Economic indices of absolute and relative riskiness

Abstract

Following Aumann and Serrano (J Polit Econ 116:810–836, 2008) who characterize by axioms an index of riskiness defined on absolute returns, we characterize a new index of riskiness defined on relative returns. Both indices are characterized by a similar principle of duality between risk and risk aversion, but while the index of absolute riskiness refers to absolute risk aversion, the index of relative riskiness refers to relative risk aversion. The similarities and differences between the two indices are studied.

Appendix

1.1 Proofs

Since $A$ is basically the index $A^*$ applied to absolute returns, many of the statements in relation to $A$ and their proofs appear already in Aumann and Serrano (2008). Here we mostly focus on the proofs of statements in relation to $R$.

In this section, investors $i$ and $j$ have utility functions $u_i$ and $u_j$ and Arrow–Pratt coefficients $\varrho _i$ and $\varrho _j$ of relative risk aversion. Since utilities may be modified by additive and positive multiplicative constants, we assume throughout that

$$\begin{aligned} u_i(1)=u_j(1)=0\quad \text {and}\quad u'_i(1)=u'_j(1)=1. \end{aligned}$$

(13)

Lemma 7.1

For some $\delta >1$, suppose that $\varrho _i(w)>\varrho _j(w)$ at each $w$ with $1/\delta <w<\delta$. Then $u_i(w)<u_j(w)$ whenever $1/\delta <w<\delta$ and $w\ne 1$.

Proof

Let $y$ be a number, $1/\delta <y<\delta$. If $y>1$, then, by Eq. (13),

$$\begin{aligned} \log u'_i(y)&= \log u'_i(y)-\log u'_i(1)=\int \limits _1^y{\left[ \log u'_i(z)\right] '\mathrm{d}z}=\int \limits _1^y{\frac{u''_i(z)}{u'(z)}\mathrm{d}z} \\&= \int \limits _1^y{-(\varrho _i(z)/z})\mathrm{d}z<\int \limits _1^y{-(\varrho _j(z)/z})\mathrm{d}z=\log u'_j(y). \end{aligned}$$

If, on the other hand, $y<1$, the reasoning is similar, but the inequality is reversed, because then $\int _1^y=-\int _y^1$. Thus, when $y>1,\,\log u'_i(y)<\log u'_j(y)$ and also $u'_i(y)<u'_j(y)$, and when $y<1\,\log u'_i(y)>\log u'_j(y)$ and also $u'_i(y)>u'_j(y)$. So if $w>1$, then, by (13),

$$\begin{aligned} u_i(w)=\int \limits _1^w{u'_i(y)\mathrm{d}y}<\int \limits _1^w{u'_j(y)\mathrm{d}y}=u_j(w); \end{aligned}$$

and if $w<1$, then

$$\begin{aligned} u_i(w)=-\int \limits _w^1{u'_i(y)\mathrm{d}y}<-\int \limits _w^1{u'_j(y)\mathrm{d}y}=u_j(w). \end{aligned}$$

$\square$

Corollary 5

If $\varrho _i(w)\le \varrho _j(w)$ for all $w>0$, then $u_i(w)\ge u_j(w)$ for all $w>0$.

Lemma 7.2

For any security $s$, its relative riskiness $R(s)$ is well defined.

Proof

For a given security $s$, we denote by ${\hat{s}}$ the relative return of $s,\,{\hat{s}}=s_1/s_0$. We define the function $f_s$ as follows:

$$\begin{aligned} f_s(\beta )\equiv {{\mathrm{E}}}{\hat{s}}^\beta =\Sigma p_i{\hat{s}}_i^\beta , \end{aligned}$$

(14)

where $\beta$ is a real number. The first and second derivatives of $f_s$ are

$$\begin{aligned} f_s'(\beta )&= \Sigma p_i {{\hat{s}}}_i^\beta \log {\hat{s}}_i, \end{aligned}$$

(15)

$$\begin{aligned} f_s''(\beta )&= \Sigma p_i {{\hat{s}}}_i^\beta \left( \log {\hat{s}}_i\right) ^2. \end{aligned}$$

(16)

Since by definition at least one of the values of ${\hat{s}}$ is greater than one and at least one of the values is less than one,

$$\begin{aligned} \lim _{\beta \rightarrow \pm \infty }f_s(\beta )=\infty . \end{aligned}$$

(17)

In addition, since $f_s''$ is positive for all $\beta ,\,f_s'$ increases with $\beta$, which implies that $f_s$ has a single minimum point. It follows from (14) that $f_s(0)=1$. If $f_s'(0)\ne 0$, there should be another value of $\beta$, for which $f_s(\beta )=1$. Based on this insight, we define $\beta ^*$ as follows:

• 1. If $f_s'(0)>0$, then there is only one additional value of $\beta ,\,\beta =\beta ^*$, in which $f_s(\beta ^*)=1$ and $\beta ^*<0$.

