Why Active Funds Can Outperform Passive Funds Without Violating Sharpe

# zero-sum active management riddle


# zero-sum active management riddle












Bill Sharpe proves that the average alpha from active management is zero (and after transaction costs the average active manager returns less than passive funds). One active manager's gain is offset by equal losses among some other active managers.

So I stumbled across this paper "Does Active Management Pay? New International Evidence" by Alexander Dyck, Karl Lins and Lukasz Pomorski. They find that on average, net annual returns for active equity strategies exceed those of passive equity strategies by about 1.1% in markets outside the U.S.

Does this empirical finding contradict Sharpe's equilibrium argument that average active management returns are zero?

Update: As soon as I finished preparing the question, I realized how the two arguments can both be true without contradiction. Still this makes for an fun riddle. I'll post my answer in 24 hours or award it to whomever gets it first.

## Answer by Tal Fishman (score 2, accepted)

https://quant.stackexchange.com/a/2450

Thanks for pointing out the article. Their findings make for very interesting food for thought, particularly from a marketing and business strategy perspective.

I'm not sure what you realized, but I think the way to reconcile the two statements is to recognize that the universe of equity investors does not break down precisely into active and passive fund managers. There are a whole host of other investors (particularly in non-US markets), such as retail investors, governments, and other companies. Some of these, particularly government, are not profit-motivated investors, and so may offer active returns to those managers betting against them. Furthermore, retail investors have been shown in many contexts (such as Barber and O'Dean's papers) to have various systematic biases that give them a lower return than the "average" of all investors, and sophisticated active investors may profit from those biases as well.

## Answer by Patrick Burns (score 3)

https://quant.stackexchange.com/a/2451

Tal is correct that "other" investors can remove the zero-sum constraint.

But that is not the only possibility. As the article linked to by http://www.portfolioprobe.com/2011/11/07/some-new-ideas-in-financial-mathematics/ points out, an index is just a trading strategy. There is no reason to suppose that it is an optimal trading strategy.

One reason we are deluded into thinking that something like the S&P 500 is optimal is because of the academic coercion to think of it as the "market portfolio" (which by definition is optimal).

## Answer by Richard Herron (score 3)

https://quant.stackexchange.com/a/2454

@Tal and @Patrick point out that there are more than two subsets of investors. If we divide the active investors into smart and dumb, we can reconcile their results (i.e., they only look at active, institutional investors).

But even if there are only two subsets, their measure is still different than Sharpe's. Sharpe's argument is an identity for stock returns. If we divide the value-weighted market into two parts, then both parts must have the same return. But the other authors look at $\alpha$, which requires a model. True, they use an accepted international risk-adjusting model, but we could add an "institutional investor" factor and reduce the active $\alpha$ to zero. This is the unexplained risk story. But Sharpe's isn't a risk story, it's just a raw returns story. I don't think they contradict Sharpe's story.