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

5. [15pm] There are tvvo states of nature [s1, s:] 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- 3F, c533 and 5:1. 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 [LEI], security 3 has payoffs [I11], and security 4 is a riskfree bond that pays [1, 1]. [a] What are the risk-neutral prohahilities? 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] 3;; 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 Eli] - BE 31[E[le - Br] where Ema] is the expected gross return of market portfolio, which we calculate in part [b], and g a measure of systematic risk for security i, The formula says that the expected risk premium of any security [is proportional to its measure of systematic risk 31. Show that the beta of Security 2 is 4, and the beta of Security 3 is -2