Risk aversion

The higher the curvature of ${\displaystyle u(c)}$, the higher the risk aversion. However, since expected utility functions are not uniquely defined (are defined only up to affine transformations), a measure that stays constant with respect to these transformations is needed. One such measure is the Arrow–Pratt measure of absolute risk aversion (ARA), after the economists Kenneth Arrow and John W. Pratt,[4][5] also known as the coefficient of absolute risk aversion, defined as

${\displaystyle A(c)=-{\frac {u''(c)}{u'(c)}}}$

where ${\displaystyle u'(c)}$ and ${\displaystyle u''(c)}$ denote the first and second derivatives with respect to ${\displaystyle c}$ of ${\displaystyle u(c)}$.

The following expressions relate to this term:

• Exponential utility of the form ${\displaystyle u(c)=1-e^{-\alpha c}}$ is unique in exhibiting constant absolute risk aversion (CARA): ${\displaystyle A(c)=\alpha }$ is constant with respect to c.
• Hyperbolic absolute risk aversion (HARA) is the most general class of utility functions that are usually used in practice (specifically, CRRA (constant relative risk aversion, see below), CARA (constant absolute risk aversion), and quadratic utility all exhibit HARA and are often used because of their mathematical tractability). A utility function exhibits HARA if its absolute risk aversion is a hyperbolic function, namely

${\displaystyle A(c)=-{\frac {u''(c)}{u'(c)}}={\frac {1}{ac+b}}}$

The solution to this differential equation (omitting additive and multiplicative constant terms, which do not affect the behavior implied by the utility function) is:

${\displaystyle u(c)={\frac {(c-c_{s})^{1-R}}{1-R}}}$

where ${\displaystyle R=1/a}$ and ${\displaystyle c_{s}=-b/a}$. Note that when ${\displaystyle a=0}$, this is CARA, as ${\displaystyle A(c)=1/b=const}$, and when ${\displaystyle b=0}$, this is CRRA (see below), as ${\displaystyle cA(c)=1/a=const}$. See [6]

• Decreasing/increasing absolute risk aversion (DARA/IARA) is present if ${\displaystyle A(c)}$ is decreasing/increasing. Using the above definition of ARA, the following inequality holds for DARA:

${\displaystyle {\frac {\partial A(c)}{\partial c}}=-{\frac {u'(c)u'''(c)-[u''(c)]^{2}}{[u'(c)]^{2}}}<0}$

and this can hold only if ${\displaystyle u'''(c)>0}$. Therefore, DARA implies that the utility function is positively skewed; that is, ${\displaystyle u'''(c)>0}$.[7] Analogously, IARA can be derived with the opposite directions of inequalities, which permits but does not require a negatively skewed utility function (${\displaystyle u'''(c)<0}$). An example of a DARA utility function is ${\displaystyle u(c)=\log(c)}$, with ${\displaystyle A(c)=1/c}$, while ${\displaystyle u(c)=c-\alpha c^{2},}$ ${\displaystyle \alpha >0}$, with ${\displaystyle A(c)=2\alpha /(1-2\alpha c)}$ would represent a quadratic utility function exhibiting IARA.

• Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.[8]
• Contrary to what several empirical studies have assumed, wealth is not a good proxy for risk aversion when studying risk sharing in a principal-agent setting. Although ${\displaystyle A(c)=-{\frac {u''(c)}{u'(c)}}}$ is monotonic in wealth under either DARA or IARA and constant in wealth under CARA, tests of contractual risk sharing relying on wealth as a proxy for absolute risk aversion are usually not identified.[9]

The Arrow-Pratt measure of relative risk aversion (RRA) or coefficient of relative risk aversion is defined as[10]

${\displaystyle R(c)=cA(c)={\frac {-cu''(c)}{u'(c)}}}$.

Like for absolute risk aversion, the corresponding terms constant relative risk aversion (CRRA) and decreasing/increasing relative risk aversion (DRRA/IRRA) are used. This measure has the advantage that it is still a valid measure of risk aversion, even if the utility function changes from risk averse to risk loving as c varies, i.e. utility is not strictly convex/concave over all c. A constant RRA implies a decreasing ARA, but the reverse is not always true. As a specific example of constant relative risk aversion, the utility function ${\displaystyle u(c)=\log(c)}$ implies RRA = 1.

