eco331-06-CompRAandPortfolioChoice

eco331-06-CompRAandPortfolioChoice - Northwestern...

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Unformatted text preview: Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 RISK AVERSION AND PORTFOLIO CHOICE 1. Introduction We continue our analysis of choice under risk. The main focus of this lecture is on comparing risk attitudes of different individuals: when can we say that one person is more risk-averse than another? As usual, we start with a simple, behavioral definition, and then provide convenient analytical characterizations. We then turn to an application of EU theory, portfolio choice. We shall introduce the basic setup and apply our results on comparing risk attitudes to investment decisions. Finally, we briefly analyze the so-called equity premium puzzle . Its interesting in its own right, and it should remind us that our theories rely on many assumptionsand many things can go wrong in practice. 2. Comparing Risk Aversion We continue to assume that the set of prizes is some set of real numbers; in particular, most often we will assume either X = R or R + (the non-negative reals); we restrict attention to the collection F of random variables that take up finitely many values in X , and hence have a finite probability distribution. In particular, recall that F contains all degenerate r.v.s, denoted x for x X : the probability distribution of any such x is simply ( x, 1). We also continue to assume that our individuals preferences are consistent with EU, with Bernoulli utility u : that is, for all X,Y F , X < Y iff E[ u ( X )] E[ u ( Y )]. Furthermore, we assume that the individuals utility function u is increasing and continuous: our person prefers more to less, and there are no holes in her preferences. Recall that we define the certainty equivalent C[ X ] of a r.v. X F as the certain sum such that u (C[ X ]) = E[ u ( X )]. As we noted last time, the existence of certainty equivalents is guaranteed if u is increasing and continuous. Also recall that a decision maker is risk-averse if, for every r.v. X F , X 4 E[ X ] or, equivalently, C[ X ] E[ X ]. Also recall that we proved the following result: a utility function displays risk aversion if and only if it is concave . In light of these observations, the following definition seems reasonable. Definition 1. Let u and v be two continuous and increasing Bernoulli utility functions, and denote by C v [ X ] and C u [ X ] the corresponding certainty equivalents for any X F . Then v is more risk- averse than u if and only if, for all X F , C v [ X ] C u [ X ] . There are other, preference-based ways to define comparisons of risk-aversion, but they reduce to the intuitive condition given above. We now provide simple analytical characterizations of Definition 1. Notation: I will denote by u ( X ) the range of the utility function u the set of values it can take. For instance, if u ( x ) =- e- x , then u ( X ) = R- , the non-positive reals. Formally: u ( X ) = { r R : u ( x ) = r for some x X} ....
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This note was uploaded on 11/17/2009 for the course ECONOMICS 331 taught by Professor Marciano during the Spring '09 term at Northwestern.

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eco331-06-CompRAandPortfolioChoice - Northwestern...

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