Question

3. In an APT world, there may be multiple factors. Suppose there are two - industrial production and balance of payments. The expected excess return on these factors are 0.1 and 0.05 and the riskless rate is 0.05. The betas for the first stock are 2 and 0.5 and for the second stock they are 1 and 3 respectively.

   1. Find the expected return on the stocks if the APT is true.

   2. Find a portfolio with a zero beta on industrial production.

   3. What is the expected return on this portfolio?

Answer 1

Expected return on first stock -

R_a = R_f + Beta_i *(Risk Premium_i ) + Beta_b *(Risk Premium_b) = .05 + 2*.1 + .5*.05 = .275 = 27.5 %

Expected return on second stock -

Rb = R_f + Beta_i *(Risk Premium_i ) + Beta_b *(Risk Premium_b) = .05 + 1*.1 + 3*.05 = .3 = 30 %

Since none of the beta's are negetively correlated with the market, a portfolio with o beta is not possible with these two stocks.

(Assuming no shorting)

Assuming shorting -

Since zero beta on industrial production required -

a. Long 2 stocks of the second one and short one stock of the first one

R_e = 2*(.05 + 1*.1 + 3*.05) - (.05 + 2*.1 + .5*.05) = .05 + .275 = 32.5%