Question

1)Consider the multifactor APT with two factors. The risk premium on the factor 1 portfolio is 3%. The risk-free rate of return is 6%. The risk-premium on factor 2 is 7.75%. Suppose that a security A has an expected return of 18.4%, a beta of 1.4 on factor 1 and a beta of .8 on factor 2. Is there an arbitrage portfolio? If not, prove it, if yes exhibit it?

2)In the APT model, what is the nonsystematic standard deviation of an equally-weighted portfolio that has an average value of σ(ei) equal to 25% and 50 securities?

Answer 1

Part 1:

Yes there is an arbitrage portfolio.

Given conditions:

Risk free rate=6%

Risk premium on factor 1 (say Rp1) is 3%

Risk premium on factor 2 (say Rp2) is 7.75%

Beta on factor 1 (Beta1)=1.4

Beta on factor 2 (Beta2)=.8

We can calculate the return using the formula:

Risk free rate +Beta1*Rp1+Beta2*Rp2

6+1.4*3+.8*7.75=16.4%

But, expected return on security A is 18.4%

So there is a spread of 2% which provides arbitrage opportunity.

Part 2:

Given that the average value of standard deviation (ei) =25% and the number of securities =50