Restricted increases in risk aversion and their application

Abstract

This paper proposes additional definitions of what it means for one decision maker to be more risk averse than another. These definitions build on the strongly more risk averse definition presented by Ross (Econometrica 49:621–663, 1981). Using examples from portfolio choice, self-protection and insurance demand, it is shown that these definitions of increased risk aversion facilitate clear-cut comparative statics analysis in decision models where traditional concepts of increased risk aversion are insufficient.

Appendix

1.1 Ross more risk aversion is not sufficient for proposition 4

Let v(x) be a quadratic utility function that is increasing and concave in the relevant interval, and let $u(x)\equiv kv(x)+\phi (x)$, where ${k} >0$ and $\phi (x)$ is a chosen cubic function with ${\phi }'(x)\le 0,\,{\phi }''(x)\le 0$ and ${\phi }'''(x)>0$ in the relevant interval. Further assume that v(x), k and $\phi (x)$ are chosen in such a way that u(x) is still increasing in the relevant interval (the concavity of u(x) is guaranteed due to ${\phi }''(x)\le 0)$. From Ross (1981), it is the case that this u(x) is Ross more risk averse than v(x).

Let $\tilde{x}_1$ and $\tilde{x}_2$ be independent, with $E\tilde{x}_1 =E\tilde{x}_2 =\mu ,\,\hbox {Var}(\tilde{x}_1 )=\hbox {Var}(\tilde{x}_2 )=\sigma ^{2}$ and $E(\tilde{x}_1^3 )>E(\tilde{x}_2^3 )$. Because v(x) is quadratic and risk averse, ${Ev} [\lambda \tilde{x}_1 +(1-\lambda )\tilde{x}_2 ]$ is maximized at $\lambda _v =\frac{1}{2}$.

Therefore, although u(x) is Ross more risk averse than v(x), the optimally chosen portfolio for u(x), represented by $\lambda _u$, cannot be closer to perfect diversification than that chosen by v(x). Indeed, we can show that $\lambda _u >\frac{1}{2}$. To see this, note that given the requirements for the construction of u(x) we can express $u(x)=ax^{3}+bx^{2}+cx+d$ with $a>0$. We prove $\lambda _u >\frac{1}{2}$ by showing that $d\left\{ { {Eu} [\lambda \tilde{x}_1 +(1-\lambda )\tilde{x}_2 ]} \right\} /d\lambda >0$ at $\lambda =\frac{1}{2}$. Indeed,

$$\begin{aligned} \frac{\hbox {d}\left\{ {{Eu} [\lambda \tilde{x}_1 +(1-\lambda )\tilde{x}_2 ]} \right\} }{\hbox {d}\lambda }= & {} 3aE\left[ {\left( {\lambda \tilde{x}_1 +(1-\lambda )\tilde{x}_2 } \right) (\tilde{x}_1 -\tilde{x}_2 )} \right] \\&+2bE\left[ {\left( {\lambda \tilde{x}_1 +(1-\lambda ) \tilde{x}_2 } \right) (\tilde{x}_1 -\tilde{x}_2 )} \right] + cE(\tilde{x}_1 -\tilde{x}_2 ). \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\hbox {d}\left\{ {{Eu}[\lambda \tilde{x}_1 +(1-\lambda )\tilde{x}_2 ]} \right\} }{\hbox {d}\lambda }\left| {_{\lambda =\frac{1}{2}} } \right.= & {} \frac{3a}{4}E\left[ {(\tilde{x}_1 +\tilde{x}_2 )^{2}(\tilde{x}_1 -\tilde{x}_2 )} \right] \\= & {} \frac{3a}{4}E(\tilde{x}_1^3 -\tilde{x}_2^3 )>0. \end{aligned}$$

1.2 Ross more risk aversion and insurance with default risk (Sect. 3.3)

We follow Schlesinger (2000) to assume fair pricing ($\lambda =0$) and an interior solution. From (6), the optimal coverage for u(x), denoted by $\alpha _u$, satisfies the equation

$$\begin{aligned} \frac{\mathrm{d}Eu\left( {\tilde{w}(\alpha )} \right) }{\mathrm{d}\alpha }=pqL\left\{ {-(1-p){u}'\left( {Y_1 } \right) +(1-pq){u}'\left( {Y_2 } \right) -p(1-q){u}'\left( {Y_3 } \right) } \right\} =0, \end{aligned}$$

where

$$\begin{aligned} Y_1= & {} w_0 -\alpha pqL \\ Y_2= & {} w_0 -\alpha pqL+(\alpha -1)L \\ Y_3= & {} w_0 -\alpha pqL-L \end{aligned}$$

