Two coin flips demonstrate the geometric asymmetry. Three out of four pathways to lose money. After 100 coin flips on the same basis your starting $100 would have eroded close to zero (you may have a few cents left). Model it and you'll see for yourself.
If your downside risk on each flip is 40% loss, you need winning flips to yield a 66.67% gain just to break even over a large number of flips (formula in the post). In our example the upside gain was only 50%, so the erosion of capital is assured.
Folks, I found this article extremely informative -- even though I thought I would learn nothing new going in. However a couple of quibble/genuine questions: I am not sure where the "path dependency" figures in the calculus. It appears that both the Win/Lose and the Lose/Win paths end up at the same place. The "gold" is in the geometric mean -- I haven't thought it through and would love the intuition behind why this is the key measure of an attractive trade. RK
Great piece again. The stock market returns are definitely non-ergodic just like your roulette example.
Also, inadvertently this article reminded me of Richard Thaler's article regarding bet sizing and behavioural biases study: Giving students $100, and their risk reward appetite.
Hey James, I have to correct you. Most people would not play that game. How do I know? Because it's the result of a previous psychology study (I don't have the reference but it's a fairly well known phenomenon). It's called loss aversion. The interesting thing about the loss aversion is that people decide not to bet or miss the optimum gain even when they are offered multiple coin tosses and positive geometrical returns.
Also, there is an important point you do not mention. Geometrical returns only apply when you can only put all of your eggs into the same basket. You can calculate the geometrical returns of your whole portfolio and try to maximize that, but when you have multiple stocks or different asset classes, you need to use the arithmetical average to calculate the return for each time period. The Kelly bet is an extension of that: sizing your bet is like having a mix of cash and one stock, and the volatility for cash is 0, but the volatility of that stock is much higher. So having an allocation to something with a neutral or even negative arithmetical return (like buying puts) can improve geometrical returns. A good book to understand this is “Safe Haven” from Mark Spitznagel.
The problem here is that on a vacuum, what this tells us is that diversification is always better than concentration because it allows you to transform geometrical into arithmetical returns. But as value investors, we do not see all investments as having equal probabilities of success or equal pay offs. Then, we need to concentrate into our best ideas, contrary to what the mathematics of geometrical returns would tell us. On the opposite end of the spectrum, there are “academic” style portfolios that aim to maximize returns by using modern portfolio theory, and they use asset classes and strategies you have probably never heard of like trend following or managed futures, and lots of leverage (but very diversified). An example of that is what they call “return stacking”, which means using swaps to lever up two or more asset classes which are supposed to go in opposite directions, like stocks and bonds. The problem is that even though the strategies sound logical, when you look at the results of these funds, using complicated mathematics to determine what to buy and what to sell, results are underwhelming to say the least.
There are also a lot of differences between the nature of a coin toss and the stock market. If the stock market is a random walk, the longer you play, the more risk you are assuming. Value investors do not see it that way. I am concentrated into not only the best ideas I have, but the ideas I know will achieve very good returns even if I need to wait for 10 years to see it. That is the fundamental difference between risk and volatility. Many people see the headline, accounting earnings, or the price chart, while I am looking at the earning power and moat of the business. That shows a clear way of sizing bets: when time is on your side and you can achieve a superior result, you need to bet strongly. When you are speculating on events, even if the reward is really high, you need to bet less. Example to that is a company trading below cash but with weak management, compared to a company trading at low valuations but with high returns on invested capital and a long runway to grow. This also highlights a key trait to be a successful investor: you need to be persistent, rather than consistent.
Finally, another fundamental difference between the coin toss and the stock market is the nature of the distribution. Coin toss returns follow a log normal distribution. The stock market returns are distributed into a shape that resembles more a Lorentzian function. If you don't know what that is, its basically a more skewed version of the normal distribution.
Thank you for your well-articulated comment. The points you raise are thoughtful and well made.
Among them, you state that “what this tells us is that diversification is always better than concentration because it allows you to transform geometrical into arithmetical returns.”
The point I was making is slightly different. Kelly becomes relevant when the geometric mean is working against you, even while the arithmetic mean remains positive. In those circumstances, Kelly helps by spreading bets in a way that pulls expected outcomes away from the negative geometric mean and closer to the positive arithmetic mean.
That framework may work well at the blackjack table, the roulette wheel, or in repeated coin flips, where probabilities are known or can be estimated objectively. But it is far more difficult to apply to equity investing. This is why I say: “When buying a stock, there is no objective probability of success. No formula can tell us the precise odds that a company will outperform, rerate, or compound over time. At best, we are making informed judgements. That makes the pure Kelly formula more elegant in theory than useful in practice for investors.”
