3. 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).

(d) The Capital Asset Pricing Model holds when the representative agent has quadratic utility. Its formula is E[Ri] Rp = Bi(E[RM] - Rf) where E(RM) is the expected gross return of market portfolio, which we calculate in part (b), and Bi 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 Bi. Show that the beta of Security 2 is 4, and the beta of Security 3 is -2. (d) The Capital Asset Pricing Model holds when the representative agent has quadratic utility. Its formula is E[Ri] Rp = Bi(E[RM] - Rf) where E(RM) is the expected gross return of market portfolio, which we calculate in part (b), and Bi 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 Bi. Show that the beta of Security 2 is 4, and the beta of Security 3 is -2