), having in mind that the function ${Q}_{\alpha}(X;\xb7)$ is a generalized inverse to the function ${P}_{\alpha}(X;\xb7)$.

**Theorem 3.3.**For all $\alpha \in (0,\infty ]$

**Proof of Theorem 3.3.**The proof is based on the simple observation, following immediately from the definitions (2.9) and (3.9), that the dual level sets for the functions ${\tilde{A}}_{\alpha}(X;x)$ and ${B}_{\alpha}(X;p)$ are the same:

**Theorem 3.4.**The following properties of the quantile bounds ${Q}_{\alpha}(X;p)$ are valid.

**Model-independence:**- ${Q}_{\alpha}(X;p)$ depends on the r.v. X only through the distribution of X.
**Monotonicity in**X:- ${Q}_{\alpha}(\xb7\phantom{\rule{0.166667em}{0ex}};p)$ is nondecreasing with respect to the stochastic dominance of order $\alpha +1$: for any r.v. Y such that $X\stackrel{\alpha +1}{\u2a7d}Y$, one has ${Q}_{\alpha}(X;p)\u2a7d{Q}_{\alpha}(Y;p)$. Therefore, ${Q}_{\alpha}(\xb7\phantom{\rule{0.166667em}{0ex}};p)$ is nondecreasing with respect to the stochastic dominance of any order $\gamma \in [1,\alpha +1]$; in particular, for any r.v. Y such that $X\u2a7dY$, one has ${Q}_{\alpha}(X;p)\u2a7d{Q}_{\alpha}(Y;p)$.
**Monotonicity in**α:- ${Q}_{\alpha}(X;p)$ is nondecreasing in $\alpha \in [0,\infty ]$.
**Monotonicity in**p:- ${Q}_{\alpha}(X;p)$ is nonincreasing in $p\in (0,1)$, and ${Q}_{\alpha}(X;p)$ is strictly decreasing in $p\in [{p}_{*},1)\cap (0,1)$ if $\alpha \in (0,\infty ]$.
**Finiteness:**- ${Q}_{\alpha}(X;p)$ takes only (finite) real values.
**Concavity in ${p}^{-1/\alpha}$ or in $ln\frac{1}{p}$:**- ${Q}_{\alpha}(X;p)$ is concave in ${p}^{-1/\alpha}$ if $\alpha \in (0,\infty )$, and ${Q}_{\infty}(X;p)$ is concave in $ln\frac{1}{p}$.
**Stability in**p:- ${Q}_{\alpha}(X;p)$ is continuous in $p\in (0,1)$ if $\alpha \in (0,\infty ]$.
**Stability in**X:- Suppose that $\alpha \in (0,\infty ]$ and a sequence $({X}_{n})$ is as in Proposition 2.4. Then ${Q}_{\alpha}({X}_{n};p)\to {Q}_{\alpha}(X;p)$.
**Stability in**α:- Suppose that $\alpha \in (0,\infty ]$ and a sequence $({\alpha}_{n})$ is as in Proposition 2.3. Then ${Q}_{{\alpha}_{n}}(X;p)\to {Q}_{\alpha}(X;p)$.
**Translation invariance:**- ${Q}_{\alpha}(X+c;p)={Q}_{\alpha}(X;p)+c$ for all real c.
**Consistency:**- ${Q}_{\alpha}(c;p)=c$ for all real c; that is, if the r.v. X is the constant c, then all of the quantile bounds ${Q}_{\alpha}(X;p)$ equal c.
**Positive sensitivity:**- Suppose here that $X\u2a7e0$. If at that $\mathsf{P}(X>0)>0$, then ${Q}_{\alpha}(X;p)>0$ for all $\alpha \in (0,\infty ]$; if, moreover, $\mathsf{P}(X>0)>p$, then ${Q}_{0}(X;p)>0$.
**Positive homogeneity:**- ${Q}_{\alpha}(\kappa X;p)=\kappa {Q}_{\alpha}(X;p)$ for all real $\kappa \u2a7e0$.
**Subadditivity:**- ${Q}_{\alpha}(X;p)$ is subadditive in X if $\alpha \in [1,\infty ]$; that is, for any other r.v. Y (defined on the same probability space as X) one has:$${Q}_{\alpha}(X+Y;p)\u2a7d{Q}_{\alpha}(X;p)+{Q}_{\alpha}(Y;p).$$
**Convexity:**- ${Q}_{\alpha}(X;p)$ is convex in X if $\alpha \in [1,\infty ]$; that is, for any other r.v. Y (defined on the same probability space as X) and any $t\in (0,1)$ one has$${Q}_{\alpha}\left((1-t)X+tY;p\right)\u2a7d(1-t){Q}_{\alpha}(X;p)+t{Q}_{\alpha}(Y;p)$$

