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.
— breakingthemarket (@breakingthemark) September 25, 2020A lottery usually has 50 million participants paying $1 to play and pays out 30 million to the winner
You question the lottery’s payout, and want to prove its paying out $30M, so you sample 100 people’s winnings.
What average winnings will be closest to your findings:
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.
— breakingthemarket (@breakingthemark) September 25, 2020There is a lottery with 50 million participants. You don't know the number of winners or how much they win, but you know its a reasonably typical lottery.
How many people do you have to sample before you have a reasonable chance of estimating the arithmetic average payout?
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.