Position Sizes, Kelly Criterion And Prudent Capital Allocation

Rule One: Never lose money

Rule Two: Never forget rule one

First Off, What Is the Kelly Criterion?

It’s a formula Bell Labs scientist John Kelly devised in the 1950s for maximizing the long-term growth rate of capital. It was a borrowed idea from the application of information theory (Claude Shannon) and applied to gambling. It tells you how to allocate your money among the choices available, and how much to invest as your edge increases and the risk decreases. It also avoids the over-betting that can ruin an investor who otherwise has an edge. Kelly imagined a system where you have an edge, a set of expectations that differs from those of the market. He then developed a formula, based on Shannon’s work, showing the exact amount of your bankroll you should bet in order to maximize your capital over the long term. Consistent with the theory, the maximum rate of return comes when you know something the market doesn’t.

  • The chance of ruin is small. Because the Kelly system is based on proportional bets, or relative bets and losing all of your capital is theoretically impossible. A small chance of a significant loss remains.
  • It is arguably the best system to reach a specified goal within the lowest period of time.

Assume you can participate in a coin toss game where heads pays $2 and tails costs $1. You start with a $100 bankroll and can play for 40 rounds. What betting strategy will give you the greatest probability of the most money at the end of the 40th round?

The payoff is $2 for heads and $1 for tails, giving us 2-1 odds. We know the odds of the coin to be 1-1 or a 50% chance.

How much do you bet each round?

100% ???
50% ???
40% ???
25% ???
10% ???

Well…

Kelly Criterion States

F = PW – (PL/W)

where

F = Percent of Proportional Capital Allocated

W = Dollars won per dollar wagered (i.e., win size divided by loss size)

PW = Probability of winning

PL = Probability of losing

Now if we plug the values into the formula we find:

F = 0.50 – (0.5/2)

F = 0.25 or 25% of capital

We can rearrange the formula to end up with

Edge/Odds = f

Where

Odds = $2
Edge = Expected return or (0.5 x 2) + (0.5 x -1)

0.5/2 = 25%

This style of betting is rather counterintuitive at first glance but upon thinking about it for a few moments, we realize this is because of the probability distribution and chance of continual (parlay style) losses in the short-run.

Mohnish Pabrai (TradesPortfolio), PIMCO, Warren Buffett (TradesPortfolio), Legg Mason and a few others seem to utilize the Kelly system in various unique respects. Buffett has not officially stated he used the system although he had the following to say during an interview with business students:

I have two views on diversification. If you are a professional and have confidence, then I would advocate lots of concentration. For everyone else, if it’s not your game, participate in total diversification.

If it’s your game, diversification doesn’t make sense. It’s crazy to put money in your 20th choice rather than your 1st choice. ‘LeBron James’ analogy. If you have LeBron James on your team, don’t take him out of the game just to make room for some else.

Charlie and I operated mostly with 5 positions. If I were running 50, 100, 200 million, I would have 80% in 5 positions, with 25% for the largest. In 1964 I found a position I was willing to go heavier into, up to 40%. I told investors they could pull their money out. None did. The position was American Express after the Salad Oil Scandal. In 1951 I put the bulk of my net worth into GEICO. With the spread between the on-the-run versus off-the-run 30 year Treasury bonds, I would have been willing to put 75% of my portfolio into it. There were various times I would have gone up to 75%, even in the past few years. If it’s your game and you really know your business, you can load up.”

Pabrai on the other hand talks about the Kelly formula extensively through out The Dhandho Investor and recommends using a more conservative approach of a 1one-fourth Kelly, one-third Kelly or one-half Kelly, that is dividing the recommended total capital allocation by two, three or four. This is due to an important factor of over-betting and the loss of wealth it can cause as well as to subdue the emotional effects of losses. Because probabilities are usually estimates and are not known and the effects of over betting are worse than under betting, it is best to be conservative and bet less than Kelly suggests.

You have to make sure that you don’t over-bet. Suppose you have a 5% edge over your opponent when tossing a coin. The optimal thing to do, if you want to get rich, is to bet 5% of your wealth on each toss — but never more. If you bet much more you can be ruined, even if you have a favorable situation.” — Ed Thorp

(Note: Thorp had 20% average returns over a 20-year span at Princeton-Newport Partners)

The following picture illustrates the distribution of 40 coin flips using a range of f values from the example above [borrowed from Legg Mason]. The multiple of original bankroll is a function of percent of proportionate capital allocated. 