**Proposition 3.5.**The upper bound $\alpha +1$ on γ in the statement of the monotonicity of ${Q}_{\alpha}(X;p)$ in X in Theorem 3.4 is exact in the following rather strong sense. For any $\alpha \in [0,\infty )$, there exist r.v.’s X and Y in ${\mathcal{X}}_{\alpha}$ such that $X\stackrel{\gamma}{\u2a7d}Y$ for all $\gamma \in (\alpha +1,\infty ]$, whereas ${Q}_{\alpha}(X;p)>{Q}_{\alpha}(Y;p)$.

**Proposition 3.6.**Suppose that an r.v. Y is stochastically strictly greater than X (which may be written as $X\stackrel{\mathrm{st}}{<}Y$; cf., (2.24)) in the sense that $X\stackrel{\mathrm{st}}{\u2a7d}Y$ and for any $v\in \mathbb{R}$ there is some $u\in (v,\infty )$ such that $\mathsf{P}(X\u2a7eu)<\mathsf{P}(Y\u2a7eu)$. Then ${Q}_{\alpha}(X;p)<{Q}_{\alpha}(Y;p)$ if $\alpha \in (0,\infty ]$.

**Proposition 3.7.**There are r.v.’s X and Y such that for all $\alpha \in [0,1)$ and all $p\in (0,1)$ one has ${Q}_{\alpha}(X+Y;p)>{Q}_{\alpha}(X;p)+{Q}_{\alpha}(Y;p)$, so that the function ${Q}_{\alpha}(\xb7;p)$ is not subadditive (and, equivalently, not convex).

**Proposition 3.8.**If the r.v. X has a pdf of the form (3.11), then

**Remark 3.9.**It is shown in [8] that for small enough values of p the quantile bounds ${Q}_{\alpha}(X;p)$ are close enough to the true quantiles ${Q}_{0}(X;p)={\phantom{\rule{0.166667em}{0ex}}VaR}_{p}(X)$ provided that the right tail of the distribution of X is light enough and regular enough, depending on α see Proposition 2.7 in [8].

## 4. Computation of the Tail Probability and Quantile Bounds

#### 4.1. Computation of ${P}_{\alpha}(X;x)$

**Proposition 4.1.**For any $\alpha \in (0,1)$, $p\in (0,1)$, and $x\in \mathbb{R}$, there is a r.v. X (taking three distinct values) such that ${P}_{\alpha}(X;x)=p$ and the infimum ${inf}_{\lambda \in (0,\infty )}{A}_{\alpha}(X;x)(\lambda )$ in (2.5) is attained at precisely two distinct values of $\lambda \in (0,\infty )$.

**Example 4.2.**Let X be a r.v. taking values $-\frac{27}{11},-1,2$ with probabilities $\frac{1}{4},\frac{1}{4},\frac{1}{2}$; then ${x}_{*}=2$. Also let $\alpha =\frac{1}{2}$ and $x=0$, so that $x\in (-\infty ,{x}_{*})$, and then let ${\lambda}_{max}$ be as in (4.3) with $y={x}_{*}=2$, so that here ${\lambda}_{max}=\frac{3}{4}$. Then the minimum of ${A}_{\alpha}(X;0)(\lambda )$ over all real $\lambda \u2a7e0$ equals $\frac{\sqrt{3}}{2}$ and is attained at each of the two points, $\lambda =\frac{11}{54}$ and $\lambda =\frac{1}{2}$, and only at these two points. The graph $\left\{\left(\lambda ,{A}_{1/2}(X;0)(\lambda )\right):0\u2a7d\lambda \u2a7d{\lambda}_{max}\right\}$ is shown here in Figure 2.