• 2. If $f_s'(0)<0$, then there is only one additional value of $\beta ,\,\beta =\beta ^*$, in which $f_s(\beta ^*)=1$ and $\beta ^*>0$.

• 3. If $f_s'(0)=0$, then there is no other value of $\beta ,\,\beta \ne 0$, in which $f_s(\beta )=1$. In this case we set $\beta ^*=0$.

Since we assumed that the weighted geometric mean of the relative return of securities is greater than one, $f_s'(0)=\Sigma p_i \log {\hat{s}}_i>0$, and the first case, in which $\beta ^*<0$, is satisfied. Defining $R(s)=-1/\beta ^*$ shows the existence of $R(s)$ and also that $R(s)>0$. This completes the proof. $\square$

Lemma 7.3

For any two securities $s$ and $r$,

$$\begin{aligned} R(r)>R(s) \Leftrightarrow f_s(-1/R(r))<1. \end{aligned}$$

Proof

We use the definition of $f_s$ of the previous proof. Since $f_s'(0)>0,\,\beta ^*=-1/R(s)<0$, and the minimum point of $f_s$ is between $-1/R(s)$ and 0 [scenario 1 in the proof of (7.2)]. This, together with the continuity of $f_s$, implies that for any $\beta ,\,-1/R(s)<\beta <0,\,f_s(\beta )<1$. Since $-1/R(s)<-1/R(r)<0,\,f_s(-1/R(r))<1$. $\square$

Lemma 7.4

For any utility function $u_\alpha$ and value of $\delta >1$, there is a security $s=s(\alpha , \delta )$, such that $u_\alpha ({\hat{s}})=0$ and $\forall i,\ 1/\delta <{\hat{s}}_i<\delta$, where ${\hat{s}}=s_1/s_0$ and ${\hat{s}}_i$s are the values that ${\hat{s}}$ takes.

Proof

Let $f(\epsilon )$ be defined as $f(\epsilon )=\epsilon u_\alpha (\sqrt{1/\delta }) +(1-\epsilon )u_\alpha (\sqrt{\delta })$. It is easy to see that if $\epsilon =0$, then $f(\epsilon )>0$, and if $\epsilon =1$, then $f(\epsilon )<0$. Since $f$ is continuous in $\epsilon ,\,f(\epsilon ^*)=0$ for some $\epsilon ^*$ between zero and one. The desired security is the one whose relative return takes the value $\sqrt{1/\delta }$ with probability $\epsilon ^*$ and the value $\sqrt{\delta }$ with probability $1-\epsilon ^*$. $\square$

Lemma 7.5

If $\varrho _i(w_i)>\varrho _j(w_j)$, then there is a security $s$ that $j$ R-accepts at $w_j$ and $i$ R-rejects at $w_i$.

Proof

Without loss of generality, $w_i=w_j=1$, and so $\varrho _i(1)>\varrho _j(1)$.²⁰ Let $\varrho$ be a number between $\varrho _i(w)$ and $\varrho _j(w),\,\varrho _i(w)>\varrho >\varrho _j(w)$. Since $u_i$ and $u_j$ are twice continuously differentiable, it follows that there is a number $h>1$ such that $\varrho _i(w)>\varrho >\varrho _j(w)$ at each $w$ with $1/h<w<h$. By Lemma 7.4, there is a security $s(\varrho , h)$ such that $u_\varrho$ is indifferent between R-accepting or R-rejecting it. Therefore, by Lemma 7.1,

$$\begin{aligned} u_i(w)<u_\varrho (w)<u_j(w)\quad \text {whenever}\quad 1/\delta <w<\delta \quad \text {and}\quad w\ne 1 \end{aligned}$$

(18)

implies that $u_i({\hat{s}}(\varrho ,h))<0<u_j({\hat{s}}(\varrho ,h))$, where ${\hat{s}}=s_1/s_0$. Hence, $i$ R-rejects the security but $j$ R-accepts it. $\square$

Proof of Lemma 3.1

The proof of the first part of the lemma appears in Aumann and Serrano (2008). Here we prove the second part. We have to show that $\varrho _i(w)\ge \varrho _j(w)$ for all wealth levels $w$ if and only if $i$ is no less uniformly relative risk averse than $j$.

