Post date: Mar 31, 2015 7:44:26 AM
I have finally gone through your answers to Homework 2. As was the case of Homework 1, the copies are in general of good quality. Nevertheless, I would like to comment on a few common mistakes, I hope clarifying them will help you to prepare your exam.
Part I (A change in risk): A lot of people mention Proposition 5 on page 20. Unfortunately, the applicability of Proposition 5 is very limited, on page 19 you have the hypotheses of the problem. The perfect diversification strategy means that we are choosing with equal probability random variables that are iid (see page 19, first item-point). The abbreviation iid means that they are identically and independently distributed. In the case of the exercise, if (1/3,1/3,1/3) would be a perfect diversification strategy it must be defined over lotteries that are iid. I can accept that a final outcome can be interpreted as a "degenerated" lottery (i.e., the outcome with probability one, and the rest of outcomes with prob. 0). But then the assumption of iid would mean that it is always the same outcome with probability one that is realizing with probability 1/3 at the diversification strategy. Note that the first outcome means 0 (so 0 with prob. 1), the second one means 50 (50 with prob. 1) and the third one means 250 (250 with prob. 1). Hence, the random variables 0 with prob. 1, 50 with prob. 1, and 250 with prob. 1 are not iid and we cannot apply Proposition 5 when we are analyzing 1/3 on first outcome, 1/3 on second outcome, and 1/3 on third outcome.
Part II (The standard portfolio problem): Some of you gave the expression of the expected utility as a function of alpha, V(alpha), instead of the optimal alpha as a function of x+ and W, which is what I meant when I asked for the demand for investment.
A couple of people complaint in their copy that the chosen utility function was not appropriate. Nevertheless, on slide 28 of Chapter 2 you have a discussion about the quadratic utility function, where the domain is restricted to be bounded so that the first derivative is positive (and don't tell me these are not the same preferences, you just have to add a constant and be careful about the sign: u(z) = 1250000-0.05(z-5000)^2). It is ok to write a comment like that in your homework, but if you write it in your exam then you are showing me that you did not read Chapter 2 (or at least the end of it), and I will be in a position to discount points for that. Recall that an exam is a discriminating mechanism (or a separating device if you prefer) and if all copies are too good, the difference in grade must come from discounts like this.
Part III (The equilibrium price of risk): Some of you found negative risk premium for the second sector, I guess mainly due to approximation errors with the decimal numbers. You should know that if the representative agent is risk averse (which he was) the risk premium is always positive. Given that you were working at home, if you find a negative result, I expect you to go back and check your computations a bit better since you should know that there is a mistake. The same then applies the day of the exam.
On the last part, a lot of people took the marginal distributions of y1 and y2 as if they were independent and re-computed the joint distribution as the product of the two. Unfortunately, the statement of the question told us that we were given the joint distribution (so no need to compute the joint distribution). On page 3 we mention that these variables are not stochastically independent. The day of the exam remember to read well the statements of the exercises before starting to work them out.