The Arithmetic Return Doesn’t Exist. It’s a dream that isn’t real.
Like waves crashing against an ocean cliff, the relentlessness of time simply overwhelms the arithmetic return.
In the short term the cliff exists. But with enough repetitions, with enough crashes of the waves along its surface, it will grow smaller until ultimately the cliff disappears.
The arithmetic return is the same as the ocean cliff. Each repetition of volatility chews away at its foundation until its dissolves into the geometric return.
Let’s find out why by first exploring the math of a lottery.
What Do You Expect To Win In A Lottery?
If you asked 100 people who played the lottery how much they won, what do you think the average answer would be? Do you think it would match the “true average” of the lottery itself?
I asked this question on twitter.
Most people said you would find $0. This makes sense because there is only 1 winner out of 50 million people. So it’s very, very unlikely that the people you talked with actually won. You would have to talk to many more people to have confidence of hearing an answer other than $0. You could probably take a sample your whole life and never find anyone who won.
The second most popular answer was $0.6. This is the arithmetic average of winnings. $30M / $50M = $0.6. We are taught to expect this value in school.
But the payout here is so skewed with only 1 winner out of 50 million, that it’s very hard to ever take a sample of the population and actually “find” the average. When the average depends on rare events happening, you need a large enough sample in proportion to the population to realize the average.
People understand this. My readers know that you need a large sample to experience the true average of a skewed population.
So does the arithmetic average make any sense here? If you don’t have the ability to draw enough samples from the population, does the arithmetic average ever show up? If it never shows up, is it real?
Time Moves the Arithmetic Average To Irrelevancy
To explore this further, we’ll continue forward using the coin flip example from the prior post with this payout:
- Heads, up 50%
- Tails, down 33.33%
Here the arithmetic return is 8.33% ({50% – 33.33% }/2), and the geometric return is zero ({1.5*.6667}^0.5 – 1).
Let’s run 100 samples of this game forward through time and see what happens to the results. There’s no rebalancing, we’re just letting the returns compound.
The red straight line is the arithmetic average (the y-axis is log scale which is why it’s straight). Notice early on there are a number of random samples which keep up with and even beat the arithmetic return.
But over time, fewer and fewer of our trials stay above the arithmetic average. By the time the flips reach 200 repetitions, all 100 samples have fallen below the arithmetic return.
And then as time continues on further with more repetitions the arithmetic return just takes off away from the samples leaving them in the dust.
With more repetitions, the arithmetic return doesn’t have anything to do with any of our realized samples. It’s become a value so rare, one that requires so much luck, that you can’t really expect any of the paths to achieve it.
The values however do seem to be clustering around the geometric return of zero (flat line at 1). Maybe the average compound growth rate is a better expectation of the randomness through time?
What’s Happening with the Distribution Through Time?
Lets see what’s happening with the distribution of returns of this game through time.
At one round the potential outcomes look like this:
That’s pretty straight forward. The average is right in the middle, with half the potential returns above the average, and half below. The arithmetic return exists here.
What about after 10 rounds:
Now the average isn’t in the middle of the distribution any longer. It’s moved a bit to the right. Sixty three percent of all returns are now below the average. This is because the distribution is now skewed to the right (you can’t see this on the chart because of the log scale on the x axis). Compound returns always do this. The arithmetic return is starting to feel the waves of repetition.
Lets go further now. What about 20 rounds:
There are more than a million possibilities here now (2^20). Seventy five percent of the returns now fall below the Arithmetic return. You can probably sense a pattern here. So lets go to 50 flips:
A quadrillion possibilities, and 90% are less than the arithmetic return. The effect of the waves are building.
Lets go further. 100 flips:
Down to 97% of all results below the arithmetic average. 150 flips:
At 150 flips, you will receive less than the “average” 99.1% of the time. So its not a coincidence that all 100 samples fall below the arithmetic average soon after 150 flips in the example above.
The waves of time have now seriously eaten into the arithmetic return. It’s slowly disappearing.
