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Kelly Criterion: Risk Management Theory – Warren Buffett, Bill Gross

Investing is often compared to gambling, and the two have many similar traits. You make bets on the unknown, in poker what cards you receive, in investing the company’s performance. The Kelly Criterion is a method of analyzing your odds and assigning a number to those odds.

Big-time investors such as Warren Buffett and Bill Gross have recently revealed that they use a form of the Kelly Criterion in their investment process.

The famous coin-flipping exercise that many investors reference to measure the performance of chance and risk is a form of the Kelly Criterion; it helps people understand the likelihood of outcomes involved in betting and investing.

Investing involves buying or selling the stocks of companies whose outcomes are unknown; it is a form of gambling. But studying the markets, companies, and the many different avenues for investigation allow for limiting the gambling aspect. But there is no way to avoid the unknown aspect completely.

In today’s post, we will learn:

  • What is the Kelly Criterion?
  • A Story to Explain How the Kelly Criterion Works
  • Kelly Criterion and Asset Management
  • Investor Takeaway

Okay, let’s dive in and learn more about the Kelly Criterion.

What is the Kelly Criterion?

The Kelly Criterion is a mathematical formula created by John L. Kelly, Jr., which relates to the long-term growth of capital. Kelly developed the formula while working at the AT&T Bell Laboratory.

As I mentioned earlier, the formula is a mainstay of the gambling and investing worlds to help manage risk in asset management.

The Kelly Criterion helps determine what percentage of capital should be used in each bet/investment to maximize that bet’s long-term growth.

The formula is also known as the Kelly strategy, and the formula is as follows:

Kelly % = W – [ (1 – W)/R ]

The inputs as follows:

  • Kelly % = percent of investors capital to make on each investment or bet
  • W = historical win percentage of the investment strategy
  • R = investors historical win/loss percentage

There are two components to the formula:

  • Winning probability factor – this factor is the probability or odds that an investment will have a positive return.
  • Win/loss ratio – this factor is the ratio of winning investments compared to losing investments.

The results of the formula help tell investors what amount of capital they should allocate to each bet. For example, if the Kelly Criterion tells us, we should bet 5% on Investment A and 12% on Investment B. It helps you determine the likelihood of success of each investment, and it outlines a chart giving you the odds of each choice.

Let’s look at a simple example, the chance of thrown dice lands on a 1, 2, or 3 is 50%. Similarly, 4, 5, and 6 are also 50%. Imagine that the dice thrower cheats and loads the dice so that the chances for 1, 2, and 3 are now 60%.

That means:

  • W = 0.60
  • R = 2 – 1 which equals 1

Plugging in the numbers:

Kelly % = (0.60 – [ (1 – 0.60 ) / 1 ] = 0.2

All of which suggests the formula that we bet 20% of our capital on the results turning out with our hoped-for result. If we continued betting 20% of our money on a low dice roll, eventually, we will go bankrupt. But if we bet less than 20%, over time, we should make a modest gain.

The above calculations are a simple version of the formula, which contains much higher math than I can explain. If you are interested in a more in-depth conversation on the math behind the Kelly Criterion, here are some links to feed that need.

A Story to Explain How the Kelly Criterion Works

Okay, let’s explore a short story to explain how the Kelly Criterion can work for an investor.

Imagine we are walking down the street one day, and we stumble across a chest, and after opening the chest out, pops a leprechaun. Forgive me; I am Irish.

The leprechaun looks you up and down and determines that you like to invest, so he sets up a brokerage account with Schwab and finances with a huge sum, $1. I know, disappointing, but hey, the luck of the Irish.

While opening your brokerage account, the leprechaun also enrolls us in the fractional share program but says no to margin or options trading—just the good, old-fashioned investing.

Next, the leprechaun gives us a stock tip, Luck of the Irish Corp (LOIC). And LOIC is a special company, each month, the stock either doubles exactly or halves exactly in value. There is a 50/50 chance of either outcome and no way to predict it in advance.

At the beginning of each month, we can rebalance our Schwab account between LOIC and cash, depending on our mood. At the end of 20 years, all the stock in the account will be sold, the account closed, and we will receive the entire balance in cash.

Our goal is to maximize the end-value of the Schwab account in 20 years. And the only control we have on the account is the rebalancing at the beginning of each month.

The big question then remains, what should our rebalancing strategy be?

Let’s start by looking at the first month’s results.

During the month, LOIC can either double or halve, which means that for every $1 invested in LOIC at the start of the month, we will either receive $2 or $0.50 at the month’s end.

On average, we’ll have ($2 + $0.50)/2 = $1.25

That first month’s result is a 25% gain, on whatever money we invest in LOIC, compared to 0% to a cash allocation. It seems like a no-brainer; put all the money in LOIC. Plainly, we should go all-in on LOIC and invest our entire $1 in LOIC over the next 20 years.

Next question, how will this “all” in strategy work for the next 20 years? Best case scenario: LOIC doubles every month, which leaves us with many billions of dollars; it seems like a good outcome. Worst case scenario: LOIC halves every month which leaves us with many fractions of a penny, not an ideal outcome.

But, neither outcome is likely.

Expecting the above outcomes is like flipping a coin 240 times for 20 years and expecting to get heads or tails on every coin flip. Not likely, but what is the average of these scenarios?

It turns out, if we employ the “all” in strategy for LOIC, the average is a pretty good return. Remember that we make 25% returns per month on average, which compounded over 20 years is many millions from our $1. I don’t know about you, but I would take that!

Not so fast, buckaroo!

