The divine answer to ‘Shannon’s Demon’ – with ‘10-K Diver’

Twitter phenomenon and recent podcast guest 10-K Diver uses his knack of breaking down complex ideas into easily digestible morsels to explain how to turn a zero expected return into a positive one

23/06/2022

Andrew Evans

Andrew Evans

Fund Manager, Equity Value

Juan Torres Rodriguez

Juan Torres Rodriguez

Fund Manager, Equity Value

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‘Get a cup of coffee.’ So starts each thread from 10-K Diver, the ‘nom de Tweet’ of an otherwise anonymous Twitter account that aims to explain complex financial and mathematical concepts in digestible and engaging ways. Having attracted almost 250,000 followers in just two years, you would have to say the man behind the account has succeeded and we were delighted to welcome him recently to The Value Perspective podcast.

 

You can hear the episode in full by clicking on the above link but here we will focus on 10-K Diver’s take on a concept known as ‘Shannon’s Demon’. Not quite as diabolical as it sounds – the ‘Demon’ part essentially indicates it is a thought experiment – this is the idea put forward by US mathematician Claude Shannon that two uncorrelated assets with a zero expected return can produce positive returns, if combined in the right way.

“Let’s say you have a stock that is extremely volatile and, every day, it either doubles or it halves – so, if the stock is at $100 today, it may be at $200 tomorrow or it may be at $50,” begins 10-K Diver. “And let’s also say each day is independent of all the others that came before so there is always a 50% chance of either outcome. The question is – can you make money from the stock over a long period of time?

“Several interesting concepts come out of this – the first of which is you can calculate the expected outcome. If the stock is at $100 today, we know tomorrow’s price could be $200 or $50 – and $200 plus $50 divided by two reaches an expected value of $125. And if today the stock is at $100 and tomorrow the expected value is $125, your expected return is 25% – in one day – which is phenomenal.”

Long-term considerations

This 25% ‘arithmetic return’ suggests we go all-in on the bet but, as 10-K Diver warns, we also need to think about the longer term. “In that case, because there is a 50/50 chance of each, the doublings and halvings will be approximately equal to each other and each doubling will cancel out each halving,” he continues. “If the stock doubles and then it halves or it halves and then it doubles, you are left exactly where you started.

“So if you put whatever money you have into the stock and wait for a very, very long time, what is the most likely result? It is that, if you play this game 1,000 times, you will get 500 doublings and 500 halvings in some order. And if you get 500 doublings and 500 halvings overall, you are left with exactly the same amount of money you started with. In other words, the most likely return you will get over these 1,000 days is 0%.

“There may be a positive arithmetic expectation of 25% but, if you try to compound a sequence of such bets, you end up with a 0% return. And, in situations where each bet works on the previous bet successively and all compound together – that latter ‘geometric return’ is far more representative of what you will get than the arithmetic return.”

Most likely scenario

10-K Diver acknowledges there may be times when an investor who goes all-in on this stock over the longer term will end up making a lot of money but the point remains – the most likely outcome of this scenario is you end up with nothing. Where it gets really interesting, he adds, however, is with Shannon’s solution to the problem, which shows – regardless of the geometric return – you can actually make money from the stock.

“Even though it seems like, over a long period of time, you will get a 0% return, it does not have to be 0% because you do not have to put all your money into the stock each time,” 10-K Diver explains. “Shannon’s policy for betting on the stock is very simple – at each turn, you rebalance your portfolio. You have a certain amount of your money in cash and you put a certain amount of your money into the stock.

“In this particular case, you keep 50% of your portfolio in cash and you put 50% of the portfolio into the stock. The next day, if the stock were to double, the stock part of the portfolio would increase and the cash part of the portfolio would remain exactly the same and then what you do is rebalance – you sell some of the stock and get the money back into cash.

Power of rebalancing

“On the other hand, if the stock were to halve, then, to bring the stock portion of the portfolio back to 50%, you would have to spend some of your cash to buy the stock. So on days the stock halves, you go and buy more of it; and on days the stock doubles, you sell. And this idea of rebalancing is super-powerful because it lets you escape this difference between the arithmetic return and the geometric return.

“Remember – the arithmetic return is 25% and the geometric return is 0% but, if you follow Shannon’s strategy, the most likely return you will get over a long period of time is 6% per day. So you can’t do as well as the arithmetic return – you can’t compound at 25% per day over a long period of time – but you do not have to settle for 0%. And the way you do that is through rebalancing the portfolio.”

This idea is particularly important for investors in companies because stocks can be so volatile – and indeed the difference between the arithmetic return and the geometric return is known as the ‘volatility tax’. In Lumping it and liking it, we will take advantage of 10-K Diver’s handy knack of breaking down complex ideas to further explore this point – whenever you fancy another cup of coffee ...

Author

Andrew Evans

Andrew Evans

Fund Manager, Equity Value

I joined Schroders in 2015 as a member of the Value Investment team and manage the European Value and European Yield funds. 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 and hold a Economics degree.

Juan Torres Rodriguez

Juan Torres Rodriguez

Fund Manager, Equity Value

I joined Schroders in January 2017 as a member of the Global Value Investment team and manage Emerging Market Value. 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.  

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