#### 4.2. Computation of ${Q}_{\alpha}(X;p)$

**Proposition 4.3.**

**(Quantile bounds: Attainment and bracketing).**

- (i)
- If $\alpha \in (0,\infty )$, then ${inf}_{t\in {T}_{\alpha}}{B}_{\alpha}(X;p)(t)={inf}_{t\in \mathbb{R}}{B}_{\alpha}(X;p)(t)$ in (3.8) is attained at some ${t}_{\phantom{\rule{0.166667em}{0ex}}opt}\in \mathbb{R}$ and hence$${Q}_{\alpha}(X;p)=\underset{t\in \mathbb{R}}{min}{B}_{\alpha}(X;p)(t)={B}_{\alpha}(X;p)({t}_{\phantom{\rule{0.166667em}{0ex}}opt});$$$$s\in \mathbb{R}\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\tilde{p}\in (p,1),$$$${t}_{\phantom{\rule{0.166667em}{0ex}}opt}\in [{t}_{min},{t}_{max}],$$$${t}_{max}:={B}_{\alpha}(X;p)(s),\phantom{\rule{1.em}{0ex}}{t}_{min}:={t}_{0,min}\wedge {t}_{1,min},$$$${t}_{0,min}:={Q}_{0}(X;\tilde{p}),\phantom{\rule{1.em}{0ex}}{t}_{1,min}:=\frac{{(\tilde{p}/p)}^{1/\alpha}\phantom{\rule{0.166667em}{0ex}}{t}_{0,min}-{t}_{max}}{{(\tilde{p}/p)}^{1/\alpha}-1}.$$
- (ii)
- Suppose now that $\alpha =\infty $. Then ${inf}_{t\in {T}_{\alpha}}{B}_{\alpha}(X;p)(t)={inf}_{t\in (0,\infty )}{B}_{\alpha}(X;p)(t)$ in (3.8) is attained, and hence$${Q}_{\infty}(X;p)=\underset{t\in (0,\infty )}{min}{B}_{\infty}(X;p)(t)$$$${x}_{*}<\infty \phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}p\u2a7d{p}_{*},$$$${Q}_{\infty}(X;p)=\underset{t>0}{inf}{B}_{\infty}(X;p)(t)={B}_{\infty}(X;p)(0+)={x}_{*}.$$

**Proposition 4.4.**

- (i)
- If $\alpha \in [1,\infty ]$, then ${B}_{\alpha}(X;p)(t)$ is convex in the pair $(X,t)\in {\mathcal{X}}_{\alpha}\times {T}_{\alpha}$.
- (ii)
- If $\alpha \in (1,\infty )$, then ${B}_{\alpha}(X;p)(t)$ is strictly convex in $t\in (-\infty ,{x}_{**}]\cap \mathbb{R}$.
- (iii)
- ${B}_{\infty}(X;p)(t)$ is strictly convex in $t\in \{s\in (0,\infty ):\mathsf{E}{e}^{X/s}<\infty \}$, unless $\mathsf{P}(X=c)=1$ for some $c\in \mathbb{R}$.

**Proposition 4.5.**For any $\alpha \in (0,1)$, $p\in (0,1)$, and $x\in \mathbb{R}$, there is a r.v. X (taking three distinct values) such that ${Q}_{\alpha}(X;p)=x$ and the infimum ${inf}_{t\in {T}_{\alpha}}{B}_{\alpha}(X;p)(t)={inf}_{t\in \mathbb{R}}{B}_{\alpha}(X;p)(t)$ in (3.8) is attained at precisely two distinct values of t.

