In addition, the five players with the most twenty-game hitting streaks in history—Pete Rose, Ty Cobb, Tris Speaker, Heinie Manush, and Chuck Klein—have a combined lifetime batting average of .333. Over time, the batting average in baseball has hovered around .260.4
The one streak in sports that defies the probabilities is Joe DiMaggio’s fifty-six-game hitting streak in 1941. (The longest streak after DiMaggio’s is 44 games, 80 percent of DiMaggio’s record, achieved by both Pete Rose and Wee Willie Keeler.) Ed Purcell, a Nobel laureate in physics, combed baseball’s streak and slump records and concluded that everything that has happened in baseball was within the realm of probability—except DiMaggio’s streak.5
Granted, DiMaggio was a great hitter—his lifetime batting average is the twenty-seventh best in baseball history—but the likelihood of his streak was less than a one in a million, even for him.6 For this reason, most statistically oriented baseball fans believe that DiMaggio’s streak is the record least likely to be broken.7
Toss Out the Coin Toss
Most finance professionals attribute money manager streaks (consecutive years of benchmark outperformance) to luck. For example, finance teachers enthusiastically invoke a coin-tossing metaphor to demonstrate market efficiency.8 The basic idea is that if you start with a sufficiently large sample of money managers, the probabilities tell you a priori that some will have a streak of outperformance. Start with a group of, say, 1,000 funds, assume a fifty/fifty chance of beating the market, and roughly thirty funds will outperform five years in a row: (0.5)5 x 1,000.
There is nothing wrong with this logic as far as it goes. The problem is that not all fund managers are of equal skill—the money-management industry has its versions of Sally Swish and Allen Airball. So attributing any fund streak to chance misses the point that skilled participants are the most likely to post a streak.
The streak that has garnered the most attention in the mutual fund world is that of Legg Mason’s Bill Miller, whose Value Trust fund managed to outperform the S&P 500 for fifteen consecutive years through 2005. No other fund has ever outperformed the market for that long in the last forty years. What are the odds of that?
Some pundits are perfectly satisfied to chalk up Miller’s record to chance. For example, Gregory Baer and Gary Gensler write: “While we are happy for Legg Mason and its manager, Bill Miller, we view that outcome as roughly in line with random chance and as an indictment of active management.”9 More incredible is the comment (quoted at top) by well-regarded bond manager Bill Gross. In 2003, when the streak was at twelve years, Gross “snarled” that Miller’s performance is equivalent to rolling twelve sevens in a row with a pair of dice. We can only hope that Gross, who has a great investment track record and familiarity with gambling, was misquoted: The odds of rolling twelve sevens in a row are approximately 1 in 2.2 billion.
We can look at Miller’s streak two ways. The first assumes that a constant percentage of funds outperform the market each year. We can then select a percentage and calculate the probability of one fund outperforming each and every year (see exhibit 7.1). For example, if you assume that mutual fund performance is essentially a coin toss—half of all funds beat the market and half underperform—the odds of one fund beating the market for fifteen consecutive years is 1 in 32,768. Given that there were only 900 comparable mutual funds at the beginning of Miller’s streak, his performance looks impressive.
EXHIBIT 7.1 Probability That One Fund Will Outperform Each Year
Source: Author analysis.
The problem with this analysis, though, is that outperforming the market is not a fifty/fifty proposition for the average mutual fund. In fact, the average percentage of outperformance over the past fifteen years was 44 percent. If we assume a 44 percent ratio, the probability of one fund outperforming for fifteen years is roughly 1 in 223,000.10
The second way of looking at Miller’s streak is to look at the actual percentages of funds that beat the market in each year (see exhibit 7.2). This allows us to determine the cumulative probability given what actually happened. This calculation shows that the probability of beating the market fifteen years in a row (ended 2005) was about 1 in 2.3 million. A quick glance at the numbers shows why the odds are so low. Two years, 1995 and 1997, create the camel-through-the-needle’s-eye probability, as only about 10 percent of all funds beat the market in those two years.
EXHIBIT 7.2 Percentage of Funds That Beat the S&P 500, 1991-2006
YearFundsPercent that Beat S&P 500
1991 889 47.7
1992 1,018 50.9
1993 1,289 72.0
1994 1,733 24.0
1995 2,325 12.6
1996 2,894 20.7
1997 3,761 7.9
1998 4,831 26.1
1999 5,873 51.4
2000 6,966 62.2
2001 8,460 49.7
2002 9,749 58.7
2003 10,780 56.7
2004 11,466 54.9
2005 11,329 67.1
2006 12,500 38.3
Source: Lipper Analytical Services and author analysis.
Streaks and Luck
In money management, the magnitude of market outperformance (adjusted for risk) is the true bottom line. But streaks are intriguing because they are without exception—they allow no bad years. Further, as the streak lengthens, the tension and pressure mount.
Was Miller lucky along the way? Without a doubt. But as Stephen Jay Gould says, long streaks are extraordinary luck imposed on great skill.11 The central message is that across domains, long streaks typically indicate skill. And since humans have a hard time relating to all but the easiest probabilities, we often fail to see the significance of streaks.
