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Ergodicity – The odd word with important implications for investors

A useful concept, if odd word, ergodicity is a lot easier to grasp through examples than definitions, which brings us to card-counting at Caesar’s Palace and Russian roulette

07/09/2020

Juan Torres Rodriguez

Juan Torres Rodriguez

Research Analyst, Equity Value

Andrew Evans

Andrew Evans

Fund Manager, Equity Value

If someone offered you an investment opportunity where 99 out of 100 people would make a 50% return in a day but one person was sure to lose all their money, would you take it? On balance, of course, that looks a pretty attractive proposition but, if you happen to be the one loser from the group, you are unlikely to gain much comfort in knowing your 99 fellow investors are out and about celebrating their good fortune.

In addressing that question, you have just taken a step into the world of ‘ergodicity’, which can actually be very helpful for investors – just so long as you can deal with the deeply off-putting name. Ergodicity is one of those concepts that is a lot easier to grasp through examples than definitions – hence our hypothetical investment offer at the start – so let’s dig a little deeper into the idea by way of a trip to Las Vegas.

This illustration comes courtesy of author and investor Taylor Pearson who, when he appeared as our guest on The Value Perspective podcast, kindly took us through some ideas we first came across in his article A Big Little Idea Called Ergodicity. As he notes, the classic examples of ergodicity tend to involve gambling because it is easier to measure, which is why we now are heading through the doors of Caesar’s Palace.

Card-counting strategy

There we find 100 gamblers who are all raring to play blackjack – not least because they have developed a card-counting strategy that gives them an edge over the dealer. So good is this technique, in fact, that they are each staking $1,000 and are confident, on average, of turning that into $1,500 a head by the end of the night. On the downside, the strategy carries the risk that one out of the 100 loses all their cash.

In short, then, after some serious card-playing, 99 of our gamblers emerge blinking from Caesar’s Palace 50% ahead on the night but one ends up broke. That is very hard luck for that person but, in the nicest possible way, it is not a problem for the other 99. Say it was gambler 28 who lost all their money, it does not affect gambler 27 or gambler 29 in any way.

That overall 50% upside is what is known as the ‘ensemble’ average (the average of an event happening many times concurrently), which can be contrasted with the ‘time average’ (what happens when you do something a lot of times consecutively). If that rings a bell, it may be on account of a piece we wrote six years ago, Law of averages, which touched on the idea of ergodicity in all but name.

What a difference a day makes

Translating our blackjack example into a time average would see just one gambler visiting Caesar’s Palace day after day with the same card-counting technique and the same initial stake. As Pearson notes: “On day one they have $1,000, on day two they have $1,500 dollars and by day 18 they have $1m. By day 27, they have $56m – and then, on day 28, they lose everything and that is the end of the strategy.”

A more usual – and brutal – way of explaining ergodicity is by way of Russian roulette. “Say you offered six people $1m each to play Russian roulette and then asked them about the experience afterwards,” suggests Pearson, tongue firmly in cheek. “Five out of the six would probably recommend it as an exciting and profitable pursuit – the sixth person, however, will be in no position to reply.

“In ensemble average terms, with six players, you win a lot of money 83.33% of the time, which makes it seem a very attractive game. At a time average level, however, where one player pulls the trigger six times, they are guaranteed to lose in a very dramatic way. You might just roll the dice and take $1m to play one time, but there is no amount of money that would make it smart for one person to play six times in a row.”

‘Non-ergodic’ systems

In an ‘ergodic’ scenario, the average outcome of the group is the same as the average outcome of the individual over time – for example, where 100 people flip a coin once or one person flips a coin 100 times. With our Caesar’s Palace and Russian roulette examples, however, the ensemble average and the time average of the series are not equal and this makes them ‘non-ergodic’ systems.

The same goes for investment as, again, the ensemble average and the time average do not equal each other. “Even if you get the probabilities right in investment, you cannot achieve the returns of the market unless you have infinite pockets,” points out Pearson. “You do not care about the returns ‘the average investor’ sees from a particular portfolio or strategy – you only care about the returns that you see yourself.

“So if I blow up on day 28, I am in no way comforted to know that someone else did not. If an investor eventually has to reduce their exposure because of losses or margins calls, because they retire or a loved one falls sick or whatever it is, the returns they see will eventually become divorced from the market. It is a non-ergodic system.” And we will consider the practical implications of that in our next piece.

Author

Juan Torres Rodriguez

Juan Torres Rodriguez

Research Analyst, Equity Value

I joined Schroders in January 2017 as a member of the Global Value Investment team. Prior to joining Schroders I worked for the Global Emerging Markets value and income funds at Pictet Asset Management with responsibility over different sectors, among those Consumer, Telecoms and Utilities. Before joining Pictet I was a member of the Customs Solution Group at HOLT Credit Suisse.  

Andrew Evans

Andrew Evans

Fund Manager, Equity Value

I joined Schroders in 2015 as a member of the Value Investment team. Prior to joining Schroders I was responsible for the UK research process at Threadneedle. I began my investment career in 2001 at Dresdner Kleinwort as a Pan-European transport analyst. 

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