EF8904Financial Theory

Fall 2013Ryerson University

Problem Set 2Due Thursday, October 3rd

1. [Lotteries] There is an individual with a well-behaved utility function, and initialwealth Y . Let a lottery offer a payoff of G with probability π and a payoff of B withprobability 1− π.

(i) If the individual already owns this lottery denote the minimum price he wouldsell it for by Ps. Write down the expression Ps has to satisfy.

(ii) If he does not own it, write down the expression Pb (the maximum price hewould be willing to pay for it) has to satisfy.

(iii) Assume now that π = 1/2, Y = 10, G = 6, B = 26, and the utility functionis U(Y ) =

√Y . Find buying and selling prices. Are they equal? Explain why

not. Generally, can they ever be equal?

2. [Stochastic Dominance] You are offered the following two investment opportunities:

Investment A Investment B

Payoff Probability Payoff Probability2 1/4 1 1/34 1/2 6 1/39 1/4 8 1/3

(i) Apply the concept of first order stochastic dominance

(ii) Apply the concept of second order stochastic dominance

(iii) Illustrate (i) and (ii) with a graph

(iv) Apply the concept the mean variance criterion

(v) Bernie is an expected utility maximizer with Bernoulli utility function U(x) =log(x). Which investment does Bernie prefer?

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EF8904Financial Theory

Fall 2013Ryerson University

3. [Minimum Variance Portfolios] Let there be two securities with (random) returnsri and rj. Suppose that these securities have identical expected rates of return andidentical variances. The correlation coefficient between ri and rj is ρ. Show that anequally-weighted portfolio of assets i and j achieves the minimum possible varianceregardless of the value of ρ.

4. [Mean-Variance Utility Hypothesis] Erin is an expected utility maximizer. Her util-ity function u is strictly increasing, strictly concave, twice differentiable and bounded.Currently, Erin is evaluating an asset with stochastic outcome R which is normallydistributed with mean µ and variance σ2. Thus, its density function is given by

f(R) =1

σ√

2πexp

[−1

2

(R− µσ

)2]

(i) Show that Erin’s expected utility from R is a function of µ and σ2 alone. Thus,show that E [u(R)] = φ(µ, σ2).

(ii) Show that φ(·) is increasing in µ.

(iii) Show that φ(·) is decreasing in σ2.

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