In finance, a business that revolves around intricate calculations, few equations appear to have stood the test of time as well as the straightforward Sharpe ratio.

Mathematically simple but intuitively profound, the ratio gives investors a way to gauge how much return they can expect to reap for each unit of risk they take in any investment -- stock, bond, mutual fund or hedge fund -- and then to compare this telling number from one investment to another. The higher the ratio, the better: the more return for the risk.

The math is Finance 101: Take the expected return on an investment, subtract the "risk free" rate of return of Treasury bills, and divide the remainder by the standard deviation of the returns. QED: You've got a Sharpe ratio. A former Stanford University finance professor, William Sharpe, came up with the concept almost 40 years ago.

Professional and savvy amateur investors alike rely on Sharpe ratios as an essential tool in making portfolio decisions. Pension plans and consultants use them to pick money managers. California Public Employees' Retirement Systems, the largest pension fund in the U.S., regards the Sharpe ratio as the baseline measure for calculating risk-adjusted returns. Credit Suisse Asset Management offers its institutional and high-net-worth clients up-to-date Sharpe ratios online. Morningstar publishes Sharpe ratios for each of the 15,000 mutual funds it covers worldwide. Mutual Funds magazine gives its "best of class" awards only to the funds with the highest Sharpe ratios in their investment categories. And last month the Hong Kong Securities and Futures Commission released proposed guidelines for hedge fund reporting that recommend Sharpe ratios.

"Sharpe ratios are an important part of evaluating managers," says Guy Miller, a senior consultant with financial advisory firm Barra. "They are used constantly."

Arguably, Sharpe ratios exert greater influence today than ever. Plunging stock markets have turned risk from an abstract financial planning concept into a painful reality for the first time in decades. Fearful of losses but eager for gains, pension funds in particular have been embracing exotic alternative investments, most conspicuously hedge funds. And no investment group boasts louder about its impressive Sharpe ratios than hedge funds. "Sharpe ratios are treated with almost religious significance in the hedge fund business," says one hedge fund marketing executive. The oldest hedge fund data service -- New York-based MAR Hedge Fund Database -- ranks funds in its publications by their Sharpe ratios.

THERE IS JUST ONE PROBLEM WITH ALL THIS. Sharpe ratios don't necessarily do what they're supposed to do: indicate a hedge fund's, or any other investment's, return relative to its riskiness. And the most prominent of the skeptics on this score happens to be a man named Sharpe.

"The Sharpe ratio is oversold," confides Sharpe, who in 1990 won the Nobel Prize for his work on the capital asset pricing model and now runs a risk management consulting firm, Financial Engines, in Palo Alto, California. "We don't use it in any direct sense here."

But don't take Sharpe's word for the shortcomings of his eponymous ratio. A growing group of academics and some sophisticated investment professionals have reached startlingly negative conclusions about the reliability of this cornerstone of modern finance. Says Barra's Miller: "You want to know if a number is very fuzzy or just has a little peach fuzz on it. Sharpe ratios tend to be buried in hair." Massachusetts Institute of Technology finance professor Andrew Lo studied ten hedge funds and found their Sharpe ratios to be overstated by as much as 60 percent. "It's amazing how easy it is to manipulate these things," Lo says. Concludes Vanguard Group founder Jack Bogle, "In terms of how the Sharpe ratio has done in evaluating mutual funds, I would say the answer is poorly."

What's wrong with the Sharpe ratio anyway? The disparagers say that, for one thing, it can be easily distorted -- or manipulated -- by complex modern trading strategies that didn't exist when the ratio was conceived. Two academic papers on the subject are titled: "Sharpening Sharpe Ratios" and "How to Game Your Sharpe Ratio." Hilary Till, a derivatives specialist who runs Chicago-based hedge fund Premia Capital Management, acknowledges that "you need models that help you simplify reality" but warns that "if individuals and institutions are now considering investments that include a lot of dynamic trading strategies and options, then the Sharpe ratio is probably not sufficient as a performance metric."

Sharpe ratios can also camouflage the enormous uncertainty that is built into their implicit predictions of future returns and thus give a deceptive sense of certitude. Many mutual funds showed superior Sharpe ratios just before they plummeted in the first half of 2000. Janus Twenty, for instance, a concentrated, technology-heavy fund, had an attractive Sharpe ratio on December 31, 1999, of 1.47 based on three-year historical returns -- three times the long-term Sharpe ratio of the Standard & Poor's 500. Nevertheless, Janus Twenty proceeded to lose 32 percent in 2000, 29 percent in 2001 and 24 percent this year through the end of September, for a cumulative loss of 65 percent.

