Many, even those that think they know mathematics, will do this simple but misleading mistake.
We've even seen experienced investment people getting this wrong.
The mistake of mixing up what kind of average to use.
If you're interested in estimating growth, historical growth, comparing performance of different strategies (and we all are interested in those things) - then avoiding this mistake become supremely important.
Let's go through three examples: The Merchant of St Petersburg, Chekov's gun (an easy example), and an Investment Portfolio.
We will see that the arithmetic average is misleading for evaluating an investment, and yet, this is a prevalent measure that is commonly presented with investment advice, and as we will see, the arithmetic average brushes risk under the carpet of miscalculation.
1. Slightly trickier example: The Merchant of St Petersburg
A paradox borrowed from Mark Spitznagel that in turned borrowed the example from Daniel Bernoulli.
Here's an adapted version of Bernoulli's paradox.
A 18th century merchant wants to ship merchandise from Amsterdam to St Petersburg, over a pirate-infested Baltic sea.
Wrong century, but still seems looks like a very savvy merchant.
Jan Gossaert, Portrait of a Merchant, c. 1530
Our merchant has 11 000 florins, buys merchandise and pays shipment costs for 8000 florins, and sell the merchandise in St Petersburg for 13000 florins, making a 18% gain on the trade.
Unfortunately, every 20 ship is lost to pirates, which means that all the invested 8000 florins are lost, and the merchant is left with 3000 florins. Our merchant makes a -73% loss. So, there is a 95% possibility of making a 18% gain, and a 5% risk of making a 73% loss.
The arithmetic average of this trade would be 95% * 18% + 5% * (-73%) = 13.45%
But pirates are frustrating. Is there a way to get around these infrequent setbacks? The Merchant ponders his options, and finds an insurance company that can insure his cargo for 800 florins.
This would change his expected gain to only 12200 florins each time the ship arrives in St Petersburg, changing the profit to only 10.9%. When the ship is captured by pirates, he only loses his insurance money, that is, 800 florins, which, on that trade, limit the loss to 7,3% instead of 73%.
Let's do the arithmetic average once more: 95% * 10,9% + 5% * (-7.3%) = 9,9%
So the expected average outcome is lower than going without the insurance.
What should our rational merchant do?
The insurance seems expensive, but is it really?
The crux here is that one can't add percentages. A loss of -10% is not the same as a win of +10%. It's not meaningful to add them together. What our Merchant is really after is how his wealth is expected to grow.
So if he does the trade between Amsterdam and St Petersburg a hundred times, reinvesting what he earned, he could expect, without insurance, a growth of 1.18 of his wealth if the ship arrives, and otherwise a setback to only 0.27 (=100%-73%) of what his wealth was when he sent that unfortunate shipment.
To evaluate the uninsured case then, we need to look the trade occuring again and again, multiplying the trade to the power of 95, and the negative pirate-outcome to the power of 5: 1.18^95 * 0.27^5, which corresponds to a growth of 9674 times his initial investment after 100 shiptments.
As this corresponds to 100 times the trip, we need to take the 100th root out of 9674 to see what it would correspond to as growth per shipment if there were no pirates. The 100th root of of 9674 is 1.096, which is called an geometric (not arithmetic) average, which is then 9.6% per shipment.
This is clearly lower than the initial 13.45% we calculated!
There's something going on here!
If we evaluate the insured case, the return is 1.109 in the 95 shipments when things go well, and 0.927 in the 5 shipments when things go bad. So his wealth will increase with 12498 after 100 shipments, corresponding to an expected 9.99% growth per trade (close, but not exact to the arithmetic average).
That's a 30% higher return in the end for our Merchant.
So our Merchant will be a lot more wealth over time with the insured alternative, in stark contrast to the first simplistic calculation using averages as we learned to calculate them in, well, kindergarden?
It seems that, especially for large losses, the arithmetic average gets things dangerously wrong and underestimate the risks.
Interesting.
2. Easy example: Chekov's gun
Another example.
Three russian frenemies, Alexei, Boris and Chekov, play russian roulette (with a three barrel gun, to make it simpler). They all put 100 roubles on the table, and after the game, Chekov ends up dead.
Their average outcome on the investment of this would be (+50% + 50% - 100% ) / 3 = 0 %
A zero sum game when it comes to wealth distribution, so no surprises so far.
So there would be, on average, no expected change in wealth playing this game, but would Chekov agree?
Let's look close at what happens.
