I'm having trouble understanding pg. 93 of Cochrane's Asset Pricing textbook. As seen in equation 5.23, σ(m)E(m)≥∣E(Re)∣σ(Re)\frac{\sigma(m)}{E(m)} \ge \frac{|E(R^e)|}{\sigma(R^e)} the Sharpe ratio on excess returns bounds the discount factor. However, to find a lower bound on σ(m)\sigma(m) for a given value of E[m]E[m] , it seems like the author is varying the value of E[m]E[m] , using the value to get a hypothetical risk-free rate ( 1/E[m]1/E[m] ), finding the maximum Sharpe ratio that can be constructed by subtracting the hypothetical risk-free rate from a return in the return space, and using that maximum Sharpe ratio to bound σ(m)E(m)\frac{\sigma(m)}{E(m)} . I don't get why this is a valid approach. The maximum Sharpe ratio of excess returns in the excess return space is a constant, and the maximum Sharpe ratio that can be constructed by subtracting the hypothetical risk-free rate from a return in the return space varies with the assumed value of E[m]E[m] . So at best it must be that the maximum Sharpe ratio that can be constructed by subtracting the hypothetical risk-free rate from a return in the return space is an upper bound for the Sharpe ratio of excess returns, for any value of E[m]E[m] . Is this so? If so, how, and if not, where did I make a mistake?