8
Time Is on My Side
Myopic Loss Aversion and Portfolio Turnover
The attractiveness of the risky asset depends on the time horizon of the investor. An investor who is prepared to wait a long time before evaluating the outcome of the investment as a gain or a loss will find the risky asset more attractive than another investor who expects to evaluate the outcome soon.
—Richard H. Thaler, Amos Tversky, Daniel Kahneman, and Alan Schwartz, “The Effect of Myopia and Loss Aversion on Risk Taking: An Experimental Test”
Loss aversion . . . can be considered a fact of life. In contrast, the frequency of evaluations is a policy choice that presumably could be altered, at least in principle.
—Shlomo Benartzi and Richard H. Thaler, “Myopic Loss Aversion and the Equity Premium Puzzle”
One or One Hundred
In the early 1960s, economist Paul Samuelson offered his lunch colleagues a bet where he would pay $200 for a correct call of a fair coin toss and he would collect $100 for an incorrect call. But his partners didn’t bite. One distinguished scholar replied, “I won’t bet because I would feel the $100 loss more than the $200 gain. But I’ll take you on if you promise to let me make 100 such bets” (emphasis added).
This response prompted Samuelson to prove a theorem showing that “no sequence is acceptable if each of its single plays is not acceptable.” According to economic theory, his learned colleague’s answer was irrational.1
Even though the lunch bet has a positive expected value, Samuelson’s proof doesn’t feel quite right to most people. The concept of loss aversion explains why. One of prospect theory’s main findings, loss aversion says that given a choice between risky outcomes we are about two times as averse to losses than to comparable gains.2
So Samuelson’s theoretical proof notwithstanding, most people intuitively agree with his lunch partner: The prospective regret of losing $100 on a single toss exceeds the pleasure of winning $200. An opportunity to take the bet repeatedly, on the other hand, seems sensible because there are lower odds of suffering regret.
One significant difference between expected-utility theory (the basis for Samuelson’s proof ) and prospect theory is the decision frame. Expected-utility theory considers gains and losses in the context of the investor’s
total wealth (broad frame). In contrast, prospect theory considers gains and losses versus isolated components of wealth, like changes in a specific stock or a portfolio price (narrow frame). Experimental studies show that investors use price, or changes in price, as a reference point when evaluating financial transactions. Said differently, investors pay attention to the narrow frame.3
If prospect theory does indeed explain investor behavior, the probabilities of a stock (or portfolio) rising and the investment-evaluation period become paramount. I want to shine a light on the policies regarding these two variables.
Explaining the Equity-Risk Premium
One of finance’s big puzzles is why equity returns have been so much higher than fixed-income returns over time, given the respective risk of each asset class. From 1900 through 2006, stocks in the United States have earned a 5.7 percent annual premium over treasury bills (geometric returns). Other developed countries around the world have seen similar results.4
In a trailblazing 1995 paper, Shlomo Benartzi and Richard Thaler suggested a solution to the equity risk premium puzzle based on what they called “myopic loss aversion.” Their argument rests on two conceptual pillars:5 1. Loss aversion. We regret losses two to two and a half times more than similar-sized gains. Since the stock price is generally the frame of reference, the probability of loss or gain is important. Naturally, the longer the holding period in a financial market the higher the probability of a positive return. (Financial markets must have a positive expected return to lure capital, since investors must forgo current consumption.)
2. Myopia. The more frequently we evaluate our portfolios, the more likely we are to see losses and hence suffer from loss aversion. Inversely, the less frequently investors evaluate their portfolios, the more likely they are to see gains.
Exhibit 8.1 provides some numbers to illustrate these concepts.6 The basis for this analysis is an annual geometric mean return of 10 percent and a standard deviation of 20.5 percent (nearly identical to the actual mean and standard deviation from 1926 through 2006).7 The table also assumes that stock prices follow a random walk (an imperfect but workable assumption) and a loss-aversion factor of 2. (Utility = Probability of a price increase - probability of a decline x 2.)
EXHIBIT 8.1 Time, Returns, and Utility
Source: Author analysis.
A glance at the exhibit shows that the probability of a gain or a loss in the very short term is close to fifty/fifty. Further, positive utility—essentially the avoidance of loss aversion—requires a holding period of nearly one year.
If Benartzi and Thaler are right, the implication is critical: Long-term investors (individuals who evaluate their portfolios infrequently) are willing to pay more for an identical risky asset than short-term investors (frequent evaluation). Valuation depends on your time horizon.
This may be why many long-term investors say they don’t care about volatility. Immune to short-term squiggles, these investors hold stocks long enough to get an attractive probability of a return and, hence, a positive utility.
