Apply the Kelly Criterion to Investing and Your Portfolio Sizing


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(This article was written by Brian Flores and Jae Jun)

Decisions, decisions, decisions.

As an investor, you’re making decisions all the time.

After all the analysis is done, you still have to decide how much capital to deploy for the idea.

One thing great value investors run away from is the widespread concept of diversification with many holdings.

 

I can’t be involved in 50 or 75 things. That’s a Noah’s Ark way of investing – you end up with a zoo that way. I like to put meaningful amounts of money in a few things. – Warren Buffett.

Charlie Munger has touched on this point too.

The wise ones bet heavily when the world offers them that opportunity. They bet big when they have the odds. And the rest of the time, they don’t. It’s just that simple.

Another great investor that supports this view is Monish Pabrai.

In his book The Dhandho Investor, he dedicates a whole chapter on a formula to sharing how to tailor the amount of capital you should put to work in a certain idea.

That formula is called the Kelly Criterion.

Introducing the Kelly Criterion

Developed by John Kelly, who worked at Bell labs, the Kelly Formula was created to help calculate the optimal fraction of capital to allocate on a favorable bet.

The great thing about the formula is that it’s flexible enough to work where information or skills can give you an advantage by estimating the outcome probabilities.

  • horses
  • blackjack
  • and obviously the stock market

As simplified by Pabrai in his book, the Kelly criterion applied to the stock market is:

Edge / Odds = Fraction of capital that should be allocated

  • The edge is calculated by the total expected value, obtained by adding up the multiplication of each scenario’s possible outcome by its corresponding probability.
  • The odds are directly the positive outcome that can be obtained.

But first, it’s important to realize that there is a big difference between “probability” and “expected value”.

Probability measures how certain a particular event will happen in a specific instance.

Expected Value represents the average outcome of a series of random events with identical odds being repeated over a long period of time. – source

If you flip a coin 10 times where you get one point for heads and zero for tails, the deviation from the expected value of 0.5 is going to be small.

But in percentage terms, the difference is huge.

Now if  you did 1,000 coin flips, the absolute deviation will be larger. That is, there will be more flips outside the 0.5 expected value.

However, the percentage deviation will be much smaller.

Do you see what I’m getting at?

Over a small sample size, the Kelly Criterion doesn’t work well, especially if you don’t trade often.

The formula is intended to work best for the long term growth of your bankroll over thousands of trades.

That’s where it’s different and difficult to apply to investing.

Unless you are day trader making tens or hundreds of trades a day, you won’t reach the number required for optimal allocation.

But there is a way for value investors to use it.

Kelly Criterion Examples

With the above discussion in mind, here’s an easy example taken from The Dhandho Investor.

You’re offered a coin toss where heads you get 2 dollars and tails you lose 1 dollar.

How much should you allocate to this bet?

Kelly Criterion

Kelly Formula Scenario

Primarily, the Kelly formula can provide you with a range of upper and lower bounds to determine the size of our investment.

Let’s take it one step further and apply it to a stock portfolio.

The Kelly Formula in Action with AAPL

Do you remember the AAPL roller-coaster last year?

At the low point, the stock was selling for about 400 dollars.

Even Jae decided to jump in and buy AAPL.

So with the analysis at the time, how much of your portfolio could you have allocated to AAPL a year ago?

When talking about stocks, there is a need to think in terms of the price differential compared to the actual values displayed by the market.

In other words: alpha.

Kelly Formula Example with AAPL

Kelly Formula Example with AAPL

Now, please remember the Kelly criterion isn’t necessarily a “Do-as-I-say” formula. It is best used as a guideline to find upper and lower values.

This is a clear example of where there are difference with investing. The Kelly Criterion clearly sends the message that this is a stock you should bet a lot on. But I wouldn’t allocate 87% of my capital to this idea.

It just shows that the odds are heavily in my favor.

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In the book Hedge Fund Market Wizards: How Winning Traders Win one of the hedge fund traders talks about allocation using the Kelly formula.

if you bet half the Kelly amount, you get about three-quarters of the return with half the volatility. So it is much more comfortable to trade. I believe that betting half Kelly is psychologically much better. – source

So instead of betting 87%, it’s more realistic to bet less than half. With AAPL at those prices last year, hindsight tells you that even a 30% allocation would have been very good.

But as always, the dots connect and your circle of competence will play a major role in determining the probabilities of outcomes.

The more you know the business and industry, the more precise the numbers will be. It’s always wiser to be conservative in your outcomes and probabilities, since it implies that even an adverse scenario can produce good results.

Summing Up

When the odds are heavily in your favor, don’t panic, but profit from the opportunity Mr. Market is giving you. You don’t want to follow the Kelly formula by the book. Use it as an indicator of how good the odds are and apply 25% to 50% of the recommended sizing.

And follow Pabrai’s recommendation.

