There are two states of nature (s₁, s₂) with equal probabilities. Suppose there is a representative agent who is endowed with 1 unit of consumption today, and (2, 1) tomorrow. The agent has quadratic utility

u(c) = - (c - 3)² , c3

and =1. In the asset market, there are three securities. Security 1 is a comprehensive stock index that is a claim to the entire endowment of the economy. By definition, the stock index is the market portfolio with payoffs (2,1). Security 2 has payoffs (1,0), security 3 has payoffs (0,1), and security 4 is a riskfree bond that pays (1, 1).

a. What are the risk-neutral probabilities? Explain intuitively why the risk- neutral probability in one state is higher than that in the other state.

b. What is the expected gross return (i.e. expected payoffs over price) of security 2? Is the expected risk premium (E[R₂]-R_f ) of this security positive or negative? Please explain intuitively why this is the case.

c. What is the expected gross return of the market portfolio? What is the expected risk premium of the market portfolio (aka. the market risk premium)?

d. The Capital Asset Pricing Model holds when the representative agent has quadratic utility. Its formula is

_i (E[R_M] R_f)

where E(R_M) is the expected gross return of market portfolio, which we calculate in part (b), and _i is a measure of systematic risk for security i. The formula says that the expected risk premium of any security i is proportional to its measure of systematic risk _i. Show that the beta of Security 2 is 4, and the beta of Security 3 is -2.