Let’s look at a table outlining the wealth buckets and likelihood of ending up with each bucket in our “all” in strategy:

End Wealth

Chances

< = $0.01

32.57%

$0.01 to $0.10

9.75%

$0.10 to $1

10.25%

$1 to $10

5.10%

$10 to $100

9.75%

$100 to $1k

4.51%

$1k to $10k

8.00%

$10k to $100k

6.45%

$100k to $1m

2.63%

$1m to $10m

4.12%

$10m to $100m

2.82%

$100m to $1b

1.01%

  • $1b

3.05%

As we can observe from the above chart, there is a more than 50% chance that we will end up with less than $1 after our 20 years expires, and a 67.4% chance that we will have less than $100, and only a 3% chance that we will end up a billionaire.

It doesn’t seem all that great as we first thought, but how is this possible?

It’s all because of outliers skewing the average, meaning the billion-dollar outcome plus is so good it lifts the rest of the average.

A great example is the winning lottery ticket; compared to the millions of duds, the winning average lifts the overall average.

If we think about it a little deeper, we realize the most likely outcome is that LOIC doubles in some months and halves in others. The doubling and halving help offset each other, leaving us with our original $1.

A big idea to take away from this, the most likely outcome can differ drastically from the average one. Outliers do not sway most likely outcomes, but do sway averages.

How do we escape the lottery ticket scenario, where averages hold power? We need a strategy that helps give us a better chance of turning our $1 into $100k or $1M. And here is where the genius of the Kelly Criterion comes into play.

In our investing situation, the solution is a simple one. A the beginning of each month, we rebalance our portfolio so that exactly half of it is in cash and the other half in LOIC.

Below is how the rebalancing portfolio strategy works on our odds:

End Wealth

All in Strategy

Kelly

<= $0.01

32.57%

0.02%

$0.01 to $0.10

9.75%

0.10%

$0.10 to $1

10.25%

0.28%

$1 to $10

5.10%

0.78%

$10 to $100

9.75%

2.87%

$100 to $1k

4.51%

4.70%

$1k to $10k

8.00%

7.89%

$10k to $100k

6.45%

15.93%

$100k to $1m

2.63%

14.85%

$1m to $10m

4.12%

15.22%

$10m to $100m

2.82%

17.28%

$100m to $1b

1.01%

9.07%

  • $1b

3.05%

11.00%

As we can see from the side-to-side comparisons that the results are night and day. For example, the “all in” strategy gives us a 50% plus chance of receiving less than $1, not optimal, where the Kelly strategy reduces that outcome to 0.4%.

Likewise, the Kelly strategy gives us a 50+% chance of becoming a millionaire and a11% chance of becoming a billionaire on the upside. Now, those are odds I can get behind.

That, in a nutshell, is how the Kelly Criterion works. It helps tells us the odds or likelihood of each outcome and helps us determine what size bet to place on each outcome.

Kelly Criterion And Asset Management

Published in 1956, the Kelly Criterion was quickly adopted by gamblers for use in horse racing. Much later, investors adopted the idea, with investors such as Warren Buffett, Charlie Munger, Jim Simons, and Mohnish Pabrai embracing some of the ideas for their processes.

Many investors, including those above, use the formula to help them grow their capital because it assumes the investor will reinvest their capital and use it for future investments. The goal of the Kelly Criterion is to determine the best amount to place in any investment.

So how do we put this to work?

Let’s follow the below steps to determine our portfolio performance based on the Kelly Criterion.

  1. Look at our last 50 to 60 trades; we can do this either through our broker or your latest tax returns, whichever is easier.
  2. Calculate the “W” – the winning probability of each of our positions. Doing this divides the number of our winning trades by our losing trades, which we can define as those with a positive return versus a negative return. The higher, the better, the closer it approaches 1. For example, any number above 0.5 is good.
  3. Calculate “R” – the win/loss ratio, which we do by dividing the average gain of our positive investments by the average loss of our negative investments. We should have a number higher than one if your average gains are better than your average losses.
  4. Plug our results in the Kelly formula from above.
  5. Record the Kelly Criterion percentage from the formula.

So, now that we have our percentage, what do we do with this number?

The percentage, less than one, that the formula spits out represents the size of each position we should take in our portfolio. For example, a 0.05 indicates we should take a 5% position for each company’s portfolio.

The Kelly Criterion helps define our measure of diversification; if the formula tells us we should have 2% positions, that means we can carry 2% positions in our portfolio. But each position higher than that will significantly impact the overall results and skew the “average.”

The Kelly Criterion implies some common sense; for example, if you carry 20% to 25% positions in your portfolio, you are carrying a higher risk of underperforming or are taking more investment risk.

Investor Takeaway

The Kelly Criterion is a method of determining investment risk and how likely the odds of your “bet” winning. Many great investors have embraced the probalistic thinking discussed in papers such as the one concerning the Kelly Criterion.

Many much deeper thinkers than me, such as Nicholas Nassim Taleb, have embraced these ideas and have written many great books about these ideas.

As we mentioned earlier, investing has an element of gambling to it, and it makes sense to think of buying companies as making bets. The better we understand the probabilistic outcomes, the better able we are to understand the outcomes of our investments.

Is this a perfect method?

Nope, no idea or method is perfect or without faults. But the Kelly Criterion helps determine a margin of safety in our investments by putting percentages on the likely outcomes. It helps us make better decisions about the companies we want to own.

There are many calculators on the web to help you determine what the perfect portfolio allocation for you is; here are some I like:

With that, we will wrap up our discussion today. As always, thank you for taking the time to read today’s post. I hope you find something of value in your investing journey. If I can be of any further assistance, please don’t hesitate to reach out.

Until next time, take care and be safe out there,

Dave