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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 Measures of risk aversion, 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 xu xwe'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. 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.
So we simply change the sign, so that a larger number indicates a more risk-averse consumer. Risk-aversion measure of what? 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.
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 Sadirajworking 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.
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.
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.
We can also classify the type of risk-aversion within these two main categories.Get Full Text in PDF. Table of Contents.
Introduction; Tools and Measures; Measures of National Income; Need for New Theory; Measures and Indicators; Characteristics of a Successful Indicator. this paper will highlight some of the most pertinent issues that need to be addressed when competing in the international business environment pertaining to risk management.
1 Introduction Keywords: Risk aversion, Arrow-Pratt risk aversion, multivariate risk aversion, comparative risk aversion.
Behavior under uncertainty and measurement of . 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.
It is sometimes important to know how averse to risk a certain individual is. To this effect there are a set of tools to measure risk in a quantitative way.
The most common and frequently used measure of risk aversion are the Arrow-Pratt measures of absolute and relative risk-aversion. The general level of risk aversion in the markets can be seen in two ways: by the risk premium assessed on assets above the risk-free level and by the actual pricing of risk-free assets, such as.