Let’s open with a question- what is the mean excess return earned by US equities (over short-term interest rates) since 1870?

*Data Source: Professor Shiller, http://www.econ.yale.edu/~shiller/*

The simple answer is to take the average, which is 5.4%- this is the

*arithmetic*mean (AM). The trouble is, that’s not what you’d have earned. An investment that returned 5.4% excess every year would have earned 7.5 times as much as the equities over the period. The equities earned the same as an investment that returned the much lower 3.9%- this is the

*geometric*mean (GM). Clearly, which you use makes a big difference.

what do they mean? The arithmetic mean of some data points is the sum of all of them divided by the number of them; in context, it is the average of the different returns earned every year. The geometric mean of n data points is the n

^{th}root of the product of those values (1); in context, it is the total return earned over the period, expressed as a constant annual rate.

Formally, if we have n years’ returns, called r

_{1}, r

_{2}, ..., r

_{n}, then

and

To illustrate the difference, suppose a fund doubles in value every year for three years, then goes bankrupt. The returns are then 100%, 100%, 100%, -100%. The arithmetic mean return is

While the geometric mean is

No matter what you earned before, if the fund goes bust the GM will be -100%; it is simply a reflection of what you earned over the whole period, whereas the arithmetic mean reflects what happens in between.

Now consider the following 4 scenarios, with 2 years of returns:

Two things become apparent: firstly, the geometric mean is lower than the arithmetic mean; it can be proved that the geometric mean will always be lower, unless the returns for every year are the same. Secondly, the larger the moves are, the bigger the difference. This leads us to the natural insight that the higher the volatility of returns, the greater the difference between arithmetic and geometric means. In fact, one commonly used approximation is (2)

So, for equities with a volatility of around 20%, an arithmetic mean assumption of 5% is roughly the same as a geometric mean estimate of just 3% (in our example above, the volatility was 17.5%, which leads to an accurate GM estimate of 3.9%). As might be guessed, this is one of the biggest causes of difference between quoted estimates; without knowing which mean is estimated, the figure is arguably of very little value.

In fact, some theoretical estimates are of the arithmetic mean; this is because it has favourable statistical properties (such as being unbiased), and is a natural property to estimate when fitting a distribution to historical values. For complex models, it may make more sense to use this figure. That said, in my last blog I highlighted the difficulties involved in effectively predicting equity returns, and some of the pitfalls and risks of complex models.

Moreover, any long-term investor should be more concerned with geometric mean returns, since they are what the asset will actually earn. Either way though, whichever you choose, anyone who uses an expected return assumption should be very clear about exactly what they are estimating.

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(1) NB- the values cannot be negative. For returns r

_{1}, r

_{2}, etc., we take the geometric mean of (1+ r

_{1}), (1+ r

_{2}), etc, and subtract 1. As such, the geometric mean is the annualized return earned.

(2) Although other, more sophisticated approximations are also used, such as:

http://www.cdiadvisors.com/papers/CDIArithmeticVsGeometric.pdf

*Please note that all opinions expressed in this blog are the author’s own and do not constitute investment advice. Click here for full disclaimer*