The forever elusive α

Note: individuals wanting more information on BRK may be interested in this information here.

Humans have been repeating this inefficient ritual for over 700 years, with the first known origins then in Europe.  There sprung lenders and insurers who assessed the relative merits of individual commercial risk and were able to loan coins in exchange for sometimes usury interest rates.  The methods were somewhat more crude versus the resources available to people today, but none-the-less this is the humble birthplace from where modern investment speculation gets its origin.  What should be the effective interest rate to lend an emerging company wanting to complete a construction project?  What should an insurer charge to protect a ship voyaging across a stormy sea, so that the premium pricing is both attractively profitable yet competitive?

Over time, more information was rapidly made available concerning those who needed capital market resources.  And more ordinary people were able to invest in companies and products.  Through the distribution of personal wealth and technological progress, society experienced episodic bouts of speculations and manias.  The conversion of defined benefit plans in the U.S. to one where American workers invest their own contributions, combined with draining real median wage growth, created a force for even greater heterogeneity of outcomes in the desperate and greedy individual pursuit of α (alpha).  And then the digital age took these advances to another level, now allowing virtually everyone to more quickly and easily trade however they want.  But how can these seeming innovations be good for society, if there is a slimmer portion of risk-adjusted beneficiaries?  Let’s explore the outcomes and difficulties in the great, inefficient search for exceptional alpha.

The true statistical test for outperformance relative to a highly liquid, investable, and specified a-priori benchmark fully takes into account how likely such performance could have been attained by luck (this is a favorable outcome by a small segment of generally haphazard performers).  This of course would be a stricter, though parallel, test therefore than just alpha.  Afterall over any period of time, there will be separation in the market fates of individual stocks within a basket.  Concurrently, some purely lucky stock holders will too own specific stocks that just happens to outperform the underlying index over this same period of time.

Nonetheless it is worth noting that the difficult statistical standard necessary to warrant the concept of skill over a long career, or life, has a smaller side effect.  And that is that only minorities of those who speculate will actually have, over time, outperformed the broader stock index with skill.  While we don't aim for precision in this note at any one data point, we attempt to provide the basic sense of less outpeformance is needed to claim a fixed rank of skill within a population, as one's investment experience is lengthier.  And at the same time, how the probability of outperformance reduces over time, for a fixed level of outperformance.  Put together, even with a small reduction in the amount of outperformance needed, as one continues to speculate in the markets: the portion of the population who outperform reduces, more than countering the greater share (of this reduced portion) that can be attributed to skill versus luck.  As a different extreme example, if we state flipping a fair coin and having it land on heads 100% of the time suggests outperformance (we're suspending the break-out here of the portion with skill), then 25% of people can still do this after 2 flips, however well less than 1/2% can do this after 50 flips.  On the other hand, if we reduce the requirement to stating that landing on heads 54% of the time suggests outperformance, then still 25% of people can do this after 50 flips.  Hence the requirements are more strict at the short-end of the experiment number of trials, and there could even be a blend of a slightly greater portion of people demonstrating outperformance at this short-end if the performance requirements are strengthened but not by enough.

Let’s show how this works, using the time since the recent financial crisis as a baseline frame for this analysis.  From there we’ll statistically expand to a broader set of applications and timeframes.  The market has gone through a large hockey-stick pattern since the height of the financial crisis, 5.5 years ago.  Equity markets initially plummeted through early 2009, but have since smoothly rallied to new highs.  Nearly all holdings have gone up.

If you and your friends had all tried your hand at stock selection and market-timing along the way, then there is a good chance that you are feeling pretty good right now.  Making money is always a welcome relief, but emotional ego perilously inflates disproportionately with the rise of one’s portfolio.  Even more, in the case of the vast majority of people (those who basically doubled their investments alongside the market index, instead of perhaps quadrupling it), feeling too good is simply unwarranted.  Humility must substitute for hubris, since if one outperformed but this was his or her only investment experience, then luck accounts for a great deal of post-crisis performance.  It's a great start of course, but the odds going forward are simply stiffer than one may realize.

How likely is it that an investor (or speculator) in U.S. equities over the past 5.5 years has demonstrated significant investment skills in this asset class?  For our test we reduce the investable universe to a mapping of the current 30 Dow Jones Industrial Average (DJIA) stocks.  We're cognizant that money-weighted returns are much more difficult to achieve (versus the time-weighted we look at) only if one's portfolio grows rapidly through one's career (though this is not a viable reason for prolonged periods of underperformance.)  Still those just starting to work and save can certainly pick from a broader basket of smaller stocks, though very quickly if one were outperforming on investment savings from career wage accumulation, then invariably they would have a greater portion of their wealth exposed to risk similar to that of large, blue-chip stocks.  Note that we'll show by the end of this note that there are several multiple more American millionaires (excluding value of their primary residential real estate), as there are skilled investors.  See pages 468-469 for the most recent broad government statistics on same, which they show right next to the tables where they tabulate the poorest Americans.  So with this background, we start with a performance threshold of selecting a basket of any of the top quarter of these 30 stocks for the past 5.5 years and then expand to a broader sample over the past 25 years.  These top 8 stocks had a minimal monthly outperformance of 1.1% (this is of course normal risk-adjusted and 14% annualized), with a 0.3% standard deviation.  This implies a significantly low, 1% chance of straying that far from the rest of the DJIA by "luck" alone.

