> From this, it follows (as the night from the day) that the return on the average actively managed dollar must equal the market return. Why? Because the market return must equal a weighted average of the returns on the passive and active segments of the market. If the first two returns are the same, the third must be also.

Let me try to poke some holes in the argument.

1. Market return (M) is a weighted combination of passive (P) and actively managed portfolio (A) returns:

M = (1-w)P+(w)A.

2. Passive investor achieves market return M by holding the whole market, so P = M.

3. To satisfy the equation, A must also be M, regardless of w.

The logical error is in the assumption that P = M. To make this a concrete programming problem: let's say at time t=1, immediately prior to the start of a trading period, a passive investor decides to construct a portfolio. The passive investor makes investment allocations based on the current set of market prices.

Subsequently, how can the passive investor possibly match market returns without w being 0, or the passive investor knowing exactly how the active investors will allocate their holdings? That information lies in the future - one only needs to go though the exercise of simulating market returns on a discrete time basis to realize that the passive investor cannot possibly allocate to achieve exactly market returns.

M[1] = (1-w[1])P[1]+w[1]A[1]