Benartzi and Thaler, using a number of simulation approaches, estimate that the evaluation period consistent with the realized equity-risk premium is about one year. It is important to note that the evaluation period is not the same as the investor’s planning horizon. An investor may be saving for retirement thirty years from now, but if she evaluates her portfolio (or more accurately, experiences the utility of the gains and losses) annually or quarterly, she is acting as if she has a short-term planning horizon.8
I will now make a leap (and hopefully it’s not too far) and suggest that for most funds, portfolio turnover is a reasonable proxy for the evaluation period. High turnover would be consistent with seeking gains in a relatively short time, and low turnover suggests a willingness to wait to assess gains and losses. For many successful funds (and companies), the evaluation period is a policy choice. And as Warren Buffett says, you eventually get the shareholders you deserve.
The Value of Inactivity
We now turn to the empirical data on the relationship between portfolio turnover and performance. We separate mutual funds into four types based on portfolio-turnover rate. The data consistently show that the low-turnover funds (which imply two-year-plus investor holding periods) perform best over three-, five-, ten-, and fifteen-year time frames (see exhibit 8.2).
We may be able to attribute this performance difference to lower costs—a reason in and of itself to reduce turnover for many portfolios—but we would note that transaction costs tend to represent only about one-third of total costs for the average mutual fund.
Despite consistent evidence supporting the performance benefits of a buy-and-hold strategy, the average actively managed mutual fund has annual turnover nearly 90 percent. What gives? First off, an efficient stock market requires investor diversity—across styles and time horizons. Not everyone can, or should, be a long-term investor. This fallacy of composition is the flaw behind the “Dow 36,000” theory, which argues that if all investors adopt a long-term horizon, the equity-risk premium will dissipate and the market will enjoy a onetime rise.9 Changing the nature of the investors changes the nature of the market. If all investors were long-term oriented, the market would suffer a diversity breakdown and hence be less efficient than today’s market.
EXHIBIT 8.2 Portfolio Turnover and Long-Term Performance
Source: Author analysis, Morningstar, Inc.
EXHIBIT 8.3 Aggregate Return and Standard Deviation
Source: Author analysis.
A second and much more profound reason for high turnover is agency costs. Studies show that a portfolio of stocks trading below expected value will outperform the market (adjusted for risk) over time. But because there is such a focus on outcome versus process, most institutional investors have time horizons that are substantially shorter than what an investment strategy requires to pay off.
Portfolio managers who underperform the market risk losing assets, and ultimately their jobs.10 So their natural reaction is to minimize tracking error versus a benchmark. Many portfolio managers won’t buy a controversial stock that they think will be attractive over a three-year horizon because they have no idea whether or not the stock will perform well over a three-month horizon. This may explain some of the overreaction we see in markets and shows why myopic loss aversion may be an important source of inefficiency.
EXHIBIT 8.4 Ratio of Standard Deviation to Return
Source: Author analysis.
EXHIBIT 8.5 Time and the Probability of Gain
Source: Author analysis.
EXHIBIT 8.6 Utility Index
Source: Author analysis.
Pictures Worth a Thousand Words
Exhibits 8.3 through 8.6 recreate some of investment sage William Bernstein’s pictures to help quantify the key ideas behind myopic loss aversion.11
Exhibit 8.3 shows the relationship between risk and reward. Because risk (measured as standard deviation) increases as a function of the square root of time but reward (measured as return) compounds with time, there is a sharp inflection point in the risk and reward trade-off. Note that the axes are on a log scale.
Another way to look at the same picture is to show a plot of the ratio of risk to reward—that is, standard deviation divided by return (exhibit 8.4).
We can now look at the probability of a positive outcome. Given the assumed underlying statistical properties, exhibit 8.5 shows how the probability that the investment will be up increases with time. If investors use gains and losses versus a purchase price as their frame of reference, this picture reveals the relationship between time and regret.
Based on the probabilities in exhibit 8.5, and assuming that losses have twice the impact of comparable gains, we can plot a simple utility function (exhibit 8.6). The scale ranges from -2.0 (a 100 percent chance of a loss x 2) to 1.0 (a 100 percent chance of a gain).
9
The Low Down on the Top Brass
Management Evaluation and the Investme
nt Process
At our annual meetings, someone usually asks “What happens to this place if you get hit by a truck?” I’m glad they are still asking the question in this form. It won’t be too long before the query becomes: “What happens to this place if you don’t get hit by a truck?”
—Warren E. Buffett, Berkshire Hathaway Annual Letter to Shareholders, 19931
Level 5 leaders channel their ego needs away from themselves and into the larger goal of building a great company. It’s not that Level 5 leaders have no ego or self-interest. Indeed, they are incredibly ambitious—but their ambition is first and foremost for the institution, not themselves.
—Jim Collins, Good to Great2
Management Counts
“Is assessing management important in the investment process?” is one of the most frequent questions I get from clients and students.
The answer is a qualified, but emphatic, yes. I suggest three areas for careful consideration: management’s leadership, incentives, and capital allocation skills. I do not purport that this discussion is all encompassing—indeed, dedicated scholars have written countless articles and books on each of these topics. My more modest goal is to stimulate thought in this vital, yet often overlooked, area.
More Than You Know Page 6