And of course as time continues on, the likelihood of your path achieving the arithmetic return nears closer and closer to zero. You only get one “sample”. Time is going to win and that one sample isn’t going to “find” the arithmetic return, just like a sample wasn’t going to find the lottery’s arithmetic return.
The Arithmetic Return of Compounding Games is a Lottery
With enough time the arithmetic return becomes pure luck. It’s theoretically possible to receive the arithmetic return, but you need to get lucky. You are betting on flipping a lot more heads than tails. This is possible but unlikely, and you only get one play. The more repetitions, the more time that goes by, the more the arithmetic return become a lottery.
I don’t know about you, but I don’t want my hard earned wealth invested in a strategy built around lottery winnings.
The Law of Large Numbers Doesn’t Work the Same Through Time
The law of large numbers works against you here. We’ve been taught that with more trials, the sample moves closer towards the “average”. But which average in this case?1
What if you play this game in series? Three 50 flip games in a row is the same as the 150 game. The likelihood of receiving the average return at 150 flips is worse than at 50 flips.
Playing the games more often, one after each other doesn’t make the arithmetic average more likely. It makes it less likely.
Historical Returns Have The Same Issue
Let’s think about monthly investment returns.
How many days do they compound over? 21
How many hours do they compound over? 136
How many minutes do they compound over? 8190
So, is a monthly return a geometric return or an arithmetic return?
Is a weekly return an arithmetic return or a geometric return?
Is a daily return an arithmetic return or a geometric return?
Is any return truly arithmetic?
Not only does the arithmetic return not exist in the future, it doesn’t really exist in the past either.
Now there is a difference between the coin flip example and real life investments. First off, investments are not binary outcomes like a coin. Second, the size of the volatility is like the size of the waves crashing into the cliff. The coin flip was volatile and is closer to repeated tsunamis when compared to the tamer waves of everyday investing. So it takes a lot more waves to create the same sized effect. But the principle is the same no matter the size of the volatility.
Everything We See is a Geometric Return
When we “receive” a return from the market it is a sample of a nearly infinite large set of possible returns because it compounds over many time intervals.
We’ve seen that a “sample” taken from multiple compounded coin flips is very, very unlikely to “find” the arithmetic return. It’s far more likely that it tracked toward the geometric return.
Therefore, when you think of a monthly return–aka 8,190 compounded minutes–think of the single data point as a sample from an enormous population. That sample pulled from the total population of potential returns for the month will more often then not fall below the average. It’s likely “found” something closer to the average compound growth rate.
Therefore, all returns are geometric, or some hybrid of the two returns. No return, past or future, is purely arithmetic.
Time Trends Toward the Geometric
Time is relentless. It wears down the arithmetic return like the waves wear down an ocean cliff.
For a short while the cliff exists, but it can’t hold back the tide. It will slowly wear away, ultimately disappearing into the sea.
For a short while the arithmetic return exists, but it can’t hold back the waves of volatility. It too slowly wears away disappearing into the inevitability of the geometric return.
Which return are you going to base your investment decisions around?
Addendum
This isn’t meant to be a rebalancing post. But I have to to take the time to point something out. The remanences of the arithmetic return come back to life when you rebalance.
If you take our 100 trial example from above, and rebalance those returns every round, you get a return stream which nearly matches the arithmetic return.
The dark green line just below the red arithmetic return is the 100 coin flips rebalanced back to equal weight each round.
Without rebalancing, the coin flip’s arithmetic return disappears through time. However, rebalancing each “round” between multiple versions of the coins reduces the negative effects of time. The portfolio nearly tracks the arithmetic return.
Rebalancing saves you from playing a lottery, and gives you hope that maybe the arithmetic return is obtainable, not just a fleeting dream destroyed through the relentlessness waves of time.
1-The average compound growth rate however does work with the law of large numbers.
