Suppose two assets satisfy a statistical factor model with a single factor: R 1 = E[R 1] + f + 1 , R 2 =

Suppose two assets satisfy a statistical factor model with a single factor:

R˜ 1 = E[R˜ 1] + ˜f + ˜ε1 , R˜ 2 = E[R˜ 2] − ˜f + ˜ε2, where E[˜f] = E[ ˜ε1] = E[˜ε2] = 0, var(˜f) = 1, cov(˜f ,ε˜1) = cov(˜f ,ε˜2) = 0, and cov(ε˜1,ε˜2) = 0. Assume var(ε˜1) = var(ε˜2) = σ2

. Define R˜ ∗

1 = R˜ 1 and R˜ ∗

2 = πR˜ 1 + (1−π )R˜ 2 with π = 1/(2+ σ2).

(a) Show that R˜ ∗

1 and R˜ ∗

2 do not satisfy a statistical factor model with the single factor ˜f .

(b) Show that R˜ ∗

1 and R˜ ∗

2 satisfy a statistical factor model with zero factors, that is, R˜ ∗

1 = E[R˜ ∗

1]+˜ε∗

1 , R˜ ∗

2 = E[R˜ ∗

2]+˜ε∗

2 , where E[ε˜∗

1 ] = E[˜ε∗

2 ] = 0 and cov(ε˜∗

1 ,ε˜∗

2 ) = 0.

(c) Assume exact APT pricing with nonzero risk premium λ for the two assets in the single-factor model, that is, E[R˜i] −Rf = λcov(R˜i,˜f) for i = 1,2. Show that there cannot be exact APT pricing in the zero-factor model for R˜ ∗

1 and R˜ ∗

2 .