eco331-07-ArrowPratt

eco331-07-ArrowPratt - Northwestern University Marciano...

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Unformatted text preview: Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 MORE ON THE ARROW-PRATT MEASURE(S) OF RISK AVERSION 1. Introduction Recall that, when comparing risk attitudes, the Arrow-Pratt measure of (absolute) risk aversion , A ( w ) =- u 00 ( w ) u ( w ) plays an important role. We proved that an individual is more risk-averse than another if the former’s Arrow-Pratt 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 risk-averse 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 risk-averse. This lecture delves deeper into the Arrow-Pratt 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 Arrow-Pratt 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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eco331-07-ArrowPratt - Northwestern University Marciano...

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