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Unformatted text preview: Chapter 12 Risky Asset In this chapter we provide an alternative and simplified model of behavior under uncertainty. 12.1 Mean-Variance Utility In the last chapter we examined the expected utility model of choice under uncertainty. Another approach to choice under uncertainty is to describe a consumers preference by considering just a few summary statistics about the probability distribution of the consumer’s wealth. Let us suppose a random variable w takes on values w s , for s = 1 ,...,S with probability π s . Then the mean of this probability distribution is μ w = S X s =1 π s w s The variance of the probability distribution is σ 2 w = S X s =1 π s ( w s- μ w ) 2 The mean-variance model assumes that the utility of a probability dis- tribution that gives the investor wealth w s with probability of π s can be 1 2 CHAPTER 12. RISKY ASSET expressed as a function of the mean and variance of that distribution, u ( μ w ,σ 2 w ), or if you prefer, u ( μ w ,σ w ), where we use the standard devia- tion rather than the variance to measure the “spread” of the distribution. The “spread” of the distribution is a measure of riskiness of the wealth distribution. Other things equal, a higher expected return is good, while a larger variance is bad. This is just another way to state the assumption that people are typically averse to risk....
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