Suppose there is no risk-free asset and the minimum-variance return is different from the constant-mimicking return, that is, bm = bc. From Section 5.5, we know that there is an SDF that is an affine function of the minimum-variance return:

m˜ = γ + β(R˜ p +bme˜p) (6.37)

for some γ and β. From Section 6.2, we know that there is no factor model with the minimum-variance return as the factor. However, because there is an SDF that is an affine function of the minimum-variance return, we also know from Section 6.2 that there is a factor model with the minimum-variance return as the factor unless E[ ˜m] = 0. So it must be that E[ ˜m] = 0 for the SDF m˜ in (6.37).

Calculate E[ ˜m] to demonstrate this.