Updated 6/7/2023
Many compare Investing to gambling, and the two have many similar traits. In investing in the company’s performance, you bet on the unknown in poker, what cards you receive. 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 the Kelly Criterion in their investment process.
The famous coin-flipping exercise that many investors use 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 allows for limiting gambling. 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
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, this formula is a mainstay of the gambling and investing worlds to help manage risk and 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 are 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:
- The winning probability factor is the probability or odds that an investment will have a positive return.
- The win/loss ratio is the ratio of winning and 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.
For example, the chance of throwing 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 = 2 – 1, which equals 1
- R = 0.60
Plugging in the numbers:
Kelly % = (1 x 0.60 – (1 – 0.60 ) / 1 = 0.2
All of this suggests that we bet 20% of our capital on the results turning out with our hoped-for result. If we continue 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 want a more in-depth conversation on the math behind the Kelly Criterion, here are some links to feed that need.
Kelly Capital Growth Investment Criterion
Ole P Peters – Blog lecture notes on Kelly Criterion
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 a huge sum of $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.
We can rebalance our Schwab account between LOIC and cash at the beginning of each month, depending on our mood. At the end of 20 years, all the stock in the account will be sold, the account will be 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, meaning that for every $1 invested in LOIC at the start of the month, we will 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, leaving us with many fractions of a penny, which is 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?
If we employ the “all” strategy for LOIC, the average is a pretty good return. Remember that we make an average of 25% returns per month, 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 the likelihood of ending up with each bucket in our “all” 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% |
| 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 as great as we first thought, but how is this possible?
It’s all because outliers skew 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 averages do.
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. At 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% |
| 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, which is not optimal, whereas the Kelly strategy reduces that outcome to 0.4%.
Likewise, the Kelly strategy gives us a 50+% chance of becoming a millionaire and an 11% 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 they will reinvest and use it for future investments. The Kelly Criterion aims to determine the best amount to place in any investment.
So how do we put this to work?
Follow the steps below to determine our portfolio performance based on the Kelly Criterion.
- Look at our last 50 to 60 trades; we can do this through our broker or your latest tax returns, whichever is easier.
- Calculate the “W” – the winning probability of each position. Doing this divides the number of winning trades by 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.
- 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.
- Plug our results in the Kelly formula from above.
- 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, 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 process of assessing investment risk and how likely the odds of your “bet” winning. Many great investors have embraced the probabilistic 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, and it makes sense to think of buying companies as making bets. The better we understand the probabilistic outcomes, the better we can 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:
Kelly Criterion Bet Calculator
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 further assist, please don’t hesitate to reach out.
Until next time, take care and be safe out there,
Dave
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