Arrow Pratt measure

The Arrow-Pratt measure named after Kenneth Arrow and John W. Pratt is a measure of the risk aversion of a decision maker, whereby a distinction must be made between the Arrow-Pratt measure of the absolute risk aversion ARA and that of the relative risk aversion RRA .

The Arrow-Pratt measure of absolute risk aversion

Let be a twice differentiable, monotonically increasing utility function , then the Arrow-Pratt measure of the absolute risk aversion for this utility function is defined as follows: ${\displaystyle u(x)}$ ${\displaystyle {\mathit {A}}RA(x)}$

${\displaystyle {\mathit {ARA}}(x):=-{\frac {u''(x)}{u'(x)}}}$.

Negative values ​​imply risk appetite ( risk affinity ) and positive values ​​risk aversion ( risk aversion ). If the measure finally assumes the value zero, the decision maker is risk-neutral .

A decision maker is more risk averse than another decision maker if, based on their utility functions and their Arrow-Pratt measures, the following applies: ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle u_{i}(x)}$ ${\displaystyle u_{j}(x)}$

${\displaystyle \forall x:{\mathit {ARA}}_{i}(x)\geq {\mathit {ARA}}_{j}(x)}$.

The derivation of the measure of absolute risk aversion ${\displaystyle ARA'(x)}$

${\displaystyle {\mathit {ARA}}'(x)=\left(-{\frac {u''(x)}{u'(x)}}\right)'=-{\frac {u'(x)\cdot u'''(x)-(u''(x))^{2}}{(u'(x))^{2}}}={\mathit {ARA}}^{2}(x)-{\frac {u'''(x)}{u'(x)}}}$

indicates the change in risk attitudes with increased income. If, for example, all possible incomes that can result from the decision-making situation are increased by a constant value, a positive value of the derivation of ARA allows the statement that the decision maker will increase his risk aversion or risk appetite, depending on the value of ARA, a negative one Value that he will be less risk-averse or risk-averse, and a value of zero that increasing all possible incomes will not affect his decision-making behavior.

Selected examples

Utility functions can be differentiated according to whether their ARA is constant, increasing or decreasing.

• CARA (constant absolute risk aversion or English constant absolute risk aversion ):

${\displaystyle u(w)={\frac {e^{-kw}-1}{e^{-k}-1}}\quad {\text{und}}\quad w(u)={\frac {\ln(u(e^{-k}-1)+1)}{-k}}}$ (Function and its inverse functions)

• IARA (increasing absolute risk aversion or English Increasing absolute risk aversion ):

${\displaystyle u(w)=-ax^{2}+bx\quad x<{\frac {b}{2a}}}$

• DARA (decreasing absolute risk aversion or English Decreasing absolute risk aversion ):

${\displaystyle u(w)=\ln x}$

There is also a widely used class of hyperbolic ARA functions (hence HARA), for example:

• HARA (hyperbolic absolute risk aversion): ${\displaystyle u(W)={\frac {1-\gamma }{\gamma }}\left({\frac {aw}{1-\gamma }}+b\right)^{\gamma }}$

The Arrow-Pratt measure of relative risk aversion

The Arrow-Pratt measure of relative risk aversion is calculated as follows:

${\displaystyle {\begin{aligned}{\mathit {RRA}}(x)={\mathit {ARA}}(x)\cdot x=-{\frac {u''(x)}{u'(x)}}\cdot x=-{\frac {\mathrm {d} u'(x)}{\mathrm {d} x}}\cdot {\frac {x}{u'(x)}}=\eta _{u'(x),x}\end{aligned}}}$

It therefore corresponds to the marginal utility elasticity of the possible income, which expresses a change in the willingness to take risks with changed possible income from the decision. If the degree of relative risk aversion is constant, the decision maker will not change his decision if all possible incomes are transformed evenly and linearly. A linear relative risk aversion means a decreasing or increasing risk aversion with an increase in the possible profits, depending on whether the measure of the RRA is negative or positive - the derivation of the RRA provides information on this.

Selected examples

Utility functions with constant relative risk aversion measure and their inverse functions are for example the functions ${\displaystyle {\mathit {RRA}}(x)=1/2}$

• CRRA: ${\displaystyle u(w)=k\cdot {\sqrt {w}}\quad {\text{und}}\quad w(u)={\frac {u^{2}}{k^{2}}}}$
• IRRA: ${\displaystyle u(w)=-e^{-a\cdot x}}$

In the general case of constant relative risk aversion (CRRA) , the isoelastic utility function results . ${\displaystyle {\mathit {RRA}}(x)=\eta }$

meaning

The Arrow-Pratt measures are invariant to a positive linear transformation of the utility function and are therefore suitable for the Neumann / Morgenstern theory .

The risk aversion observed via the Arrow-Pratt measures often decreases with increasing wealth. In the insurance market this means, for example, that wealthy people demand less insurance (or have a lower risk premium). For the capital market , this means that as wealth increases, investors increasingly invest in risky assets.

RRA recognizes that wealth and risk may be related to some extent. This means that in the event that a larger asset is exposed to a greater risk than a smaller asset, the RRA would be the appropriate metric to measure risk aversion.

Individual evidence

1. ^ Doubt, Peter, and Roland Eisen. Insurance economics. Springer Science & Business Media, 2012. p. 49.

literature

• Arrow, Kenneth J .: Essays in the Theory of Risk-Bearing . Amsterdam, 1970, ISBN 072043047X .
• Pratt, John W .: Risk Aversion in the Small and in the Large . Econometrica, Vol. 32, No. 1/2, pp. 122-136, 1966.