We can see that when we over bet it leads to complete ruin while if we under bet we are potentially leaving money on the table or in the market. Kelly System is a parlay style betting system and requires an investor to maximize geometric return versus a simple arithmetic return.

A quick illustration of why to use GeoMean. Imagine we had five years of data.

2013 = 25% return

2012 = -60% return

2011 = 25% return

2010 = 18% return

2009 = 90% return

The arithmetic average would be 19.6% annually. Is this measurement truthful?

No it is actually very misleading as most of us can probably see based on the 2012 returns of a 60% loss. The Geometric mean would be roughly 7% over five years, or a difference of 12.6% in expected value.

(Side Note: It is absolutely incredible how many managers report arithmetic average returns instead of geometric average returns, this is clearly due to incentives, as the arithmetic return must always be higher or equal to geometric return. This is not surprising given the negative incentive of losing assets if returns are not expectable, leading to lower compensation due to AUM loss).

Prospect Theory and Utility Theory

Both theories suggest that an investor’s utility, (as a function of investment return) is relative to proportionate wealth. Meaning the more money we have the less likely a small gain or loss is to effect us. The less money we have the more utility the returns and losses have.

These theories introduced by Kahneman and Tversky also introduce us to the evidence that humans are hard wired to be loss averse. Imagine for a second about winning $1000 and losing $1000 tomorrow. The dollars that are lost effect us asymmetrically compared to gains, usually 2-2.5 times as much as winning the same amount, dependent on the random variable, investor wealth.

Cognitive biases aside, lets review an example that can be applied to the value-investingdomain. Both examples are borrowed from Mohnish Pabrai (TradesPortfolio), the first being a gas station and the second being Warren Buffett (TradesPortfolio)’s investment in Washington Post. He gives more detailed examples regarding American Express, the gas station (when a loss occurs) and “Papa Patel’s” motel investments.

Real World Examples

First thing is first, before capital allocation should even be considered, the investment should have passed our investment checklist. The downside is minimal, it is a business that is within our circle of competence and we understand it very well. We know how the cash flows are likely to change or what they are to be in 5-10 years time. The business is priced at a discount from intrinsic value. It is a business we would be willing to commit a large portion of capital to. The management is honest, able and sound. The business has a durable moat that is expanding.

Present Value ($) of future cash flow
Year Free Cash Flow 10% Discount Rate
2007 100,000 90,909
2008 100,000 82,645
2009 100,000 75,131
2010 100,000 68,301
2011 100,000 62,092
2012 100,000 56,447
2013 100,000 51,315
2014 100,000 46,650
2015 100,000 42,410
2016 100,000 38,554
2017 Sale Price 1,000,000 385,543
Total $1,000,000 (rounded)

We find that a gas station goes on sale at the end of 2006 for $500,000.

Should we buy? Well yes.

The intrinsic value we calculated was $1,000,000 based on cash flows it will produce over ten years and the sale price (or terminal value). Lets assume now that two years have passed and someone offers us $950,000 for the gas station. What should we do?

Well first we re-calculate and analyze the intrinsic value, assuming everything has stayed constant, a sale should be made. We have received $200,000 in dividends or cash flow on top of the $950,000 sale price.

Funds Invested: $500,000

Total proceeds: $1,150,000

Years: 2

Annualized Return: 51.66%

Although this is a great return, it is dependent upon how much we invested proportionate to what was available. If a 51% return is achieved on only 3-5% of the portfolio, a total or portfolio return of only 1.5-2.5% is achieved. If 60% of the portfolio was invested, the total return would about 30%. A difference of 27.5% excess return.

Re-Enter the Kelly System

Odds of a 2-times return in three years = 80%

Odds of breaking even in three years = 15%

Odds of total loss in three years = 5%

The Kelly system would suggest we invest 92% of our available bankroll. We can reduce this suggestion by ½, 1/3 or 1/4 to 23% to 46% of available bankroll for reasons outlined previously. (f = 0.95 – (0.05/1.55)

The return of 50% would then be translated into a total return of 11.5% – 23%.