**Example 4.6.**

#### 4.3. Optimization of the Risk Measures ${Q}_{\alpha}(X;p)$ with Respect to X

**Theorem 4.7.**(Optimization shortcut.) Take any $\alpha \in (0,\infty ]$ and any $p\in (0,1)$. Let ${\mathcal{Y}}_{\alpha}$ be any subset of the set ${\mathcal{X}}_{\alpha}$ of r.v.’s defined by Formula (2.14). Then, for any $\alpha \in (0,\infty ]$ and any $p\in (0,1)$, the minimization of the risk measure ${Q}_{\alpha}(X;p)$ in $X\in {\mathcal{Y}}_{\alpha}$ is equivalent to the minimization of ${B}_{\alpha}(X;p)(t)$ in $(t,X)\in ({T}_{\alpha},{\mathcal{Y}}_{\alpha})$, in the sense that

#### 4.4. Additional Remarks on the Computation and Optimization

## 5. Implications for Risk Assessment in Finance and Inequality Modeling in Economics

#### 5.1. The Spectrum ${\left({Q}_{\alpha}(X;p)\right)}_{\alpha \in [0,\infty ]}$ Contains $\phantom{\rule{0.166667em}{0ex}}VaR$ and $\phantom{\rule{0.166667em}{0ex}}CVaR$.

#### 5.2. The Spectrum Parameter α as a Risk Sensitivity Index

- (i)
- one of the portfolios is clearly riskier than the other;
- (ii)
- this distinction is sensed (to varying degrees, depending on α) by all the risk measures ${Q}_{\alpha}(X;p)$ with $\alpha \in (1,\infty )$;
- (iii)
- yet, the values of ${\phantom{\rule{0.166667em}{0ex}}CVaR}_{p}={Q}_{1}(X;p)$ are the same for both portfolios.

**Figure 4.**Sensitivity of ${Q}_{\alpha}(\xb7;p)$ to risk, depending on the value of α: graphs $\left\{\left(p,{Q}_{\alpha}(X;p)\right):0<p<1\right\}$ (blue) and $\left\{\left(p,{Q}_{\alpha}(Y;p)\right):0<p<1\right\}$ (red) for $\alpha =1$ (

**left**); $\alpha =2$ (

**middle**); and $\alpha =5$ (

**right**).

#### 5.3. Coherent and Non-Coherent Measures of Risk

**Corollary 5.1.**For each $\alpha \in [1,\infty ]$ and each $p\in (0,1)$, the quantile bound ${Q}_{\alpha}(\xb7;p)$ is a coherent risk measure, and it is not coherent for any pair $(\alpha ,p)\in [0,1)\times (0,1)$.

#### 5.4. Other Terminology Used in the Literature for Some of the Listed Properties of ${Q}_{\alpha}(\xb7;p)$

#### 5.5. Gini-Type Mean Differences and Related Risk Measures

**Proposition 5.2.**The risk measure ${R}_{H}(X)$ is nondecreasing in X with respect to the stochastic dominance of order 1 if and only if the function H is $\frac{1}{2}$-Lipschitz: $|H(x)-H(y)|\u2a7d\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}|x-y|$ for all x and y in $[0,\infty )$.

**Proposition 5.3.**The risk measure ${R}_{H}(X)$ is coherent if and only if $H=\kappa \phantom{\rule{0.166667em}{0ex}}id$ for some $\kappa \in [0,\frac{1}{2}]$.

