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Sum Games - Negative, Positive, & Zero Sum Games

True or false: Buying a single company's stock usually provides a safer return than a stock mutual fund.
  1. True
  2. False
  3. Don't Know

     In economic theory, a zero-sum game refers to a situation in which all the players' gains and losses offset each other and sum to zero. In other words for someone to have gains, someone else needs to lose. In contrast, a non-zero-sum game describes a situation in which the interacting parties' aggregate gains and losses can be less than, or more than zero.

     Casino games are generally negative sum games. Yet there are professional gamblers that consistently (or persistently) make money at the expense of other gamblers. This occurs only in games of skill like poker, where weak or suboptimal players provide enough of an opportunity for professionals to overcome the costs (or vigorish). But there are no professional gamblers in negative sum games like craps and roulette that don't involve skill.

     In the July/August 1975 issue of Financial Analysts Journal, Charles Ellis published an article titled “The Loser's Game,” which was followed by his book titled Investment Policy: How to Win the Loser's Game. Ellis cited Simon Ramo's analysis of tennis, in which he argued that professional tennis is a winner’s game (where the winner tends to be the player that can hit winning shots), while amateur tennis tends to be a loser’s game (where the winner is the player that makes the fewest unforced errors or mistakes). Therefore in a loser's game the goal should be to keep the ball in play until your opponent makes a mistake, rather than trying to hit unreturnable winners. Ellis then compared tennis to investing (specifically active investors) and argued that investing has become a loser's game. Thus he argued the logical goal for investors should be to use passive strategies, keep costs to a minimum, and avoid mistakes, rather than trying to beat the market.

     The argument that active investing is speculation relative to passive investing is not a new argument and there are multiple proofs in prior writings (some of which I discussed in chapter five). For instance, William Sharpe's “The Arithmetic of Active Management”1 summarized the mathematics and Steven Thorley elaborated on the argument in a paper titled “The Inefficient Market Argument for Passive Investing.”2.

     Similarly Richard Ferri argued in Forbes in 2010 that Active Investing is “uncompensated risk.” Active managers take additional risks in attempting to outperform the market or index, but they arguably do not increase their expected returns while incurring additional costs.3 Others argue they do increase their expected returns because different risk factors have higher expected returns, but let’s hold off on that discussion for later when we discuss anomalies and factors in more depth (in chapters 20-24).

     Sum games can be viewed as analogous to the discussion of investment, speculation, and gambling. Investing is analogous to participating in a positive sum game, while participating in a zero-sum game is speculating. Participating in a negative sum game where no skill is involved is gambling, unless it is for insurance purposes, which is a separate, but related discussion.

     Gamblers intentionally assume risk for thrills, excitement, or other reasons. Speculators only assume risk when they believe they have identified an advantage (or an opponent’s weakness) that compensates for the risk and their costs. Or alternatively, gamblers and speculators get some pleasure or value from the process of speculating, which compensates for the negative expected returns. Investors however avoid risk and don't allocate any entertainment value to the process.

“Although the stock market may be somewhat of a casino, it’s a lot better than Atlantic City or Las Vegas because the odds are in your favor-—there’s a long-term uptrend in the stock market. Do I buy some individual stocks? Yes, because it’s fun. I also go to Las Vegas and Atlantic City. But as a trustee for family trusts, or a member of foundation investment committees, and in my own 403(b) account, I believe in indexing stocks, bonds, and real estate.”
Burton Malkiel4

     Purchasing a security, a stock for example, is effectively an active investment that can be measured against the performance of the stock market itself (and purchasing a corporate bond can be measured against an index of comparable corporate bonds). When compared to a passive investment in a stock index, the purchase of an individual stock can be viewed as a combination of an asset allocation to stocks and an active investment in that stock with the belief that it will outperform the stock index.

     Arguments can be made for both active and passive investing but a much larger percentage of institutional investors invest passively relative to individual investors. The arguments for passive investing include reduced costs, tax efficiency, and the fact that historically, passive funds outperform a majority of active funds. The arguments for active investing are that there are anomalies in securities markets that can be exploited to outperform passive investments and the fact that some investors and managers have outperformed passive investing for long periods of time (although the odds of being able to identify them in advance are not good). Active management is ultimately a negative sum game since it has associated costs, and is thus speculative. But, since active management also involves skill, perhaps some active managers can overcome the costs, just like some professional poker players seem to consistently make money at the expense of other poker players.