In intertemporal choice problems, the elasticity of intertemporal substitution often cannot be disentangled from the coefficient of relative risk aversion. The isoelastic utility function

${\displaystyle u(c)={\frac {c^{1-\rho }-1}{1-\rho }}}$

exhibits constant relative risk aversion with ${\displaystyle R(c)=\rho }$ and the elasticity of intertemporal substitution ${\displaystyle \varepsilon _{u(c)}=1/\rho }$. When ${\displaystyle \rho =1,}$ using l'Hôpital's rule shows that this simplifies to the case of log utility, u(c) = log c, and the income effect and substitution effect on saving exactly offset.

A time-varying relative risk aversion can be considered.[11]

The most straightforward implications of increasing or decreasing absolute or relative risk aversion, and the ones that motivate a focus on these concepts, occur in the context of forming a portfolio with one risky asset and one risk-free asset.[4][5] If the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the number of dollars of the risky asset held in the portfolio if absolute risk aversion is decreasing (or constant, or increasing). Thus economists avoid using utility functions such as the quadratic, which exhibit increasing absolute risk aversion, because they have an unrealistic behavioral implication.

Similarly, if the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the fraction of the portfolio held in the risky asset if relative risk aversion is decreasing (or constant, or increasing).

In one model in monetary economics, an increase in relative risk aversion increases the impact of households' money holdings on the overall economy. In other words, the more the relative risk aversion increases, the more money demand shocks will impact the economy.[12]

In modern portfolio theory, risk aversion is measured as the additional expected reward an investor requires to accept additional risk. Here risk is measured as the standard deviation of the return on investment, i.e. the square root of its variance. In advanced portfolio theory, different kinds of risk are taken into consideration. They are measured as the n-th root of the n-th central moment. The symbol used for risk aversion is A or A_n.

${\displaystyle A={\frac {dE(c)}{d\sigma }}}$

${\displaystyle A_{n}={\frac {dE(c)}{d{\sqrt[{n}]{\mu _{n}}}}}}$

Children's services such as schools and playgrounds have become the focus of much risk-averse planning, meaning that children are often prevented from benefiting from activities that they would otherwise have had. Many playgrounds have been fitted with impact-absorbing matting surfaces. However, these are only designed to save children from death in the case of direct falls on their heads and do not achieve their main goals.[21] They are expensive, meaning that less resources are available to benefit users in other ways (such as building a playground closer to the child's home, reducing the risk of a road traffic accident on the way to it), and—some argue—children may attempt more dangerous acts, with confidence in the artificial surface. Shiela Sage, an early years school advisor, observes "Children who are only ever kept in very safe places, are not the ones who are able to solve problems for themselves. Children need to have a certain amount of risk taking ... so they'll know how to get out of situations."[22]

A vaccine to protect children against the three common diseases measles, mumps and rubella was developed and recommended for all children in several countries including the UK. However, a controversy arose around fraudulent allegations that it caused autism. This alleged causal link was disproved by Madsen et al.,[23] and the doctor who made the claims was expelled from the General Medical Council. However, years after the claims were disproved, some parents wanted to avert the risk of causing autism in their own children and instead chose to spend significant amounts of their own money on alternatives from private doctors. These alternatives carried their own risks which were not balanced fairly, most often that the children were not properly immunized against the more common diseases of measles, mumps and rubella.

Mobile phones may carry some small[24][25] health risk. While most people would accept that unproven risk to gain the benefit of improved communication, others remain so risk averse that they do not. (The COSMOS cohort study continues to study the actual risks of mobile phones.)

One experimental study with student-subject playing the game of the TV show Deal or No Deal finds that people are more risk averse in the limelight than in the anonymity of a typical behavioral laboratory. In the laboratory treatments, subjects made decisions in a standard, computerized laboratory setting as typically employed in behavioral experiments. In the limelight treatments, subjects made their choices in a simulated game show environment, which included a live audience, a game show host, and video cameras.[26] In line with this, studies on investor behavior find that investors trade more and more speculatively after switching from phone-based to online trading[27][28] and that investors tend to keep their core investments with traditional brokers and use a small fraction of their wealth to speculate online.[29]