Question

2. Suppose the uncertainty involves two possible states (0- = 1,2) with equal probability 0.5. The investment A produces a state-contingent rate of return 3% at the state 1 and 5% at the state2. The investment B produces a state-contingent rate of return 2% at the state 1 and 8% at the state 2

a. Based on the state-by-state dominance, which investment project would you prefer?

b. Based on the mean-variance criterion, which investment project would you like better?

c. Which investment project would you rather have? Tell me your criterion

Answer 1

a. Based on the state-by-state dominance, which investment project would you prefer?

Investment A produces better return in state 1 but a lower return than B in state 2. Hence none of the two investments have a state by state dominance. Here, neither dominates the other. Hence, there is no ranking possible on the basis of the dominance criterion.

b. Based on the mean-variance criterion, which investment project would you like better?

Investment A:

Expected (mean) return = E(R_A) = 0.5 x 3% + 0.5 x 5% = 4%

Variance = 0.5 x (3% - 4%)² + 0.5 x (5% - 4%)² = (1%)²

Hence, standard deviation, σ_A = 1%

Investment B:

Expected (mean) return = E(R_B) = 0.5 x 2% + 0.5 x 8% = 5%

Variance = 0.5 x (2% - 5%)² + 0.5 x (8% - 5%)² = (3%)²

Hence, standard deviation, σ_B = 3%

Again there is no clear dominance. Investment A has lower expected return and lower standard deviation. B has higher expected return but then it has higher risk indicated by higher standard deviation. So there is no mean variance dominance.

c. Which investment project would you rather have? Tell me your criterion

We create a new criterion: Expected return per unit of risk i.e E(R) / σ

Hence For investment A: E(R) / σ = 4% / 1% = 4

For B: E(R) / σ = 5% / 3% = 5/3

Since this criterion is better for Investment A, I would have Investment A.