Fortune's Formula (24 page)

This may or may not sound plausible. There is a point where this power-of-ten business becomes hard to swallow, though. Is there any advantage to having $10 billion instead of a just $1 billion? Not if you’re just concerned with “living well.” Is there then any further glory in possessing $10 trillion over $1 trillion? Not if you’re just interested in being the richest person on earth.

Logarithmic utility is not a good model of poverty, either. It implies that losing 90 percent of your last million is just as painful as losing 90 percent of your last dime. That’s absurd.

In 1936 economist John Burr Williams published an article in the
Quarterly Journal of Economics
titled “Speculation and the Carryover.” The article was about cotton speculators, people who buy excess cotton at a cheap price in hopes of selling it a year or more later at a profit. Speculators “bet” that the next year’s crop will be poor, causing prices to rise. Williams notes the strong element of chance in this activity. No one can predict the weather, for instance. He observes that the successful speculator must have an edge. He must know something that the market does not.

In a “Note on Probability” at the end of the article, Williams says that “if a speculator is in the habit of risking his capital
plus
profits (or losses) in each successive trade, he will choose the geometric rather than the arithmetic mean of all the prices…as the representative price for the distribution of possible prices” in his calculations. Williams does not elaborate on this somewhat cryptic statement. It has much to do with both Bernoulli’s and Kelly’s ideas. Williams was a prominent economist, known for the (now-quaint) idea that stocks can be valued by their dividends. Despite Williams’s reputation, this statement did not get much attention and was quickly forgotten.

T
HE
J
ANUARY
1954
ISSUE
of
Econometrica
carried the first English translation of Bernoulli’s 1738 article mentioning the St. Petersburg wager. Few Western economists read the original, so the full content of the article was not widely known. The translation showed that Bernoulli’s achievement had long been distorted and underrated.

The article was not really about the St. Petersburg wager or utility. Both were mentioned only as asides. Bernoulli’s thesis was that risky ventures should be evaluated by the geometric mean of outcomes.

You may remember from school that there are two kinds of “averages.” The arithmetic average (or mean) is the plain-vanilla kind. It’s what you get when you add up a list of values and divide by the number of values in the list. It’s what batting averages are, and what an Excel spreadsheet calculates when you enter the formula =AVERAGE ( ).

The geometric mean is the one that most people forget after high school. It is calculated by multiplying a list of
n
values together, then taking the
n
th root of the product.

Not many people enjoy taking
n
th roots if they can help it, so the geometric mean is left mostly to statisticians. Of course, nowadays no one computes either kind of average by hand. There is an Excel formula for computing the geometric mean, =GEOMEAN ( ).

The point of any average is to simplify life. It is easier to remember that Manny Ramirez has a batting average of .349 than to memorize every fact about his entire career. A batting average may be more informative about a player’s abilities than a mountain of raw data.

In baseball and much else, the ordinary, arithmetic mean works well enough. Why should we bother with a geometric mean?

Bernoulli starts with gambling. A “fair” wager is one where the expectation, computed as an arithmetic mean of equally likely outcomes, is zero. Here’s an example of a so-called fair wager. You bet your entire net worth on the flip of a coin. You play against your neighbor, who has the same net worth. It’s double or nothing. Winner gets the loser’s house, car, savings,
everything
.

Right now you have $100,000, say. After the coin toss, you will either have $200,000 or $0, each an equally likely outcome. The arithmetic mean is ($200,000 + $0) /2, or $100,000. If you adopt $100,000 as the fair and proper value of this wager, then it might seem you should be indifferent to taking this wager or not. You’ve got $100,000 now, and you expect the same amount after the coin toss. Same difference.

People don’t reason this way. Both you and your neighbor would be nuts to agree to this wager. You have far more to lose by forfeiting everything you have than to gain by doubling your net worth.

Look at the geometric mean. You compute it by multiplying the two equally possible outcomes together—$200,000 times $0—and taking the square root. Since zero times anything is zero, the geometric mean is zero. Accept that as the true value of the wager, and you’ll prefer to stick with your $100,000 net worth.

The geometric mean is almost always less than the arithmetic mean. (The exception is when all the averaged values are identical. Then the two kinds of mean are the same.) This means that the geometric mean is a more conservative way of valuing risky propositions. Bernoulli believed that this conservatism better reflects people’s distaste for risk.

Because the geometric mean is always
less
than the arithmetic mean in a risky venture, “fair” wagers are in fact unfavorable. This, says Bernoulli, is “Nature’s admonition to avoid the dice altogether.” (Bernoulli does not allow for any enjoyment people may get from gambling.)

In Bernoulli’s view, a wager can make sense when the odds are slanted in one’s favor. It can also make sense when the wagering parties differ in wealth. Bernoulli thus solved one of Wall Street’s oldest puzzles. It is said that every time stock is traded, the buyer thinks he’s getting the better of the deal, and so does the seller. The implied point is that they can’t both be right.

