Editing Risk aversion (section)

==<span id="Measures of risk aversion"></span> Measures of risk aversion under expected utility theory==
There are various measures of the risk aversion expressed by those given utility function. Several functional forms often used for utility functions are represented by these measures.

===Absolute risk aversion===
The higher the curvature of <math>u(c)</math>, 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 rather than just the second derivative of <math>u(c)</math>. One such measure is the '''Arrow–Pratt measure of absolute risk aversion''' ('''ARA'''), after the economists [[Kenneth Arrow]] and [[John W. Pratt]],<ref name="Arrow">{{cite book |last=Arrow |first=K. J. |year=1965 |title=The Theory of Risk Aversion|chapter-url=https://books.google.com/books?id=hnNEAAAAIAAJ|chapter=Aspects of the Theory of Risk Bearing |publisher=Yrjo Jahnssonin Saatio |location=Helsinki }} Reprinted in: [https://books.google.com/books?id=KkMoAQAAMAAJ ''Essays in the Theory of Risk Bearing''], Markham Publ. Co., Chicago, 1971, 90–109.</ref><ref name="Pratt">{{cite journal |last1=Pratt |first1=John W. |title=Risk Aversion in the Small and in the Large |journal=Econometrica |date=January 1964 |volume=32 |issue=1/2 |pages=122–136 |doi=10.2307/1913738 |jstor=1913738 }}</ref> also known as the '''coefficient of absolute risk aversion''', defined as

:<math>A(c)=-\frac{u''(c)}{u'(c)}</math>

where <math>u'(c)</math> and <math>u''(c)</math> denote the first and second derivatives with respect to <math>c</math> of <math>u(c)</math>. For example, if <math> u(c)=  \alpha +  \beta ln(c),</math> so <math>u'(c) = \beta/c</math> and <math>u''(c) =  -\beta/c^2,</math> then <math>A(c) = 1/c.</math> Note how <math>A(c)</math> does not depend on <math>\alpha</math> and <math>\beta,</math> so affine transformations of <math>u(c)</math> do not change it.

The following expressions relate to this term:
* [[Exponential utility]] of the form <math>u(c)=1-e^{-\alpha c}</math> is unique in exhibiting ''constant absolute risk aversion'' (CARA): <math>A(c)=\alpha</math> 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 [[hyperbola]], namely

:<math> A(c) = -\frac{u''(c)}{u'(c)}=\frac{1}{ac+b}</math>

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

:<math> u(c) = \frac{(c-c_s)^{1-R}}{1-R}</math>

where <math> R=1/a </math> and <math> c_s = -b/a </math>.
Note that when <math> a = 0 </math>, this is CARA, as <math> A(c) = 1/b = const </math>, and when <math> b=0 </math>, this is CRRA (see below), as <math> c A(c) = 1/a = const </math>.
See <ref>{{cite web |url=http://leeds-faculty.colorado.edu/zender/Fin7330/1-RiskAversion.doc |title=Zender's lecture notes }}</ref>
* ''Decreasing/increasing absolute risk aversion'' (DARA/IARA) is present if <math>A(c)</math> is decreasing/increasing. Using the above definition of ARA, the following inequality holds for DARA:

:<math>\frac{\partial A(c)}{\partial c} = -\frac{u'(c)u'''(c) - [u''(c)]^2}{[u'(c)]^2} < 0</math>

and this can hold only if <math>u'''(c)>0</math>. Therefore, DARA implies that the utility function is positively skewed; that is, <math>u'''(c)>0</math>.<ref>{{cite book |first=Haim |last=Levy |year=2006 |title=Stochastic Dominance: Investment Decision Making under Uncertainty |location=New York |publisher=Springer |edition=2nd |isbn=978-0-387-29302-8 }}</ref> Analogously, IARA can be derived with the opposite directions of inequalities, which permits but does not require a negatively skewed utility function (<math>u'''(c)<0</math>). An example of a DARA utility function is <math>u(c)=\log(c)</math>, with <math> A(c)=1/c</math>, while <math>u(c)=c-\alpha c^2,</math> <math>\alpha >0</math>, with <math>A(c)=2 \alpha/(1-2 \alpha c)</math> would represent a quadratic utility function exhibiting IARA.
*Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.<ref>{{cite journal |last1=Friend |first1=Irwin |last2=Blume |first2=Marshall|author-link2=Marshall E. Blume |year=1975 |title=The Demand for Risky Assets |journal=[[American Economic Review]] |volume=65 |issue=5 |pages=900–922 |jstor=1806628 }}</ref>
* 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 <math>A(c)=-\frac{u''(c)}{u'(c)}</math> 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.<ref>{{cite journal |last1=Bellemare |first1=Marc F. |last2=Brown |first2=Zachary S. |title=On the (Mis)Use of Wealth as a Proxy for Risk Aversion |journal=American Journal of Agricultural Economics |date=January 2010 |volume=92 |issue=1 |pages=273–282 |doi=10.1093/ajae/aap006 |hdl=10161/7006 |s2cid=59290774 |hdl-access=free }}</ref>

=== Relative risk aversion ===
The '''Arrow–Pratt measure of relative risk aversion''' (RRA) or '''coefficient of relative risk aversion''' is defined as<ref name="SB">{{cite book|author1=Simon, Carl and Lawrence Blume|title=Mathematics for Economists|year=2006|publisher=Viva Norton|isbn=978-81-309-1600-2|pages=363|edition=Student}}</ref>
: <math>R(c) = cA(c)=\frac{-cu''(c)}{u'(c)}</math>.

