Pratt, John W. “Risk Aversion in the Small and in the Large.” Econometrica 32.1/2 (1964): 122-136.

For the utility functions for money, a local risk measure is defined. (equation (3))

• The risk premium can be locally represented using the local risk aversion. (equation (2))
• This measure is closely related to global risk-averse preferences.

Definitions and notations

• $u_{1}(x)\sim u_{2}(x)$: two functions are equivalent as utilities (up to increasing linear transformation)
• $\pi({x,\tilde{z}})$: the risk premium (elaborated below)

The risk premium $\pi$

Given assets $x$ and utility function $u$, a decision maker is indifferent between receiving a risk $\tilde{z}$ and the non-random amount $E[\tilde{z}]-\pi$. Mathematically, \(u(x+E(\tilde{z})-\pi(x,\tilde{z}))=E[u(x+\tilde{z})].\tag{1}\)

• From equation (1) the risk premium is uniquely defined.
• It follows from (1) that for any constant $\mu$, $\pi(x,\tilde{z})=\pi(x+\mu,\tilde{z}-\mu)$. Therefore, we may only consider any actuarially neutral risk.

There are other related concepts like the cash equivalent and insurance premium. These concepts should be distinguished from the bid price.

Local risk aversion

Consider $\pi(x,\tilde{z})$ for an actuarially neutral risk $\tilde{z}$ with infinitesimal variance $\sigma_z^2{\rightarrow}0$. Locally expand equation (1) on both sides shows \(\begin{equation} \begin{split} u(x-\pi)&=u(x)-\pi u'(x) + O(\pi^2)\\ E[u(x+\tilde{z})]&=E[u(x)+\tilde{z}u'(x)+\frac{1}{2}\tilde{z}^2u''(x)+O(\tilde{z}^3)]\\ &=u(x)+\frac{1}{2}\sigma_z^2u''(x)+o(\sigma_z^2) \end{split} \end{equation}\)

Setting equal these equations gives \(\pi(x,\tilde{z})=\frac{1}{2}\sigma_z^{2}r(x)+ o(\sigma_z^2),\tag{2}\) where \(r(x)=-\frac{u''(x)}{u(x)}=-\frac{d}{dx}\log u'(x). \tag{3}\)

• There is a similar interpretation with discrete risks and probability premium $p(x,h)=p(\tilde{z}=h)-p(\tilde{z}=-h)$.

Relation with utility function

The local risk aversion function $r$ associated with any utility function $u$ contains all essntial information about $u$.

Equation (3) implies \(u\sim\int e^{-\int r} \tag{4}\)

• It can be convenient to preserve $u$ since it determines ordinary (as against infinitesimal) risk preferences.

Concavity

$u$’’$(x)$ is not in itself a meaningful measure of concavity in utility theory

• the sign of $u$’’$(x)$ implies general attitude towards risk
• the absolute magnitude is not meaningful

Comparative (global) risk aversion

If $r_1(x)>r_2(x)$, then $u_1$ is more risk-averse than $u_2$ at $x$ not only locally, but also globally. Thm. Let $r_i(x),\pi_i(x)$ be the local risk aversion and risk premium according to the utility function $u_i,\,i=1,2$. Then the following conditions are equivalent:

1. $r_{1}(x)\geq r_2(x)$ for all $x$
2. $\pi_1(x,\tilde{z})\geq \pi_2(x,\tilde{z})$ for all $x$ and $\tilde{z}$
3. $u_1(u_2^{-1}(t))$ is a concave function of $t$
4. $\frac{u_1(y)-u_1(x)}{u_1(w)-u_{1(v)}} \leq \frac{u_2(y)-u_2(x)}{u_2(w)-u_2(v)}$ for all $v<w\leq x<y$.

• condition 1. requires local risk-aversion to be larger for any asset.

Special family of risk aversion

Constant risk aversion

If the local risk aversion is constant $r(x)=c$, then

• If the risk aversion is constant locally, then it is also constant globally. For any $k,\,u(x+k)\sim u(x)$.

Decreasing risk aversion

Decreasing risk aversion describes a decision maker, who (1) attaches positive risk premium (risk-averse) to any risk, but (2) attaches smaller risk premium the greater his assets $x$. Formally,

1. $\pi(x,\tilde{z})>0$ for all $x$ and $\tilde{z}$
2. $\pi(x,\tilde{z})$ is a strictly decreasing function of $x$ for all (given) $\tilde{z}$

Decreasing global risk aversion is equivalent to decreasing local risk aversion,i.e. the following conditions are equivalent

1. The local risk aversion $r(x)$ is decreasing
2. The risk premium $\pi(x,\tilde{z})$ is a decreasing function of $x$ for all $\tilde{z}$