#### 5.6. A Lorentz-Type Parametric Family of Risk Measures

**Proposition 5.4.**If the r.v. X is nonnegative, then

#### 5.7. Spectral Risk Measures

#### 5.8. Risk Measures Reinterpreted as Measures of Economic Inequality

#### 5.9. “Explicit” Expressions of ${Q}_{\alpha}(X;p)$

## 6. Conclusions

- ${P}_{\alpha}(X;x)$ and ${Q}_{\alpha}(X;p)$ are three-way monotonic and three-way stable – in α, p, and X.
- The monotonicity in X is graded continuously in α, resulting in varying, controllable degrees of sensitivity of ${P}_{\alpha}(X;x)$ and ${Q}_{\alpha}(X;p)$ to financial risk/economic inequality.
- $x\mapsto {P}_{\alpha}(X;x)$ is the tail-function of a certain probability distribution.
- ${Q}_{\alpha}(X;p)$ is a $(1-p)$-percentile of that probability distribution.
- For small enough values of p, the quantile bounds ${Q}_{\alpha}(X;p)$ are close enough to the corresponding true quantiles $Q(X;p)={\phantom{\rule{0.166667em}{0ex}}VaR}_{p}(X)$, provided that the right tail of the distribution of X is light enough and regular enough, depending on α.
- In the case when the loss X is modeled as a normal r.v., the use of the risk measures ${Q}_{\alpha}(X;p)$ reduces, to an extent, to using the Markowitz mean-variance risk-assessment paradigm – but with a varying weight of the standard deviation, depending on the risk sensitivity parameter α.
- ${P}_{\alpha}(X;x)$ and ${Q}_{\alpha}(X;p)$ are solutions to mutually dual optimizations problems, which can be comparatively easily incorporated into more specialized optimization problems, with additional restrictions on the r.v. X.
- ${P}_{\alpha}(X;x)$ and ${Q}_{\alpha}(X;p)$ are effectively computable.
- Even when the corresponding minimizer is not identified quite perfectly, one still obtains an upper bound on the risk/inequality measures ${P}_{\alpha}(X;x)$ or ${Q}_{\alpha}(X;p)$.
- Optimal upper bounds on ${P}_{\alpha}(X;x)$ and, hence, on ${Q}_{\alpha}(X;p)$ over important classes of r.v.’s X represented (say) as sums of independent r.v.’s ${X}_{i}$ with restrictions on moments of the ${X}_{i}$’s and/or sums of such moments can be given; see e.g. [7,54] and references therein.
- The quantile bounds ${Q}_{\alpha}(X;p)$ with $\alpha \in [1,\infty ]$ constitute a spectrum of coherent measures of financial risk and economic inequality.
- The r.v.’s X of which the measures ${P}_{\alpha}(X;x)$ and ${Q}_{\alpha}(X;p)$ are taken are allowed to take values of both signs. In particular, if, in a context of economic inequality, X is interpreted as the net amount of assets belonging to a randomly chosen economic unit, then a negative value of X corresponds to a unit with more liabilities than paid-for assets. Similarly, if X denotes the loss on a financial investment, then a negative value of X will obtain when there actually is a net gain.

## Acknowledgment

## Appendix

## A. Proofs

**Proof of Proposition 2.1.**This proof is not hard but somewhat technical; it can be found in the more detailed version [8] of this paper; see the proof of Proposition 1.1 there. ☐

**Proof of Proposition 2.2.**This too can be found in [8]; see the proof of Proposition 1.2 there. ☐

**Proof of Proposition 2.3.**Let α and a sequence $({\alpha}_{n})$ be indeed as in Proposition 2.3. If $x\in [{x}_{*},-\infty )$, then the desired conclusion ${P}_{{\alpha}_{n}}(X;x)\to {P}_{\alpha}(X;x)$ follows immediately from part (i) of Proposition 2.2. Therefore, assume in the rest of the proof of Proposition 2.3 that

**Proof of Proposition 2.4.**This is somewhat similar to the proof of Proposition 2.3. One difference here is the use of the uniform integrability condition, which, in view of (2.3), (4.1), and the condition $X\in {\mathcal{X}}_{\alpha}$, implies (see e.g. Theorem 5.4 in [15]) that for all $\lambda \in [0,\infty )$

**Proof of Theorem 2.5.**The

**model-independence**is obvious from the definition (2.5). The

**monotonicity in X**follows immediately from (2.23), (2.10), and (2.7)–(2.9). The

**monotonicity in α**was already given in (2.13). The

**monotonicity in x**is Part (i) of Proposition 2.1. That ${P}_{\alpha}(X;x)$ takes on only

**values**in the interval $[0,1]$ follows immediately from (2.16). The

**α-concavity in x**and

**stability in x**follow immediately from parts (iii) and (i) of Proposition 2.2. The

**stability in α**and the

**stability in X**are Propositions 2.3 and 2.4, respectively. The

**translation invariance**,

**consistency**, and

**positive homogeneity**follow immediately from the definition (2.5). ☐

**Proof of Proposition 3.1.**

**Proof of Theorem 3.4.**The

**model-independence**,

**monotonicity in X**,

**monotonicity in α**,

**translation invariance**,

**consistency**, and

**positive homogeneity**properties of ${Q}_{\alpha}(X;p)$ follow immediately from (3.2) and the corresponding properties of ${P}_{\alpha}(X;x)$ stated in Theorem 2.5.

