Skill, Luck, & the Kelly Criterion

How do we know if we have skill or are just plain lucky?  Unfortunately for our self-esteem, many outcomes are more determined by luck than by skill.  How do you know when an activity is driven by luck versus skill?  Easy: if there’s a way to purposely lose, then skill plays a large role.  If there isn’t, then luck clearly plays a large role.  Understanding the difference is crucial, as skill-based activities allow you to assess potential outcomes and anticipate results.  Luck-based activities are much more difficult to assess – you are just as likely to be misled by the data and there’s no way to determine if your process is well-constructed or poorly-constructed.

Luck versus skill plays out a great deal in the investment world.  Active management is one key area where luck and skill are constantly debated.  How do we know what is signal (a skilled investor), and what is just noise (a lucky investor)?  There is a big difference between a champion chess player and someone who just won the lottery.  Beyond active management is something far more important.  Our chosen asset allocation is also driven by luck and skill.  How do we know if we’re any good at allocating our capital appropriately?  We can purposely make “bad” decisions, but those may actually work in our favor.  Conversely, we can work hard to make appropriate decisions and still show poor results.

Investing falls somewhere on the spectrum between skill and luck.  The market is not perfectly efficient and there is skill out there that allows outperformance over time.  Sadly, participating in the skilled outperformance is one of the hardest parts of investing.  The odds of you either being that skilled individual, or at least finding them, are exceedingly low.  According to Elizabeth MacBridge at the Stanford Business School, mutual fund managers are persistently skilled.  The market is so hyper competitive and assets pour in to high performing managers so that skill gets diluted.  As discussed in a previous post, size and fees often negate any benefits from skill.  Moreover, even the best investors can be overwhelmed by market events, as during the Global Financial Crisis in 2008 and 2009.  Regardless of how skilled you were, there was little chance of dodging that bullet.

Luck and skill are impacted by a number of factors – most important of which is timeframe.  Luck can overwhelm skill in the short run, and in both directions.  Investors tend to overestimate skill and underestimate the role of luck.  For example, someone with a crappy investment process and no skill could do better over short timeframes than someone with a high quality investment process and strong skills.  However, the role of luck diminishes through time while the role of skill increases.  Making investment decisions even more difficult is the role of competitive parity.  Competitive parity is an environment where most participants are reasonably comparable to each other in skill, which describes the investment world pretty well.  In these environments, luck plays a much greater role in separating the winners from the losers.

Skill versus luck impacts not just small time investors like you and me, but also big institutional investors as well.  It’s not just active managers either, although there are a lot of assets chasing the hottest hedge funds and private equity managers.  Large institutional money tends to follow asset classes that have performed well, just as many individuals put money in stocks after they’ve run up and pull it out after they’ve sold off.

Knowing that luck often plays a greater role in performance than we care to admit, how do we make investment and asset allocation decisions in a way that tilts the playing field in our direction?  Enter the Kelly Criterion.

Kelly Criterion

John Kelly worked at AT&T and originally developed the Kelly Criterion to deal with noise problems for long distance calling.  Since then, it’s been used for long term corporate investments, litigation, investing, and gambling.  In fact, gambling was where the Kelly Criterion really made a name.  The Kelly Criterion, and corresponding Kelly Ratio, are well-known statistical techniques that help investors or gamblers determine how to size their bets.  In investment management, it’s used to determine proper diversification.  It’s used widely in hedge funds to help determine single stock allocations, but can be used in asset allocation decisions as well.  The Kelly Criterion gives you a way to turn both skill and luck (probability) in your favor.

The Kelly Criterion consists of two basic components: win probability, or the probability that any given trade will return a positive number, and win-loss ratio, or the average gain divided by the average loss.  The basic formula for the Kelly Ratio is: 

Another way to think of it is your edge over your odds.

Let’s use an example to illustrate this.  Assume your win probability is 55% (anything above 50% is good) and your average winner returns 11% versus your average loser returning -10%, for a win-loss ratio of 1.1 (ignore the negative).  In this case, your Kelly Ratio is: .55 – ((1-.55)/1.1) = 14.1%.  Each position in your portfolio should be a 14% allocation based on your history.  This is the benefit of the Kelly Criterion – it guides you on how much to diversify.  A better win probability and win-loss ratio will lead to higher allocations and less diversification.  Of course, if you’ve shown that level of skill, then you should run a more concentrated portfolio.  Conversely, less skill and lower probability would lead to greater diversification, such as using an index instead of picking individual stocks.