Bernoulli challenges that idea. “It may be reasonable for some individuals to invest in a doubtful enterprise and yet be unreasonable for others to do so.” Though he does not mention the stock market, Bernoulli discusses a “Petersburg merchant” who must ship goods from overseas. The merchant is taking a gamble because the ship may sink. One option is to take out insurance on the ship. But insurance is always an unfavorable wager, as measured by the arithmetic mean. The insurance company is making a profit off the premiums.

Bernoulli showed that a relatively poor merchant may improve his geometric mean by buying insurance (even when that insurance is “overpriced”) while at the same time a much wealthier insurance company is also improving its geometric mean by selling that insurance.

Bernoulli maintained that reasonable people are always maximizing the geometric mean of outcomes, even though they don’t know it: “Since all of our propositions harmonize perfectly with experience it would be wrong to neglect them as abstractions resting upon precarious hypotheses.”

There is a deep connection between Bernoulli’s dictum and John Kelly’s 1956 publication. It turns out that Kelly’s prescription can be restated as this simple rule:
When faced with a choice of wagers or investments, choose the one with the highest geometric mean of outcomes.
This rule, of broader application than the
edge
/
odds
Kelly formula for bet size, is the Kelly criterion.

When the possible outcomes are not all equally likely, you need to weight them according to their probability. One way to do that is to maximize the expected logarithm of wealth. Anyone who follows this rule is acting as if he had logarithmic utility.

In view of the chronology, it is reasonable to wonder whether Kelly knew of the Bernoulli article. There is no evidence of it. Kelly does not cite Bernoulli, as he almost certainly would have had he known of Bernoulli’s discussion. As a communications scientist, it is unlikely that Kelly would have read
Econometrica
.

Bernoulli’s article was, however, a direct influence on Henry Latané. It was Latané, not Kelly, who would introduce these ideas to economists.

H
ENRY
L
ATANÉ HAD
the interesting fortune to enter the job market, armed with a Harvard M.B.A., in the grim year of 1930. He claimed to be the last man hired on Wall Street before the Depression. Latané worked as a financial analyst in the 1930s and 1940s. He was the type of person whom Samuelson thought should get a real job, and in a way he took Samuelson’s advice. Well into middle age, Latané quit his Wall Street job and went back to school to earn a Ph.D. He spent the rest of his life as an educator and theorist.

In 1951 Latané began doctoral work on portfolio theory at the University of North Carolina. He read the translated Bernoulli article and realized that its ideas could be applied to stock portfolios. Latané later met Leonard Savage. He convinced Savage that the geometric mean policy made a lot of sense for the long-term investor.

Latané presented this work at a prestigious Cowles Foundation Seminar at Yale on February 17, 1956. Among those attending was Harry Markowitz.

Markowitz was the founder of the dominant school of portfolio theory, known as mean-variance analysis. Markowitz used statistics to show how diversification—buying a number of different stocks, and not having too much in any one—can cut risk.

This idea is so widely accepted that it is easy to forget that sensible people ever thought otherwise. In 1942 John Maynard Keynes wrote, “To suppose that safety-first consists in having a small gamble in a large number of different [companies] where I have no information to reach a good judgment, as compared with a substantial stake in a company where one’s information is adequate, strikes me as a travesty of investment policy.”

Keynes was afflicted with the belief that he could pick stocks better than other people could. Now that Samuelson’s crowd had tossed
that
notion in the dustbin of medieval superstition, Markowitz’s findings had special relevance. You may not be able to beat the market, but at least you can minimize risk, and that’s something. Markowitz used statistics to show, for instance, that by buying twenty to thirty stocks in different industries, an investor can cut the overall portfolio’s risk by about half.

Markowitz saw that even a perfectly efficient market cannot grind away all differences between stocks. Some stocks are intrinsically riskier than others. Since people don’t like risk, the market adjusts for that by setting a lower price. This means that the average return on investment of risky stocks is higher.

As the name indicates, mean-variance analysis focuses on two statistics computed from historical stock price data. The mean is the average annual return. It is a regular, arithmetic average. The variance measures how much this return jumps around the mean from year to year. No equity investment is going to have the same return every year. A stock may gain 12 percent one year, lose 22 percent the next, gain 6 percent the next. The more volatile the stock’s returns, the higher its variance. Variance is thus a loose measure of risk.

For the first time, Markowitz concisely laid out the trade-off between risk and return. His theory pointedly refuses to take sides, though. Risk and return are apples and oranges. Is higher return more important than lower risk? That is a matter of personal taste in Markowitz’s theory.

Consequently, mean-variance analysis does not tell you which portfolio to buy. Instead, it offers this criterion for choosing: One portfolio is better than another one when it offers higher mean return
for a given level of volatility
—or a lower volatility
for a given level of return
.

This rule lets you eliminate many possible portfolios. If portfolio A is better than portfolio B by the rule above, then you can cross out B. After you eliminate as many portfolios as possible, the ones that are left are called “efficient.” Markowitz got that term from a mentor who did efficiency studies for industry.

Markowitz made charts of mean vs. variance. Any stock or portfolio is a dot in the chart. When you erase all the dots rejected by the above rule, the surviving portfolios form an arc of dots that Markowitz called the “efficient frontier.” It will range from more conservative portfolios with lower return to riskier portfolios with higher return.

