Ergodicity In Investing

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During their latest episode of the VALUE: After Hours Podcast, Boyar, Taylor, and Carlisle discussed Ergodicity In Investing. Here’s an excerpt from the episode:

Tobias: JT, do your you got some veggies for us today?

Jake: Yes, sir, I do. So, this segment is– it’s actually just some lessons from a simple little dice game that we’re going to construct. It’s adapted from a section of Mark Spitznagel’s book, Safe Haven, which is really about tail risk hedging, ostensibly. So, let’s play this very simple little game of dice. First off, let’s get this out of the way that the dice are fair, each side will come up one six of the time, so there’s not any tricks there. Here’s how we’re going to construct it.

If you roll a 1, you lose half of your money that you bet. If you roll a 2, 3, 4, or 5, you get plus 5% on your bet. And if you roll a 6, you get plus 50%. This, to me, it looks roughly like what the odds you might face in public equities, where a lot of times, you just kind of do okay, maybe get your cost of capital, and then some winners and some losers. We can easily calculate the expected outcome here. Add up all of the outcomes and then divide it by six. And so, that ends up being a 3.3% expected outcome or an edge. That’s actually pretty solid. Any casino would take a 3.3% edge in a heartbeat, if you can run enough into that.

Now, in our particular game here, you were allowed to roll the dice or die 300 times. If you did that with your 3.3% expected edge, your wealth then would be 19,000 times your starting amount. A 3% edge is a ton, especially when you have 300 rolls. Really any number that’s over 1.0 to the 300th power is going to be a pretty big ass number. That’s just the math. Now, first, would you want to play this game?

Jonathan: Okay. With leverage.


Jake: Yes.

Tobias: What’s the first? Number One, is it lose 50% or lose 100%?

Jake: Lose 50%.

Tobias: Okay. Yeah.

Jake: So, 50% down, 50% up, and then middle is a 5% gain, which is where your 3.3 really comes from. Let’s do a little reality check on this game first though, and let’s just imagine that we’re going to roll the dice six times and then see what is our outcome there. Let’s assume that it’s going to land on each face one time, because it’s one-sixth the chance of probability, right? So, let’s just walk through that and say that it went like 3, 6, 1, 5, 4, 2, okay?

Now, we’d expect our return to be somewhere around that 3.3% value that we were talking about. However, if you multiply out one times, 1.05 times, and then 1.5 and then 0.5, you actually end up with 0.912. You’re actually losing. So, what’s going on there? How did we end up losing here when we rolled after six tries? And also– [crosstalk]

Tobias: Isn’t that 50% up is not enough to compensate for 50% down?

Jake: That’s part of it. Yes. So, what’s going on? How are we going to get to 19,000 times our money in the next 294 rolls, if we’re already down after six rolls? Now, there’s an important assumption here that we need to understand that’s embedded in that average that we used. That 3.3% edge on every bet is the arithmetic average, okay? It’s calculated as if we allowed you to gather 299 of your friends and you would all have one die that you would roll at the same time. And then we took those and totaled those results and then gave you 19,000 times your money.

But that’s not how the real world works most of the time. Usually, you have to take rolls of the die in a sequential order through time and then multiply them out, right? So, there’s a compounding that happens. And that changes the underlying math of everything here. So, by the way, this is what non-ergodicity is. We’ve talked about this on the show a little bit before, but you’ve probably heard the term. It’s the difference between the ensemble average, which is you and 299 of your friends rolling at the same time, and the time average, which is you throwing the die 300 times in a row by yourself.

So, the time average has compounded your wealth after each roll, and it’s much closer to how the real-world works. So, this is probably what we should really be paying attention to. If you run a Monte Carlo where you play this game 10,000 times of 300 rolls of the dice and then you look at the cloud of outcomes that come out of that 10,000 runs, what you’ll find is rather shocking. Your probability of ending up with 19,000 times your money is somewhere around a half of a percent. So, it’s like 50 out of the 10,000 rolls are what are mapped out on the paths that you take through time. Almost never do you end up with your arithmetic mean.