Then being satisfied with our critical threshold, we next solve the probability of continuously selecting a basket of the annual top quarter of DJIA stocks by chance alone.  This is an elementary, compounded Bernoulli problem, and it comes to less than 1%.

We then use Bayesian probability (see equality below) to determine the portion of the population that has skill near the required 1.1% monthly outperformance, in order to compensate for the probability of attaining these results by luck alone.  And this portion of the population comes to the low 20s% (see addendum at bottom).  Sometimes those with skill can not outperform due to (bad) luck, just as those with no skill (the haphazard performer) can sometimes outperform due to (good) luck, hence some stable threshold needs to be set a-priori.

p(outperform) = p(outperform|luck)*p(luck) + p(outperform|skill)*p(skill)

Which rearranges to the following.

p(skill) = [p(outperform) - p(outperform|luck)*p(luck)] / p(outperform|skill)

While there are empirical differences that would ensue from, not the β (beta) of the 30 DJIA stocks, but rather from the component of the typical correlation and dispersion components of beta.  For example, when the correlation is high and the dispersion is high, then greater than typical portion of the investing population at that time would be able to skillfully outperform.  And when the opposite parameters define the investment regime, then less than the typical portion of the investment population would be able to skillfully outperform.

Theoretically expanding this example to different timeframes, we get the following results.  Note that these examples work for the most common approach to equities speculation: market-timing with a discretionary allocation towards individual stocks.  For 2 years, instead of 5.5 years, the portion of the population with skill increased to the high 30s%.  This is because it is significantly less difficult to outperform monthly at the same stated 1.1% level, but for a far briefer time period.  Instead of focusing on a precise level for any one person in the middle of the chart, one should again simply appreciate in this note we are providing a strong sense of the directionally correct pattern for attaining outperformance that essentially erodes over the long haul.  And we discuss some of the confidence levels to practically consider these probabilities, particularly at the shortest-end of the timeframe where we need to demand a higher outperformance threshold or perhaps one may want a slightly greater information ratio.

On the other end of the time spectrum, for 25 years and 50 years of speculation, the portion of speculators who are able to maintain the same level of statistical evidence of investment skills rapidly decreases to 1%, and less than 1/2%, respectively.  This is shown in blue, on the left axis of the chart below.

We can also skills-adjust these data, so that we can solve for the level of outperformance that a 2, 25, and 50 years investment career would need to equate to the same level of difficulty for different time periods, we adjust to statistically balance for the same difficulty (since we are now looking at briefer or much lengthier measurement periods), as the 1.1% monthly outperformance associated with a 5.5 years sample.  This comes to 1.5%, 0.5%, and 0.4%, respectively.  See the red data below.  Incidentally these monthly outperformances equate to an annual outperformance of about 19%, 7%, and 5%.

We also identify in green, a proxy performance of an extraordinary investor, Warren Buffett.  While not precise for this study, it gives a significant contextual idea of how this research note's conclusions square with Warren's performance as he often has.  Now Warren does obfuscate some risk-adjusted comparisons in the annual report, by using book value of BRK versus the market value of the S&P 500, though making small risk adjustments here would not help in the broader context of outperforming the benchmark market by a statistically large amount.  And while not required, one could imagine a concave function, connecting the two green triangles, in terms of cumulative differences working backwards through time.  In 2001 Warren expressed in Berkshire Hathaway’s (BRK) Chairman’s Letter:

Investors should remember that excitement and expenses are their enemies. And if they insist on trying to time their participation in equities, they should try to be fearful when others are greedy and greedy when others are fearful.

To some degree this turned out to be even more sage advice in 2008-2009, then it was in 2000-2001.  Yet despite his extraordinarily favorable private placements, which he was able to negotiate during the depths of the financial crisis, on a market price he barely underperformed during the past 5.5 years (note that John Bogle has often said that investment success isn't guaranteed to be permanent).  Market price was used since BRK’s recent months of valuation accounting data is not yet available.  We also see in this chart that he has significantly outperformed over his lengthier, 45-plus year history (which he has generally illustrated in the initial pages of the annual report), and doesn’t really need the skills-adjusted handicap we show for those investing for such a long period of time.  This reconciles with the fact over the past 48 dazzling years, Buffett's BRK has outperformed the market in 38 years (roughly besting the market 75%-80% over a very lengthy time).