“If”: Assume that there are $w_i$ and $w_j$ with $\varrho _i(w_i)< \varrho _j(w_j)$. By Lemma 7.5, there is a security that $i$ R-accepts at $w_i$ and $j$ R-rejects at $w_j$, thereby contradicting $i$ being less uniformly relative risk averse than $j$.

“Only if”: Assuming that $\varrho _i(w_i)\ge \varrho _j(w_j)$ for all wealth levels $w_i$ and $w_j$, we must show that for both wealth levels, $w_i$ and $w_j$, and security $s$, if $i$ R-accepts $s$ at $w_i$, then $j$ R-accepts $s$ at $w_j$. Without loss of generality, $w_i=w_j=1$, and so we must show that

$$\begin{aligned} \text {if}\,i\,\text {R-accepts}\,s\,\text {at 1, then}\,j\,\text {R-accepts}\,s\,\text {at 1}. \end{aligned}$$

From Corollary 5 (with $i$ and $j$ reversed), we conclude that $u_j(w_j)\ge u_i(w_i)$ for each $w_i$ and $w_j$, and so ${{\mathrm{E}}}u_j({\hat{s}})\ge {{\mathrm{E}}}u_i({\hat{s}})$, where ${\hat{s}}=s_1/s_0$. That yields the above claim. $\square$

Lemma 7.6

An agent i has a CRRA utility if and only if for any security $s$ and any two wealth levels, i either R-accepts $s$ at both levels or R-rejects $s$ at both levels.

Proof

We denote by ${\hat{s}}$ the relative return of $s$, i.e., ${\hat{s}}=s_1/s_0$. Recall that all CRRA utility functions have the form

$$\begin{aligned} u_\alpha (x) = \left\{ \begin{array}{ll} \frac{(x^{1-\alpha }-1)}{1-\alpha } &{} \quad \text {if}\,\alpha \ne 1\\ \log (x) &{} \quad \text {if}\,\alpha = 1\\ \end{array} \right. \end{aligned}$$

(19)

for $\alpha >0$.

“Only if”: Let $u_\alpha (x)$ be a CRRA utility with parameter $\alpha$. $u_\alpha$ R-accepts $s$ at $w$ if and only if ${{\mathrm{E}}}u_\alpha (w{\hat{s}})>u_\alpha (w)$, that is, if and only if ${{\mathrm{E}}}u_\alpha ({\hat{s}})>u_\alpha (1)$.

“If”: It follows from Lemma 7.5; just take $j=i$. $\square$

Proof of Theorem 1

The first part of the theorem appears in Aumann and Serrano (2008). Here we prove the second part.

For $\alpha >0$, let $u_\alpha (x)$ be the CRRA utility function with parameter $\alpha$. The functions $u_\alpha$ satisfy (13), and so by Lemma 7.1 (with $\delta$ arbitrarily large) their graphs are nested; that is,

$$\begin{aligned} \text {if}\,\alpha >\beta ,\,\text {then}\,u_\alpha (x)<u_\beta (x)\,\text {for all}\,x> 0, x\ne 1. \end{aligned}$$

(20)

The existence of $R(s)$ is proved in Lemma 7.2.

To see that R satisfies the duality axiom, let $i,j,r,h$, and $w$ be as in the hypothesis of that axiom; without loss of generality, $w=1$. Set $\gamma \equiv 1+1/R(s),\,\eta \equiv 1+1/R(h),\,\alpha _i=\inf \varrho _i$ and $\alpha _j=\sup \varrho _j$. For a given security $s$, we denote by ${\hat{s}}=s_1/s_0$ the relative return of $s$. Thus,

$$\begin{aligned} {{\mathrm{E}}}u_\gamma ({\hat{s}})=0\text { and }{{\mathrm{E}}}u_\eta (\hat{h})=0. \end{aligned}$$

(21)

By hypothesis, $R(s)>R(h)$, so $\eta >\gamma$. By Corollary 5,

$$\begin{aligned} u_i(x)\le u_{\alpha _i}(x)\quad \text {and}\quad u_{\alpha _j}(x)\le u_j(x)\text { for all x}. \end{aligned}$$

