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Decision-Making Under Uncertainty - Advanced Topics

Measuring Risk-Aversion

From 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 curvature of their Bernoulli utility function. As a matter of fact, the more "curved" a concave utility function is, the lower will be a consumer's certainty equivalent, and the higher their risk premium - the "flatter" the utility function is, the closer the certainty equivalent will be to the expected value of the gamble, and the smaller the risk premium. The question is, now - how do we measure the amount of curvature of a function?

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-aversion

If 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 what?

Arrow and Pratt's original measure used wealth as the argument in the Bernoulli function, so for wealth w, the Arrow-Pratt measure of risk-aversion is -u"(w)/u'(w). This has, in fact, become the traditional way in which the measure is used. However, it is not the only way, and the expected utility axioms do not specify whether the argument of the utility function should be wealth (a stock) or income (a flow). William Vickrey (1945) used income as the argument of the utility function, so for income y, the Arrow-Pratt measure of risk-aversion is -u"(y)/u'(y).

In fact, the Arrow-Pratt measure of risk-aversion can be even more flexible than that, due to the nature of the VNM utility function. James Cox and Vjollca Sadiraj (2004, working paper) use both income and wealth as arguments for the VNM utility function. In this case, wealth represents the fixed portion of an individuals assets, while income is the portion which is subject to change. An individual's Arrow-Pratt measure of risk-aversion is then -uyy(w,y)/uy(w,y). Here, uyy(w,y) refers to the second-order partial derivative of the Bernoulli utility function with respect to income, and uy(w,y) refers to the first-order partial derivative with respect to income.

For a discussion of experiments testing risk aversion, see the risk-aversion section under Experiments.

Absolute v/s Relative Risk-aversion

In simple terms, what we are measuring above is the actual dollar amount an individual will choose to hold in risky assets, given a certain wealth level w. For this reason, the measure described above is referred to as a measure of absolute risk-aversion.

If we want to measure the percentage of wealth held in risky assets, for a given wealth level w, we simply multiply the Arrow-pratt measure of absolute risk-aversion by the wealth w, to get a measure of relative risk-aversion, i.e.:

The Arrow-Pratt measure of relative risk-aversion is = -[w * u"(w)]/u'(w).
We can also classify the type of risk-aversion within these two main categories.

How Absolute Risk-Aversion Changes with Wealth

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)

How Relative Risk-Aversion Changes with Wealth

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 - cw2
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 -e2w-1/2

References

• 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.