**monotonicity of ${Q}_{\alpha}(X;p)$ in p**: that ${Q}_{\alpha}(X;p)$ is nondecreasing in $p\in (0,1)$ follows immediately from (3.3) for $\alpha =0$ and from (3.8) and (3.9) for $\alpha \in (0,\infty ]$. That ${Q}_{\alpha}(X;p)$ is strictly decreasing in $p\in [{p}_{*},1)\cap (0,1)$ if $\alpha \in (0,\infty ]$ follows immediately from Part (v) of Proposition 3.1, and the verified below statement on the stability in p: ${Q}_{\alpha}(X;p)$ is continuous in $p\in (0,1)$ if $\alpha \in (0,\infty ]$.

**finiteness of ${Q}_{\alpha}(X;p)$**was already stated in Part (i) of Proposition 3.1.

**concavity of ${Q}_{\alpha}(X;p)$ in ${p}^{-1/\alpha}$**in the case when $\alpha \in (0,\infty )$ follows by (3.8), since ${B}_{\alpha}(X;p)(t)$ is affine (and hence concave) in ${p}^{-1/\alpha}$. Similarly, the

**concavity of ${Q}_{\infty}(X;p)$ in $ln\frac{1}{p}$**follows by (3.8), since ${B}_{\infty}(X;p)(t)$ is affine in $ln\frac{1}{p}$.

**stability of ${Q}_{\alpha}(X;p)$ in p**can be deduced from Proposition 3.1. Alternatively, the same follows from the already established finiteness and concavity of ${Q}_{\alpha}(X;p)$ in ${p}^{-1/\alpha}$ or $ln\frac{1}{p}$ (cf. the proof of [2, Proposition 13]), because any finite concave function on an open interval of the real line is continuous, whereas the mappings $(0,1)\ni p\mapsto {p}^{-1/\alpha}\in (0,\infty )$ and $(0,1)\ni p\mapsto ln\frac{1}{p}\in (0,\infty )$ are homeomorphisms.

**stability of ${Q}_{\alpha}(X;p)$ in X**, take any real $x\ne {x}_{*}$. Then the convergence ${P}_{\alpha}({X}_{n};x)\to {P}_{\alpha}(X;x)$ holds, by Proposition 2.4. Therefore, in view of (2.19), if $x\in {E}_{\alpha ,X}(p)$ then eventually (that is, for all large enough n) $x\in {E}_{\alpha ,{X}_{n}}(p)$. Hence, by (3.2), for each real $x\ne {x}_{*}$ such that $x>{Q}_{\alpha}(X;p)$ eventually one has $x\u2a7e{Q}_{\alpha}({X}_{n};p)$. It follows that ${lim\; sup}_{n}{Q}_{\alpha}({X}_{n};p)\u2a7d{Q}_{\alpha}(X;p).$ On the other hand, by Part (vi) of Proposition 3.1, for any $y\in \left(-\infty ,{Q}_{\alpha}(X;p)\right)$, one has ${P}_{\alpha}(X;y)>p$ and, hence, eventually ${P}_{\alpha}({X}_{n};y)>p$, which yields $y\notin {E}_{\alpha ,{X}_{n}}(p)$ and, hence, $y\u2a7d{Q}_{\alpha}({X}_{n};p)$. It follows that ${lim\; inf}_{n}{Q}_{\alpha}({X}_{n};p)\u2a7e{Q}_{\alpha}(X;p)$. Recalling now the established inequality ${lim\; sup}_{n}{Q}_{\alpha}({X}_{n};p)\u2a7d{Q}_{\alpha}(X;p)$, one completes the verification of the stability of ${Q}_{\alpha}(X;p)$ in X.

