
From the discussion on riskaversion in the Basic Concepts section, we recall that a consumer with a von NeumannMorgenstern utility function can be one of the following:
 Riskaverse, with a concave utility function;
 Riskneutral, with a linear utility function, or;
 Riskloving, with a convex utility function.

Knowing this, it seems logical that the degree of riskaversion 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 widelyused measure of riskaversion called, unsurprisingly, the ArrowPratt measure of riskaversion.

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 riskaversion? 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 riskaversion 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 riskaverse 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 riskaverse consumer.
The ArrowPratt measure of riskaversion is therefore = u"(x)/u'(x).

Arrow and Pratt's original measure used wealth as the argument in the Bernoulli function, so for wealth w, the ArrowPratt measure of riskaversion 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 ArrowPratt measure of riskaversion is u"(y)/u'(y).
In fact, the ArrowPratt measure of riskaversion 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 ArrowPratt measure of riskaversion is then u_{yy}(w,y)/u_{y}(w,y). Here, u_{yy}(w,y) refers to the secondorder partial derivative of the Bernoulli utility function with respect to income, and u_{y}(w,y) refers to the firstorder partial derivative with respect to income.
For a discussion of experiments testing risk aversion, see the riskaversion section under Experiments.

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 riskaversion.
If we want to measure the percentage of wealth held in risky assets, for a given wealth level w, we simply multiply the Arrowpratt measure of absolute riskaversion by the wealth w, to get a measure of relative riskaversion, i.e.:
The ArrowPratt measure of relative riskaversion is = [w * u"(w)]/u'(w).
We can also classify the type of riskaversion within these two main categories.

Type of RiskAversion  Description  Example of Bernoulli Function 
Increasing absolute riskaversion  As wealth increases, hold fewer dollars in risky assets  w^{cw2} 
Constant absolute riskaversion  As wealth increases, hold the same dollar amount in risky assets  e^{cw} 
Decreasing absolute riskaversion  As wealth increases, hold more dollars in risky assets  ln(w) 

Type of RiskAversion  Description  Example of Bernoulli Function 
Increasing relative riskaversion  As wealth increases, hold a smaller percentage of wealth in risky assets  w  cw^{2} 
Constant relative riskaversion  As wealth increases, hold the same percentage of wealth in risky assets  ln(w) 
Decreasing relative riskaversion  As wealth increases, hold a larger percentage of wealth in risky assets  e^{2w1/2} 

 Cox, James C., and Sadiraj, Vjollca (2004), "Implications of Small and LargeStakes 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. 122136.
 Vickrey, William (1961), "Counterspeculation, Auctions, and Competitive Sealed Tenders", The Journal of Finance, Vol. 16, No. 1. (Mar., 1961), pp. 837.
 Vickrey, William (1945): "Measuring Marginal Utility by Reactions to Risk", Econometrica, Vol. 13, No. 4. , pp. 319333.




