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The difference between arithmetic and geometric investment returns

The most commonly quoted statistics in investing are historical average investment returns. But are we talking about arithmetic means or geometric means of those returns, is it uniform across the industry and does it matter? It’s vital to understand this to analyse past results correctly.

First, some definitions:

  • Arithmetic returns are the everyday calculation of the average. You take the series of returns (in this case, annual figures), add them up and then divide the total by the number of returns in the series.
  • Geometric returns (also called compound returns) involve slightly more complicated maths. The geometric mean is calculated by multiplying all the (1+ returns), taking the n-th root and subtracting the initial capital (1). The result is the same as compounding the returns across the years.

The arithmetic mean can never be less than the geometric mean.

A simple way to explain the difference is by taking the numbers 2 and 8. The arithmetic average is 5, being (2 + 8)/2 = 10/2 = 5. The geometric mean, on the other hand, is 4: exactly 20 per cent lower. This is calculated as v(2 x 8) = v16 = 4.

The last 33 years of the S&P/ASX 200 accumulation index provides a relevant example of investment returns:

The arithmetic mean of these returns is 13.9% per annum. The geometric mean can be calculated from the index levels of 1000 on 31 December 1979 and 37,134.5 on 31 December 2012 and is 11.6% per annum. In other words, if the investment return were 11.6% every year from 1980 onwards, and you compounded the result, you would have grown your capital to the same extent as the index over the same period (ignoring cash flows, taxes, fees and so on).

This annual 2.3% gap between the arithmetic (13.9%) and geometric (11.6%) means is a big difference! If the index in fact grown at 13.9% each year compounded, it would have finished 2012 at 73,330.2, nearly double the actual value of 37,134.5.

Volatility, risk and average returns

The gap is caused by volatility. The more volatile a stream of investment returns, the greater the difference between the two measures. Let’s calculate the gap over two years for three hypothetical investment return scenarios:

  • two years of zero returns (0, 0)
  • up 10% in the first year and down 10% in the second (+10, -10)
  • up 20% in the first year and down 20% in the second (+20, -20).

The arithmetic average of each of these scenarios is 0% per annum (over-weighting the effect of gains and under-weighting the effect of losses). The geometric mean of each is different, being:

  • 0% per annum
  • minus 0.5% per annum (your capital goes from 100 at the start to 110 in year one to 99 in year two, so you have lost money)
  • minus 2.0% per annum (you have lost even more money).

If you are looking at a share investment, where the standard deviation of volatility can approach 20% per annum, the gap between arithmetic and geometric means can be significant. A 7.5% arithmetic average annual return, with 20% per annum volatility, will translate into a compound return of 5.9% a year (ie what actually ends up in your pocket over the longer term). This gap is what volatility costs the investor.

Not the risk premium

Don’t jump to the wrong conclusion that the gap is part of the risk premium. If you are comparing the returns on a risky asset with those of a risk-free asset, you need to consider the end result for both assets; that is, use the geometric return.

Any risk premium, which the investor demands to be paid to accept risk, needs to be above the compound return of the risk-free asset. Also, if you measure historical risk premiums (what was actually received) you should be careful to use compounding and not take a simple average.

The calculation of the historical risk premium can be problematic. Consider the same 33-year period from December 1979 until the end of 2012, this time for the return on Australian bond investments. Using a combination of the Commonwealth Bank Bond Index and the UBS Composite, bonds returned a compound 9.6% per annum over that period. The arithmetic average return was 9.9% per annum.

The trap for the unwary, in looking at the equity risk premium, is to calculate the return difference every year, and then average that risk premium. This is the same as estimating the equity risk premium from our 33-year sample period as 4.0% per annum (from 13.9 - 9.9 = 4.0) when it is only 2.0% per annum, using the geometric returns (11.6 – 9.6 = 2.0). If our sample 33-year period is anything to go by, this 2% is the risk premium that will compound over time.

The right set of scales for ‘weighing’ returns

Self-directed investors are well-advised to ask themselves, or their advisers, how their investments have performed over the preceding 12 months and longer periods of 3, 5 and 10 years (or longer). Take an SMSF trustee without access to the necessary advice or tools. How do they do this? If they compile a spreadsheet with each return from the relevant periods and then simply average them, chances are they are over-estimating their returns.

A point-to-point measure of how an index has moved over a 12-month period is what it is. However, it is easy enough for a self-directed investor to average these 12-month measures incorrectly (ie using a simple arithmetic mean) over multiple periods and, even worse, using the result to estimate future wealth accumulation.

Investing is a journey across many financial periods and calls for a way to ‘weigh’ those returns properly. The geometric mean is the appropriate set of scales for this job, at least until you look at the difference between time-weighted and money-weighted returns, but that is a topic for another day.

 

Aaron Minney is Head of Retirement Income Research at Challenger Limited. He was assisted in preparing this article by Senior Research Analyst, Liam McCarthy.

 

11 Comments
Joe Bob
April 26, 2018

Why precisely is geometric always lower?