"I take the point," says Sharpe, "that Sharpe ratios can give a false sense of precision and lead people to make predictions unwisely." Past performance is no guarantee of future results, as the boilerplate investment warning goes. But the level of the equation's uncertainty -- which rarely gets reported along with the specific Sharpe ratio -- might shock many investors. Consider the expected return, or numerator, of the relatively staid S&P 500's Sharpe ratio. "You would need 1,600 years of annual returns from the S&P 500 to predict future expected returns within 1 or 2 percentage points," calculates Financial Engines research chief Christopher Jones.

Sharpe ratios overlook certain risks altogether. Illiquidity is one of the biggest hazards with investments that are outside the mainstream, such as small-capitalization stocks. Small-cap funds were one of the year's trendier investments. How much of their very real risk gets captured by this classic risk barometer? None. "Illiquidity is not measured in the Sharpe ratio," notes MIT's Lo.

Moreover, the Sharpe ratio has become a crutch for many investors in the current troubled markets. They place too much reliance on its risk calculus and don't do their investing homework. "If the choice is between a single average measure and a Sharpe ratio, I'd choose the latter," says the ratio's inventor. "But it would be far better to use more than one measure and to make predictions that take into account rudimentary notions of market clearing, equilibrium and the like."

"It's the Enron of ratios," contends Kelsey Biggers, a longtime Wall Street risk management specialist who now works for the fund-of-hedge-funds K2 Advisors. "It's kind of scary that you have an industry built around a ratio that, when it comes to hedge fund strategies, no one believes in."

The trouble is that many people do believe uncritically in the Sharpe ratio -- especially investors in hedge funds. In a series of published and unpublished studies, academics have been warning with mounting urgency that investors who use Sharpe ratios to pick hedge funds could be taking on frightening amounts of unknown risk. "What's new is not the academic concerns," points out Premia Capital's Till. "What's new is what people are investing in." Says Leola Ross, an analyst with Tacoma, Washington­based money management consulting firm Frank Russell Co., "There is a disconnect" between academics' misgivings and investors' credulity.

One of the concerned academics is William Goetzmann, Edwin J. Beinecke professor of finance and management studies and director of the Yale School of Management's International Center for Finance. He and three colleagues, Jonathan Ingersoll, Matthew Spiegel and Ivo Welch, examined the circumstances of the Art Institute of Chicago's nearly $20 million loss last year in a hedge fund run by Integral Investment Management that had a very high Sharpe ratio. The fund, Integral Hedging, reported Sharpe ratios of between 3.9 and 14.1 in 1999 and 2000, according to MAR.

In a paper circulating in draft form, the Yale professors conclude that Integral Hedging appeared to have been selling out-of-the-money puts on U.S. equity indexes, a strategy that could have generated a very high Sharpe ratio that understated the risks of the fund. The tumbling stock market after September 11 apparently hit the fund with huge losses. (A lawyer for Integral has not returned calls; a spokeswoman for the Art Institute declines to comment.)

"For some hedge fund strategies," asserts Goetzmann, "the Sharpe ratio doesn't give you a true sense of the risk-reward profile."

The Yale professor takes pains, however, to emphasize that "we are not attacking Bill Sharpe." Such is the deference researchers accord the well-liked Nobelist, who appears to take himself and his ratio less seriously than do most of the critics. Pressed on the irony of seeming to disown his famous equation, Sharpe dryly notes, "I didn't call it the Sharpe ratio." But then he never designed the ratio to certify the future performance of investments -- and certainly not of hedge funds.

BORN IN CAMBRIDGE, Massachusetts, in 1934, Sharpe earned a BA and, in 1961, a Ph.D. in finance from the University of California at Los Angeles. After graduation he got a job as an assistant professor at the University of Washington, where he formulated the capital asset pricing model. In 1970 he moved to Stanford, where he remained on the faculty until 1999. Sharpe has consulted extensively for such financial firms as Merrill Lynch & Co., UBS and Wells Fargo & Co., and in 1996 he co-founded Financial Engines. The 175-person Internet company (www.financialengines.com) provides financial planning advice to individuals. Sharpe and his wife, Kathy, a well-regarded San Francisco painter, along with their bichon frise, Cloud Monet, have a house on the beach in Carmel, California.

