Abstract: The conventional wisdom of employing a fixed percentage position size, ex. 2% of one’s portfolio, in trading, might not be the most effective strategy for optimal returns. This article looks into the potential benefits of utilizing the Kelly Criterion as a dynamic position-sizing mechanism to optimize leverage in real-time based on trailing performance.
Kelly Criterion – a perfect, mathematical proof for adrenaline junkies to over-leverage themselves and have an excuse later on. At least that’s what I understood from the first encounter with it. Before reading further, I would advice to learn about Kelly Criterion, if you don’t know about it. Investopedia has a nice article: Kelly Criterion (investopedia.com)
Have you ever read a trading book? There is a high chance it had a section called “Risk Management” or “Position Sizing”. And the whole chapter was likely about risking 2% from your portfolio value to a single trade. Maybe adding an ATR indicator to calculate where to place a Stop Loss? That is how far most books will go with a small majority of them being honorable exceptions. But is it really ideal? Is having a fixed position size (in this case a proportion from a portfolio) a good idea?
In my opinion, having a fixed position size, which is one of the most important variables in trading, does us no good job. What if we are having the worst month? Jost job, girlfriend broke up, broke a rib! Still willing to risk those 2%, knowing damn well you will lose them? What about being in the zone? Every trade is profitable, mind is sharp, efficiency over the roof! Maybe I’d want to risk more than 2% then.
What if we just drop the 2% or whatever proportion from the portfolio whatsoever? What if I determine where my Stop Loss should be, and then instead of risking a specified percentage, I just buy some leverage from my account balance, and then if I want, I will calculate the percentage of the portfolio that I just risked. Maybe it will be 1%, maybe 3%? Depending on the leverage.
Hence, I will be exploring this idea with the Kelly Criterion:
Kelly % = Win Rate/Avg. Loss – Loss Rate/Avg. Win
This formula calculates the ideal position size for the given parameters. Let’s take an example shown in Fig. 1
The output is 100%, meaning if we have a trading system which has a probability of winning of 60% and every time we win- we get 20%, that means the idea position size is to go all in every time! But what if our Win rate is 40% as opposed to 20%? Well then the position size is 200%. We leverage ourselves! This is assuming we always have up to one position per account.
Before continuing, read this misplaced note: This is designed specifically for one-position-per-portfolio systems. If your strategy is to pick stocks and you want to have 5 stocks, this is not going to work, as Kelly formula does not take into account any of the other positions.
Now if we look again at the table, we will see that there are 4 variables (actually 3, because Loss Probability is 1 – Win Probability). So, to use this formula for our trades, one needs to know its own metrics. These can be calculated easily from the trade log of the terminal. The problem arises- do we calculate these statistics and then just use continuously? Or do we re-calculate them once every while? We change as traders, after all. Especially when we are still finding our style of trading. We become better or worse.
Once Every Month. Imagine we calculate a trailing Kelly of month-to-month. We take a predetermined Profit/Loss statistic (we will get to it) and for the Probability- we use our month-to-month returns to calculate it. Then if we become better traders, Kelly will give us a bigger position size to maximize our returns. What if we suck? Then Kelly will give us a smaller position size. That sounds ideal!
There are several problems to solve:
- We need to assume our Average Profit and Loss %
- If we are being unprofitable some month, Kelly will give us a negative leverage (go short on ourselves!)
Let’s address these two:
We need to assume our Average Profit and Loss %. There are two ways for us to make a reasonable assumption. We can always (ALWAYS) use a Stop Loss and a Take Profit and never trade gaps. Then we will absolutely know our risk/reward (Average Profit and Loss %). And the Kelly will calculate accurately the ideal position size. Another way is to estimate our risk/reward based on our historic results. For example, if you have been trading for a year and the risk/reward is 1.5, then we can put that number into Kelly. We cannot, however, use a trailing risk/reward to calculate Kelly- that would result in a circular reference.
If we are being unprofitable some month, Kelly will give us a negative leverage. Unless we want to short ourselves (would be an interesting experiment), we have to limit the Kelly to not give us a negative leverage. My advice is: set the most conservative leverage and if Kelly gives us a value below that- go for the predetermined value. Additionally, if we know that we have a maximum allowed leverage (for EU citizens it’s 1:30 for the CFD instruments), we can do the same with the upper bound. And so with this approach we will always have a optimal leverage, which is calculated on our past months’ performance.
Example:
I have simulated a random result from forex trading, which I designed to have big drawdowns and where at first, the trader is losing money, but then he learns something, continues to get better and his account value starts to grow.
Blue line is the original trading with a constant leverage of 1:10, the red line is with leverage that is calculated with Kelly Criterion. The lower bound is 1:1 while the upper bound is 1:30. Results shown in Fig. 2
A small note, this chart is made from the second month, as I used the first month’s values for initial calculation of Kelly.
So the difference is easy to see! When we are doing badly, Kelly disallows us to take big positions. However, when we start to make progress and get the win rate under control; when our risk/reward becomes better, Kelly quickly gives us a bigger leverage and send us to trade.
Drawbacks:
- Notice the drawdown on 2023-07-01. That is- if we are doing well and Kelly gives us a large leverage to trade with, any unlucky (or not careful!) trades will have a big impact, thus we must employ rigid stop losses to prevent painful accidents to happen.
- It is also obvious that the better we do, the bigger drawdown we will have. Also, the leverage changes rather slowly- based on months data, so if we are lucky one month, then the next month we are unlucky or make a ton of mistakes- it can introduce a massive drawdown as we might be trading at our limits.
Further risk management should involve imposing daily/weekly drawdown limits by proportion (like max drawdown of 5% a week).
Conclusion:
Having a trailing Kelly Criterion as our position sizing algorithm can optimize our leverage real time, based on our performance. This can be an ideal solution to experienced traders who have a good self-control and are consistent with their trading. Novice traders can also use the system, however they should employ very rigid risk
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