**stability of ${Q}_{\alpha}(X;p)$ in α**is proved quite similarly, only using Proposition 2.3 in place of Proposition 2.4. Here the stipulation $x\ne {x}_{*}$ is not needed.

**positive sensitivity**property. First, suppose that $\alpha \in (0,1)$. Then, for all real $t<0$, the derivative of ${B}_{\alpha}(X;p)(t)$ in t is less than $D:=1-{(\mathsf{E}{Y}^{\alpha})}^{-1+1/\alpha}\mathsf{E}{Y}^{\alpha -1}$, where $Y:={(X-t)}_{+}=X-t>0$. The inequality $D\u2a7d0$ can be rewritten as the true inequality $\frac{\tau}{\tau +1}L(-1)+\frac{1}{\tau +1}L(\tau )\u2a7eL(0)$ for the convex function $s\mapsto L(s):=ln\mathsf{E}exp\{(1-\alpha )slnY\}$, where $\tau :=\frac{\alpha}{1-\alpha}$. Therefore, the derivative is negative and hence ${B}_{\alpha}(X;p)(t)$ decreases in $t\u2a7d0$ (here, to include $t=0$, we also used the continuity of ${B}_{\alpha}(X;p)(t)$ in t, which follows by the condition $X\in {\mathcal{X}}_{\alpha}$ and dominated convergence). On the other hand, if $t>0$, then ${B}_{\alpha}(X;p)(t)\u2a7et>0$. Also, ${B}_{\alpha}(X;p)(0)>0$ by (3.9) if the condition $\mathsf{P}(X>0)>0$ holds. Recalling again the continuity of ${B}_{\alpha}(X;p)(t)$ in t, one completes the verification of the positive sensitivity property – in the case $\alpha \in (0,1)$.

**subadditivity**property is easy to see to be equivalent to the convexity; cf. e.g. Theorem 4.7 in [56].

**convexity**property. Assume indeed that $\alpha \in [1,\infty ]$. If at that $\alpha <\infty $, then the function ${\parallel \xb7\parallel}_{\alpha}$ is a norm and hence convex; moreover, this function is nondecreasing on the set of all nonnegative r.v.’s. On the other hand, the function $\mathbb{R}\ni x\mapsto {x}_{+}$ is nonnegative and convex. It follows by (3.9) that ${B}_{\alpha}(X;p)(t)$ is convex in the pair $(X,t)$. So, to complete the verification of the convexity property of ${Q}_{\alpha}(X;p)$ in the case $\alpha \in [1,\infty )$, it remains to refer to the well-known and easily established fact that, if $f(x,y)$ is convex in $(x,y)$, then ${inf}_{y}\phantom{\rule{0.166667em}{0ex}}f(x,y)$ is convex in x; cf. e.g. Theorem 5.7 in [56].

_{X1+⋯+Xn})

^{*−1}≤ given in the course of the discussion in [23] following Corollary 2.2 therein. However, a direct proof, similar to the one above for $\alpha \in [1,\infty )$, can be based on the observation that ${B}_{\infty}(X;p)(t)$ is convex in the pair $(X,t)$. Since $t\phantom{\rule{0.166667em}{0ex}}ln\frac{1}{p}$ is obviously linear in $(X,t)$, the convexity of ${B}_{\infty}(X;p)(t)$ in $(X,t)$ means precisely that for any natural number n, any r.v.’s ${X}_{1},\cdots ,{X}_{n}$, any positive real numbers ${t}_{1},\cdots ,{t}_{n}$, and any positive real numbers ${\alpha}_{1},\cdots ,{\alpha}_{n}$ with ${\sum}_{i}{\alpha}_{i}=1$, one has the inequality $tln\mathsf{E}{e}^{X/t}\u2a7d{\sum}_{i}{\alpha}_{i}{t}_{i}ln\mathsf{E}{e}^{{X}_{i}/{t}_{i}}$, where $X:={\sum}_{i}{\alpha}_{i}{X}_{i}$ and $t:={\sum}_{i}{\alpha}_{i}{t}_{i}$; but the latter inequality can be rewritten as an instance of Hölder’s inequality: $\mathsf{E}{\prod}_{i}{Z}_{i}\u2a7d{\prod}_{i}{\parallel {Z}_{i}\parallel}_{{p}_{i}}$, where ${Z}_{i}:={e}^{{\alpha}_{i}{X}_{i}/t}$ and ${p}_{i}:=t/({\alpha}_{i}{t}_{i})$ (so that ${\sum}_{i}\frac{1}{{p}_{i}}=1$). (In particular, it follows that ${B}_{\infty}(X;p)(t)$ is convex in t, which is useful when ${Q}_{\infty}(X;p)$ is computed by Formula (3.8).)

