From the January 2008 issue of Wealth Manager Web • Subscribe!

January 1, 2008

Rethinking Zero

Alpha is widely viewed as a zero sum game, and so for every investor who beats the market, someone must trail it. If this balancing act accurately describes how the money game works, there's a limited supply of alpha--which is to say something other than beta. And of this finite quantity, then only half is positive alpha.

A world where alpha sums to zero implies a particular set of laws that constrain the productivity of investment strategies overall. One example is the expectation that indexing will fare reasonably well over time compared to a relevant pool of active funds. Owning beta for the long run, in other words, looks like a compelling alternative if zero sum alpha is the rule.

But what if alpha doesn't sum to zero? What if it sums positively? Or negatively? Is it possible that, say, 80 percent of managers can beat their benchmark? If so, how does that alter the outlook for indexing relative to active management? Or, more ominously, what if 80 percent of managers could fall short of the market's return over some time frame?

Clearly, whether investing is, or isn't, a zero sum game matters. A casual review of the topic turns up an abundance of opinion, with most of it arguing that zero summing reigns supreme. A widely quoted source for this claim is "The Arithmetic of Active Management," a 1991 article by Nobel Laureate William Sharpe. He explained that simple mathematics offer the proof, which leads to some fairly basic and enduring principles in the business of money management.

"Properly measured," Sharpe wrote, "the average actively managed dollar must underperform the average passively managed dollar, net of costs." The reason is that "the market return must equal a weighted average of the returns on the passive and active segments of the market."

By that standard, the index is destined for above-average performance relative to the appropriate universe of active managers. Yes, some managers will win the race, but the victories will be offset by the relative-performance losers. So it goes when alpha adds up to naught.

You would expect indexing's disciples to say as much. But active managers and their supporters generally agree, too. Does that mean the subject is closed? Yes and no. The issue is being debated anew in the 21st century. Helping stoke the fires of deliberation is an intriguing paper by Joanne Hill of Goldman Sachs. In a 2005 research report, "Alpha as a Net Zero-Sum Game," she laid out a case for why the alpha pie is not as limited as Sharpe and others would have us believe.

No, Sharpe isn't wrong, at least as he defined the terms. Rather, Hill and others question Sharpe's underlying assumptions as the one and only way to consider alpha. "The alpha game in practice (rather than theory) is not a closed system, where there are a set of identical and finite chips available at the start and end, so that the returns delivered by winners must come at the expense of losers," Hill wrote. The reason is that investors in the aggregate have different time horizons, behavioral biases, risk preferences, capital constraints, skill levels and so on, she explained--none of which are addressed in Sharpe's article. In other words, the rules required for zero sum alphas give way after you factor in the complexities of the real world.

Alpha may not sum to zero once you consider risk in context with return, says Max Darnell, chief investment officer and partner at First Quadrant, a money manager in Pasadena, Calif.

Yes, returns alone sum to zero, he concedes, but risk-adjusted returns are another matter. "If you're living in a return-only world and ignore the differences between investors, which are risk differences, then return is a zero sum game," Darnell tells Wealth Manager. But assuming that everyone shares the same tolerance and objectives for risk doesn't match reality. "It's on the risk side that we all distinguish ourselves." (For Darnell's extended comments on the subject see "A Question of Supply" on page 66.)

As an example, Darnell points to this past summer's liquidity crisis, which triggered a sharp sell-off in the stock market. Reaction to the crisis varied, depending on risk tolerance, investment horizon, etc. Some hedge funds were selling because of short-term trading mandates that prevented them from sitting idle during a surge in market volatility driven by falling prices. Such hedge funds are a natural seller to pension funds, which often have long-term horizons and look to raise risk exposure at moments of crisis. In other words, sellers and buyers can both be "winners" in risk-adjusted terms even though one or even both may have losses for a moment in time.

Thinking of alpha in risk-adjusted terms can trace its intellectual origins to 1738, when Dutch mathematician Daniel Bernoulli laid out the foundations of expected utility theory. As he put it, "...the value of an item must not be based on its price, but rather on the utility it yields. ...the dependent on the particular circumstances of the person making the estimate." That leads to the notion that people may place a diminishing value on additional wealth under risky conditions, as the graph on page 64 illustrates.

