Write any return R˜ as R˜ p +(R˜ −R˜ p) and use the fact that 1− ˜ep is orthogonal to excess returns—because e˜p represents the expectation operator on the space of excess returns—to show that x˜ def

= 1 E[R˜ p]

(1 − ˜ep)

is an SDF. When there is a risk-free asset, x˜, being spanned by a constant and an excess return, is in the span of the returns and hence must equal m˜ p. Use this fact to demonstrate (5.25).