To clarify this previous point and returning to the main story here for everyone else, we see in that chart above that having only a third, of the recent 5.5-year skilled monthly outperformance (e.g., from 1.1%, to 0.4%), is needed to correct for the precipitous drop in the odds of outperforming at that level for over 25 to 50 years.  Conversely, a spike higher in monthly outperformance is needed, during a more brief investment period, to statistically perform the same as the nearly 24% of people have been capable of showing skill during a briefer 5.5 years.

The confidence interval here is tightest about the 10-year area of the chart above due to the broader sample going back 25 years, and the large probability errors when estimating at the shortest-end of the timeframe.  And going forward in this analysis, we assume that there are tens of millions of Americans who are in the labor force, saving, and actively invest (also the major online brokers generally have less then ten million accounts each so this estimate is in the correct magnitude).  This is just to give a rough sense of how these probability proportions filter through the population and not a deep demographic analysis on personal financial balance sheets.  

Given this, the probability of outperformance would suggest roughly a couple million Americans in their 20s have this sort of investing experience and can feel comfortable with their initial results, even though a small fraction of this group will actually continue to bear out skillful outperformance over the long-run.  And even early on nearly ¾ of their peers, are already doomed in any pursuit of a 25 to 50 year statistical outperformance (equivalent to going up against roughly 10 additional economic cycles).  Since we showed above that only 20s% have succeeded.  And Warren Buffett’s recent unraveling in both performance and active investment confidence isn't a compelling counter-example for the wishful possibilities of the reverse: a miraculous, late-career revival!

Now on the other end of the age spectrum, nearly a hundred thousand people with 20-30 years of investing experience have shown the same skill level over this lengthier time frame.  And finally of those in Warren’s age group (45-55 years of investing experience), the number who have also proven to have skill is just in the thousands.  So not a literally a handful, but highly rare in the bigger scheme of things.  Spreading this talent level equally across the 50 states, this latter group would show to be a few dozen people per state.

Do these numbers such as 50,000 skilled performers aged in their 50s (or 30-40 years of investment experience), seem like a lot?  Well to put this into some perspective, there 8 million Americans  who work in finance related activities.  If half of these 50,000 skilled performers were represented by a spread of the top 10% of those senior workers in finance (now out of 800 thousand), then that would imply 97% from this top 10% still have not proven investment skill over this period.  While obviously the details of individual performance and strategies are impossible to assess at all times, this statistical analysis should at least provide a directional picture for the sheer improbability of maintaining skill over a lengthy period, for all but a few.  This is even accounting for a generous handicap on the necessary outperformance needed for those with longer investment histories.

With such daunting odds, what advice is there for people who dimly choose to speculate anyway, tying up large amounts of their human capital?  There are five specific advice here to impart. 

• The first advice is that this age-old ritual is exceptionally difficult, and perhaps brutally more so as more people attempt to acquire alpha.  With the rise of low-cost index funds, there can be less broad opportunities to achieving exceptional alpha (this is the practical, and very high risk-adjusted outperformance).  Just as additional people playing the lottery (or stories about the "nearby store" over the decades having sold multiple winning tickets) can never increase your ability to attain the winning ticket, nor to outperform the population who more judiciously invests their money.  
• The second advice is that simply learning the rules of finance or working in the industry hardly increases your absolute chance of outperforming the market (see quote at bottom).  This chance we showed in the note is fairly established in probability theory, and unfortunately it's super low (and likely even lower if one is uncertain within a few years of trying whether they have strong investment skill).  Stocks are simply too connected to one another to make distinguishing their fates not as likely.  The overall advice here is akin to knowing how to throw a javelin or play chess in junior high school doesn’t imply we should think we can then effectively compete in the Olympics nor play chess against a computer, respectively.  It's great to try for a several years, but one should also know that quitting if one doesn't succeed and moving on to Plan B is sometimes more wise.
• The third advice is that much more often it is better to simply buy appropriate index funds (and thereby be guaranteed to outperform most of the people who are generally unsuccessful in their attempt to smartly outperform the market), and know that investment capacity is often dear and that human capital are often better spent only entertaining some other pursuits.  Better of course to buy the broadest risk exposures (with maybe a small adjustment relative to their appetite and tolerance) to different asset classes.  
• The fourth advice is that the very small number of people are skilled investors share some rare talents.  They are gifted with an unusual and recognizable super-ability to seamlessly connect specific dots within changing investment problems, over long periods of time, well beyond the abilities of normal smart people.  The skills could be in a subset of understanding behavioral finance, consumer sentiment, technical analysis, international public policy, global macro economics, risk, statistics, derivatives, valuation accounting, etc.  Of greater importance, they know the many areas of investment knowledge where they are not personally skilled on a global stage, and nimbly have the sense then to avoid those investment risks which trap others.  
• And the fifth advice is that selecting world-class stocks or a world-class investment manager are both generally difficult, and anyway inefficient.  If one can’t properly select the former, then one usually can't skillfully select the latter.  And the risk factors are mostly the same.  Simply selecting an investment manager based on past results, such as BRK (which at least can proudly prove their long-term record), can often provide a false value for what to infer about an investment manager's future performance.  Just see how the past 5.5 years of BRK were, as they were the most wickedly disastrous streak for the company, since 1965!  Another example could be one of my college professors (Merton) who won a Nobel prize in economics, yet then went on to co-lead the destruction of a master hedge fund.