My suggestion, therefore, is to avoid investments where the geometric mean is unfavourable in the first place. If you do that, you do not need Kelly at all. Instead, I offer a formula and mental framework to help investors identify situations where the geometric mean is favourable. Those are the investments worth making. In my view, that is what enables strong returns from a concentrated portfolio.
Very critical line of thought when it comes to making decisions in life. It is not about making right decisions all the time — it is avoiding the mistakes that hurt your life massively. Also, the advice on containing the downside is absolutely practical in real life and it protects from self-delusion.
nowhere i say that you have to hold your losers nor to cash in your win but only to cash in the winning part of your win you keep your original position and continue.
Aha, I understand what you are saying now. Sorry for the misunderstanding.
Your suggestion would de-risk the portfolio, but also completely destroy any prospect of compound returns (which is what drives true wealth creation).
I think that this method makes winning the game more difficult over the long term. You are cutting off the long tail on the outcome distribution curve.
That means three paths to lose, and the one winning path can never achieve outsized returns. So the median is still unfavourable, and the arithmetic mean is significantly reduced.
You need to optimize your wins and cut your losses.
If you cash in your wins early, and hold your losers hoping that they will recover, you end up with a portfolio of losers.
Water the flowers, cut the weeds.
The Kelly approach follows this methodology.
It was adopted by the Turtle traders with exceptional results. They would scale into their winning positions a maximum of four times, amplifying the wins, while cutting loses quickly.
Very interesting article, thank you. Just one small pushback: why 2 coin flips? If it were 100 coin flips everyone
would tend to land 50 “right” ones and make money
Two coin flips demonstrate the geometric asymmetry. Three out of four pathways to lose money. After 100 coin flips on the same basis your starting $100 would have eroded close to zero (you may have a few cents left). Model it and you'll see for yourself.
If your downside risk on each flip is 40% loss, you need winning flips to yield a 66.67% gain just to break even over a large number of flips (formula in the post). In our example the upside gain was only 50%, so the erosion of capital is assured.
understood your point, thanks.
loss > profit - it seems
Folks, I found this article extremely informative -- even though I thought I would learn nothing new going in. However a couple of quibble/genuine questions: I am not sure where the "path dependency" figures in the calculus. It appears that both the Win/Lose and the Lose/Win paths end up at the same place. The "gold" is in the geometric mean -- I haven't thought it through and would love the intuition behind why this is the key measure of an attractive trade. RK
4 paths on 2 flips: WW, WL, LW and LL
You say you haven't thought it through. When you do, it'll make sense.
Fantastic article to begin the week with!
Great piece again. The stock market returns are definitely non-ergodic just like your roulette example.
Also, inadvertently this article reminded me of Richard Thaler's article regarding bet sizing and behavioural biases study: Giving students $100, and their risk reward appetite.
https://business.columbia.edu/sites/default/files-efs/pubfiles/1154/thaler_and_johnson.pdf
Students usually increase bets after wins but more importantly, losing bets led to bigger bets in desperation to recover losses.
I personally, find the Kelly Criterion in some shape or form very useful, although some people diminish it as being too blunt a tool for bet sizing.
Which is why I think the Edward Thorpe version is a lot stronger, incorporating risk free rate/cost of capital.
f* =(μ - r )/ σ²
f* = bet size
where : μ is your expected return
r is your risk free rate or cost of capital
σ² is your variance/volatility.
Still I think I would like to apply a conviction coefficient to skew bet sizes.
And there are more portfolio level adjustments such as covariance analysis that can help adapt that bet size even further.
Thank you for your kind words, additional insights and the link to the Richard Thaler article.
My intention in writing these posts is to create a community in which ideas are discussed and shared, so thank you once again!
Hey James, I have to correct you. Most people would not play that game. How do I know? Because it's the result of a previous psychology study (I don't have the reference but it's a fairly well known phenomenon). It's called loss aversion. The interesting thing about the loss aversion is that people decide not to bet or miss the optimum gain even when they are offered multiple coin tosses and positive geometrical returns.
Also, there is an important point you do not mention. Geometrical returns only apply when you can only put all of your eggs into the same basket. You can calculate the geometrical returns of your whole portfolio and try to maximize that, but when you have multiple stocks or different asset classes, you need to use the arithmetical average to calculate the return for each time period. The Kelly bet is an extension of that: sizing your bet is like having a mix of cash and one stock, and the volatility for cash is 0, but the volatility of that stock is much higher. So having an allocation to something with a neutral or even negative arithmetical return (like buying puts) can improve geometrical returns. A good book to understand this is “Safe Haven” from Mark Spitznagel.
