Assume there are three possible states of the world: 1, 2, and 3. Assume there are two

Question:

Assume there are three possible states of the world: ω1, ω2, and ω3. Assume there are two assets: a risk-free asset returning Rf in each state, and a risky asset with return R1 in state ω1, R2 in state ω2, and R3 in state ω3. Assume the probabilities are 1/4 for state ω1, 1/2 for state ω2, and 1/4 for state ω3. Assume Rf = 1.0, and R1 = 1.1, R2 = 1.0, and R3 = 0.9.

(a) Prove that there are no arbitrage opportunities.

(b) Describe the one-dimensional family of state-price vectors (q1, q2, q3).

(c) Describe the one-dimensional family of SDFs m˜ = (m1,m2,m3), where mi denotes the value of the SDF in state ωi. Verify that m1 = 4, m2 = −2, m3 = 4 is an SDF.

(d) Compute the projection of SDFs onto the span of the risk-free and risky assets by applying the formula (3.33) for the projection of a random variable y˜ onto the span of a constant and a random variable x˜. Take y˜ to be the SDF m1 = 4, m2 = −2, m3 = 4 and x˜ to be the risky asset return R1 = 1.1, R2 = 1.0, R3 = 0.9.

(e) The projection in part

(d) is by definition the payoff of some portfolio.

What is the portfolio?

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