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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 +˜....
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online