The problem here is that on a vacuum, what this tells us is that diversification is always better than concentration because it allows you to transform geometrical into arithmetical returns. But as value investors, we do not see all investments as having equal probabilities of success or equal pay offs. Then, we need to concentrate into our best ideas, contrary to what the mathematics of geometrical returns would tell us. On the opposite end of the spectrum, there are “academic” style portfolios that aim to maximize returns by using modern portfolio theory, and they use asset classes and strategies you have probably never heard of like trend following or managed futures, and lots of leverage (but very diversified). An example of that is what they call “return stacking”, which means using swaps to lever up two or more asset classes which are supposed to go in opposite directions, like stocks and bonds. The problem is that even though the strategies sound logical, when you look at the results of these funds, using complicated mathematics to determine what to buy and what to sell, results are underwhelming to say the least.
There are also a lot of differences between the nature of a coin toss and the stock market. If the stock market is a random walk, the longer you play, the more risk you are assuming. Value investors do not see it that way. I am concentrated into not only the best ideas I have, but the ideas I know will achieve very good returns even if I need to wait for 10 years to see it. That is the fundamental difference between risk and volatility. Many people see the headline, accounting earnings, or the price chart, while I am looking at the earning power and moat of the business. That shows a clear way of sizing bets: when time is on your side and you can achieve a superior result, you need to bet strongly. When you are speculating on events, even if the reward is really high, you need to bet less. Example to that is a company trading below cash but with weak management, compared to a company trading at low valuations but with high returns on invested capital and a long runway to grow. This also highlights a key trait to be a successful investor: you need to be persistent, rather than consistent.
Finally, another fundamental difference between the coin toss and the stock market is the nature of the distribution. Coin toss returns follow a log normal distribution. The stock market returns are distributed into a shape that resembles more a Lorentzian function. If you don't know what that is, its basically a more skewed version of the normal distribution.
Thank you for your well-articulated comment. The points you raise are thoughtful and well made.
Among them, you state that “what this tells us is that diversification is always better than concentration because it allows you to transform geometrical into arithmetical returns.”
The point I was making is slightly different. Kelly becomes relevant when the geometric mean is working against you, even while the arithmetic mean remains positive. In those circumstances, Kelly helps by spreading bets in a way that pulls expected outcomes away from the negative geometric mean and closer to the positive arithmetic mean.
That framework may work well at the blackjack table, the roulette wheel, or in repeated coin flips, where probabilities are known or can be estimated objectively. But it is far more difficult to apply to equity investing. This is why I say: “When buying a stock, there is no objective probability of success. No formula can tell us the precise odds that a company will outperform, rerate, or compound over time. At best, we are making informed judgements. That makes the pure Kelly formula more elegant in theory than useful in practice for investors.”
My suggestion, therefore, is to avoid investments where the geometric mean is unfavourable in the first place. If you do that, you do not need Kelly at all. Instead, I offer a formula and mental framework to help investors identify situations where the geometric mean is favourable. Those are the investments worth making. In my view, that is what enables strong returns from a concentrated portfolio.
Very critical line of thought when it comes to making decisions in life. It is not about making right decisions all the time — it is avoiding the mistakes that hurt your life massively. Also, the advice on containing the downside is absolutely practical in real life and it protects from self-delusion.
I agree. Could you please develop some examples of what comes to mind when thinking about “downside in life”?
Exactly! This is very applicable to life in general
Very good article
we are talking about your bet example.
nowhere i say that you have to hold your losers nor to cash in your win but only to cash in the winning part of your win you keep your original position and continue.
Aha, I understand what you are saying now. Sorry for the misunderstanding.
Your suggestion would de-risk the portfolio, but also completely destroy any prospect of compound returns (which is what drives true wealth creation).
I think that this method makes winning the game more difficult over the long term. You are cutting off the long tail on the outcome distribution curve.
That means three paths to lose, and the one winning path can never achieve outsized returns. So the median is still unfavourable, and the arithmetic mean is significantly reduced.
Just my view. I accept that I could be wrong.
interesting.
it shows that you have to take your win from the table and continue the bet.
You need to optimize your wins and cut your losses.
If you cash in your wins early, and hold your losers hoping that they will recover, you end up with a portfolio of losers.
Water the flowers, cut the weeds.
The Kelly approach follows this methodology.
It was adopted by the Turtle traders with exceptional results. They would scale into their winning positions a maximum of four times, amplifying the wins, while cutting loses quickly.
You can read about the Turtle traders as one of the three stories in this post: https://rockandturner.substack.com/p/three-stories-every-investor-should?utm_source=publication-search