(22)

Now assume ${{\mathrm{E}}}u_i({\hat{s}})>0$; we must prove that ${{\mathrm{E}}}u_j(\hat{h})>0$. From ${{\mathrm{E}}}u_i({\hat{s}})>0$ and (22), it follows that ${{\mathrm{E}}}u_{\alpha _i}({\hat{s}})>0$. So by (21), $E_\gamma ({\hat{s}})=0<{{\mathrm{E}}}u_{\alpha _i}({\hat{s}})$. So by (20), $\gamma > \alpha _i$. By Lemma 3.1 $\alpha _i\ge \alpha _j$ so $\eta >\gamma$ yields $\alpha _j<\eta$. Since (21), (20) and (22) yield $0<{{\mathrm{E}}}u_\eta (\hat{h})<{{\mathrm{E}}}u_{\alpha _j}(\hat{h})<{{\mathrm{E}}}u_j(\hat{h})$, it follows that R satisfies the duality axiom.

That $R$ satisfies the scaling axiom is immediate, and so, indeed, $R$ satisfies the two relative axioms.

In the opposite direction, let Q be an index that satisfies the relative axioms. We first show that

$$\begin{aligned} \text {Q is ordinally equivalent to R.} \end{aligned}$$

(23)

If this is not true, then there must exist $s$ and $r$ that are ordered differently by Q and R. This means either that the respective orderings are reversed, that is,

$$\begin{aligned} Q(s)>Q(r)\,\text {and}\,R(s)<R(r), \end{aligned}$$

(24)

or that the equality holds for exactly one of the two indices, that is,

$$\begin{aligned} Q(s)>Q(r)\,\text {and}\,R(s)=R(r) \end{aligned}$$

(25)

or

$$\begin{aligned} Q(s)=Q(r)\,\text {and}\,R(s)>R(r). \end{aligned}$$

(26)

If either (25) or (26) holds, then by the scaling axiom, replacing $s$ by $s^\delta$ for sufficiently small $\delta >1$ leads to reversed inequalities. So without loss of generality, we may assume (24).

Now let $\gamma \equiv 1+1/R(s)$ and $\eta \equiv 1+1/R(r)$; then (21) holds. By (24), $\gamma >\eta$. Choose $\mu$ and $\nu$ so that $\gamma >\mu >\nu >\eta$. Then $u_\gamma (x)<u_\mu (x)<u_\nu (x)<u_\eta (x)$ for all $x\ne 0$. So by (21) ${{\mathrm{E}}}u_\mu ({\hat{s}})>{{\mathrm{E}}}u_\gamma ({\hat{s}})=0$ and ${{\mathrm{E}}}u_\nu (\hat{r})<{{\mathrm{E}}}u_\eta (\hat{r})=0$. So if $i$ and $j$ have utility functions $u_\mu$ and $u_\nu$, respectively, then $i$ R-accepts $s$ and j R-rejects $r$. But from $\mu >\nu$ and Lemma (3.1), it follows that $i\succ j$, contradicting the duality axiom for Q. So (23) is proved.

To see that Q is a positive multiple of R, let $s^*$ be an arbitrary but fixed security and set $\lambda \equiv Q(s^*)/R(s^*)$. If $s$ is any security and $t\equiv Q(s)/Q(s^*)$, then $Q((s^*)^t)=tQ(s^*)=Q(s)$, and so $tR(s^*)=R((s^*)^t)=R(s)$ by the ordinal equivalence between Q and S, and $R(s)/R(s^*)=t=Q(s)/Q(s^*)$, and $Q(s)/R(s)=Q(s^*)/R(s^*)=\lambda$, and $Q(s)=\lambda R(s)$. This completes the proof of Theorem A. $\square$

Needless to say, both duality and scaling are essential to Theorem 1. Thus, the mean log $E\log s$ satisfies scaling but violates duality, while the index $[R(s)]$, where $[x]$ denotes the integer part of $x$, satisfies duality but violates scaling. Neither $E\log s$ nor $[R(s)]$ is even ordinally equivalent to $R$.

