# Arithmetic and geometric mean returns

This article appeared in Viewpoint: Franck on 16 March 2008.

Edited on 19 March 2008: see Correction, 19 March 2008 below.

One of the many issues that has been confusing debates about the cost of capital, and the return on capital to be allowed in price controls, has been whether historical returns on investment should be measured as an arithmetic average or a geometric average.

This is the sort of generic question which the UK economic regulators tried to address through joint working some years ago. In 2002, they commissioned a study on the cost of capital, and one of the issues that they asked their researchers to focus on was the measurement of parameters such as the risk-free rate, equity risk premium and total return on equity.

The resulting Smithers & Co report was published in February 2003. It addresses the question of geometric and arithmetic averages concisely and fairly clearly (I thought).

So, five years on, are we witnessing well-informed debates between regulators and interested parties in price controls based on a clear understanding of the basics?

Judging from the CAA's decision on Heathrow and Gatwick landing charges, it appears not.

## What the Smithers & Co report said

The 2003 Smithers & Co report (prepared by Stephen Wright, Robin Mason and David Miles) addressed the question of geometric and arithmetic averages as follows:

Before examining the data, we note that care should be applied as to whether returns are being measured using arithmetic or "geometric" averaging. The former is conceptually superior, though possibly less stable. The most crucial thing is to be aware that the difference between the two measures can be significant — as much as two percentage points or more.

Standard theory requires that the appropriate measure of any given return used in deriving the cost of capital should be [the expectation value of the return], i.e. the true arithmetic mean. This requirement holds whatever the nature of the process that generates [the return].

In contrast, historical studies frequently quote two alternative, but closely related, measures. One is what is often rather loosely described as the "geometric mean"; the other is the arithmetic mean of the logarithmic return.

The geometric mean return is a natural metric of returns viewed from the perspective of an investor: an investment with a positive geometric mean return will grow over time.

Diversion: what is the intuition behind the difference?

I don't find it easy to see. The maths are simple enough, but they do not have an obvious interpretation.

One possible way of describing them is to say that a high return appears to be lower when it is seen as part of a compound average growth rate (i.e. the geometric mean), because the investor will be measuring the rate of growth by reference to a measure of capital employed during the period that includes part of the growth itself.

By contrast, if the return is seen as a risk or insurance premium, i.e. part of income or consumption — as is the case in setting price controls — then the only relevant measure of capital is the capital employed at the beginning of the period.

End of diversion.

Having established that the aim was to measure an arithmetic average (over a probability distribution of future returns), the authors of the Smithers & Co report then considered how it might be measured using historical data.

They estimated that the effect of volatility in total market returns is equivalent to something like 2 percentage points between the geometric and arithmetic averages, and then consider in more detail the factors that might affect that gap, concluding that:

... the relationship between geometric and arithmetic average returns:

• will only be constant over time if volatility of returns is constant;

• will only be constant across different return horizons if returns are unpredictable.

... if the above conditions do not hold, any presumption that, e.g., the arithmetic mean return has been stable over time must, logically imply that the geometric mean return has not been stable over time; and vice versa. ... Campbell and his various co-authors typically assume lognormality ... and hence stability of the mean log return and the geometric average, as implicitly, do Dimson et al. In contrast, eg, Fama and French have, in various papers, worked on the assumption that the arithmetic mean return is stable.

Our (not very strong) preference would be to side with Campbell, since the assumption of lognormality of returns is consistent with the feature of financial returns that they cannot fall below -100%, but are unbounded in the opposite direction.

So the suggested recipe is to:

- use historical data to calculate a geometric mean, which will look quite stable over time; and
- adjust it to an arithmetic mean by reference to historical evidence or other beliefs about volatility, accepting that there is quite a lot of variability and uncertainty about volatility and in arithmetic returns.

That recipe is reflected in the summary of the report, which quotes a single figure (5.5 per cent above inflation) for the geometric mean of equity returns, and then says that this might correspond to an arithmetic average somewhere in between 6.5 and 7.5 per cent above inflation.

If anyone at the CAA in 2007 or 2008 was unsure that this was the approach, then they will probably have looked at the most obvious practical application of the recipe, in the Smithers & Co 2006 report for Ofgem. No need to read further than the executive summary, which includes in Table 1.1 a calculation of the cost of equity as:

Real Market Return: Compound Average: 5.5 per cent

Adjustment to Arithmetic Average: 1 to 2 per cent

Real Market Return: Arithmetic Average: 6.5 to 7.5 per cent

Risk-Free Rate: 2.5 per cent

Estimated "Value Effect": 0 to 1.25 per cent

Implied Real Cost of Equity with beta=0.5: 4.5 to 6.25 per cent

The ranges even work in this table: with an assumed beta of 0.5, the real cost of equity excluding the "value effect" element is the mid point between the real risk-free rate and the real market return, i.e. 4.5 to 5 per cent. Add the "value effect" at the top end to get 6.25. As expected, the calculations are based on the market return adjusted for the use of an arithmetic mean.

