## Measuring Risk-AversionFrom the discussion on risk-aversion in the Basic Concepts section, we recall that a consumer with a von Neumann-Morgenstern utility function can be one of the following: - Risk-averse, with a concave utility function;
- Risk-neutral, with a linear utility function, or;
- Risk-loving, with a convex utility function.
Knowing this, it seems logical that the degree of risk-aversion a consumer displays would be related to the Simple - using the function's second derivative. For a Bernoulli utility function over wealth, income, (or in fact any commodity x), u(x), we'll represent the second derivative by u"(x). A linear function has a second derivative of zero, a concave function has a negative second derivative, and a convex function has a positive second derivative. Using these facts, Kenneth Arrow and John Pratt developed a widely-used measure of risk-aversion called, unsurprisingly, the Arrow-Pratt measure of risk-aversion. ## The Arrow-Pratt Measure of Risk-aversionIf all the information we need about the curvature of a function is contained in its second derivative, shouldn't that be a sufficient measure of risk-aversion? Well, as it turns out, it isn't - reason being, it is not invariant to positive linear transformations of the utility function. Invariance to an affine transformation is an essential property of the VNM utility function. Given this, Arrow and Pratt had to design a measure of risk-aversion that would remain the same even after an affine transformation of the utility function. The easiest way to do this is to divide the second derivative by the first derivative, i.e. obtaining u"(x)/u'(x). However, this would give us a negative number as a risk-averse person's measure. (Note that any utility funtion must be increasing in its argument, i.e. wealth, and must have a positive first derivative - this comes from the property of monotonicity.) So we simply change the sign, so that a larger number indicates a more risk-averse consumer. The Arrow-Pratt measure of risk-aversion is therefore = -u"(x)/u'(x). ## Risk-aversion measure of |

Type of Risk-Aversion | Description | Example of Bernoulli Function |
---|---|---|

Increasing absolute risk-aversion | As wealth increases, hold fewer dollars in risky assets | w^{-cw2} |

Constant absolute risk-aversion | As wealth increases, hold the same dollar amount in risky assets | -e^{-cw} |

Decreasing absolute risk-aversion | As wealth increases, hold more dollars in risky assets | ln(w) |

Type of Risk-Aversion | Description | Example of Bernoulli Function |
---|---|---|

Increasing relative risk-aversion | As wealth increases, hold a smaller percentage of wealth in risky assets | w - cw^{2} |

Constant relative risk-aversion | As wealth increases, hold the same percentage of wealth in risky assets | ln(w) |

Decreasing relative risk-aversion | As wealth increases, hold a larger percentage of wealth in risky assets | -e^{2w-1/2} |

- Cox, James C., and Sadiraj, Vjollca (2004), "Implications of Small- and Large-Stakes Risk Aversion for Decision Theory", working paper Elton,
- Edwin J., and Gruber, Martin J.,
*Modern Portfolio Theory and Investment Analysis*, (New York: John Wiley & Sons, 2001, 5th edition) - Pratt, John W. (1964), "Risk Aversion in the Small and in the Large",
*Econometrica*, Vol. 32, No. 1/2., pp. 122-136. - Vickrey, William (1961), "Counterspeculation, Auctions, and Competitive Sealed Tenders",
*The Journal of Finance*, Vol. 16, No. 1. (Mar., 1961), pp. 8-37. - Vickrey, William (1945): "Measuring Marginal Utility by Reactions to Risk",
*Econometrica*, Vol. 13, No. 4. , pp. 319-333.

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