Warren Buffett (TradesPortfolio) and Washington Post

Mr. Buffett bought his Washington Post (WPO) stake for about $6.15 per share in 1973 and had believed the business to be worth $25 per share. Let us imagine now that the business will grow intrinsic value by about 10% annually ($25 x 1.1 x 1.1 x 1.1 = $33.28). A sale of the business will be made in three years when intrinsic value reaches 90%, or roughly $30 per share. We would achieve an annualized return of just under 70%.

Odds of making 4 times of better return in three years = 80%

Odds of making 2-4 times or better return in three years = 15%

Odds of breaking even to 2 times = 4%

Odds of total loss = 1%

How much bankroll would Kelly tell us to allocate?

F = PW – (PL/W)

where

F = Percent of Proportional Capital Allocated

W = Dollars won per dollar wagered (i.e., win size divided by loss size)

PW = Probability of winning

PL = Probability of losing

F = 0.99 – (0.01/3.68) = 98.7%

Take 1/2 , 1/3 or ¼ Kelly and we end up with a suggested range of 24.675% to 49.35%.
As Pabrai outlines in his book “At the time, Berkshire Hathaway had a total market capitalization of about $60 million. Available cash was likely a small fraction of this number. I’d estimate that Mr. Buffett likely used well over 25% of his available bank roll.”

The keys are the psychology of losses and the emotional component of investing, the edge that the investor can identify that is different from the markets views and capitalizing by concentration when the odds present themselves.

But do not be fooled. There is no “perfect” system to avoid all loses. All we can do is minimize losses, maximize gains, and optimize bankrolls. The Kelly Formula insures that you’ll never lose everything but it doesn’t guarantee that you won’t lose sometimes. The point is not to use the stated Kelly allocation amount and mathematical models when determining investments. Spend your time reading annual reports, analyzing businesses and looking to find bargains that you believe to be in the 95%+ of Kelly recommended capital allocation.

When you find these types of bargains, buy as much as you are comfortable with. What the Kelly system tells us is that is ok to concentrate our holdings in 4-20 great businesses we deem to be undervalued. Is the point of investing and the main function of any investor, whether growth or value to outlay capital at the present in return for additional capital as well as the originally outlaid capital, in the future?

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. – Charlie Munger(TradesPortfolio)

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2 thoughts on “Position Sizes, Kelly Criterion And Prudent Capital Allocation

  1. There is a video from a year or two ago taped at Columbia where Mohnish told the students to basically forget what he wrote about the Kelly Formula. His argument was that the Kelly works best on repeated bets and since investment decisions are not repeated bets it does not work.

    As for concentration: Amen. I like to listen to Buffett’s address to Florida MBAs over and over and he told them: “if you can find 6 wonderful businesses that is all the diversification you need and I can guarantee you if you invest in a 7th one, rather than putting more into your first one, that’s a serious mistake. Very few people get rich on their 7th best idea.”

    Thanks for your articles, I really enjoy reading and learning from them.

    PS: I share your heroes! But I see you’re Canadian? Where is Prem Watsa, Francis Chou and Peter Cundill?

    • Yes you are correct and I believe you are referring to the compounding presentation? https://www.youtube.com/watch?v=zXRSVvKf0Bc (Since 2009)

      1st 75% of cash – minimum 2x in 2-3 years
      Next 10% of cash – minimum 3x in 2-3 years
      Next 5% of cash – minimum 4x in 2-3 years
      Next 5% of cash – minimum 5x in 2-3 years
      Last 5% of cash – more than 5x in 2-3 years

      And yes I was discussing this with someone on Linkedin yesterday that we have a martingale problem and that is likely the suggestion of the 1/2, 1/3, and 1/4 kelly. Also the probabilities are subjective and not clearly defined.

      Yes! I love the diversification video by Buffett in Flordia. https://www.youtube.com/watch?v=wbjPiYE-F4Y

      (Just watched it again now, haha)

      I use a simple mind experiment thinking if there was no access to the stock market like it is today and I was John D Rockefeller or another business owner, how in the world could I manage and understand over 20 businesses? I personally have never had over 6 companies in my portfolio at any one time, this is due to both a gear towards heavy concentration and lack of funds.

      Thanks for reading the articles and glad to hear you have learned a thing or two.

      The list is by no means complete and is missing Pascal, Watsa, Schloss, Cundill, Templeton, Gould, Hume and others that come to mind. I should update and include more names, especially Canadian ones like you mentioned.

      Cheers!

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