**Proof of Proposition 3.5.**Take indeed any $\alpha \in [0,\infty )$. Let then Y be a r.v. with the density function f given by the formula $f(y)={c}_{\alpha}{y}^{-\alpha -1}{(lny)}^{-2}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathsf{I}\{y>2\}$ for all $y\in \mathbb{R}$, where ${c}_{\alpha}:=1/{\int}_{2}^{\infty}{y}^{-\alpha -1}{ln}^{-2}y\phantom{\rule{0.166667em}{0ex}}\text{d}y$. Then $Y\in {\mathcal{X}}_{\alpha}$ and, by the finiteness property stated in Theorem 3.4, ${Q}_{\alpha}(Y;p)\in \mathbb{R}$. Thus, one can find some real constant $c>{Q}_{\alpha}(Y;p)$. Let now $X=c$, for any such constant c. Then, by the consistency property stated in Theorem 3.4, ${Q}_{\alpha}(X;p)=c>{Q}_{\alpha}(Y;p)$. On the other hand, for any $\gamma \in (\alpha +1,\infty ]$ one has $\mathsf{E}{g}_{\gamma -1;t}(X)={g}_{\gamma -1;t}(c)<\infty =\mathsf{E}{g}_{\gamma -1;t}(Y)$ for all $t\in {T}_{\gamma -1}$ (letting here $\gamma -1:=\infty $ when $\gamma =\infty $), so that, by (2.23), $X\stackrel{\gamma}{\u2a7d}Y$. ☐

**Proof of Proposition 3.6.**Consider first the case $\alpha \in (0,\infty )$. Let r.v.’s X and Y be in the default domain of definition, ${\mathcal{X}}_{\alpha}$, of the functional ${Q}_{\alpha}(\xb7;p)$. The condition $X\stackrel{\mathrm{st}}{<}Y$ and the left continuity of the function $\mathsf{P}(X\u2a7e\xb7)$ imply that for any $v\in \mathbb{R}$, there are some $u\in (v,\infty )$ and $w\in (v,u)$ such that $\mathsf{P}(X\u2a7ez)<\mathsf{P}(Y\u2a7ez)$ for all $z\in [w,u]$. On the other hand, by the Fubini theorem, $\mathsf{E}{(X-t)}_{+}^{\alpha}={\int}_{\mathbb{R}}\alpha {(z-t)}_{+}^{\alpha -1}\mathsf{P}(X\u2a7ez)\text{d}z$ for all $t\in \mathbb{R}$. Recalling also that X and Y are in ${\mathcal{X}}_{\alpha}$, one has ${B}_{\alpha}(X;p)(t)<{B}_{\alpha}(Y;p)(t)$ for all $t\in \mathbb{R}$. By Proposition 4.3, ${Q}_{\alpha}(Y;p)={B}_{\alpha}(Y;p)({t}_{\phantom{\rule{0.166667em}{0ex}}opt})$ for some ${t}_{\phantom{\rule{0.166667em}{0ex}}opt}\in \mathbb{R}$. Therefore, ${Q}_{\alpha}(X;p)\u2a7d{B}_{\alpha}(X;p)({t}_{\phantom{\rule{0.166667em}{0ex}}opt})<{B}_{\alpha}(Y;p)({t}_{\phantom{\rule{0.166667em}{0ex}}opt}$