Sharpening the Arithmetic of Active Management

Abstract

I challenge Sharpe’s (1991) famous equality that “the return on the average actively managed dollar will equal the return on the average passively managed dollar.” This equality does not hold in general. It is based on the implicit assumption that the market portfolio never changes, which does not hold in the real world because new shares are issued and others are repurchased. Hence, even “passive” investors must trade regularly in order to maintain their market-weighted portfolios. Since passive investors may trade at less favorable prices than active managers, Sharpe’s equality is broken. The changes of the market portfolio are large enough that active managers can add noticeable returns. Hence, active managers can be worth positive fees, which allow them to provide an important, beneficial, role in the economy — helping to allocate resources efficiently. Passive investing also plays a useful role, especially since the average active mutual fund manager has charged larger fees than their value added.