You can have data without information, but you cannot have information without data
Daniel Keys Moran
I must admit that I had not heard of a concept called the Kelly Criterion until recently stumbling over it by chance. The Kelly Criterion is a formula first described by J. L. Kelly Jr, a researcher at Bell Labs, in 1956 and widely considered a significant contribution to gambling theory. Given specific odds, the formula provides a gambler with the optimal amount to wage in a repeated bet. A 2016 paper illustrated this with an excellent example.
In an experiment, participants including financial professionals from leading asset managers were given $25 and asked to place 300 subsequent bets on the outcome of a coin toss with a coin that has a 60% probability of showing heads each time. It is clear that this game has a positive expected payout. Nevertheless, 28% of participants ended up losing everything, and on average, participants ended up with a payout of only 91$. Only 21% of the participants reached the maximum payout capped at $250.
It seems that without further guidance, humans are pretty bad at placing bets resulting in poor outcomes even if the odds are in their favour. The chart below shows a single simulation of the experiment for differently sized bets. In this example, risking up to 40% of the capital in each bet resulted in the maximum payout of $250, while gamblers who risked 50% or 60% were eventually wiped out.
Obviously, the single simulation contains an element of change which is why I repeated it 10.000 times and derived the mean outcome. The chart below shows the output. As the coin toss has a positive expected return, average wealth increases over time even if larger bets are placed, but some strategies clearly dominate.
The mathematically derived Kelly Criterion formalises this problem and provides gamblers with a simple formula that describes the optimal size as a function of the given odds.
Here, p is the probability of winning, q is the probability of losing, and b denominates the ratio between the gain obtained when winning and the loss realised when losing. The optimal size is given as f*. For the previously shown example where the potential profit equals the possible loss while the probability of winning is 60%, the formula yields 0.6-0.4/1 = 0.2. In the particular case where the maximum gain is capped at 250, the optimal approach is a bit more defensive. It risks only 12% of the capital in each bet, an outcome my simulation already indicated.
While we did expect to observe poorly conceived betting strategies from our subjects, we were surprised by the fact that 28% of our subjects went bust betting on a coin that they were told was biased to come up heads 60% of the time. Before this experiment, we did not appreciate just how ill-equipped so many people are to appreciate or take advantage of a simple advantageous opportunity in the presence of uncertainty. The straightforward notion of taking a constant and moderate amount of risk and letting the odds work in one's favour just doesn't seem obvious to most people.
Haghani et al. 2016
Not surprisingly, the formula has since attracted attention in the investment community, and there is a sizable body of literature promoting its application in the stock market or portfolio construction. Some sources claim that famous investors, including Warren Buffet, use the Kelly Criterion to determine the size of their single stock picks.
Unfortunately, this is where things become more tedious and sometimes misleading.
Investopedia, for instance, simply applies the original Kelly formula to financial assets, entirely omitting the fact that most investments are not all-or-nothing bets but come with partial losses.
This was called out by a 2018 article in the Entrepreneurial Investor on the website of the CFA Institute. The article correctly identifies the shortcoming and suggests the application of a different formula that allows partial losses.
Again, p denominates the probability of winning (or a positive return), q describes the converse probability, a is the arithmetic mean of the loss incurred when returns are negative. B is the arithmetic mean of the gain realised when returns are positive.
The article illustrates the formula with a simple example, assuming an asset that gains 20% with a 60% probability and loses 20% with a 40% probability results in an optimal wager of 100% per bet. Indeed, simulating 300 such bets 10.000 times shows that this strategy results in the highest median wealth.
We can also solve the equation for p and determine the odds necessary to make a given payoff profile attractive. This can be useful when assessing trading strategies. Assuming for instance a trader who either earns 10% or loses 20%. Using the Kelly formula we can derive that the trader needs to be correct at least 67% of the time to make a profit on average.
However, what is often overlooked, the median tells only part of the story. Coming back to the arbitrary assumption of a 20% gain/loss with a 60%/40% probability: The chart below illustrates the worst outcome obtained for each bet size during the 10.000 simulations. As can be seen, an investor who follows the optimal approach still loses almost everything in the worst case. Maximizing log wealth can be very risky.