We close with a 1998 quote from Warren Buffett.  May the wisdom prove promising to those who still choose to toil away as the many generations before them have, in pursuit of that magically elusive thing.

Success in investing doesn't correlate with I.Q. once you're above the level of 25. Once you have ordinary intelligence, what you need is the temperament to control the urges that get other people into trouble in investing.


March 2014, some additional details:

For the 12 months of 2013, the top quartile security among large companies had an average monthly relative performance of 1.1%.  In a 20% random sample from the past few decades, a top quartile security in any year, also outperformed the index by about, at least 1.1% monthly average.  For this note, we set a baseline ouperformer performance, which is partially correlated from year to year.  This comes to a probability of outperformance of 11%.  There make inetresting modeling assumptions in the baseline, in terms of probability of outperformance and degree of correlation, which could tweak the results slightly to edge the monthly outperformance up or down, to between 1.0% to 1.2%.  

The probability of selecting a top quartile security in any year is just greater than 25% due to discrete approximation.  So over 5.5 years such a haphazard individual would succeed roughly 25%^5.5.  Also a baseline critical probability level for a skilled investor to outperform the market with typically top quartile securities is set to 50%, but as well noted in article above this could be reduced slightly for one's personal utility and it still doesn't change the storyline of rapidly vanishing skill probability over one's lifetime.  If the hurdle were set too low, then more haphazard people would be considered skilled (a false positive in the statistical sense).  If the hurdle were set too high, then more skilled people would not get credit for good performance relative to a haphazard investor (a false negative in the statistical sense).

So p(outperform)=p(outperform|luck)*p(luck)+p(outperform|skill)*p(skill). 

Or 11% =0.1%*p(luck)+50%*p(skill).

And p(luck)~75-80%+p(skill)~20-25%=100% would solve this.  One can follow this data on the blue line of the illustration above.  Bear in mind that personal circumstances vary per individual so one can calibrate this analysis slightly to their own appetite, though the interlocking movement of the trends in probability and performance over time would still fit the patterns shown here.

Now the statistical “penalty” associated with sample sizes is ∝(1/√time).  We see this in the greater difficulty of outperforming by 1.1% monthly over lengthier periods, or the direct proportional reduction in the needed α over a lengthier period.  So to stretch from 5.5 years, to 22 years, this would be ~[1/√(22/5.5)] ratio of outperformance required.  Or from 1.1%, to about 0.5%-0.6%.  One can follow this data on the red line of the illustration above.  The actual data in the illustration was solved precisely; we are just describing an approximate, quick rule-of-thumb here in this paragraph.

The above illustration and note provides a direction on the statistical difficulty in outperforming the market, and maintaining this for a lengthy period by consistently picking the top securities over multi-year periods.  It works by wrapping theoretical (Bayesian) modeling on top of an empirical sample of data. 

Also other more empirical mathematical analysis, aiming to study specific investment managers’ outperformance, has shown results that in the middle years (e.g., about 10 years) come to roughly similar small fractions of outperformers.  Many of these popular studies, by design, focus on mostly credibility rules to drive at usually a different type of result: p(outperform), as opposed to p(skill) versus p(luck).

Now we'll see a different extreme illustration to make this more broad then just one person or application.  We show how the requirement to demonstrate outperformance rises (above 50/50), somewhat related to the exponential style penalty function noted in earlier paragraphs, as the number of trials is reduced.  See this illustration below concerning flipping a constant fair coin (so now completely unhinged from empirical stock market analysis and the break-out of haphazardness versus skill).  We see the probability of flipping this 50/50 coin consistently with one outcome (e.g., heads) is inversely proportional to the number of chained attempts (and greatly reduces well under 1/2% if done over 50 times).  Notice the small jaggedness in the red line, due to the discrete approximation error with the number of coin flips.