Post date: Mar 16, 2015 7:17:10 PM
This second week has been a very intense week: nine hours of class!! We have covered three chapters, an average of a chapter per 3 hours, namely chapters 3, 4, and 5 in the book by Gollier. In chapter 3 we have discussed how to compare two random variables in terms of the behavior they would induce on decision makers. We saw that second order stochastic dominance means that all risk averse agents prefer the dominant risk to the dominated one, while first order stochastic dominance means that all decision makers, risk averse or not, prefer the dominant risk to the dominated one. Because first order stochastic dominance is more difficult to hold, we also should know that if a lottery 1 dominates lottery 2 in the sense of FSD then lottery 1 also dominates lottery 2 in the sense of SSD while the opposite is not necessarily true. We have also defined an increase in risk to be a dominated risk in the sense of SSD that preserves the mean, or a mean-preserving spread (equal risks, higher variance), or a risk that is obtained after adding a white noise to each possible outcome. We have seen, both theoretically and by means of an exercise, how to identify stochastic dominance by just looking at the probabiity structure of the random variables that need to be compared. As an application, we saw that a risk averse agent always prefers a perfect diversification investment strategy because the perfect diversification investment strategy SSD-dominates any other portfolio decision. In Chapter 4 we studied the standard portfolio decision, which is a simple model of how a decision maker, usually risk averse, decides how to divide his money into investing either in a risk-free asset and investing in a risky asset. We saw how to compute the optimal demand for investment in the risky asset by means of a couple of examples, and we saw theoretically that the the amount of money to be invested in the risky asset gets smaller the higher the risk aversion of the investor. We saw that the risk averse agent will invest a positive amount of money in the risk asset as far as the excess return of the risky asset is positive on average, and that the risk aversion will play a role of "second order", in the sense of determining the amount and fixing the second order condition of the maximization problem. We also saw that the same risk averse investor reduces his demand for the optimal investment if the probability structure of the risky asset worsens in the sense of stochastic dominance and certain conditions are met about the risk aversion structure: for FSD we need the coefficient for relative risk aversion to be smaller than one, while for SSD we need the coefficient to be smaller than one and increasing, and the Arrow-Pratt coeficient of absolute risk aversion to be decreasing. An interesting proposition was the one stating that a reduction in wealth always reduces the investment in the risk asset if and only if the utility function is DARA (decreasing absolute risk aversion). Finally, Chapter 5 dealt with a simple model for asset pricing. We consider the case of a two-sector economy and saw that the price of the asset in each sector depends on the covariance of the profits in the sector with the marginal utility of the per capita value of total production. We said that intuitively, the sector that manages to produce profit when the economy is low is typically the one with a higher price. In order to compute the risk premium in the economy, we need to divide the expected profit of the economy divided by its price and then deduct one unit. It might therefore be that a sector with a high asset price has a low risk premium due to its low profitability on average. The equity premium is then defined as the average risk premium of the economy, where we have weighted each sector's risk premium by their relative prices. We just mentioned fast that the equity premium is non negative if the representative agent is risk averse, and that it is an increasing function of risk aversion. We finish the week by computing an example on asset pricing on the board. We will meet again on March 24 and 26. On March 24 we will talk about Chapter 13, "the demand for contigent claims". On March 26 we will solve the second homework on the board and solve your doubts. Have a nice week.