The Kelly Criterion is a mathematical formula to determine the optimal dollar amount to bet in a given wager or investment. Say you’re offered a bet where you are a 60% favorite to win and it pays 2:1 in your favor—Kelly suggests betting 40% of your net worth. If you were offered this exact bet a million times, betting 40% of your net worth each attempt (adjusting as you go) would net you the most amount of money in the end. Thus, 40% is the optimal bet size given those odds and payouts.
I’m betting no one (myself included) would actually bet 40% of their net worth in the above scenario—even if they knew the odds and payouts were legit. Most of you probably read that previous paragraph and thought 40% was insanity. I think one of the more interesting takeaways from the Kelly Criterion is how few people live their life in a way that optimizes their net worth over the long-term (again, myself included). And most of us aren’t a little off the mark, we’re not even in the ballpark of what’s optimal. If the above offer was made to all Americans, the average bet size would probably be less than 1% of each person’s net worth (with many doing a nominal amount like $10 per bet).
The Kelly Criterion is ideally used in situations that are highly repeatable and you have a good idea of what your odds are to win (in many situations, like investing, the odds and payouts are a guess). One good use for Kelly would have been back when single deck blackjack was commonly offered at casinos and professional blackjack players were actually favored vs the house. A more current example could be professional sports bettors who have data on every baseball game in history and can pretty accurately measure their advantage in each of the 2,430 baseball games every season.
The Kelly Criterion has also been recommended as a way to help investors with position sizing. The problem is there are far more unknown variables when it comes to investing in a public company. On top of that, thanks to all our cognitive biases, we’re more likely to overestimate an investment’s upside vs downside potential. Nonetheless, going through some Kelly Criterion examples can give us some goalposts on what position sizes are ideal and what is clearly not good (too small or too large).
First, a quick note on half Kelly. While sizing bets according to the Kelly Criterion is optimal when you know the odds and payouts, doing so with many unknowns is higher risk. The main reason is that overbetting is far more detrimental than underbetting, but our cognitive biases make us more likely to overbet than underbet. Thus, most advocates of the Kelly Criterion suggest betting half of what the Kelly formula recommends. This protects us from our overconfidence and it significantly decreases the volatility. Also, the Kelly Criterion assumes each bet is completely independent of every other bet. This obviously is not the case in the stock market where different stocks are often correlated to each other. Again, this supports the use of half Kelly as a conservative way to account for that.
With that being said, let’s think about what position sizes will grow our investment portfolio optimally over the long-term. If you have a new investment that you think has 100% upside, 25% downside, and the upside scenario is 70% to happen, half Kelly suggests a position size of 31.3%. If that upside scenario is 80% to happen instead of 70%, the position size increases to 37.5%. In my opinion, either of those is a fantastic scenario that doesn’t come around very often. Most Kelly calculations I’ve done for what I consider the best public market opportunities suggests a half Kelly position size somewhere in the 30-40% range. To be even more conservative, we can round that down to 30% as the top of the range for those rare, fantastic investments.
Onto the low end of that position sizing range. If you have an investment that you think has 25% upside, 25% downside, and the upside scenario is 60% to happen, half Kelly suggests a position size of 10%. Alternatively, if you have an investment that has 50% upside, 25% downside, and the upside scenario is only 50% to happen, half Kelly suggests a 12.5% position. I was pretty shocked at those two position size suggestions. I’m betting a lot of readers wouldn’t even invest in a stock if they thought those were the actual odds and payouts, yet half Kelly suggests significant position sizes. If you want to round those numbers down similar to the above, the bottom of the range for any position might be mid to high single digits.
When it comes to investing and all the unknown variables that come with it, I think the Kelly Criterion is best used at a high level to provide a range of position sizes. In my own experience, I ran a bunch of Kelly calculations (similar to the past two paragraphs) and came to the conclusion that it’s hard to justify a position size less than 7-10% or more than 25-30%.
