# 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.

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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.

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Yep. Put in other words, it's perfectly fine for active managers to have positive alpha in the international market. However, it requires that average alpha be negative - on average - for the rest of the investment universe. –  Quant Guy Nov 30 '11 at 4:50

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).

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I believe the paper corrects both active and passive strategies for the risk they take, so I'm not sure differences between the "passive" portfolio and the "optimal market portfolio" can explain this difference. Furthermore, they examine many different types of passive strategies, not just market-weighted. –  Tal Fishman Nov 28 '11 at 18:24
I only glanced through the paper so certainly I missed a lot, but I'm guessing that if they had a passive strategy of investing in the half of the universe with the lowest (ex ante) volatility, then they would see outperformance over most periods and universes. –  Patrick Burns Nov 29 '11 at 10:08
Investing in only half the universe is, by definition, an active strategy. For every active investor who invests exclusively in low volatility stocks, there must be an equal and opposite active investor who invests disproportionately in high volatility stocks. After costs, the passive guys will still outperform the active guys. That is Sharpe's insight. –  Tal Fishman Nov 29 '11 at 15:05
Tal, I can see how my lower-half strategy could be considered active. However, S&P 500 invests in something like 1/12th of US stocks but is "passive". –  Patrick Burns Dec 1 '11 at 9:44
Sure, but then the comparison to an active strategy should be to a large cap domestic active manager benchmarked to the S&P 500. I would suspect these managers as a group do a bit worse than the S&P 500 after fees. I don't think anybody who tackles this problem in a serious manner, including the referenced paper, would suggest the S&P 500 is the right passive benchmark for all active managers. –  Tal Fishman Dec 1 '11 at 16:32

@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.

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interesting angle, thank you –  Quant Guy Nov 30 '11 at 4:51
@QuantGuy -- Other than the case of value-weighted portfolios and value-weighted market factor (i.e., CAPM), I don't know of a requirement for $\sum_i w_i \alpha_i = 0$ and $\sum_i w_i \beta_i = 1$, where $w_i$ is the value weighting for each portfolio $i$. –  Richard Herron Nov 30 '11 at 12:26
(As well, I would guess that the Roll critique of these pricing models is particularly strong in these less-sophisticated markets.) –  Richard Herron Nov 30 '11 at 12:31