Law of averages - Investors need to distinguish between 'time' and 'ensemble' averages


Kevin Murphy

Kevin Murphy

Fund Manager, Equity Value

Let’s play a game we will call ‘Russian dice’, the rules of which were invented by a physicist/economist called Ole Peters. They are pretty simple – roll a dice and, if it comes up ‘one’, I will shoot you. Do you fancy playing? It does not sound very appealing but, if you were a nihilistic mathematician, you might be tempted because – in the very strictest terms – on average you will be absolutely fine.

The word ‘average’, however, can be somewhat misleading if not defined very precisely. If 100 people roll the dice instantaneously, the average result is ‘three and a half’ – that is, (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 – so, if you are judging the game on the average result, everyone survives. However, if I asked you to roll the dice 100 times in a row, it is extremely unlikely you could do so without at some point seeing a ‘one’ come up. Bang. You do not get a chance to see the result of the 100-roll average over time.

Intuitively, we all know there is a difference between these scenarios without any complicated maths or concepts. Mathematically speaking, it is known as the difference between an ‘ensemble average’ (the average of an event happening many times concurrently) and a ‘time average’ (what happens when you do something a lot of times consecutively). However, this concept and its implications are not well understood in investment.

This distinction between time and ensemble averages is critical – and not just for those who may need reminding not to play dice with someone holding a gun. Here on The Value Perspective, we struggle to think of anything investment-related that does not function in terms of time averages, whether it is the stockmarket, any other asset class and indeed anything where the system is dynamic – which is to say, most things.

Unfortunately, an awful lot of financial theory is based on ensemble averages – beginning with the theory of expected return. This theory smoothes out fluctuations in the price of an investment before they happen because, in effect, it assumes a dice will always come up ‘three and a half’ and a coin toss will always split exactly 50/50 between heads and tails.

The ability of ensemble averages to smooth out fluctuations often encourages investors to introduce leverage into their portfolios – in other words, to take on debt to invest more in the hope of achieving higher returns. If the theory says the dice will always come up ‘three and a half’, many investors use this successful result to justify using leverage. Once again, however, this is just not how the world works – if you leverage up ahead of a dice roll and it comes up ‘one’, then the consequences for your portfolio could be every bit as nasty as in our game of ‘Russian dice’ above. In short, leverage and expected return do not mix.

Our final point on the distinction between time and ensemble averages is arguably the most controversial because it concerns its impact on the widely-used Sharpe ratio. Simply put, this is a measure of risk-adjusted performance and, the higher the ratio, the better a portfolio’s risk-adjusted performance either has been or is expected to be.

While The Value Perspective has plenty of issues with the very concept of the Sharpe ratio – it is after all an incoherent calculation that does not work either mathematically or conceptually – we will confine ourselves here to just the one. That is the fact the ‘ex ante’ Sharpe ratio – the one used ahead of time in a predictive fashion – uses expected return as its numerator and, as such, ends up as a hostage to ensemble averages. In this context, the Sharpe ratio is collateral damage in the ongoing war between time and ensemble averages.


Kevin Murphy

Kevin Murphy

Fund Manager, Equity Value

I joined Schroders in 2000 as an equity analyst with a focus on construction and building materials.  In 2006, Nick Kirrage and I took over management of a fund that seeks to identify and exploit deeply out of favour investment opportunities. In 2010, Nick and I also took over management of the team's flagship UK value fund seeking to offer income and capital growth.

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