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Unformatted text preview: Northwestern University Marciano Siniscalchi Fall 2009 Econ 3310 MORE ON THE ARROWPRATT MEASURE(S) OF RISK AVERSION 1. Introduction Recall that, when comparing risk attitudes, the ArrowPratt measure of (absolute) risk aversion , A ( w ) = u 00 ( w ) u ( w ) plays an important role. We proved that an individual is more riskaverse than another if the former’s ArrowPratt measure is greater than the latter’s for every wealth level w . Although this was not stated explicitly, it is also evident that an individual is riskaverse if and only if A ( w ) ≥ for all w . This follows from our characterization of risk aversion in terms of the concavity of the utility function u : since u > 0 because we assume that utility is strictly increasing (more money is strictly better), it follows that A ( w ) ≥ 0 if and only if u 00 ≤ 0, i.e. if and only if u is concave, i.e. if and only if the DM is riskaverse. This lecture delves deeper into the ArrowPratt measure. First, we show that it also appears in a related, but essentially independent analysis of risk aversion. Second, we introduce the related notion of relative risk aversion. 2. Risk Premium We assume that the set of prizes is X = R , and that our individual’s utility function u is increasing and continuous: our person prefers more to less, and there are no “holes” in her preferences. In fact, we will assume that u is twice continuously differentiable (otherwise the ArrowPratt measure is not even defined!) The notion of risk premium is closely related to that of certainty equivalent. Suppose that an investor is subject to a “risk”: formally, her initial wealth W is augmented or reduced according to the realization of a random variable ˜ with probability distribution ( 1 ,p 1 ; ... ; n ,p n ), such that E[˜] = 0. [we could generalize this to arbitrary cdf’s, but we won’t bother to do so here]. Then her final wealth is the random variable ( W + 1 ,p 1 ; ... ; W + n ,p n ) which can of course we viewed simply as the probability distribution of the random variable W +˜....
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 Spring '09
 Marciano
 Economics

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