The derivation that in total the passives outperform the actives is not a statistical result from historical data and doesn't require efficient market assumptions at all. It's just basic arithmetic that holds over any time period that the passives as a group will have the same returns as the actives as a group but spend less on fees. Sharpe's webpage on that gives a simple rundown of the calculations:
That doesn't mean there haven't been long-running active strategies that have worked. But those have overperformed by grabbing returns from other actives underperforming, not the passives.
> 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.
> 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.
Because all he has to do is do absolutely nothing and he achieves that. The passive investor doesn't make choices, he buys a portfolio that has all the components of the index in the proportions at that point. Then the active traders trade among themselves and the prices move. And now the passive investor still holds all the components in the index at the proportions at t=2 because he hasn't traded, only the assets have changed value. You can observe this in practice. Index funds track their index incredibly well.
The mistake you make is to think of two groups, when there are actually three. Active, passive and "outiders", who got their stock outside of the stock market. Maybe they are early investors or maybe they get stock as part of their compensation package.
If passive investors make worse trades with outsiders than the active investors do, they will underperfom.
https://web.stanford.edu/~wfsharpe/art/active/active.htm
That doesn't mean there haven't been long-running active strategies that have worked. But those have overperformed by grabbing returns from other actives underperforming, not the passives.