Proof of (6) in Section 2.3

It is enough to show that

$$\begin{aligned} \lim _{\alpha \rightarrow 0} R(s(\alpha ))/\alpha =A(s)/s_0. \end{aligned}$$

(27)

Indeed, following Eq. (3), $R(s)=A^{*}(\log (s_1/s_0))$; hence

$$\begin{aligned} R(s(\alpha ))/\alpha =A^{*}(\log (1+\alpha (s_1/s_0-1)))/\alpha , \end{aligned}$$

which equals $A^{*}(\log (1+\alpha (s_1/s_0-1))/\alpha )$ (by scaling). Since $A^{*}$ is continuous, the limit of this expression as $\alpha$ goes to zero equals $A^{*}(s_1/s_0-1)=A(s)/s_0$. $\square$

Proof of Lemma 3.2

For the proof of the first part of the lemma, see Aumann and Serrano (2008). Here we prove only the second part.

An agent with a CRRA utility with parameter $\gamma$ R-accepts security $s$ if and only if

$$\begin{aligned} f_s(1-\gamma )>1, \end{aligned}$$

where $f_s(\beta )$ is the function defined in (14). Since for all $\beta <\beta ^*,\,f_s(\beta )>1$ and for all $\beta ^*<\beta <0,\,f_s(\beta )<1$ (by the proof of Lemma 7.2), every CRRA agent with a parameter greater than $1-\beta ^*$ R-rejects $s$ and every CRRA agent with a parameter lower than $1-\beta ^*$ R-accepts $s$. $\square$

Proof of Lemma 3.3

For the proof of the first part of the lemma, see Aumann and Serrano (2008). Here we prove only the second part. Let $u_i$ be $i$’s utility and assume that $\varrho _i(x)<1/R(s)+1$ for all $x$ between $w\min {\hat{s}}$ and $w\max {\hat{s}}$, where ${\hat{s}}=s_1/s_0$. Define a utility $u_j$ as follows: When $x$ is between $w\min {\hat{s}}$ and $w\max {\hat{s}}$, define $u_j(x)\equiv u_i(x)$; when $x\le w\min {\hat{s}}$, define $u_j(x)$ to equal a CRRA utility with parameter $\varrho _i(w\min {\hat{s}})$ and $u_j(w\min {\hat{s}})=u_i(w\min {\hat{s}})$ and $u_j'(w\min {\hat{s}})=u_i'(w\min {\hat{s}})$; when $x\ge w\max {\hat{s}}$, define $u_j(x)$ to equal a CRRA utility with parameter $\varrho _i(w\max {\hat{s}})$ and $u_j(w\max {\hat{s}})=u_i(w\max {\hat{s}})$ and $u_j'(w\max {\hat{s}})=u_i'(w\max {\hat{s}})$. Let $u_k$ be a CRRA utility with parameter $[1/R(s)+1]-\epsilon$. Then

$$\begin{aligned} \min _x\varrho _k(x)>\max _x\varrho _j(x) \end{aligned}$$

for positive $\epsilon$ sufficiently small. By Lemma 3.2, a CRRA person with parameter $[1/R(s)+1]$ is indifferent between R-accepting and R-rejecting $s$. Therefore, $k$, who is less risk averse, R-accepts $s$, and so $j$ also R-accepts $s$. But between the minimum and maximum of $w{\hat{s}}$, the utilities of $i$ and $j$ are the same. So $i$ R-accepts $s$ at $w$. $\square$

Proof of Lemma 4.1

The first part of the lemma is proved in Aumann and Serrano (2008). Here we prove only the second part.

For $\gamma \ge 0$, set $f(\gamma )=E {\hat{s}}^{1-\gamma }/(1-\gamma )$ and $f_*(\gamma )=E {\hat{s}}_*^{1-\gamma }/(1-\gamma )$. Let $s$ and $s_*$ be two securities whose relative returns are ${\hat{s}}$ and ${\hat{s}}_*$, respectively. If $s$ first-order relatively dominates $s_*$, then $f(\gamma )<f_*(\gamma )$ whenever $\gamma >1$. It follows that the unique positive root of $f_*=1$ is less than that of $f=1$, an so $R(s_*)>R(s)$.

If $s$ second-order relatively dominates $s_*$, then $f(\gamma )<f_*(\gamma )$, too, because of the strict convexity of $x^{1-\gamma }/(1-\gamma )$ as a function of $x$ for all $x>0$. The remainder of the proof is as before. $\square$

Proof of Lemma 4.2

Let ${\hat{s}}=[x_1,p_1; x_2,p_2;\ldots ;x_n,p_n]$ be the relative return of a security $s$. For convenience, we denote $\epsilon ={\hat{s}}-1$, where $\epsilon _i=x_i-1$. We have to show that for any $0<\alpha <1,\,R(s)>R(s(\alpha ))$. That is the result of the following lemma (whose claim is a bit stronger). $\square$

Lemma 7.7

Every agent who R-accepts $s$ would R-accept $s(\alpha )$ for all $0<\alpha <1$.