While nominal arithmetic is not less than geometric, surely it can return less….? 7% geometric beats 8% arithmetic over 9 years. And to extract arithmetic return requires reset at start of each time period, which may require additional capital (eg +33%) in the example above. So neither the final outcome nor the starting ‘capital’ required (and hence the final return) appears to be directly comparable between arithmetic and geometric returns.
Be glad to hear your ideas on the limits of your general observation: that arithmetic is more advantageous than geometric.
By the way, excellent series of posts.
I’m not sure what you mean by “require additional capital”. My example doesn’t add any “capital” in expect to at time zero. The reset at the start of each time period is just balancing the exposure being risked on each coin. If you rebalance, the long term returns are closer to the arithmetic return. If you don’t they are closer to the geometric.
OK…..yes, I misunderstood ‘rebalance’.
I was trying to extract arithmetic return from say S&P500. Re-invest $100 each year/period. After say 30years you will end up with arithmetic return on $100. It will be ‘more’ than nominal geometric return, but could deliver less cash in hand than geometric. Also, in down years, you would need to top up to re-bet the $100. The geometric investor just needs their original $100 and then leaves it, the arithmetic investor might need access to more (if early years were losses). At the end of each year/period they take their money off the table and start again.
In other words if you are given a fixed sum to invest in S&P500 for 30 years and told that you can choose to receive either the arithmetic or geometric return, the arithmetic return under these conditions is not always going to deliver a higher end total. So while it is correct to say that geometric never beats arithmetic, actually in the real word it can do.
Be glad to hear your thoughts on this… pretty sure this equates to your game 4.
I’m not sure I would call on of them an arithmetic investor and the other a geometric investor. The geometric return can’t be higher than the arithmetic return. But rebalancing raises the geometric return of a portfolio upwards towards the arithmetic return.
Geometric return can be higher than arithmetic. US index 1929-32 down over 100% in arithmetic terms, at least geometric return can’t go below -100%. Arithmetic can.
Sorry, I do not understand how what this is saying applies to investing? If I invest in an index and the value keeps growing (inflation, dropping losers over time, keeping winners), what does that have to do with arithmetic vs. geometric return profile?
Again a very clear explanation of the difference between arithmatic and geometric returns. Thank you. However, the thing I still struggle with is ‘frequency’. Because I can calculate investment-profits over an innumerable amount of time units. From ‘tossing a coin’ once every second to once every – let’s say – year. And does not every comparison with lotteries and coin-tossing and therefore a translation to rebalancing fail as a result of the fact that I can calculate with any arbitrary frequency?
No, because if you could truly capture the distribution at a year, you’d see that its skewed. Although after a year, it probably won’t be overwhelmingly skewed. I put the coinflip charts on log scale, because if I didn’t all you’d see is the skew and not the shape. Its the size of the skew that makes the bet more like a lottery.
Thank you for educating us on the Geometric return versus arithmatic return.
We are grateful to you for that.
I do not understand how to implement the info on our website.
Assuming no midweek rebalancing for simplicity of your response,
how do we use weekly Friday closing prices and monday morning % allocations for SPY, TLT, GLD and BIL?
They are theoretically the % of your portfolio in each asset.
Great post! You may like to read the 98 paper by S. Redner (random multiplicative processes: an elementary tutorial) in which he tries to explain why the arithmetic average takes so long to be estimated in simulations. Another cool topic adjacent to your post is Jaynes’ derivation of the second law of thermodynamics using information theory. Basically the time ensemble of sequences is so thoroughly dominated by sequences with the geometric average (which happens to maximize entropy with the given constraints) that you can discard the less probable ensemble (the one with sequences responsible for the higher arithmetic averages…). This is also the basis of Shannon’s coding theorem.
Ole Peters stresses this properties when saying that you don’t need utility theory if you actually pay atention to the dynamics of the process. Your analysis of Bernoulli also makes this point I guess.
Thanks for papers. The Jaynes one looks very interesting. I’m always intrigued how aspects of finance investing seem to parallel science and engineering subjects.