Financial advisers responded to Markowitz’s model. They were growing aware of this new and threatening current in academic thought: the efficient market hypothesis. Markowitz demonstrated that all portfolios are
not
alike when you factor in risk. Therefore, even in an efficient market, there is reason for investors to pay handsomely for financial advice. Mean-variance analysis quickly swept through the financial profession and academia alike, establishing itself as orthodoxy.

Latané’s 1957 doctoral dissertation treats the problem of choosing a stock portfolio. This is something that Bernoulli did not do, and that Kelly alluded to only vaguely, in the midst of a lot of talk about horse races and entropy. With Savage’s encouragement, Latané published this work in 1959, three years after Kelly’s article, as “Criteria for Choice Among Risky Ventures.” It appeared in the
Journal of Political Economy
.

It’s unlikely that any of the article’s readers had heard of John Kelly. Latané himself had not heard of Kelly at the time of the Cowles seminar.

Latané called his approach to portfolio design the geometric mean criterion. He demonstrated that it is a
myopic
strategy. A “near-sighted” strategy sounds like a bad thing, but as economists use it, it’s good. It means that you don’t have to have a crystal ball on what the market is going to do in the future in order to make good decisions now. This is important because the market is always changing.

The “myopia” of the geometric mean (or Kelly) criterion is all-important in blackjack. You decide how much to bet now based on the composition of the deck now. The deck will change in the future, but that doesn’t matter. Even if you
did
know the future history of the deck’s composition, it wouldn’t bear on what to do now. So it is with portfolios. The best you can do right now is to choose a portfolio with the highest geometric mean of the probability distribution of outcomes, as computed from current means, variances, and other statistics. The returns and volatility of your investments will change with time. When they do, you should adjust your portfolio accordingly, again with the sole objective of attaining the highest geometric mean.

Also in 1959, Harry Markowitz published his famous book on
Portfolio Selection
.
Everyone
in finance read that, or said they did. Markowitz told me he first became aware of Latané’s work in the 1955–56 academic year, when James Tobin gave him a copy of an early version of Latané’s article. Markowitz devoted a chapter of
Portfolio Selection
to the geometric mean criterion (possibly the most ignored chapter in the book) and cited Latané’s work in the bibliography.

Markowitz was virtually the only big-name economist to see much merit in the geometric mean criterion. He recognized that mean-variance analysis is a static, single-period theory. In effect, it assumes that you plan to buy some stocks now and sell them at the end of a given time frame. Markowitz theory tries to balance risk and return for that single period.

Most people do not invest this way. They buy stocks and bonds and hang on to them until they have a strong reason to sell. Market bets ride, by default. This makes a difference because there are gambles that look favorable as a one-shot, yet are ruinous when repeated over and over. Any type of extreme “overbetting” on a favorable wager would fit that description.

The geometric mean criterion can also resolve the
Hamlet
-like indecision of mean-variance analysis. It singles out one portfolio as “best.” Markowitz noted that the geometric mean can be estimated from the standard (arithmetic) mean and variance. The geometric mean is approximately the arithmetic mean minus one-half the variance. This estimate may be made more precise by incorporating further statistical measures.

One additional name must be added as codiscoverer/midwife of the Kelly or geometric mean criterion. In 1960 statistician Leo Breiman published “Investment Policies for Expanding Businesses Optimal in a Long-Run Sense.” This appeared in a publication as unlikely as the
Bell System Technical Journal
, namely the
Naval Research Logistics Quarterly
. Breiman was the first to show that maximizing the geometric mean minimizes the time to achieve a particular wealth goal. Who wants to be a millionaire? Breiman showed that a gambler/investor will reach that (or any other) wealth goal faster using the geometric mean criterion than by using any fundamentally different way of managing money.

Because of this complex lineage, the Kelly criterion has gone by a welter of names. Not surprisingly, Henry Latané never used “Kelly criterion.” He favored “geometric mean principle.” He occasionally abbreviated that to the catchier “G policy” or even, simply, to “G.” Breiman used “capital growth criterion,” and the innocuous-sounding “capital growth theory” is also heard. Markowitz used MEL, for “maximize expected logarithm” of wealth. In one article, Thorp called it the “Kelly[-Breiman-Bernoulli-Latané or capital growth] criterion.” This is not counting the yet-more-numerous discussions of logarithmic utility. This confusion of names had made it relatively difficult for the uninitiated to follow the idea in the economic literature.

The person most shortchanged by this nomenclature is probably Daniel Bernoulli. He had 218 years’ priority on Kelly. The unique and unprecedented part of Kelly’s article is the connection between inside information and capital growth. This is a connection that could not have been made before Shannon rendered information measurable. Bernoulli considers a world where all the cards are on the table, so to speak, and all the probabilities are public knowledge. There is no hidden information. Kelly treats a darker, more ambiguous world where some people know the probabilities better than others and attempt to profit from that knowledge over time. It is this feature particularly that has much to say about the financial markets.