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Edge, Paradox, Bravery and the Stock Market

Updated: May 17, 2022

I have written in the past about the Kelly Betting Criterion, named after the Bell Laboratory researcher John L. Kelly. In short, ‘the Kelly’, as it is known in rarefied circles, is a very simple mathematical formula for determining the percentage of your purse to bet on, say, a horse. It can also be applied to investing, and famed investors Warren Buffett , Bill Gross, and Jim Simons, have all used it to help determine position size for a particular investment – you can see the link.

"Making money legally should never be easy"

The formula is simple because the recommended position or bet size is directly proportional to your edge – an edge being some insight you have that your competitors don’t. Some insights can be gained unfairly, such as card counting or using a coin in a game of ‘toss’ that only you know is biased. If you want to play fair, your edge will generally involve taking advantage of your competitors’ tendency to be irrational. Irrational behaviour can be emotional, such as the tendency of us English to think our national football team is always going to win, thus pushing down the odds. Or, it can relate to a poor grasp of maths and logic, wherein the probability of some outcome is thought to be much higher – or indeed lower – than is actually the case.

For example, in a room of 30 people, the probability of two having the same birthday is 71% – most people guess that the probability is below 10%. This is commonly known as the ‘Birthday paradox’, though it is not really a paradox. A paradox is a “statement that, despite apparently valid reasoning from true premises, leads to an apparently-self-contradictory or logically unacceptable conclusion” – that’s one for fans of Monty Python’s ‘argument sketch’.

To turn the observation regarding people’s misperception of probabilities into money, you’d then need to construct some sort of bet that didn’t appear like a trap. This is where some ‘smarts’ and a bit of smooth talking might come in – after all, making money legally should never be easy.

An example of a true paradox is the ‘liar paradox’. The statement “I am lying” appears valid – I was able to write it and you were able to read it. But if I am indeed lying, then I am telling the truth, in which case I wasn’t lying as I said I was. This quirky contradiction may have piqued your interest, but you’ll be tempted I suspect to think it is essentially frivolous. On the contrary, it was the basis for what were arguably among the greatest ever mathematical ideas, Gödel’s incompleteness theorems.

Later in life, Kurt Gödel suffered mental illness. He had an obsessive fear of being poisoned and, after a period spent refusing to eat, died of starvation. The circular nature of paradox may have got the better of him. Wikipedia lists 289 known paradoxes, though many of these will be similar – for example, the Barber paradox is essentially the same as Russell’s paradox. Wikipedia groups the paradoxes into categories such as ‘logic’, ‘mathematics’, ‘probability’, ‘politics’, and ‘self-reference’. The last category is for those paradoxes where a contradiction arises from some sort of self, or circular, reference and it is where ‘Barber’ and ‘Russell’ can be found.

In relation to the former, a town barber only shaves people who do not shave themselves. The question is, does the barber shave himself? The barber cannot shave himself as he only shaves those who do not shave themselves. As such, if he shaves himself he ceases to be the barber. Conversely, if the barber does not shave himself, then he fits into the group of people who get shaved by the barber, and thus, as the barber, he must shave himself. Getting dizzy?

My favourite I think will always be ‘Monty Hall’. This, like ‘Birthday’ is not really a paradox but an example of unintuitive probabilities. A game show host presents a contestant with three doors, telling him that behind them are two goats and a car. He tells the contestant to point to a door, but not to open it. Next, the game show host, who knows what’s behind each door, opens one of the other two to reveal a goat, then asks the contestant if he’d like to change his choice to the third door. What should he do?

Most people think it makes no difference. After all, the odds were one in three to begin with and they’re still one in three. Right?


I spent a very late night with a sceptical friend many years ago, resorting to playing the game ourselves a hundred or so times before he could see it was advantageous to switch.

An easy way to understand the puzzle is to think of a million doors behind which are one car and 999,999 goats. You pick a door and the host opens 999,998 to reveal all goats. Where do you think the car is?

Can a knowledge of paradoxes or human misperception help you to be a better investor? Absolutely. A “40% off” sale at a department store will see people clambering over each other to get through the entrance. A 40% fall in the stock market, however, will see a stampede the other way. In the case of the department store sale, people know they are getting a bargain and that when the sale ends prices will go up again. Many cannot seem to do this with stock markets.

The analogy is not entirely fair. In 1990, the Japan stock market fell 40%. Over the next 18 months it fell another 30%. That said, the rational shopper would have realised that “pre-sale” prices were absurdly expensive – the trailing PE for the Japan stock market was 60x – so a 40% fall wasn’t going to cut it.

So, a good grasp of human nature will help, but you’ll also need a strong valuation framework. Moreover, you’ll also need to be brave. The trick to department store sales is that you’re never quite sure when they will end – they can go on a long time. With financial markets, the waiting can require great strength of mind, whether for cheap things to go up or expensive things to go down.

Published in Investment Letter, June 2019

The views expressed in this communication are those of Peter Elston at the time of writing and are subject to change without notice. They do not constitute investment advice and whilst all reasonable efforts have been used to ensure the accuracy of the information contained in this communication, the reliability, completeness or accuracy of the content cannot be guaranteed. This communication provides information for professional use only and should not be relied upon by retail investors as the sole basis for investment.

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