Kelly Criterion Formula For Optimal Position Sizing
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In this review we look at the Kelly Criterion strategy and how we can apply the principles to our trading or investing activities to maximise growth. Created by computer scientist John Kelly in a paper from 1956, often referred to as the Fortune’s Formula, and later popularised by Edward Thorp when he applied the theory to beat the game of Blackjack.
Another note-worthy investor to apply the theory is Jim Simons founder of the Renaissance Medallion hedge fund. I also use a variation of the strategy and is the reason for our review.
Let’s take a look, and as always, please hit the like button and join our growing community below.
The Kelly Criterion calculation is used to calculate the optimum stake or position size for a given event, and is made up of four factors.
The equity balance.
The expected return from a winning outcome.
The probability of winning, and the probability of losing.
Ultimately, the Kelly Criterion formula calculates the perceived edge you have over an event and varies the stake accordingly, thereby determining the optimum stake size, or in trading terms, determining the optimum position size.
Contrary to the trading community, position size should not be a predetermined input to a strategy but rather an output depending on each of these four variables.
I’ll apply this concept to my personal trading stats shortly but first let’s look at how it applies to a sporting event.
In this example we have two boxers in a tournament. Boxer A from the red corner and boxer B from the blue corner.
Boxer B is seen to have a slight advantage due to the weight difference, as such the odds of 1.9 offered by the exchanges suggest a 52.6% of winning, leaving boxer A with a 47.4% chance of winning.
However, through our own analysis we determine that boxer B not only has a weight advantage but is also much more experienced, therefore we suggest a more realistic 55% chance of winning, reflecting odds of 1.8 and leaving boxer A with a 45% chance of winning.
We decide to put money aside for the whole tournament and start with a betting bank of $1000.
With all the metrics available we can now complete the Kelly Criterion calculation and therefore determine the optimum stake.
We have, an expected return offered by the exchanges of 1.9, which equates to a 90 cent profit for every dollar wagered, and our analysed ‘true’ winning probability of 55% and losing probability of 45%.
The equation is therefore; 0.9 multiplied by 0.55 minus 0.45, divided by 0.9, which provides the sum of 0.05 or 5%.
5% multiplied by our betting bank of $1000 is $50, therefore the optimum Kelly Criterion stake is $50.
Try not to be put off by all the numbers, there are calculators and spreadsheets available to calculate the figure automatically, but its important to understand the reasoning behind the number.
Had our probability been equal to the probability offered by the market, the calculation would have provided a negative figure implying there was no edge and therefore no reason to place a wager. Or, if our probability was even higher, perhaps 60%, the calculation would have suggested a higher wager to take advantage of the higher edge.
Next we will use my trading stats as an example to show how we can apply the same Kelly principle to a trading or investing strategy.
For simplicity let us use a $20000 equity balance.
My long-term win rate (or win probability) of 59%, leaving my losing probability of 41%.
My expected profit on risk for a winning trade is 4.04. Therefore, for every dollar risked I get just over $4 dollars in profit.
Using the same Kelly calculation as before we can now determine the optimum position size for a trade.
We have 4.04 multiplied by 0.59 minus 0.41, divided by 4.04, which provides the sum of 0.488 or rounded up, 49%.
Therefore, the optimum position size recommended by the Kelly formula would be, the $20,000 equity multiplied by 49% which equals $9,800.
I’m sure many of you are quite rightly thinking that this is a large amount of capital to have allocated to a position in comparison to the equity balance. That’s why many, including myself, use what is called ‘Fractional Kelly’.
In essence, this simply means a certain fraction of the recommended position size. I personally use 33% of the recommended position, therefore in this example my position size would be $3266.
Remember this is position size, not expected risk. My risk (based on a typical stop loss) would be near 10% of this position size equating to $326. This $326 positional risk equates to a risk on equity of 1.63%.
The difficulty however in applying the Kelly Criterion formula to the stock market is the randomness of price movement, and therefore the variability of these inputs. The game of blackjack on the other hand has factors that are fixed, for example, the number of cards in a deck is fixed at 52.
With this in mind, it’s important to keep track of your performance and adjust the inputs accordingly.
Let’s assume our strike rate reduces to 50% and we only profit $2 for every $1 risked.
By applying the same formula, Kelly calculates a new position size of 25% of equity, or $5000. But again, I prefer Fractional Kelly equal to 33% of the suggested position size.
Based on an average 10% stop loss position, the expected risk on equity would now be $165 or 0.83% per trade. The theory is that the Kelly score changes with the fluctuating performance of the trading strategy.
If enough people show interest below, I’ll add the staking spreadsheet I use into the description for ease of use.
To conclude, we present the theory of Kelly Criterion in this diagram to better understand its concept.
This centre green line represents full Kelly as opposed to Fractional Kelly, which could be at 33% here.
The blue line represents portfolio return.
The left side of the Kelly line is considered more conservative, and the right side more aggressive.
The concept suggests that the higher the Kelly (or equity) used, the less return and more volatility is experienced. Whereas a smaller position provides less return and less volatility.
The optimum position size is considered to be here.
But remember, this theory is ultimately based on events with fixed variables and true probabilities, like a game of blackjack.
The stock market however has numerous variables, as such ‘Fractional’ Kelly would be the advised approach should you decide on using the concept.
It is my preferred position management tool due to its ability to reflect on my trading performance and adjust the risk accordingly. More often than not, a trending market will improve my edge and the position size will increase as a result. The opposite is also true for a ranging market, my edge will decrease as will my position size to compensate.
I hope you found this relatively short review of the Kelly Criterion formula useful, and as always please hit the like button and I look forward to seeing you in the next book review.
Thanks for listening.