It was Sharpe, characteristically, who commended perhaps his ratio's severest critic to this magazine. And a more exuberant Sharpe-ratio-basher than Harry Kat, a leading expert on hedge funds, would be hard to imagine.

"You can create portfolios that can only lose money but have high Sharpe ratios!" exclaims the expansive, 40-year-old Kat, an associate professor of finance at the University of Reading in the U.K. and onetime head of European equity derivatives for Bank of America. This January Kat, who earned his Ph.D. in finance at the University of Amsterdam, takes up a new post as professor of risk management at City University in London.

"He's doing some fantastic work," says Sharpe of his ratio's debunker, with whom he exchanges e-mails. "He told me one of his papers got rejected -- I said, 'Don't worry, the CAP-M got rejected.'"

Kat pronounces Sharpe ratios virtually worthless for selecting most alternative investments, such as hedge funds. "They are completely meaningless," he says. "They are nonsense." The Sharpe ratios of hedge funds, he explains, are typically calculated from a small set of monthly returns that almost invariably miss the rare but catastrophic losses that haunt the hedge fund business.

He also expresses considerable skepticism that Sharpe ratios alone can be effective at evaluating certain more conventional investments, because risks like illiquidity don't get factored into the ratios. Concludes Kat, "I would probably only trust them for large diversified portfolios investing in large names." There the potential risk is more likely to be reflected accurately in the historical data.

Even some of those who have put out white papers touting hedge funds largely because of their high Sharpe ratios acknowledge that the ratios have major drawbacks. "We don't disagree with the academic arguments about Sharpe ratios," admits Michael Urias, a quantitative investing specialist at Morgan Stanley & Co. who coauthored "Why Hedge Funds Make Sense." That high-profile 2000 research report predicted that hedge funds would increasingly challenge conventional investments because of their "high levels of risk-adjusted returns" -- as evinced partly by Sharpe ratios. "We mentioned Sharpe ratios because they continue to be used in the hedge fund business, but we think numbers like maximum drawdown [which measures how much the fund has fallen from its high] may be more useful."

Hedge fund consulting firm Van Hedge Fund Advisors International singled out Sharpe ratios in its review of the markets last month: "Hedge fund risk [since 1988] as measured by the Sharpe ratio finds hedge funds with a Sharpe ratio of 1.35, versus 0.37 for the average equity mutual fund and 0.57 for the S&P 500." CEO George Van says that "while I agree with the academics, you use the tools that people can understand." Van Hedge, he adds, uses many measures to evaluate risk.

Plainly, investors shouldn't rely overly much on Sharpe ratios, either. The ratios' dependence on historical data is inherently problematic: Most hedge funds, after all, have been around for only a short time, and their limited trading data typically fails to capture the infrequent but drastic drops that come with the hedge fund territory. Says Kat: "It's like buying a house, and the real estate agent says you can peek into two windows. The first room looks great. The second room looks great. The plumbing is crap. The electricity is crap. But you wouldn't see it."

Compare charts of the historical returns for a stock index fund and for a merger arbitrage fund, and the problems of using the same statistic to measure any and all investments become apparent. The returns on the stock index look somewhat like the sort of bell curve your high school math teacher used to grade the final exam: The vast bulk of returns are clustered in the middle, with a handful at the high and low ends of the curve. The Sharpe ratio captures this so-called normal distribution of the index fund's returns quite nicely, giving a pretty good sense of their volatility, or risk.

But now plot the returns of the merger arbitrage fund -- often there is nothing normal about them. Like most hedge fund returns, they feature long, low curves on both ends, which statisticians call fat tails. The tail swooping into negative territory represents the rare but catastrophic events that the hedge fund industry is notoriously prone to. And it is these elongated curves that the Sharpe ratio has such a hard time taking into account in assessing a hedge fund's risk.

Thus the hedge fund world has had more than its share of high-Sharpe-ratio funds that proceeded to explode. Long-Term Capital Management boasted a Sharpe ratio of as much as 4.35 before it collapsed in 1998, nearly taking the financial system down with it. The Shetland Fund, run by former Goldman, Sachs & Co. partner Michael Smirlock, was listed by MAR Hedge as having one of the ten highest Sharpe ratios in the hedge fund business before it also fell apart that year because of severe losses on its mortgage derivatives holdings. Fund manager Smirlock was convicted of fraudulently hiding the losses and was sentenced this year to four years in prison.