     Investors can also combine active and passive investing by investing part of a portfolio passively and another part actively (for example you can invest half of your stock allocation in an index fund and the other half in active funds). Investors can also invest actively in sectors, in a passive manner. For example, you can invest in an index fund or ETF of small stocks if you think small stocks will outperform large stocks, or you can invest in a passive country fund or ETF if you believe a particular country will outperform the region or the rest of the world.

Active management will never completely go out of business because it’s far too lucrative and tempting for type-A personalities to prove themselves.
Ben Carlson5

     Why do active managers succeed or fail? There are two main factors that determine success in active management. The first is whether the manager outperforms on a pre-cost basis either due to skill or luck. Determining which is responsible takes time and many observations. The second major factor is the costs. If we start with 100 managers, on a pre-cost basis we would expect 50 to outperform due to skill/luck and 50 to underperform. With costs added, the number of active managers that underperform will increase over time. Typically over one year, roughly 8 of 100 active managers that outperform before costs will underperform after costs (more for higher costs, less for lower costs). If we go out to a much longer period, say 20 years the number that will underperform due to costs will likely rise to roughly 35. In other words, over long periods of time (like 20 years) we can expect 50% of active stock investors to underperform because they have no skill or are unlucky, and 35% (more than 1/3) will underperform because of costs. These 35% would have outperformed if not for the costs. The 50% of managers that underperformed before costs have both their losses and the costs on top of them. And finally the 15% that succeeded have their returns, but less the costs. Adding insult to injury the typical correlation between an active stock fund manager and the broad market tends to be fairly high (often more than 80%). That means the costs incurred by active managers are getting over 80% of their performance from market returns, but paying active management costs on that 80% in addition to the 20% that is less correlated with the benchmark.

     Discussions related to success rates in active management are fairly common in performance reporting. For instance, at year end a frequent discussion is the percentage of funds that beat their respective benchmarks like the S&P 500, Wilshire 5000, regional, and style benchmarks. But there have not been nearly as many thorough discussions of predicting success rates and odds in active investing before the fact.

     The earliest discussion that I'm aware of in the academic research was in “The Evolution of Passive versus Active Equity Management” by Larry Martin in The Spring 1993 issue of The Journal of Investing. The relevant probability calculations were cited and summarized in Fact and Fantasy in Index Investing by Eric Kirzner (January 2000).6 The "Probability of Active Management Outperforming an Index" for a single manager was 41% in 1 year, 29% in 5 years, 22% in 10 years, and 14% in 20 years.

     In How A Second Grader Beats Wall Street, Allan Roth also discussed his analysis of this topic. After being asked to create a Monte Carlo simulation for Jack Bogle, Roth ran a simulation of mutual funds versus index funds. Roth did not list all his assumptions (varying past performance, cost, volatility, and correlations can significantly affect the results), but Roth did state that the index fund had a 0.23 percent total expense and the average mutual fund or separately managed account had a 2.00 percent expense ratio.7

     Roth's estimate was that one active fund has a 42% chance of beating an index fund over a 1 year period, 30% over a 5 year period, 23% over a 10 year period, and 12% over a 25 year period. In other words, the Martin and Roth probabilities were virtually identical. Roth’s Monte Carlo analysis compared active funds to an index fund rather than an index, and Martin's was a statistical estimate. Yet despite those minor differences and at least partially different time periods, the similarity in the results suggests the results are good estimates and they tend to be confirmed by actual results.

     Martin and Roth both came up with even worse odds if you have multiple funds or managers. Roth also analyzed a higher cost differential to account for tax implications and emotions. Numerous studies have shown that investors get even worse dollar weighted returns than time weighted returns because they tend to buy and sell at the wrong times (more on that in Chapter 11). There are many sources of actual results and one of the most useful is Standard & Poor's Indices Versus Active reports.8

     Once in a while more than half of managers in a specific category may beat their benchmark, for instance in 2017 63% of large cap managers beat their benchmark according to S&P. But for the longer term, the data was much worse for managers in that period. Over the five-year period, 84% of large-cap managers, 85% of mid-cap managers, and 91% of small-cap managers lagged their respective benchmarks, while over the 15-year investment horizon, 92% of large-cap managers, 94% of mid-cap managers, and 95% of small-cap managers failed to outperform on a relative basis.”9

     I began researching the probabilities and odds of active management in the late nineties and the specific piece of information that kick started me was a section in the second edition of Jeremy Siegel's book Stocks for the Long Run. Siegel included a table with holding periods ranging from 1 to 30 years and expected excess return ranging from 1-5%. Siegel used the table to discuss finding skilled managers. In other words, given an expected excess return over the market and a specific number of years, his numbers project how statistically certain you could be that the performance was due to skill and not luck. Siegel's assumptions (based on data from 1971 to 1996) were a 14 Percent Expected Return, 16.6 Percent Standard Deviation, and a 0.88 correlation coefficient. The return assumption is high, but given the length of the period, the assumptions should be reasonable for projecting the future.