We have to calculate instead the geometric mean, which is a completely different math to be done. I’ll spare you some of the details, but it involves taking E to the power of a log function. But if you run the numbers in our game, the geometric average is actually minus 1.5% compound annual growth rate. So, multiply that out times 300, and you end up with ostensibly zero at that point. That’s your geometric expected outcome. So, right now, you should be fairly flabbergasted. I just showed you that a simple game where you take this average that I think most people in finance use arithmetic averages when they’re trying to figure out the world, you end up with 19,000 times your money of an expected value. But if you actually do it in reality, that median outcome from the geometric average is more like zero. So, I’m thinking of Fight Club, where it’s like I Am Jack’s Cold Sweat.

Tobias: laughs.

Jake: So, let’s change the game just a little bit to see if we can learn some things here. We’ll pretend that when you win, you win quite a bit bigger. So, it’s like Bessembinder’s idea about this small percentage of stocks that provide most of the market’s return. They’re carrying the load. We’ll call this scenario, betting for the right tail. This is going to be a good outcome. So, all the same stuff, you roll a 1, you still lose half your money. You roll 2, 3, 4, 5, you get 5% return. If you roll a 6, you get 100% return. So, you doubled that previous 50% payout, okay? And now, let’s run these scenarios. Add up all of the outcomes and divide by six and you get an 11.7% expected return per role as the arithmetic average. Multiply that times 300 rolls and you end up with a final wealth expectation of 260 trillion times your money.

Tobias: [laughs]

Jake: Okay, we’re getting into some absurd. This is ridiculous. We’re also approaching St. Petersburg paradox territory here as well, which we’ve talked about. But what about the real world of geometric returns? That actually equates to a 3.3% compound annual growth rate. So, that’s the median outcome. Kind of a quick aside. Isn’t it a little bit interesting that we had to double that big payoff to get the geometric average up to that original arithmetic average of 3.3. That’s why I picked that number. So, a 3.3 CAGR times 300 rolls gives you 17,000 times your money. Now, obviously, it’s nice to pick winners if you can figure out that skew. This is what I think a lot of people are go for. They’re looking for that upside asymmetry.

All right. Now let’s examine if were to limit the losses in a scenario that we’ll call limiting the left tail. So, in this instance, if you roll a 1, you only lose 25% of your money instead of that original 50%. So, we’re protecting more of the downside. 2, 3, 4, 5, you get your same plus 5%. And if you roll a 6, you get your same 50% from our original scenario. All right. First, the simple average. Add up all the outcomes, divide by six. That’s a 10% expected return for the arithmetic average. Multiply that out times 300 rolls and you get 2.6 trillion times your money. That’s pretty good. But notice that it’s 100 times less than what betting the right tail scenario looked like.

I think there’s a bit of fool’s gold there is what I’m trying to get at. Then in a final aha here. Let’s run the numbers and find the geometric return of limiting that left tail. It’s actually plus 5.4%. So, multiply that out by 300 rolls of the dice and you get 6.2 million times your money. So, compare that right tail ending wealth with 17,000 times, your starting point, versus 6.2 million times. We’re not even in the same universe at that point. So, hopefully, I didn’t lose you with all of these numbers. I’m going to try to summarize right here.

Using the arithmetic mean can make you feel like you’re about to win big. When in reality, you might be about to lose it all. So, you have to be very careful when applying it. And doubling the payment of the big winner, i.e., fattening the right tail outcome is helpful to the geometric average that we should care about. It really makes the arithmetic mean jump off the page, which is tricking us into thinking that there’s more going on there than really is. And in the real world, limiting the severity of the left tail outcome leads to the massively positive bet, because you don’t have to make up for those major setbacks, which also goes by this catchy name, variance strain.

So, you can start to see how incredibly dangerous it is to use the simple arithmetic averages when a geometric average is called for. You can also see how dangerous most modeling is, if you ignore the non-ergodicity of the real world. And especially, when you mix it with leverage, which is basically what happened to long-term capital. As always, we’re going to end up here. Mr. Buffett got here first, and he summarized this entire veggie segments into just six words. Rule number one, don’t lose money.

Tobias: Yeah. Well done. That was great. It’s really hard to describe non-ergodicity and ergodicity, I think. When I said that Murdoch and I, on average, are pretty wealthy.

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