Proof

By definition, for every concave function $u$ and two different numbers $x$ and $y$,

$$\begin{aligned} u(\alpha x+(1-\alpha )y)> \alpha u(x)+(1-\alpha )u(y). \end{aligned}$$

Submitting $x=w+w\epsilon$ and $y=w$, we get

$$\begin{aligned} u(w+\alpha w\epsilon )> \alpha u(w+w\epsilon )+(1-\alpha )u(w), \end{aligned}$$

and so,

$$\begin{aligned} E u(w+\alpha w\epsilon )> \alpha E u(w+w\epsilon )+(1-\alpha )u(w). \end{aligned}$$

If an agent with utility $u$ and wealth $w$ R-accepts $s$, then $Eu(w+w\epsilon )>u(w)$ implies that $Eu(w+\alpha w\epsilon )>u(w)$, which means that the agent R-accepts $s(\alpha )$. In terms of Hart (2011), $s(\alpha )$ acceptance dominates $s$. $\square$

Now, if anyone who R-accepts $s$ also R-accepts $k$ (but not vice versa), then $R(s)>R(k)$; otherwise, there would have been a CRRA agent who would R-accept $s$ but R-reject $k$. $\square$

Proof of the Portfolio Diversification Property

The proof of Eq. (9) follows from the subadditivity of $A$ (see Equation 5.8.2 in Aumann and Serrano 2008), which implies that

$$\begin{aligned} A(a_\alpha (h,k))\le \alpha A(h)+(1-\alpha ) A(k), \end{aligned}$$

and equality obtains if and only if the absolute return of $h$ is a positive multiple of the absolute return of $k$. If $A(h)\ne A(k)$ then $\alpha A(h)+(1-\alpha ) A(k)< \max (A(h),A(k))$, and it follows that unless the returns of $k$ and $h$ are equal, $A(a_\alpha (h,k))< \max (A(h),A(k))$.

To prove Eq. (10), let $\hat{h}=h_1/h_0$ and $\hat{k}=k_1/k_0$ be the relative returns of $h$ and $k$, respectively, and assume that $\hat{h}$ and $\hat{k}$ are not equal. Without loss of generality, assume that $R(h)\ge R(k)$. From the convexity of the function $f(x)=x^{(-1/R(h))}$, we get

$$\begin{aligned} E\left( \alpha \hat{h}+(1-\alpha ) \hat{k}\right) ^{-1/R(h)}< \alpha E \hat{h}^{-1/R(h)}+(1-\alpha )E\hat{k}^{-1/R(h)}\le 1, \end{aligned}$$

where the last inequality follows from Lemma 7.3. Hence, it follows from the same lemma and from the convexity of $f(x)$ that unless the returns of $h$ and $k$ are equal, $R(r_\alpha (h,k))< R(h)$.

Note that while it is shown that $A$ is convex, it follows from the proof that $R$ is quasiconvex. Many of our tests further indicate that $R$ is also convex, but at this stage we do not have formal proof of this. Therefore, the convexity of $R$ remains a conjecture. $\square$

Proof of Theorem 4

Here we prove only the second part of the theorem; for the proof of the first part see Hart (2011).

Let $r$ and $k$ be two securities such that $R(k)>R(r)$. If agent $i$ R-rejects $r$ at any wealth level, then it follows from Lemma 3.3 that for any $w_0>0$ there is $w_0'\in (w_0\min r,w_0\max r)$ for which $\varrho _i(w_0')\ge 1/R(r)+1$. Since IRRA is assumed, it follows that for all $w>0,\,\varrho _i(w)>1/R(r)+1$. That implies that for all $w>0\,\varrho _i(w)>1/R(k)+1$. So it follows from 3.3 that $k$ is R-rejected at all $w$.

The opposite direction is proved as follows. Assume that $r$ wealth uniformly dominates $k$ but that $R(r)>R(k)$. Let $x=(1/(-1+R(r))+1/(-1+R(k)))/2$. According to Lemma 3.2, a CRRA agent with parameter $x$ R-rejects $r$ but R-accepts $k$ at any $w>0$, a contradiction. $\square$