Question

2. Consider a two-factor economy. The riskfree rate is 4%. There are two well-diversified risky assets with the following information. Assume the market is arbitrage free. Asset Factor 1 sensitivity Factor 2 Sensitivity Return 1.0 0.5 0.5 1.0 14% 18% (1) What are the risk premiums of factor portfolio 1 and 2? (15 marks) (2) A well-diversified risky asset has B1-1.5 and ß2-0.5. What is its arbitrage-free expected return? (10 marks) (3) If the forecasted return of asset in (2) is 14%. Indicate an arbitrage transaction that could be used to profit from this pricing situation. (15 marks)

Answer 1

(2) Asset A:

Expected Return = Ra = 14 %, Sensitivity to Factor 1 = Fa1 = 1 and Sensitivity to Factor 2 = Fa2 = 0.5

Asset B:

Expected Return = Rb = 18 %, Sensitivity to Factor 1 = Fb1 = 0.5 and Sensitivity to Factor 2 = Fb2 = 1

Risk Free Rate = Rf = 4 %

(a) Let factor portfolio risk premums for factor 1 and 2 be RP1 and RP2 resepctively.

Therefore, by Multifactor Arbitrage Pricing Theory, we get:

Ra = Rf + Fa1 x RP1 + Fa2 x RP2

Rb = Rf + Fb1 x RP1 + Fb2 x RP2

14 = 4 + 1 x RP1 + 0.5 x RP2 ------------------ Eq(i) and 18 = 4 + 0.5 x RP1 + 1 x RP2 ------------------Eq(ii)

Solving Eq(i) and Eq(ii) for the two variables RP1 and RP2 we get

RP1 = 4 % and RP2 = 12 %

(b) Beta 1 = F1 = 1.5 and Beta 2 = F2 = 0.5

Therefore, Expected Return = Rf + F1 x RP1 + F2 x RP2 = 4 + 1.5 x 4 + 0.5 x 12 = 16 %

(c) If the forecasted return of asset 2 is 14 % then one can purchase asset 2 by borrowing say $ 100 at the forecasted rate of return of the asset. Hence, a borrowing of $ 100 would entail a payback of 100 x (1.14) = $ 114

However, the actual return as per no arbitrage theory is 18% . This would imply that the borrower makes 100 x (1.18) = $ 118 on the borrowed investment.

If the borrowing payback is removed from the investment proceeds, one gets the following riskless profit (arbitrage profit):

118 - 114 = $ 4