So far, we have worked with the simplified, stylised examples given by the various articles covering the topic. However, investors are seldom concerned about the performance of strategies under arbitrary assumptions. I, therefore, pulled the total return time series of the S&P 500 index since 2000 and derived the optimal allocation to the index using the amended Kelly formula. Over the past 22 years, the likelihood of realising a positive return on a given date in the S&P 500 was approximately 52%. The average return on those days was 0.763%, slightly below the 0.798% realised on days with negative returns. Using the Kelly formula for partial gains/losses with these numbers results in 52.2%/0.797%-47.8%/0.763% = 2.64.
Ouch, that's a pretty aggressive strategy! Indeed, holding the S&P 500 with 164% leverage would have generated phenomenal returns in the long run if investors had not gone bust on the way. As the chart below shows, between 2000 and 2022, investors holding the index with this amount of leverage would have been stopped out twice.
A more recent paper on the practical implementation of the Kelly Criterion published in frontiers in Applied Mathematics and Statistics introduces another formula for financial assets.
Here, u denominates the expected arithmetic return, r represents the risk-free interest rate, and σ is the standard deviation of returns. The authors illustrate the formula using a stylised example assuming an asset that returns 12% p.a. under the assumption of a fixed risk-free interest rate of 1% with a volatility of 40%. Utilising the formula, these assumptions yield a suggested exposure of 68.75%. Sounds fair enough.
Again, I tested the formula using real market data. I derive the realised arithmetic return of the S&P 500 and use short-term US Government Bonds as a proxy for the risk-free rate. The annualized parameters are u = 8.9%, r = 2.4% and s = 18.9% which, used in the formula yields a suggested exposure of 183.6%.
Again, using real data, the Kelly formula aggressively levers the portfolio, and between 2000 and 2022, an investor following this strategy would have experienced a maximum drawdown of 85%. In practice, this would result in a margin call.
In these examples, I used realised risk and return as input parameters for the Kelly formula and of course, one could now adjust these. Assuming lower returns and higher volatility would result in more conservative allocations based on the Kelly Criterion.
However, the previous exercises have shown the sensitivity of the formula to small changes in the input parameters. In fact, the formula proposed in Carta and Conversano 2021 turns the optimal allocation into a function of the asset's Sharpe Ratio and its volatility. As the chart below illustrates, this results in the aggressive use of leverage once the expected Sharpe Ratio is high enough.
Noting that Kelly 'optimal' portfolios tend to be riskier than solutions recommended by mean-variance optimisation, some publications propose approaches such as 'Half Kelly'. Carta and Conversano 2021 also point out the possibility of using the formula for investors with different risk appetites to construct a range from 'Half Kelly' to 'Tripple Kelly'. However, I am sure that using real market data, the latter will usually be hazardous.
Regardless of this, the Kelly Criterion is a crucial concept in decision making under uncertainty and should be much more proliferated in financial education. Apart from its usefulness in gambling (horse racing, blackjack etc.), it can provide valuable guidance to traders who have a solid understanding of their payoff and hit ratios. Nevertheless, its direct application to more traditional portfolio construction is less straightforward and riskier than frequently advertised (in this context it's worth mentioning the fierce debate between Paul Samuelson and Ed Thorp and William Ziemba over the Kelly criterion in economics).
While studying multiple publications, I came across various practical examples utilizing different versions of the concept resulting in widely different conclusions for one and the same asset. Many articles show convincing examples but tend to leave out the significant input sensitivity of the respective formula. Unfortunately, this seems to be a common phenomenon in finance-related research, and practitioners know the frustration of dealing with frameworks that are presented nicely with well-curated data but quickly clash with real market conditions.
As mentioned earlier, this is a wide field of research. There are many papers not even mentioned here that discuss the Kelly Criterion in the context of the Growth Optimal Portfolio and the heuristic as such is certainly helpful as a general rule of thumb and aid in probabilistic decision making. Nevertheless, I generally believe that quantitative models and concepts should more commonly be tested under truly realistic assumptions and presented respectively, including their shortcomings.
I have published my calculations as a markdown file to make this fully transparent and reproducible. The markdown also comes with a simple, flexible function that allows users to replicate the simulations and visualizations shown or apply their own assumptions. Beyond that, I have added an interactive calculator and simulator to the Monte Carlo application on the quantamental platform.
The underlying market data can be obtained from Google Drive. Apologies in case there are any errors or misconceptions. Happy to hear any corrections or suggestions.
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