     While Siegel's discussion is interesting regarding the luck vs. skill debate (more on that in the next chapter), my twist on the topic is that it is a good tool for evaluating probabilities of success in active management if you substitute the investment costs (negative) for the expected excess return (positive). In other words, Siegel's expected alpha relative to the market is analogous to the inverted disadvantage of active investors due to costs. I see the luck/skill analysis and probabilities of success in active management as opposite sides of the same coin.

     For example, with a 2% expected excess return, Siegel's numbers suggest that you can be 70% certain after five years (and 90% after 30 years) that outperformance was due to skill and there is a 30% chance it was due to luck. Using Siegel's data, I would say an investor (or fund, or manager) with a 2% cost disadvantage has a 30% chance of beating the benchmark after five years and 10% after 30 years.

     At a 2% expected excess return, after one year Siegel's estimate is 59%, five years 70%, 10 years 77%, and 20 years 85%. Assuming an inverse -2% for costs, we get 40%, 30%, 23%, and 15% projected probabilities of outperformance. Comparing those rates, to the Martin and Roth estimates, we find that again they are virtually identical. The Fourth Edition of Stocks For The Long Run (2014) included an updated version of the discussion (with more years of data), but does not include the assumptions. In the fourth edition, the probabilities were generally closer to 50% (+2% for 5 years changed from 70% to 67% and 30 years changed from 90% to 85%).

     The assumptions are important, but ultimately this is a game of estimation. Yet we have a lot of empirical data to verify whether the numbers pass the smell test and in general they do quite well. We can't expect these to be perfect estimates for several reasons. For one, manager behavior changes over time depending on how well they do. Managers that do well tend to attract more assets and tend to become more index-like due to either bloated assets, reduced risk tolerance from an incentive to stabilize their income, and the desire to not risk underperforming when they already have a lead on their benchmark. If a manager is ahead of the benchmark, they only need to match it going forward to have a long term track record of outperformance. Therefore so-called closet indexing is common. But managers that fall behind tend to increase their risk and position sizes in an attempt to get positive, since they are more likely to lose assets or get fired if they maintain their underperformance. Plus there are often staff departures, and the possibility of changing conditions over time. Next we'll take a look at the role of luck and whether past results predict future results.

     Regarding the question at the start of this chapter, only 46% of participants in a 2015 survey correctly chose false for the statement that buying a single company's stock usually provides a safer return than a stock mutual fund. 10% incorrectly chose true, while 44% acknowledged they did not know.10 The explanation of the correct answer to the Financial Literacy Quiz is the following.

“In general, investing in a stock mutual fund is less risky than investing in a single stock because mutual funds offer a way to diversify. Diversification means spreading your risk by spreading your investments. With a single stock, all your eggs are in one basket. If the price falls when you sell, you lose money. With a mutual fund that invests in the stocks of dozens (or even hundreds) of companies, you lower the chances that a price decline for any single stock will impact your return. Diversification generally may result in a more consistent performance in different market conditions.”11

Notes - The Footnotes in the Book are sequential and for this chapter start at #179 and end at #189.

1. William Sharpe, “The Arithmetic of Active Management” The Financial Analysts' Journal, January/February 1991
4. Burton Malkiel “Market Efficiency and Active Management: A Non-Random Talk with Burton G. Malkiel, Ph. D.” The Journal of Investment Consulting, Winter 2003/2004.
5. Ben Carlson, Debunking the Silly “Passive is a Bubble” Myth, September 5, 2019
7. Richard Ferri summarized the results in “A Winning Fund Doesn't Equal A Winning Portfolio” in Forbes (4/22/2010)

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Table of Contents and Launch Site

Last update 12/31/2019. Copyright © 2019 Gary Karz. All rights reserved.
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