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Unformatted text preview: Northwestern University Marciano Siniscalchi Fall 2009 Econ 3310 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 riskaverse 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 socalled equity premium puzzle . It’s interesting in its own right, and it should remind us that our theories rely on many assumptions—and 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 nonnegative 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 δ x for x ∈ X : the probability distribution of any such δ x is simply ( x, 1). We also continue to assume that our individual’s 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 individual’s 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 riskaverse if, for every r.v. X ∈ F , X 4 δ 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, preferencebased ways to define comparisons of riskaversion, 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 nonpositive reals. Formally: u ( X ) = { r ∈ R : u ( x ) = r for some x ∈ X} ....
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 Spring '09
 Marciano
 Economics, rf, Portfolio Choice

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