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Position Sizes, Kelly Criterion and Prudent Capital Allocation
Posted by: Tannor Pilatzke (IP Logged)
Date: March 6, 2014 05:26PM

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 (Trades, Portfolio), PIMCO, Warren Buffett (Trades, Portfolio), 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 acceptable, 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-investing domain. Both examples are borrowed from Mohnish Pabrai (Trades, Portfolio), the first being a gas station and the second being Warren Buffett (Trades, Portfolio)’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 portfolio return of only 1.5-2.5% is achieved. If 60% of the portfolio was invested, the total return would be 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 (Trades, Portfolio) 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 (without over-betting). What the Kelly system tells us is that is ok to concentrate our holdings in 4-20 great businesses we deem to be undervalued. Isn't 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 (Trades, Portfolio)




Guru Discussed: Warren Buffett: Current Portfolio, Stock Picks
Mohnish Pabrai: Current Portfolio, Stock Picks
Stocks Discussed: WPO, BRK.A, BRK.B, SPY, IWM, SPX,
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Re Position Sizes Kelly Criterion and Prudent Capital Allocation
Posted by: TannorP@twitter (IP Logged)
Date: March 7, 2014 04:16PM

No problem Cesc and thanks for reading.

Cheers. 




Guru Discussed: Warren Buffett: Current Portfolio, Stock Picks
Mohnish Pabrai: Current Portfolio, Stock Picks
Stocks Discussed: WPO, BRK.A, BRK.B, SPY, IWM, SPX,
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Re Position Sizes Kelly Criterion and Prudent Capital Allocation
Posted by: skyaboveyou (IP Logged)
Date: March 7, 2014 06:09PM

PL = Probability of losing:

About the probability of losing, wouldn't it make  sense to make the definition wider? I do agree that the main risk is losing everything but not achieving the minimal return should also be considered losing to an extent, right? Otherwise it seems to me that the kelly formula will mostly propose very high allocations to most ideas




Guru Discussed: Warren Buffett: Current Portfolio, Stock Picks
Mohnish Pabrai: Current Portfolio, Stock Picks
Stocks Discussed: WPO, BRK.A, BRK.B, SPY, IWM, SPX,
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Re Position Sizes Kelly Criterion and Prudent Capital Allocation
Posted by: TannorP@twitter (IP Logged)
Date: March 7, 2014 07:31PM

Yes any negative possibility would be considered losing, a break even would be in theory be a negative expected value based on opportunity cost. The probabilities are subjective to begin with based on the analyst's research. The Kelly system shouldn't be used explicitly as Mohnish has recently said in 2009, he does not follow the Kelly approach stickily but does utilize a similar concentration allocation method. This is due to the changing odds and inability to make continual bets. The point being, focus on concentration and load up when the odds seem very favorable, this is also the reason behind under-betting using a 1/4 or 1/3 kelly. 

Mohnish addresses his allocation method in a Columbia presentation  about compounding. 

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

Combined with Buffett's statements
"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." and "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." (Florida MBA Presentation)

and his most recent 13-F we find he has only 6 holdings. 

(GM) General Motors Warrants - 25.2% of portfolio
(BAC) Bank of America - 21.2%
(ZINC) Horsehead Holding Corporation - 18.9%
(CHK) Chesapeake Energy - 16.3%
(C) Citigroup - 14.9%
(GS) Goldman Sachs - 3.5%

Interesting to see he recently is actually using all of this advice exactly how it was stated. As he addressed in the presentation to Columbia, he is a cloner of the best. 

Cheers and thanks for reading. 


 




Guru Discussed: Warren Buffett: Current Portfolio, Stock Picks
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