Show that E[R˜ 2] ≥ E[R˜ 2 p] for every return R˜ (thus, R˜ p is the minimum second-moment return). The returns having a given second moment a are the returns satisfying E[R˜ 2] =

a, which is equivalent to var(R˜) +E[R˜]

2 = a ;

thus, they plot on the circle x2 +y2 = a in (standard deviation, mean) space. Use the fact that R˜ p is the minimum second-moment return to illustrate graphically that R˜ p must be on the inefficient part of the frontier, with and without a risk-free asset (assuming E[R˜ p] > 0 in the absence of a risk-free asset).