Gracias por tu gran trabajo, el mismo ejemplo de aleatoriedad que muestras con la media aritmética, donde salen muchas combinaciones, lo puedes hacer con los resultados del backtest? O sea ordenar aleatoriamente los retornos de los activos, aplicar el rendimiento geométrico y ver la gama de resultados, para ver cuanto se desvían del resultado ordenado cronológicamente. En teoría el orden aleatorio no debería de afectar o sí? Se deberia de cumplir rendimiento geométrico de todas las simulaciones mayor al del activo de riesgo y menor volatilidad?
If I understand you correctly, I think somewhat. Within the same asset, if all your looking at is a summary of the entire period, you can for some metrics. But if you try to mix assets, it wont work, and it won’t work for things like max drawdown. This is because the standard deviation and correlation are somewhat correlated through time with investing assets and you break that by randomizing them.
The best effort at this was something resolve asset management did that I think they called jitter analysis. They made things random but only in a tight window. The challenge is if standard deviation and correlation are somewhat correlated through time, and I think they are, then you can’t just pick a return from September 2017 and replace it with one from March 2020. But if you re-shuffle all the returns within March 2020, its going to be closer.
Muchas gracias por la respuesta, hice algunas pruebas después de revisar sus cálculos en una hoja de Google con dos activos VFINX Y VFITX con los retornos mensuales, pude comprobar que en el orden cronológico se obtenian buenos resultados. Usando distintos espacios de tiempo para dicho cálculo desde los últimos 3 retornos mensuales hasta los 20. Una media de 9% para la desviación, 10% CAgr y un Max. DD 15%. En cualquier caso se cumplía la teoría aumentando el rendimiento VS activo de riesgo y menor volatilidad,sin tener en cuenta los costes ni efecto divisa con rotaciones mensuales a fin de mes. Sin embargo como comenta, con pruebas aleatorias y comprobando que no se perdiera mucho dicha descorrelacion entre los activos. Los resultados cambian muchísimo, prácticamente igualando combinaciones aleatorias entre los dos activos sin más. El simple hecho de combinar pesos aleatoriamente de los dos activos, la desviación baja mucho con respecto al activo de riesgo,pero el reto de superar el rendimiento se pierde. Me gustaría preguntarle si hacer los cálculos de la correlación para aumentar el rango de mezcla realmente aporta gran diferencia? En mis pruebas no pude apreciar prácticamente nada en cuanto a resultados. Entiendo que al ampliar el rango existe más probabilidad de capturar ambos activos dentro de la mezcla. Gracias por sus aportes nuevamente.
This is such a good post. Is there a way to get email alerts of your posts (i like the kelly one too) but not all the portfolio updates?
Not right now. I may be changing the way I post the weekly portfolio soon though so it may work itself out. If I don’t I’ll see if I can figure out how to opt out of the portfolio updates.
I’ve been thinking about this concept for hours straight now and I’m finding it so hard to intuit. I’m getting lost.
So one thing is that if you look at this as a bet where someone is offering you odds of 50 to 33, you realise you dont need to bet your whole bankroll. And so you can check which amount to bet has the highest geometric return, and if you actually bet half your bankroll (so on bet number 1 you will either lose 17 or win 25, that has a geometric return of 1.25*.8333 = 1.041. And this is pretty much the max geometric return and if you simulate that 100 times you are pretty likely to do a lot better. (17% of your capital at risk is also the kelly optimal amount on this bet, is that by chance or is that exactly what the kelly criterion is solving for?)
I realise im not actually saying anything here or offering you any questions, i guess im rambling with my thoughts. Do you struggle with this concept intuitively? I think Nassim Taleb refers to this concept as a volatility tax.
I definitely used to, and I still do on some of the concepts. It’s not always straight forward.
Fascinating. But I think we can get to the point more simply:
1 + 0.5 -.333 = 1.167, creating the illusion of an arithmetic return.
But that equation doesn’t represent what happens. Returns don’t add, they multiply–so the sequence will converge to:
1 x 1.5 x .667 = 1, erasing the illusion.