Place few bets, big bets, infrequent bets.

Pros of the Kelly Criterion:

  • Tells your right away whether your odds of making a profit are good
  • Provides a range of upper and lower values for my capital allocation decisions
  • Comparisons across opportunities are possible

Cons:

  • Recommendation relies heavily on the accuracy to determine probabilities and outcomes
  • Data is not easily obtainable as you need to calculate the intrinsic value estimate to determine alpha
  • Potential mental mistakes in minimizing potential losses and maximizing potential wins

Questions for You

  1. Would you use the Kelly Criterion?
  2. How do you size your positions at the moment?

References and Additional Reading

Disclosure: Author does not hold a position in AAPL

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10 responses to “Apply the Kelly Criterion to Investing and Your Portfolio Sizing”

  1. Boris Marjanovic says:

    The Kelly Criterion works well in gambling situations where the probability can easily be calculated. When it comes to investing, I have my doubts about the effectiveness of such a formula.

    How do you calculate the probability that AAPL is worth $400 or $600 or any other amount? You essentially have to use arbitrary inputs to determine the probabilities. You can’t do this in most casino-style gambling situations, where exact probabilities can be calculated.

    There are simpler ways to decide how much capital to deploy for an idea. For example, you could do something like this (based on how confident you are that a stock will hit your fair value estimate):

    Low confidence (high risk): 5% of portfolio
    Moderate confidence (moderate risk): 25% of portfolio
    High confidence (low risk): 50% of portfolio

    Those percentages are based on personal preference and can be anything the individual investor desires. In investing, this method works just as well as the Kelly Criterion but it’s a lot simpler to use.

  2. Ney J Torres says:

    How did you calculate odds = 200?

  3. good points. Risk reduction, portfolio sizing, stock valuation, selling prices. It’s all art in the end.

    The thing is to just use whatever method works.

    I like your simple example too.

  4. I believe the Co-Author was basing it off the intrinsic value he calculated.

  5. Jeremy says:

    Hi Brain, this is a really interesting article. I’m going to have to break out the spreadsheets to get my head around it when I am a little less tired.
    One of the things that I have to laugh about with Buffett is his access to deals with far better terms than the average investor. During the market crash, I bought a couple of BoA mid-term corp bonds yielding 5% and I thought it was a great return. A week later, Buffett closed his deal for preferred stock with a 6% dividend (and the possibility for market appreciation). Only Buffett got that deal – not main street.

  6. Ney J Torres says:

    I think it should be as part of a %, right?.
    ex: odds 20%, or 80%

    I´m confused with the meaning of 200.

  7. packer says:

    I use Kelly as a sizing tool (esp. when a stock has risen in price but still has upside). I can give you an example that is a little different. I own a stock called Alliance Health Care (AIQ) and I have calculated an upside of 90% from its current price. So I calculate the upside as 90% the downside 100% and typically use a 50/50 as probability up and down (a conservative estimate of my win loss ratio). So my edge is .45 and my odds are 1.9 so I get a 24% allocation. So when the position size approaches 24% I will sell even if this position has the highest return potential.

    Packer

  8. JB Marwood says:

    I had this thought too and I like your idea of using confidence levels. Here is another idea:

    Instead of estimating the probability based on stock value, why not look at your own track record as an investor?

    As a simple example, if 50% of your investments have historically made money and 50% have lost money, the probability involved in your next purchase making money is 0.5. You could make it more sophisticated by calculating the historical probability of your investments hitting their target levels.

  9. Interesting point.
    But you’d also have to consider how much you made or lost with that 50/50 win.

    e.g. if you hit it big with 1 stock but lost a little on another, that’s still a 0.5 win rate but the actual profit is much different.

  10. Giuseppe says:

    Hi, I found very interesting how Ariel has approached this. Please, red ‘The Patient Investor’ Annual Report September 30, 2013 Ariel Investments.
    “(…) In
    practice, each holding is assigned a conviction rating and a valuation rating,
    both of which are determined by the portfolio managers and revisited on a
    weekly basis. Initially, we decide our level of conviction—of which there are
    only two choices: high or moderate. As already noted, we do not buy low
    conviction stocks, and if a company devolves to low conviction, it should be
    sold. Next, we determine the degree of discount: large, medium or small. We
    then assign every stock to a “bucket” with position size guidelines prescribed
    to them. For example, high conviction holdings trading at large discounts
    represent 3.5% to 5% positions in the portfolio. Moderate conviction issues
    trading at large discounts and/or high conviction names selling at medium
    discounts have weightings ranging from 2% to 3.5%. Lastly, moderate conviction
    holdings at medium discounts and high conviction stocks trading at small discounts
    represent just 1% to 2% of the portfolio. Besides low conviction names, the one
    area we also actively avoid is moderate conviction/small discount issues since
    they are much less attractive from a risk/ reward perspective.

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