Bernoulli's hypothesis challenged the era's conventional wisdom for making decisions under conditions of uncertainty. Rather than picking the strategy that offered the highest expected value, Bernoulli's model favored the highest expected utility. As a result, two people faced with the same decision and looking at the same data could reasonably come to different conclusions in Bernoulli's world. Why? Because expected utility varies depending on a person's preferences.

Fast forward several centuries and utility theory figures prominently in the explanation of why tactical asset allocation (TAA) holds out the potential for raising expected returns with little or no corresponding rise in expected volatility. At first glance, the apparent free lunch appears to run afoul of modern portfolio theory, which equates higher return with higher risk. But here's where utility theory steps in.

"The linkage between risk and reward is not inviolate [in TAA] if a higher-return strategy has lower 'utility' than a more comfortable but less-rewarding strategy," wrote Robert Arnott (head of Research Affiliates and Darnell's predecessor at First Quadrant) in The Portable MBA in Investment (1995, Wiley). TAA doesn't offer the so-called free lunch, but it "succeeds because total return and investor utility are not one and the same thing. When wealth is declining, most investors seek the solace of lower risk, hence lower exposure to risky markets," Arnott advised. "Tactical asset allocation potentially enhances long-run returns without increasing portfolio risk, but at a cost of lower comfort, hence lower utility, for many investors."

Explained another way, raw performance numbers and risk-adjusted returns exist in parallel universes. Each has a distinctive set of rules and the two measures are related. But the calculus of investing that works smoothly in one universe can malfunction in the other.

That, at least, is part of the explanation for the reason why alpha may not sum to zero. Yes, alpha ultimately balances out in the world laid out in Sharpe's 1991 paper. But if you consider a more nuanced framework, the standard assumptions may not hold.

Even so, the notion of alpha summing to zero in the return-only space provides a powerful warning that's not easily dismissed. "When Bill Sharpe wrote 'The Arithmetic of Active Management,' the way he said it is indisputable: The average manager can't outperform the average," asserts Ronald Kahn, global head of advanced equity strategies at Barclays Global Investors in San Francisco.

Kahn's view is notable for several reasons. First, he and Richard Grinold share billing for the creation of the so-called fundamental law of active management, which quantifies the idea that generating alpha is dependent on opportunity plus skill and the frequency of its application. Although Kahn works at Barclays, the world's largest indexer, where he oversees research for the firm's nearly $500 billion in active quantitative strategies.

Kahn, in short, is a card-carrying believer in active management. Yet he also recommends caution for thinking that alpha's supply may be greater than tradition suggests. In fact, he warns that after adjusting returns by some measure of risk, alpha probably sums to a negative number. The odds of producing positive alpha, in other words, may be less favorable than is widely assumed--even among the zero-sum crowd. "I think that's a healthy way to think about it," he says.

Another veteran investment strategist agrees. "You can have academic debates about whether alpha's a zero sum game," says Robert Jaeger, chief investment officer of EACM, a division of BNY Mellon that oversees $5 billion for institutional investors--more than half allocated to hedge funds. "But my working assumption and a guiding principle is that it is a zero sum game." The hedge fund business, Jaeger adds, is no exception.

Steven Foresti echoes Jaeger's caution. "I wouldn't be persuaded to abandon the zero-sum game idea," says the managing director of research for Wilshire Consulting, a unit of Wilshire Associates. "I suppose you can make theoretical arguments that investors have different objectives. Some are laying off risk while others are trying to beat the market, for example. But in the aggregate, it really does come back to the 'Arithmetic of Active Management.'"

Foresti argues too, that the more competitive markets become, the more likely that alpha will sum to zero--especially over time. That doesn't invalidate active management, but it does help keep investors sober about the nature of the game, he says.

Recognizing that returns-based alpha sums to zero boils down to common sense, suggests Laurence Siegel, director of research in the investment division of the Ford Foundation, which favors active management for its $13 billion-plus portfolio. "You should know the rules of the game before you start to play," he says. Otherwise, you shouldn't be playing, he adds.

But which set of rules? Each one is valuable for thinking about the limits and opportunities of investment strategy. One reminds that everyone can't be above average; the other asserts that risk matters. Giving up one or the other seems hopelessly misguided. Alas, the two principles aren't easily reconciled into one strategic vision. Perhaps one day a brainy financial economist will win a Nobel by figuring out how to integrate them.

JAMES PICERNO ( is senior writer at Wealth Manager.

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