Now choose an arbitrary $\phi (x)$ such that ${\phi }'(x)\le 0$ and ${\phi }''(x)\le 0$, and let $v(x)\equiv u(x)+\phi (x)$. We have that v(x) is Ross more risk averse than u(x) (Ross 1981). Then consider

$$\begin{aligned} \frac{\hbox {d}Ev\left( {\tilde{w}(\alpha )} \right) }{\hbox {d}\alpha }\left| {_{\alpha _u } } \right. =pqL\left\{ {-(1-p){\phi }'\left( {Y_1 } \right) +(1-pq){\phi }'\left( {Y_2 } \right) -p(1-q){\phi }'\left( {Y_3 } \right) } \right\} . \end{aligned}$$

Although we know that ${\phi }'\left( {Y_1 } \right) , {\phi }'\left( {Y_2 } \right) ,\hbox { and }{\phi }'\left( {Y_3 } \right)$ are all nonpositive and that ${\phi }'\left( {Y_1 } \right) \le {\phi }'\left( {Y_2 } \right) \le {\phi }'\left( {Y_3 } \right)$, these conditions are not sufficient for signing $\frac{\hbox {d}Ev\left( {\tilde{w}(\alpha )} \right) }{\hbox {d}\alpha }\left| {_{\alpha _u } } \right.$. Indeed, $\frac{\hbox {d}Ev\left( {\tilde{w}(\alpha )} \right) }{\hbox {d}\alpha }\left| {_{\alpha _u } } \right.$ can be of any sign due to the arbitrariness of $\phi (x)$.

Therefore, the Ross more risk aversion does not necessarily imply more coverage in the insurance model with default risk.

1.3 Proof of Lemma 2

Lemma 2

If a risk averse individual is also prudent (i.e., ${u}'''(x)>0$), then the optimally chosen coverage satisfies $\alpha ^ *<\frac{1-p}{1-pq}<1$.

Proof

From (6), it is readily seen that, for a risk averse $u(x),\,\frac{\hbox {d}^{2}Eu\left( {\tilde{w}(\alpha )} \right) }{\hbox {d}\alpha ^{2}}<0$. Therefore, if $\frac{\hbox {d}Eu\left( {\tilde{w}(\alpha )} \right) }{\hbox {d}\alpha }<0$ at some $\bar{{\alpha }}>0$, we would know that $\alpha ^*<\bar{{\alpha }}$.²¹

Now we complete the proof by showing that $\frac{\hbox {d}Eu\left( {\tilde{w}(\alpha )} \right) }{\hbox {d}\alpha }<0$ at $\bar{{\alpha }}=\frac{1-p}{1-pq}<1$. Indeed,

let

$$\begin{aligned} w_3= & {} w_0 -(1+\lambda )\bar{{\alpha }}pqL \\ w_2= & {} w_0 -(1+\lambda )\bar{{\alpha }}pqL+(\bar{{\alpha }}-1)L \\ w_1= & {} w_0 -(1+\lambda )\bar{{\alpha }}pqL-L. \end{aligned}$$

Because $w_3>w_2 >w_1$ and ${u}'''(x)>0$, we have from the intermediate value theorem that

$$\begin{aligned} \frac{{u}'(w_3 )-{u}'(w_2 )}{w_3 -w_2 }>\frac{{u}'(w_2 )-{u}'(w_1 )}{w_2 -w_1 }, \end{aligned}$$

which implies that

$$\begin{aligned}&\bar{{\alpha }}{u}'\left( {w_0 -(1+\lambda )\bar{{\alpha }}pqL} \right) -{u}'\left( {w_0 -(1+\lambda )\bar{{\alpha }}pqL+(\bar{{\alpha }}-1)L} \right) \\&\quad +(1-\bar{{\alpha }}){u}'\left( {w_0 -(1+\lambda )\bar{{\alpha }}pqL-L} \right) >0 \end{aligned}$$

or, replacing some $\bar{{\alpha }}$ in the above inequality with $\frac{1-p}{1-pq}$ and multiplying through by $-(1+\lambda )(1-pq)$,

$$\begin{aligned}&-(1+\lambda )(1-p){u}'\left( {w_0 -(1+\lambda )\bar{{\alpha }}pqL} \right) \nonumber \\&+(1+\lambda )(1-pq){u}'\left( {w_0 -(1+\lambda )\bar{{\alpha }}pqL+(\bar{{\alpha }}-1)L} \right) \nonumber \\&\quad -(1+\lambda )p(1-q){u}'\left( {w_0 -(1+\lambda )\bar{{\alpha }}pqL-L} \right) <0. \end{aligned}$$

(7)

From (6), and using (7) and noting $\lambda \ge 0$, we have $\frac{dEu\left( {\tilde{w}(\alpha )} \right) }{d\alpha }<0$ at $\bar{{\alpha }}=\frac{1-p}{1-pq}<1$. Therefore, $\alpha ^*<\frac{1-p}{1-pq}<1$. $\square$