14.27 Problem Set #5

Due November 6, 2001

1. Suppose that there are two risky assets in an economy. The return
r_{1} on stock in company 1 is distributed normally with mean 0.1
and variance 0.05. The return r_{2} on stock in company 2 is distributed
normally with mean 0.2 and variance 0.25. Assume that the two returns are
independent. These are the only two assets in the economy. There is no
risk free investment and there is also no cash or physical assets that
consumers can use to carry wealth over from one period to the next.

(a) Suppose consumer invests w_{1} in stock 1 and w_{2}
in stock 2 in the first period. His second period wealth W is a random
variable given by W = w_{1} (1 + r_{1}) + w_{2}
(1 + r_{2}).

What is the distribution of W?

(b) Suppose that the investor has wealth W_{0} to invest in
the first period. Suppose that he is extremely risk averse and simply wants
to minimize the variance of his return. What portfolio does he choose?
What is his expected return? What is the variance of his return?

(c) Let __v__ be the minimum variance you found in part (b). Note
that for any higher level of variance v that the investor is willing to
hold he can earn a higher expected return. Assume that the investor is
allowed to take a short position either stock, e.g. to choose w_{1}
< 0, and can thereby earn any expected return he wants. Find the maximum
possible return for a given v > __v__. Graph the expected return as
a function of v. (This graph is sometimes referred to as the efficient
portfolio frontier.) Put points corresponding to the two stocks on this
graph. Why is one point on the graph and one point below it?

(d) Suppose the investor's utility function for second period wealth
is u(w) = -e^{-Aw} with A=3. Suppose the investor has an initial
wealth of $100,000. How will the investor divide his portfolio between
the two assets? (Recall that I noted in class that when W is normally distributed
E(e^{tw}) = exp(tE(w) + t^{2}Var(w)/2) ).

2. Scott Morton, Zettelmeyer and Silva report that Autobytel customers
pay an average of about $400 less for their cars than other buyers.

(a) A crucial question for the future of referral services is whether
prices are lower because consumers used Autobytel or whether they are just
lower because the consumers using Autobytel are more savvy and would have
gotten lower prices anyway. They note that the savings of Autobytel customers
vary by car class. They are much larger for purchasers of pickups than
for purchasers of luxury goods. Why might this finding be taken to suggest
that much of the price savings is due to web customers being more savvy?

(b) Another reason why the lower prices might not be indicative of savings from using Autobytel is that prices may be lower just because the cars Autobytel users buy are worse in unobserved ways, e.g. they may have worse stereo systems. What does the evidence in the paper on the frequency with which Autobytel customers use dealer financing and buy insurance and service contracts suggest about this possibility?