Ergodicity: my next high
A stock wrapped in my monologue
Before I get into to the stock of the day. I want to take a (related) detour down my new favourite rabbit hole: ergodicity.
The best way to start this is with a dice game. It is lifted in its entirety from Value After Hours (see sources below). The first game (Game #1) is set up as follows, one die with the following return profile:
You roll a 1: you lose 50% of your capital.
You roll a 6: you make +50% return on your capital.
You roll a 2, 3, 4 or 5: you make +5% return on your capital.
For the nit pickers among you, just imagine rates are zero and inflation is zero or these returns are real returns.
Simple arithmetic shows your expected return (arithmetic mean) is +3.3% ((-50% x 1/6)+(+50% x 1/6) + (+5% *2/3)). This could be interpreted as your “edge” or for example the expected value of making an event trade. For a casino this would be a significant edge.
If I gave you the opportunity to play this game 300 times, would you?
A naive extrapolation of the edge would imply a return of 1.033^300 which is ~17,000x return. Not bad eh?
But let’s take an example of rolling each number once: 3, 6, 1, 5, 4, 2. You’d expect the return from rolling each number once to be similar to the expected return of 3.3%? In fact you’ve lost 8.8% (1.05 x 1.5 x 0.5 x 1.05 x 1.05 x 1.05 = 0.912).
So what’s going on? this is essentially the definition of ergodicity. Ergodicity is when the arithmetic average = the geometric average. If there is a difference between the average of 300 people rolling the dice simultaneously and outcome of one person rolling the dice 300 times in sequence, the system is said to be “non-ergodic”. The sequence compounds your wealth in both directions, or one could say there is path dependence.
If you run a Monte Carlo simulation (I’ve not personally done this…) where you play this game 10,000 times with 300 rolls, your probability of ending up with 17,000x return is just 0.5%. You almost never end up with your arithmetic mean. Now if you work out the geometric mean (which accounts for a sequence of bets) it is negative 1.5% (!!) (if you’re into the maths ((1.5 x 0.5 x 1.05^4)^1/6)).
So over 300 rolls… you’re actually compounding negative 1.5% 300 times. Thus on average this is going to leave you with effectively zero (0.98% of your original capital for the pedants).
So taking a step back: a simple game with a positive expected return. The arithmetic mean suggests 17,000x return and a geometric mean (accounting for sequence) actually shows you’ve got a 0.5% chance of making that money and the median outcome is virtually total capital loss.
Now if we extend this into two further games:
Game #2 - we double the upside case
You roll a 1: you still lose 50% of your capital.
You roll a 6: you now make +100% return on your capital.
You roll a 2, 3, 4 or 5: you still make +5% return on your capital.
Game #3 - we halve the downside case
You roll a 1: you now lose 25% of your capital.
You roll a 6: you still make +50% return on your capital.
You roll a 2, 3, 4 or 5: you still make +5% return on your capital.
Let’s start with Game #2 (aka maximising the right tail):
Expected outcome (arthmetic average) is +11.7% (((-50% x 1/6)+(+100% x 1/6) + (+5% *2/3)). Multiply that times 300 rolls… 1.117^300… and you end with 260 trillion times your money… truly absurd.
Real world outcome (geometric average) is +3.3%, which as per the first example gives you a 17,000x return, but in this case it is the median outcome (this is why these numbers were chosen). So you need to double the upside outcome to actually realise the arithmetic outcome in the real world.
But what about Game #3 (aka limiting the left tail):
Expected outcome (arthmetic average) is +7.5% (((-25% x 1/6)+(+50% x 1/6) + (+5% *2/3)). Multiply that by 300 rolls… 1.075^300… and you end with 2.6 billion times your money… still pretty absurd.
Real world outcome (geometric average) is +5.3%, multiply that out over 300 rolls it gives you a 6.2 million x real world return. This is the median outcome! So by limiting the left tail as opposed to maximising the right tail you get a return 365x greater.
I’m going to be unfair on economists here but I think its a classic example of intuition surpassing naive economics.
The average person on the street is never going to take this bet with any material amount of their own capital, the downside risk is too great. A naive economist would point to the positive expected return and suggest that such a person is exhibiting “loss-aversion bias”. But it is no bias, this is not how the real world works for an individual. The world from an individual’s perspective is mainly non-ergodic, it is path dependent, irreversible and you are subject to total loss. You can be left unable to continue playing the game.
