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A Game You 'Probably' Can't Lose

GAM Systematic’s Dr Ewan Kirk and Dr Tom Gillam pay a short visit to a world of odds-based binary bets and apply their findings to a more familiar financial landscape.

11 April 2017

If you were given the opportunity to participate in a game of coin flipping and told that the coin had a 60% chance of landing heads and a 40% chance of landing tails, how well do you think you would do?

Such an experiment was recently conducted with 61 students of economics or finance and young financial industry professionals. They were given USD 25 and told they could bet as much as they wanted on each toss, could continue to play the game for 30 minutes and could toss the coin as often as they liked.

Even without any deep grounding in probability theory, it should be fairly obvious that it makes sense to bet on heads most, if not all, of the time, and that any game that is so biased in favour of the participant should be played as often as possible. The issue is, therefore, how to determine the most effective staking plan.

We will spare you the algebra and formulas, but the participants should have used the famous Kelly Criterion, which was originally formulated in 1956 and is very well known in both the finance and gambling communities. This determines that, where there is a 60% chance of winning and a reward of 1-to-1 odds (the chance to double your money), the long-run growth rate is maximised when the participant stakes 20% of the bank roll at each opportunity.

So, how did the subjects do? The answer is – surprisingly badly. Seventeen participants ended the game with less than USD 2. No fewer than 18 of them bet their entire bank roll on a single flip (exposing them to a 40% probability of being wiped out), while almost half bet on tails on five or more occasions, which is clearly sub-optimal. But can we learn from this about the way we typically invest?

While a 60% chance of winning seems like quite a low probability, in the field of finance this is about as good as it gets. It would be very interesting to be able to estimate what realistic odds one can expect to have when making a financial decision. We need to apply a little bit of mathematics here, as this will allow us to shift from a world of odds-based binary bets into the vast financial landscape of trading, volatilities and Sharpe ratios.

Let’s assume that we have an asset with a known and constant annualised volatility of 10%. On a daily basis the standard deviation (a measure of volatility) of the returns is approximately 0.10/16, which is 62.5 basis points. If we are 60% correct in our call of the direction of the market, the daily expected return is 10 basis points. If this can be achieved every day for a year, the total return would be an impressive 25.5%.

These results would provide a (risk-adjusted) Sharpe ratio of 2.5, which is the stuff of dreams for any manager. We can, therefore, conclude that the probability of a typical discretionary or systematic decision being correct is much smaller than 60%. So, next time a manager discusses his or her positions, remember that the probability that they are correct with any individual call is almost certainly less than 55% and probably less than 51%.

If it is impractical to make repeated use of small edges over time, what is the alternative? Our old friend diversification comes to the rescue here. If multiple independent bets can be taken each day, we would expect to achieve better results. Furthermore, the more independent bets that are taken, the better.

Computers, however, can evaluate an entire 15,000 stock universe easily and come up with an estimate of the probability of outperformance or underperformance in milliseconds. They can then combine the individual positions in optimal ways, accounting for volatilities, correlations, liquidity and trading costs. Small signals run on thousands of positions can therefore be multiplied into attractive returns very efficiently.

With the available edge in large and mid-cap stocks quite small, the rise of the machines in investing is set to continue.

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