Post date: Feb 27, 2015 2:11:26 PM
Good afternoon. Today we have finished the first two chapters of the book by Gollier. These two chapters deal with the expected utiiity model and its limits, and the assumption and measurement of risk aversion. What I would like you to keep from the first chapter is that (1) although your instict as statistitians and actuarians is to take the average of the lottery that is representing the risky situation at hand there is enough evidence to think that this is what the average person does. Hence, we need the utility function to capture the decision maker's perceptions and choices. (2) For reasons of practicality, we want such a utility function over lotteries to be equal to the average of the utilities of the outcomes, as paralell to what the mean is as the average of the outcomes. This is what we call the expected utility property. (3) This particular utility function can be inferred by just asking the individual to compare sure amounts of money with lotteries that combine the best and worst outcome. At the probability mix that makes the individual indifferent we will find such a utility (in particular, the probability applied to the best outcome). Any other utility with the expected utility property has to keep the cardinality properties intact, therefore, only multiplying by a positive constant and adding another constant is allowed. (4) When we represent the indifference map of such utility functions (for three outcomes) in what is called the Machina triangle, we observe that the indifference curves when the utilty function satisfies the expected utility property are parallel straight lines. Regarding Chapter 2 we studied the notion of risk aversion, meaning that the individual dislikes pure risks, i.e., situations where the average or mean is equal to zero (but the variance is not). (More) Risk aversion can be measured seen in the (more) concavity of the utilty function, and it can be measured in terms of the Arrow Pratt coefficient of absolute risk aversion or by means of the risk premium. The risk premium is defined as the expected value of the lottery minus the sure amount of money that will make the individual indifferent to the lottery (in terms of the expected utility). Next to the risk premium, we find the certainty equivalent, which is the amount of money that given to the individual with certainty will make him indifferent to the lottery (in terms of the expected utility). The risk premium depends crucially on the variance of the risk, as we saw from the Arrow Pratt approximation, a formula that can be used to approximate the risk premium for small risks, or for small variance risks. We concluded the chapter by passing fast through different types of HARA utility functions. In two weeks, we will talk about changes in risk and the portfolio decision. Until then, you have a homework to do before next Friday. I will tell you more about the homework in the next post.