Once again the arithmetic average shows up. As in our Merchant example, it assumes - almost always erroneously for the kind of questions that we want to study - that units can be compared one-per-one, that is that 1% up is the same as 1% down, so they can be added together and divided to create an average.
But of course, that is not the case, especially not for our frenemy Chekov. He will suffer an even more dire fate than our Merchant from St Petersburg.
Because what he, Alexei and Boris have in common, is that they are interested in the growth of their investment (and perhaps the sudden exit of one of their competitors).
So, what is interesting for the individual is the growth one can expect from playing the game, and especially playing the game several times.
Let's say that Chekov, while still alive, decides to play the game three times, always with three players that always match what Chekov puts on the table, and, unfortunately, our friend Chekov dies the third time he pulls the trigger.
His growth will be: 150% the first game, then 150% the second game, and 0% the fatal final round. He's wiped out.
So the growth of his investment here would be 1.5 * 1.5 * 0 = 0, with a complete loss of wealth and no possibility to rejoin the game.
The total expected growth to do Chekov's little game would thus be 0, even if he earns 50% on the first two games.
The worse the outcome of a single game, the more misleading the arithmetic average becomes.
3. Stock market example
A final example.
Let's take the US stock index for the years 1970 - 2022. The arithmetic, inflation adjusted average, is around 7.5% - a number that we have seen many times, and that is being touted as the expected inflation adjusted return from the US stock market.
This inflation adjusted expected average of 7.5%, we're sorry to say, is wrong.
As we see above, it's really growth we are concerned with, and the geometric inflation adjusted average is only 6% of the US stock market. So if one invested those 52 years, one would earn 6% per year, on average, known as the cumulative average growth rate (shorted CAGR) - inflation adjusted.
It's just harder to get up when one has been fallen down, and this is more accurately reflected in the geometric average.
The very common number that is floated around is unfortunately wrong, if one wants to stay invested over time and consider one's growth.
Growth is hard.
Can we hedge?
Yes. We can hedge. If one mixes 20% gold into the mix and rebalance annually, we actually - and very counterintuitively from most investment advice out there, get 6.3% geometric average return.
So actually slightly better, and this is more astonishing and counterintuitive than one might think.
And we probably sleep better, because the swings are smaller with this 80/20 portfolio.
So adding gold doesn't cost us as the common thinking might have it - it has actually given us a higher return - just like the insurance for the St Petersburg Merchant above.
We earn more money over time. There's no other way to put it.
Another way to hedge is using options, though this requires serious expertise. Options can give insurance that is quite similar to our Merchant of St Petersburg above. If we buy an option that costs us 2% of our capital each year, but has a payoff that offset the loss each time the loss is more than 15%, we get a geometric average return of 7.3% for those 52 years.
There are actually ways to hedge an investment, that makes it less volatile, and gives it higher return over time.
What's going on?
There are three common ways to calculate means.
Arithmetic, geometric and harmonic.
Arithmetic concerns what can be added and subtracted linearly (such as adding or removing apples in the shopping cart), geometric concerns growth, and harmonic concerns velocity.
For any given set of changes, the arithmetic average will be the highest, the harmonic the lowest, and the geometric will be in between.
Almost everywhere we look we see how this is presented wrong when we see assumptions on return.
We are concerned with growth, not apples and oranges.
This has misleading consequences when making assumptions about the performance of different portfolios, the true cost and impact of downswings and risk, and what numbers people unfortunately put in when trying to calculate different "rich-by-Excel"-numbers, and misleadingly indicates especially high volatility or large drawdowns as less dangerous than they are.
Geometric averages more correctly assess the impact of growth.
And the risk of ruin and the impact of large drawdowns are very dangerous risks with big impacts, that the arithmetic average gets very wrong as we saw with our friend Chekov and Merchant above.
Still, one should not only rely on geometric averages.
One still need to be prepared for the black swans, ergodicity - ensuring that one can live through and recover also from the very bad scenarios - and understand that a close shave with an absorbing barrier is more common, likely to hit at some point during a lifetime, and hard to get out of.
Never put your life savings at unacceptable risk, thinking that the barrell will always be empty when playing russian roulette.
Remove hope out of the equation. Look at scenarios. Simulate. Be careful when seeing an average being presented. What will happen in the worst and the bleak scenario. Do you get stuck in an absorbing barrier?
And will you be able to get up again?
Farewell,
//antinous&lucilius