Sharpe ratios do a poor job of picking up on fat-tail risk for a couple of reasons. First, rare events are, well, rare, which means that they are less likely to show up in the historical returns used to calculate Sharpe ratios. Second, even if a huge loss is averaged into a fund's expected returns, this can mask the unpleasant fact that if one such event occurred, the fund would be wiped out.

Suppose you had invested your life savings in a hedge fund whose only strategy was to sell deep out-of-the-money put options on the S&P 500. Most years the fund would collect its premiums and never pay out anything on the puts. If you paid heed only to the fund's Sharpe ratio, it would seem like a dream investment. "You're going to have a very, very high Sharpe ratio," says Lo.

However, there would be a downside: "Every eight years," notes Lo, "you're going to get wiped out."

The MIT professor compares examining a hedge fund to getting a checkup: "When you go to a doctor, he does not say you're a 95. He gives you a number for blood pressure. He gives you a different number for cholesterol. It's a strange phenomenon that people try to come up with one number in the investment world that captures everything."

But how appropriate is it to compare obscure hedge funds that explicitly sell options with more conventional ones that, for instance, bet on mergers' taking place. Very, as it turns out.

Consider how such risk arbitrage funds make their money: They play the spread between what an acquiring company has offered to pay for a takeover target (per share) and the target's current stock price; in return, the fund accepts the risk that the target's stock may plunge if the deal falls through. That risk-reward scenario is not so dissimilar from selling an option on a merger that would pay out only in the unlikely event that the deal collapsed.

But what a Sharpe ratio for a merger arb fund might not reflect is the rarer risk that the whole merger market might collapse because of some common factor, such as the sudden drying-up of financing for all deals. "The Sharpe ratio is very sensitive to optionslike strategies," points out Yale's Goetzmann. "Many hedge funds essentially sell out-of-the-money options, which every once in a while go boom." As Sharpe succinctly puts it, "A fair number of people in risk arbitrage have a small probability of a really bad outcome."

It's not just merger arbitrageurs that must venture across the occasional minefield. Examine the expected returns of many hedge fund strategies -- from convertible bond arbitrage to distressed investing -- and evidence emerges of optionslike payouts. Although these hedge fund managers may not actually be buying and selling options, they are making optionslike bets on being able to trade in illiquid markets, say, or on being assured that financing will be available for takeover deals. Says one hedge fund manager, "To the extent that a hedge fund engages in any type of options strategy -- as many do either explicitly or implicitly -- its returns would tend to be much less normal than a mutual fund's."

For example, investors have long worried that convertible bond hedge funds would suffer tremendous losses if liquidity were to one day disappear amid a market plunge. In effect, investors have sold a risky option that liquidity won't evaporate.

Other serious risks can slip in under the Sharpe ratio radar. As hedge funds tend to trade illiquid instruments like convertible bonds and mortgage derivatives, the raw data that goes into the funds' Sharpe calculations is sometimes suspect. After all, determining the volatility of instruments that rarely trade leaves ample room for conjecture.

MIT's Lo, who runs his own hedge fund, has found high "serial correlation" in hedge fund returns, that is, one up month tends to be followed by another up month. This smooths returns -- and increases the funds' Sharpe ratios. But since financial assets typically don't generate so neat a statistical pattern, says Lo, investors should consider whether some managers might be using the investments' illiquidity to manage their returns.

Sharpe ratios are so faulty at measuring hedge fund risk, asserts Kat, that investors might do well to turn the theory on its head and stick with hedge funds carrying lower ratios. "You could even go so far as to say that a stellar high historical Sharpe ratio is an indication of an upcoming big loss," he says. Michael Litt of Greenwich, Connecticut­based hedge fund group FrontPoint Partners, says his firm avoids hedge funds with extreme Sharpe ratios because they may carry "undesirable short volatility or event risk exposure."

Would Bill Sharpe rely on his ratio to pick a hedge fund? "Looking at the Sharpe ratio for hedge funds is better than just looking at the average returns," he says. "But I would not go about buying hedge funds on the basis of Sharpe ratios -- it's not enough information."

Sharpe ratios are by and large better at pegging the risk of mutual funds because of their simpler investment strategies. Nonetheless, the ratios can still have a lot trouble hitting the mark here, too.

The trend toward small-cap investing in the U.S. raises special concerns for Sharpe ratio watchers. And the stakes are immense: Before June's stock market plunge sent small investors scrambling for the refuge of the bond markets, many had been redirecting a lot of their money into small-cap stock mutual funds. In 2002's first quarter $8.9 billion of fresh cash flowed into small-cap value funds alone.