It does raise another question: are you the gambler or the gamble?
A topic perhaps for another day but it reveals the misalignment of incentives between an organisation and its employees. As a trader at an investment bank, ergodicity matters, survival is paramount. Positive expected value bets are not sufficient. For the bank, subject to lack of correlation and reasonable risks limits (don’t laugh), it is a population of simultaneous bets, and thus this is fine. The trader is the gamble. The bank’s anti-fragility depends on a certain amount of fragility in the individuals who are continuously replaced and hopefully upgraded.
Why does this matter?
That was long. Well done if you got this far. There are so many implications of this idea that it could be spread over several more blog posts, but to keep it brief here’s summary of what it means to me for my investment strategy:
Positive expected value special situations are not enough. The ratio needs to be much more significantly in my favour than a simple arithmetic average would imply.
For the same quantum, wherever possible, limit downside as opposed to raising upside. Buffet’s old adage: "Rule No. 1: Never lose money. Rule No. 2: Never forget Rule No. 1.”
In the context of the above, do not use leverage.
Cheap tail-risk hedges, barbell portfolios. Limiting downside risk and variance can realise real-world benefits even beyond those implied by the mathematical scenario outlined above. For example, in large draw-downs you have capital available to invest exactly when securities are at their cheapest and the risk-reward opportunity is greatest. Secondly the emotional/psychological impact can’t be understated. The pressure of a 50% drawdown is immense and hard to sustain. As per the above, the geometric median outcomes, is is very hard to recover from a 50% loss if it is realised.
The stock rehash
So there’s a a link here about margin of safety, low chance of total capital loss and positive optionality. OK, OK.. the stock, the stock..
To give you a taster:
A hated, bombed-out sector. It has had more than 2 years of classic capital destruction, really one for the purists. Textbook case.
There are various restrictions for institutions (and individuals) investing in it. Furthermore, given its boom-bust, meme-stock, bubble-mania association it is somewhat embarrassing to even admit you’re invested in it. Hence, there are some understandable, cosmetic reasons why it’s so cheap.
It is basically a net-net and there is $848mn of cash on the balance sheet vs. a market cap. of just $814mn.
The free float is even smaller. A mega cap. shareholder has 40% of the stock (struck 5.6x higher than today’s price).
There is no need for additional capital.
The business (which you effectively get for free) grew top-line +46% in 2Q. It has a long growth run-way in the double digits. It is one of the leading brands in all its categories.
It is expected to inflect to profitability in the next 2-3Qs. FCF should be positive in 2025. While an important fundamental facet of the investment case - it will also tip the stock into various value/profitable small cap. screens.
Its smaller, poorer capitalised peers are higher-up the cost curve and are filing for bankruptcy like it’s going out of fashion.
There are several transformative potential positive catalysts (tax reform, legal reform, international expansion, accretive deployment of capital). At the current price you also get these options for free.
Management: The CEO was previously at an NYC based multi-strat investment management firm, focused on “identifying mis-priced assets across various industries, asset classes and geographies”. Capital allocation hopefully won’t be learned on the job.
And after all that I’m not even going to bother pitching it! The stock has been so eloquently and thoroughly discussed there’s little point in me rehashing it… I would simply recommend you read
’s two posts here and here.I more than doubled my position size on Friday. It is a top-5 position.
As a reminder this is primarily an investment journal, this is not financial advice or a recommendation. All of the ideas are plundered from other (cited) sources. I simply own the stock at the time of writing. Please refer to the disclaimer at the bottom or my introductory post for further details.
Sources and recommendations
I first came across Ergodicity in Nassim Nicholas Taleb’s books, I forget which one they all blur into one for me. Incidentally I’d recommend the deep dive’s into the books by Cyrus Yari and Iman Olya on their rational vc podcast.
The dice example is lifted wholesale from Value After Hours S05 E29.
I’d recommend the following books:
Deep simplicity by John Gribbin
Ergodicity by Luca Dellana
Happy hunting,
The Geez
Great piece my friend. Have not seen the math laid out that way before.