Unlike ARA whose units are in $<sup>−1</sup>, RRA is a dimensionless quantity, which allows it to be applied universally. 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 <math>u(c) = \log(c)</math> implies {{nowrap|1=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
: <math>u(c) = \frac{c^{1-\rho}-1}{1-\rho}</math>
exhibits constant relative risk aversion with <math>R(c) = \rho </math> and the elasticity of intertemporal substitution <math>\varepsilon_{u(c)} = 1/\rho</math>. When <math>\rho = 1,</math> using [[l'Hôpital's rule]] shows that this simplifies to the case of ''log utility'', {{nowrap|1=''u''(''c'') = log ''c''}}, and the [[income effect]] and [[substitution effect]] on saving exactly offset.

A time-varying relative risk aversion can be considered.<ref>{{cite journal |last1=Benchimol |first1=Jonathan |title=Risk aversion in the Eurozone |journal=Research in Economics |date=March 2014 |volume=68 |issue=1 |pages=39–56 |doi=10.1016/j.rie.2013.11.005 |s2cid=153856059 }}</ref>

===Implications of increasing/decreasing absolute and relative risk aversion===

The most straightforward implications of changing risk aversion occur in the context of forming a portfolio with one risky asset and one risk-free asset.<ref name="Arrow" /><ref name="Pratt" /> If an investor experiences an increase in wealth, he/she will choose to decrease the total amount of wealth invested in the risky asset in proportion to absolute risk aversion and will decrease the relative fraction of the portfolio made up of the risky asset in proportion to relative risk aversion. Thus economists avoid using utility functions which exhibit increasing absolute risk aversion, because they have an unrealistic behavioral implication.

In one [[economic model|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.<ref>{{cite journal |last1=Benchimol |first1=Jonathan |last2=Fourçans |first2=André |title=Money and risk in a DSGE framework: A Bayesian application to the Eurozone |journal=Journal of Macroeconomics |date=March 2012 |volume=34 |issue=1 |pages=95–111 |doi=10.1016/j.jmacro.2011.10.003 |s2cid=153669907 }}</ref>

===Portfolio theory===

In [[modern portfolio theory]], risk aversion is measured as the additional expected reward an investor requires to accept additional risk. If an investor is risk-averse, they will invest in multiple uncertain assets, but only when the predicted return on a portfolio that is uncertain is greater than the predicted return on one that is not uncertain will the investor prefer the former.<ref name=":0" /> Here, the [[Risk–return spectrum|risk-return spectrum]] is relevant, as it results largely from this type of risk aversion. 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<sub>n</sub>.

:<math>A = \frac{dE(c)}{d\sigma}</math>

:<math>A_n = \frac{dE(c)}{d\sqrt[n]{\mu_n}} </math>

=== Von Neumann-Morgenstern utility theorem ===
The [[Von Neumann–Morgenstern utility theorem|von Neumann-Morgenstern utility theorem]] is another model used to denote how risk aversion influences an actor’s utility function. An extension of the [[Expected utility hypothesis|expected utility]] function, the von Neumann-Morgenstern model includes risk aversion axiomatically rather than as an additional variable.<ref name="von Neumann 1944">{{Cite book |last1=von Neumann |first1=John |url=https://www.jstor.org/stable/j.ctt1r2gkx |title=Theory of Games and Economic Behavior |last2=Morgenstern |first2=Oskar |last3=Rubinstein |first3=Ariel |date=1944 |publisher=Princeton University Press |jstor=j.ctt1r2gkx |isbn=978-0-691-13061-3 |edition=60th Anniversary Commemorative }}</ref>

[[John von Neumann]] and [[Oskar Morgenstern]] first developed the model in their book ''[[Theory of Games and Economic Behavior|Theory of Games and Economic Behaviour]].''<ref name="von Neumann 1944"/> Essentially, von Neumann and Morgenstern hypothesised that individuals seek to maximise their expected utility rather than the expected monetary value of assets.<ref>{{Cite journal |last=Gerber |first=Anke |date=2020 |title=The Nash Solution as a von Neumann–Morgenstern Utility Function on Bargaining Games |journal=Homo Oeconomicus |language=en |volume=37 |issue=1–2 |pages=87–104 |doi=10.1007/s41412-020-00095-9 |s2cid=256553112 |issn=0943-0180|doi-access=free |hdl=10419/288817 |hdl-access=free }}</ref> In defining expected utility in this sense, the pair developed a function based on preference relations. As such, if an individual’s preferences satisfy four key axioms, then a utility function based on how they weigh different outcomes can be deduced.<ref>{{Cite web |last=Prokop |first=Darren |date=2023 |title=Von Neumann–Morgenstern utility function {{!}} Definition & Facts {{!}} Britannica |url=https://www.britannica.com/topic/von-Neumann-Morgenstern-utility-function |access-date=2023-04-24 |website=www.britannica.com |language=en}}</ref>

In applying this model to risk aversion, the function can be used to show how an individual’s preferences of wins and losses will influence their expected utility function. For example, if a risk-averse individual with $20,000 in savings is given the option to gamble it for $100,000 with a 30% chance of winning, they may still not take the gamble in fear of losing their savings. This does not make sense using the traditional expected utility model however;

<math>EU(A)=0.3($100,000)+0.7($0)</math>

<math>EU(A)=$30,000</math>

<math>EU(A)>$20,000</math>

The von Neumann-Morgenstern model can explain this scenario. Based on preference relations, a specific utility <math>u</math> can be assigned to both outcomes. Now the function becomes;

<math>EU(A)=0.3u($100,000)+0.7u($0)</math>

For a risk averse person, <math>u</math> would equal a value that means that the individual would rather keep their $20,000 in savings than gamble it all to potentially increase their wealth to $100,000. Hence a risk averse individuals’ function would show that;

<math>EU(A)\prec$20,000 (keeping savings)</math>