Fidelity Investment's Low Priced Stock Fund attracted $2 billion in just three months and in May had to be closed to new investors for six months. With its Sharpe ratio in September of 0.7 -- versus ­0.91 for Fidelity flagship Magellan Fund -- the Low Priced Stock Fund might appear to be a perfectly sensible proposition for investors chasing good risk-adjusted returns. And through September 25 it had lost 11.8 percent for the year, compared with the 19.4 percent for the S&P 500.

Unfortunately, investors consulting only the Sharpe ratios of small-cap funds like LPS may be underestimating their risk. For a start, small-cap funds appear to have the same sort of implicit options-payout structure as hedge funds. In effect, small-cap fund managers are betting that liquidity won't dry up in thinly traded stocks, causing their prices to plunge.

Taking such a gamble to earn excess returns is not unreasonable. But in small-cap funds as in hedge funds, Sharpe ratios may not capture this odd option characteristic's risk factor. Indeed, small-cap funds may deceptively appear to be no more risky than other types of funds. "Our analysis shows that this feature of small-stock returns may enhance their apparent risk-adjusted performance compared to large stocks," notes the draft paper by the Yale professors.

A Fidelity spokesman said the firm does not promote the Sharpe ratios of its mutual funds but gives investors an assortment of statistics to measure risk-adjusted returns. Nor do Sharpe ratios figure prominently in the analyses that Morningstar offers to investors, says the firm's director of research, Paul Kaplan.

Naturally, Sharpe remains a more than interested observer of his ratio, reporting that he reads many, though not all, of the papers written about it. He is philosophical about its misuse: "I take solace in convincing myself that if investors weren't using the Sharpe ratio, they wouldn't take risk into account at all."

Nevertheless, he felt compelled to spell out what he meant by the Sharpe ratio in a 1994 paper in the Journal of Portfolio Management, a publication of Institutional Investor, Inc. It had a cautionary tone. "Clearly, any measure that attempts to summarize even an unbiased prediction of performance with a single number requires a substantial set of assumptions for justification," wrote Sharpe. "In practice, such assumptions are, at best, likely to hold only approximately."

How, in the end, should one assess the risk-return characteristics of an investment? "The way to make investment decisions is to throw all the variables into a computer," says Sharpe. "Sharpe ratios may be a good place to start, but it's not where you want to end up." Or as e-mail pal and Sharpe ratio scourge Kat puts it, "Risk is one word, but it is not one number."

Staying Sharpe

More than three decades after developing the Sharpe ratio, William Sharpe, 68, is still up to his eyeballs in mutual funds and risk.

In 1996 then­Stanford University economics professor Sharpe, Stanford Law School professor Joseph Grundfest and Silicon Valley lawyer Craig Johnson launched Financial Engines, an online retirement planning service for small investors that employs advanced risk management models and user-friendly graphics.

Backed by the likes of American International Group and Goldman, Sachs & Co., the private Palo Alto, California­based company has emerged as one of the more promising independent online financial ventures. Financial Engines has assembled a client roster that includes financial firms like Merrill Lynch & Co. and Vanguard Group and corporate plan sponsors like Boise Cascade Corp. and Merck & Co. Last year the 175-person operation turned its first profit.

Sharpe serves as chairman and highbrow front man while pursuing research on mutual funds. Financial Engines' Web site features his photo with the tag line "Meet the Nobel Prize winner and founder of Financial Engines."

Sharpe's early work remains a bedrock of modern financial theory. Last year Portfolio Theory and Capital Markets, written by Sharpe in 1970, was translated into Chinese. Although he hasn't published new research in a couple of years, he is exploring behavioral finance -- how investors make choices. This past August Sharpe attended the Presidential Economic Forum in Waco, Texas.

"Forget the pie in the sky," he told Institutional Investor in January 2000, not long before the collapse of the technology stock bubble. "You're not going to make 30 percent on stocks."

Sharpe, who retired from Stanford in 1999, shouldn't have too many worries about his own nest egg. Since winning notice in the investment world for his capital asset pricing model in the early 1970s, he has consulted for numerous financial institutions, including Merrill and UBS. Besides heading Financial Engines, Sharpe serves as a trustee of the Axa Rosenberg mutual fund complex and advises a large family investment office. -- H.L.