As an educational example on the smaller position sizes, if an investment has 25% upside, 25% downside, and the upside is 55% to happen, half Kelly still suggests a 5% position. Likewise, 25% upside, 20% downside, and the upside is only 50% to happen results in that same 5% position size recommendation. I think most investors would consider both of those uninvestable. Thus, according to the Kelly Criterion it’s nearly impossible to have a situation where an investor has enough confidence to invest, but should only take a 1-3% position size.
6 thoughts on “Position Sizing with the Kelly Criterion”
Interesting post. Some other limitations of the Kelly system when it comes to investing:
– It doesn’t take into account the time it takes for an idea to work out, i.e. it assumes that all bets are settled instantly. In the real world there’s an opportunity cost if you invest in something. An investment can take years to work out.
– It assumes all your bets are uncorrelated. That isn’t quite true in the real world. In another 2009 style crisis most of your holdings will go down.
– It assumes the utility of extra money is logarithmic and it assumes you don’t care about variance. Also not quite true (at least for me). E.g. if you are retired with a nice nest egg you probably don’t care as much about doubling your portfolio as you do about not going broke.
All these points lead me to believe that even half Kelly is quite aggressive. Each to his own I guess. It’s a nice tool to play around with every now and then to get some ideas about position sizing but I wouldn’t weigh the results too much.
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Thanks for the comment writser. It’s interesting to think about how much conservativeness half Kelly incorporates. In my opinion, taking half of what is optimal is already a very large buffer that should make up for a lot of things, but it’s impossible to know.
And you are absolutely right on all your points, though I am writing from the vantage point of a younger investor who’s more worried about long-term compounding vs short-term smoothness. This blog definitely isn’t targeted at retired people with nest eggs, I doubt many of them are reading a micro-cap investing blog 🙂
If I were offered the bet you described in the first paragraph, I wouldn’t bet the Kelly amount. But that is because I am a very skeptical, suspicious person, and I would figure that either I am missing something or that I am about to be cheated.
You didn’t discuss the concept of fluctuations in net worth. Kelly optimal betting maximize long term wealth, but at the cost of enormous net worth fluctuations. Most can’t mentally handle the fluctuations of Kelly optimal betting. I couldn’t years ago as a professional blackjack player, and I can’t now as a money manager. I am certain that even if I could, my clients wouldn’t.
The other critical concept in Kelly betting is repeatability. You touched on it, but I think the concept is under appreciated. If you are offered the bet you described in the first paragraph, and you bet the Kelly optimal amount, you would go through multiple periods where you lose 90% of your net worth (it only takes 4 losses in a row when you are betting 40% of your net worth) just due to bad luck even though the odds are strongly in your favor. But, then you would make it all back and more because you can make the bet a million times. if you invest like Ed Thorpe, and you are doing some sort of statistical arbitrage, Kelly optimal betting may make sense for investing (Thorpe never had these sort of drawdowns and I don’t think made Kelly optimal bets either). If you are a long term investor, you will never make enough investments in your lifetime to overcome the role of luck and to insure that your actual results approximate your estimated odds.
Good thoughts Larry. Fluctuations are one of the biggest benefits of half Kelly (which I thought I mentioned, but maybe wasn’t clear enough). I believe half Kelly results in 75% of the upside with 50% as much volatility. Repeatability is no doubt an important part of consider, but it doesn’t change what the optimal sizing is for one individual bet. Repeatability just determines how likely you are to achieve your actual expected value in the long-term.
I believe the Kelly formula you’re using here is “Edge/Odds”. That only applies to a single risky investment with two possible outcomes where the remaining money goes into cash at 0% interest. It sounds like you’re trying to apply this calculation to position sizing within a portfolio that’s close to fully invested in stocks. That’s really not what the “Edge/Odds” formula was designed to optimize.
Even in a simplified case where the portfolio consists of binary-outcome stocks that are statistically independent, the optimal portfolio weights will probably be very different than what this formula predicts for each stock considered separately. If you’re interested I can work through an example.
Interesting. When you say half Kelly merits a 30-40% position sizing for a highly attractive opportunity, it makes me think of Buffet and his 40% position in American Express.
It’s almost like he knew what he was doing.
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