Question

2. Suppose there are two independent risk factors governing securities returns according to the two factor APT. The risk-free rate is 10%. The following well-diversified portfolios exist: beta with respect beta with respect Expected Return to factor 1 to factor 2 Portfolio #1 25% Portfolio #2 25% (a) What are the expected returns on each of the two risk factors in this economy? (b) Suppose another portfolio has a beta with respect to the first factor of 1, a beta with respect to the second risk factor of 1, and an expected return of 14%. Is this consistent with the APT? (Explain your answer) If not, what asset you would want to short in an arbitrage portfolio?

Answer 1

The Formula for the Arbitrage Pricing Theory Model Is

E(R)i​ = R_f​ + (E_R − R_f) × β

Where

E(R)i​ = Expected return on the asset

R_f ​= Risk-free rate of return = 10%

β=Sensitivity of the asset price to macroeconomic factor

E_R - R_f =Risk premium associated with factor i​

(A) Expected Return for Portfolio #1

= 10%+(25%-10%)*2+(25%-10%)*1

= 10%+30%+15%

= 55% Expected Return of portfolio #1 with two independent risk factor.

Expected Return for Portfolio #2

= 10%+(25%-10%)*1+(25%-10%)*3

= 10%+15%+45%

= 70% Expected Return of portfolio #2 with two independent risk factor.

(B) Expected Return for Another Portfolio.

= 10%+(14%-10)*1+(14%-10%)*1

= 10%+4%+4%

= 18% Expected Return of another portfolio with two independent risk factor.

Yes, it is consistent with APT model where two independent risk factor has a beta of 1 which is very low as compare to the other two portfolio risk factor which means low risk and return is also good as per market std.