Aaron
August 28, 2018

Mathematically, a geometric mean of a set of numbers is always less than or equal to the arithmetic mean.
The geometric mean equals the arithmetic mean of a set of numbers when the numbers are all the same. Thus if you use fixed returns, the arithmetic and geometric returns are the same.
When the numbers/ returns are different, consider the arithmetic mean. Some numbers are disturbed away (above and below) from the arithmetic mean. Computing the geometric return compounds this disturbance and the asymmetry of the geometric mean produces the lower result.

Timothy
June 25, 2019

That the geometric mean of positive numbers is less than or equal to the arithmetic mean, with equality only when the numbers are identical, can be proved quite quickly using natural logarithms, followed by Jensen's inequality (because natural log is convex), followed by the fact that the natural logarithm is strictly increasing. The first proof here (https://artofproblemsolving.com/wiki/index.php/Proofs_of_AM-GM) uses this approach, though it is a little terse.

George
March 17, 2018

Aaron,

That is a very nice comparison between the arithmetic and geometric mean and your point on using the geomean for calculating returns is spot on. I saw Keith's comment above about calculating negative returns so I sought a tool that can do that and many of the online tools I checked failed. MS Excel's GEOMEAN function also fails if negative values are provided.

I found this little tool: https://www.gigacalculator.com/calculators/geometric-mean-calculator.php which can handle negative returns (and input in percentages, which others did not). An input like 2% 3% 8% -4% -1% 5% was calculated properly and results in the actual average yearly return rate, so I thought it might be useful for your readers.

David Bell
June 20, 2013

It is useful to note that a quick way, not perfect but pretty accurate, to equate annualised arithmetic and geometric returns is as follows:

Rg = Ra - ((std dev)^2) / 2

So in words the geometric mean can be approximated by the arithmetic mean less half the squared standard deviation.

This relationship can also help illustrate the impact of volatility on portfolio outcomes.

Cheers,

David

Aaron
June 20, 2013

David, I agree that this is a very good tip. I would add that this gets better with time. That is, the more you compound the volatility, the closer you get to the approximation as your long term return falls.

Harry Chemay
April 09, 2013

Aaron,

Great job making a topic with a high MEGO (My Eyes Glaze Over) factor eminently readable. And it definetely needs to be read and understood, particularly by the almost 1 million Australians who are members of SMSFs.

I cringe when I (all too often) see SMSF annual reports with some form of arithmetic return calculation masquerading as a geometric return. As you rightly state, to misunderstand the two would result in an over-estimation of the performance of that SMSF relative to the actual compound return achieved.

So there are two inter-related issues that the SMSF industry should be aware of. Firstly, there is precious little empirical evidence that self-directed investors consistently beat the 'market' on a risk-adjusted basis. Second, if they are relying on arithmetic returns they may be creating a false sense of investment savvy that is not warranted.

SMSFs should be subject to the same performance calculation rigour as any APRA regulated super fund. I can't for the life of me see why SMSFs get such a huge free kick from their regulator.

Keith Brodie
March 13, 2013

How do you calculate geometric means when there are negative returns in the sequence?

(Maths was never my long suit!!)

Aaron Minney
March 13, 2013

Hi Keith
The geometric return, refers to the product of (1+r) rather than multiplying the r terms directly. In practice the easiest way is to calculate the return directly (assuming there are no cashflows). Divide the balance at N years by the balance at the start, and then take the N-th root before subtracting 1 as the initial capital. [The formula is
((End Balance)/(Start balance))^(1/N)-1
which looks better in an equation in word than in a text comment!]
Those who have maths as a long suit will know that strictly the calculations should be done in logarithmic terms, but for most purposes that is not necessary.

Aaron
March 01, 2013

Thanks Geof, I do remember the reports on the Beardstown Ladies. There are two problems involved. One is mistaking the arithmetic return for the compounding return (geometric mean) and the other is that of the money-weighted performance v time-weighted performance. Fund managers don't want cash flows distorting their performance, but to the end investor it is precisely the return on (and return of) cash flows that matter.

Geof Marsh
February 22, 2013

You might remember the story of “The Beardstown Ladies” who shot to international fame in the 1990s saying that their common-sense approach to investing in companies they understood had given them a 24.5% annual return since they established their club in 1983. Unfortunately, their fame came to an abrupt end when a team at CNN worked through the details of the investments and discovered that their actual financial return was a much more modest 9.1% per annum.

Many investors track the performance of managed funds and shares over the last calendar month, last 3 months, last 12 months, last two years and so on using published data from Morningstar or the ASX but cannot accurately measure the actual ‘financial returns’ of their own specific investments over a time period of their choosing.

This is made more difficult when investors contribute a regular amount (a percentage of earnings) into a superannuation fund, and even more so when additional contributions are made sporadically. As an example, a fund may have a total annual return of 20% as at 30th June, but an investor buying units in the fund at different times during the year may well have had a much better or much poorer experience.
Investment performance firstly needs to be separated from the increase in portfolio value as a result of adding additional funds, or withdrawing funds for accounting, administration or advice fees, insurance premiums or taxation payments.

 

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