Both Harry Markowitz (MPT) and John Kelly (when applied to bets) believed that you shouldn't bet all your money on a single bet or stock. Markowitz proposed the widely accepted concept of diversification (accept a lower expected return to get a lower standard deviation) while Kelly believed that you need to save some part of your horsepower for future opportunities.
One focuses on the simultaneous investment opportunities while the other focuses on future betting opportunities.
"We have the same Sharpe ratio whether we put 10% of our money into this portfolio or we leverage it 10 to 1. But those decisions make a big difference. If we put 10 percent of our money into this portfolio, we get only $1,132 back at the end. If we leverage it up 10 to 1, we end up with only $2; that is, we lose almost everything." - pg 83, Red Blooded Risk
The question that the Kelly criterion aims to answer is: what is the optimum amount of risk to take for every bet?
Kelly % = W – [(1-W)/R]
Where, W = Winning probability, R = Win/Loss ratio.
This Kelly% will be the 'mathematically optimum' trading size which will maximize your returns in the long run - assuming that your edge also holds up in the long run :)
Obviously, the prerequisite for applying the Kelly criterion is having a rules-based trading system for which you have determined the probabilities and risk multiples - either through a comprehensive backtest or through a walk forward/live run in the markets (preferably both). This entire process pivots on the expected returns of your system and the reliability of your estimate.
Non-Ergodicity: Sequence matters
For most people, determining the total return just by seeing the 'average annual return and the volatility' will not come very intuitively. The volatility of the returns play a big part in deciding how much you will actually make.
Any analysis of the performance of a trading system needs to consider that there is always an alternate history of realizable outcomes where things may not have turned out the way they actually did - or maybe just the sequence could have been different.
Trading and investing are 'non-ergodic' businesses. Even the results of a very good system will be determined in the short run only by the trajectory of the wins and losses. Survival is key and a good system is a good system only in the long run.
But what makes a 'good system'?
As a side note, it is important to first understand what all you should be tracking for a given type of system. The metrics that I personally use have been described in brief in the "Performance Measurement" section on www.cauchydistribution.com.If you have a system with a high probability of winning, the Kelly % might come up to a very high number. So practically speaking, we will need to filter the Kelly % with an upper cap based on the account size using another Position Sizing model along with it.
Practical application
For directional trading strategies, the simplest way to apply the Kelly Criterion is
pyramid up and pyramid down
filter the Kelly% through max loss filters based on your account size
The core idea is that - when you are right and when you win, your position size (or your position delta) should be optimally higher - as compared to when you are wrong and when you lose. The way I do it for directional plays is that I adjust deltas of my option positions as and when the system tells me that the probability of winning on this trade has increased - to reach full Kelly%.
My first entry is far from full size which ensures that if I am wrong on the entry or the direction - in both cases, the hit on the account is limited. Also, if I am playing something like a gamma acceleration on an OTM option, I have rules to convert the trade into a synthetic future once that specific play is over (once it becomes ATM) - and if the directional play is still more than valid.
Nuances and guidelines
The biggest problem with Kelly is that it assumes only two possibilities: either you win this much or you lose that much. So when you pyramid out of a position, the maths of the whole trade needs to be considered (see next section).
Every trade is one instance for the Kelly criterion. It doesn't matter whether you pyramid up or pyramid down- as long as that buy/sell is part of the same transaction. Kelly requires independent occurrences of the signals that constitute your strategy.
It maximizes your expected payoff, not your actual payoff
For a discretionary strategy, if psychological/execution issues become a hindrance, then Kelly won't add to your edge - infact a larger position size may cause harm.
Kelly will most likely overestimate the optimum position size for a bet. There are various reasons to allocate less than Kelly - but never a reason to allocate more than the Kelly%.
Mix and match
There are numerous other models for position sizing which can be used either independently or in conjunction with the Kelly criterion - and honestly most of them are much better than Kelly on a standalone basis.
Van Tharp has around 30 different models for position sizing which he describes in his book "The Definitive Guide to Position Sizing: How to Evaluate Your System and Use Position Sizing to Meet Your Objectives".
Core equity method, Total equity method, Reduced Total equity method
Units per fixed amount of equity
Equal units/Equal leverage
Percent margin, Percent volatility, Percent risk
Market's money method
etc.......
It must be noted that a stop loss is different from position sizing. Say you are holding 4 lots in a position. A "fixed open-risk to total-equity" position size model will ensure that you exit 1 lot if the to total risk to equity at any point goes above, say, 3%. This is different from raising the stop for the 4th lot - which should be a function of your trading system and not your position sizing model.
The above is an example of hedge fund manager Tom Basso's way of scaling out of a position (Van Tharp's Model no. 21) where he uses open-risk and open-volatility of the position w.r.t. his equity.
Conclusion
Most traders keep worrying about their entries and fail to pay enough attention to their exits and position sizing. In the long run, it doesn't matter if you are right most of the time or not -- as long you manage your positions correctly when you are right and manage them correctly when you are wrong.
Every indicator, every strategy and every system only works some of the time - we just have to first accept this and then find a way to manage both eventualities.
If traders pay as much attention to their position sizing instead of constantly changing their strategies in a quest to find the holy grail, they would do much better I believe. Also, for discretionary traders, taking profits on the way has a great psychological impact which cannot be discounted, given that the 'need to be right' is an ailment which affects most of us.
References
Brown, Aaron. "When Harry met Kelly" (pg 73-101), Red Blooded Risk: The Secret History of Wall Street
Tharp, Van K. The Definitive Guide to Position Sizing: How to Evaluate Your System and Use Position Sizing to Meet Your Objectives
Resources
The maths behind the Kelly criterion
https://www.youtube.com/watch?v=o7YIa1w58Yc&feature=emb_title
Prof. Sanjay Bakshi on "Non-Ergodicity and its Implications for Businesses and Investors"
https://www.dropbox.com/s/m2nu1ymugzi48vi/Non-Ergodicity%20and%20its%20Implications%20for%20Businesses%20and%20Investors.pdf?dl=0
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