Some common sense is obviously needed here.  For example, if you’ve done very little single stock trading, then your win probability and win-loss ratio are less certain.  Like any other equation, it’s garbage in, garbage out.  In fact, only the most active traders (hence, the heavy use of Kelly in the hedge fund industry) truly benefit from the Kelly Ratio.  Further, even if you’ve shown great win probabilities and win-loss ratios, exceedingly large Kelly Ratios imply little diversification.  Anything above 20-25% is considered very concentrated, and few of us have the confidence to hold a four stock portfolio.

Most importantly, remember that the Kelly Criterion is only used to help determine your level of diversification.  It can’t help you pick good positions and it shouldn’t supersede other risk management techniques.  However, with enough inputs, it can give you some level of mathematical certainly around luck and skill.

Kelly and Asset Allocation

The above example is really only useful for those investors who try their hand at stock picking.  This should only be done in your Get Rich account, once you’ve got your basics covered and a healthy start on your F-You fund.  However, the Kelly Criterion can also be used in your asset allocation decisions.

Let’s assume we’ve come into $50,000 and have a five year window when we’re likely to need the money.  Deciding between what amounts of stocks to hold relative to bonds can be a difficult choice.  Let’s apply the Kelly Ratio for guidance and look for both our edge and our odds.

For odds of stocks outperforming bonds, let’s look to the past.  Since 1973, the S&P 500 (stocks) has outperformed the Barclays Treasury Index (bonds) 73% of the time for rolling five year periods.  Although highly imperfect, let’s assume that this is a reasonable future estimate for relative performance and use this number as our odds.

Estimating the “edge” is more difficult.  What level of expected outperformance can we use for the S&P 500 versus the Barclays Treasury Index?  Historically, the S&P 500 has outperformed the Barclays Treasury index by 3.8%, on average, over rolling five year periods.  Another method would be to look at current valuations.  A quick and dirty expected future performance for the S&P 500 is to use the earnings yield plus inflation.  The earnings yield is simply the inverse of the price/earnings ratio (P/E).  Since the current P/E is around 22 or so, the earnings yield is 4.5%.  Assuming inflation runs at 2%, the expected return for the S&P 500 is 6.5%.  It’s a highly sloppy calculation, but enough for use in this example.  The five year Treasury yield is 2.2%, so we’ll use that as our expected return.  The difference is 4.3%, or pretty close to the historical average, which gives some level of comfort.  Our edge, or win-loss ratio, is 2.95 (6.5%/2.2%).

Our optimal allocation based solely on the Kelly Criterion is: .73 – ((1-.73)/2.95) = 64%.  In this case, using nothing other than the Kelly Ratio, we should allocate 64% to the S&P 500 and 36% to Treasury bonds.

Implication for Investors

We’ve meandered from skill versus luck all the way to the Kelly Criterion, but what are the implications for most normal investors?  The biggest takeaway is to focus on the intersection between things that matter and things that you control.  Asset allocation is the most important determinant of your investment performance, and is fully within your control.  Focus your time here, as opposed to figuring out whether Apple will outperform Google or not.  Market performance is not something you can control, nor is tax policy or the regulatory environment.  However, your allocation among stocks, bonds, REITs, and other investments is.

Focusing on finding quality managers may or may not matter.  Finding the right manager with the best investment practices and processes will tilt the odds in your favor.  As we’ve seen though, luck can overwhelm good managers, especially in the short term.  If you have staying power, and can handle underperformance for potentially years, then finding quality managers may be worth your time.  If not, focus on asset allocation and stick your investments in an index fund.

Trying to achieve super high returns year in and year out is not something that you can control.  Much like quality managers, you can control the process though, and improve your odds of winning.  The goal should be continual improvement, not shooting for the moon.  Find an investment strategy that works, improve it through time, and stick with it through the inevitable downturns.  Given how difficult achieving strong returns year after year is, instead of spending your time picking stocks, you should spend much of your time earning a larger paycheck and saving a big chunk of it.  These are the things in your control, not whether Tesla will hit $500 or not.

Using things like the Kelly Criterion, earnings yield, dividends and bond yields can help you tweak the asset allocation of your portfolio without making wholesale changes.  The asset allocation decision should always be more than a simple optimizer based on these inputs.  Return goals and your comfort level with potential losses always come into play.  However, it can allow you to size your bets accordingly and give yourself a tailwind.  Just as an asset allocation decision should never be left solely to an algorithm, neither should it ignore these inputs either.

Finally, avoid overconfidence, especially if you’ve had a few lucky breaks.  Remember that luck can overwhelm skill in both directions, particularly in the short run.  A few winners does not make you a stock-picking savant.  The Kelly Criteria can help you determine how much diversification you should have, but won’t help you with the underlying process used to pick stocks or other investments.

Keep building my friends.


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