## What BAA said

BAA decided to pay various consultants and professors to write its submission. Thankfully it then merged the various separate papers into a single document (135 pages, PDF).

The points about arithmetic versus geometric means are best made in the part attributed to Steward C. Myers:

The CC report's upper-end ERP of 4.5% is "consistent with historical returns" only if geometric averages are used, rather than the arithmetic averages given above. Geometric means are useful measures of past returns but not the right measure of the cost of capital.

The CC report appears to give roughly equal weight to arithmetic and geometric averages, based on only casual analysis.

The CC report gives no statistics for serial correlation and does not explain why serial correlation makes geometric averages "relevant."

It is sometimes convenient to estimate geometric means from past data, and then adjust upwards to obtain a forecast of the future arithmetic mean. But past geometric averages are in this case just an intermediate step.

The CC report says at p. F16 that "Many academics believe that past equity returns are far too high to represent rationally expected returns in the future." The Report cites the "equity risk premium puzzle" and the related "risk free rate" and the "excess volatility" puzzles. These are only puzzles, however, not empirical results. The equity risk premium puzzle does not prove that the historical ERP was too high; it's more likely that financial theory is deficient or that investors perceive future risks that are not evident in past data.

## What the CAA and its consultants say (and why it's wrong)

The CAA's decision addresses BAA's criticisms of the Competition Commission reliance on geometric means as follows:

10.37 On the second of these issues, the CAA's advisers, Europe Economics, have re- assessed the Commission's analysis of the equity market risk premium in light of BAA's criticisms. It has raised concerns over the validity of the Commission's stated reasons for focusing on geometric averages. Broadly speaking, it would consider an assumption based on an arithmetic average to be preferable in the context of five-year price cap regulation. However, it highlights a secondary issue concerning whether a simple arithmetic mean should be adopted, or rather, the arithmetic mean of the logarithms of returns. In practice, the adoption of an arithmetic mean of the logarithms of returns tends to generate similar results to the geometric mean.

10.38 As a result, whilst Europe Economics questions the basis upon which the Commission arrived at its recommended range, it does consider that there are other potentially compelling arguments to support such a range. Indeed, it notes that the Commission's position is aligned with the (tentative) recommendations of the Smithers & Co report, which was commissioned in 2002/03 by several of the UK's economic regulators (including the CAA) to inform their setting of cost of capital allowances as part of their price control functions.

I find ¶10.37 a bit hard to parse. I am not sure whether the "It" at the beginning of the second sentence of ¶10.37 refers to BAA or to the CAA's advisers (the use of the past tense would suggest that it is BAA). But it seems clear the "it" in the third and subsequent sentences refers to the CAA's advisers (even though they were plural in the first sentence).

Anyway, it seems clear enough that the CAA ignored BAA's point “as a result” of the “issue concerning whether a simple arithmetic mean should be adopted, or rather, the arithmetic mean of the logarithms of returns”.

But this seems odd.

The Smithers & Co 2003 report had mentioned the arithmetic mean of the logarithmic return and told me it was closely related to the geometric mean (specifically, the arithmetic mean of the logarithmic return is the logarithm of one plus the "geometric mean"). These two measures were together, close to each other, far away from the ordinary arithmetic mean return.

So it seems that the "secondary issue" of choosing between arithmetic mean of the logarithmic return and arithmetic mean is actually almost the same as the original issue of choosing between the geometric mean and the arithmetic mean.

But how could highlighting the secondary issue make BAA's submission irrelevant? And how did I miss the "alignment" of the arithmetic mean of the logarithmic return with "(tentative) recommendations" in the Smithers & Co 2003 report?

At that point, there was no alternative but to look for this consultants' report that the CAA relied on. It's a ~~51~~35-page PDF, but the relevant discussion is only a handful of paragraphs:

2.36 Therefore, Myers' criticisms of the Commission's justification of using the geometric mean returns as the basis of EMRP estimate seem reasonable. Nevertheless, there could be other reasons, not explored in the Commission report, to move away from using arithmetic means of historical returns.

2.37 However, it might not be correct simply to use arithmetic means of actual observed returns, but, rather, arithmetic means of logarithms of the returns. This issue was considered in the Joint Regulators study by Smither's & Co (2003). Smither's & Co point out that it is very commonly assumed that investment returns follow a lognormal distribution. [Some uncontroversial arguments for using a lognormal distribution.]

2.38 Smither's & Co (2003) show that the geometric mean of returns corresponds quite closely to the arithmetic mean of logarithms of returns. [Numerical example.] Therefore, if the arithmetic mean of log returns is the preferred measure, the geometric mean used by the Commission could be a close approximation to it.

2.39 Unfortunately, however, there is no clear cut answer or agreement on the issue. Again, the clear aim should be to derive an estimate of the arithmetic mean return. The ambiguity relates to whether the arithmetic mean of normal returns or lognormal returns should be preferred. As discussed by Smither's & Co (2003), unless volatility of returns is constant and returns are unpredictable, assuming that the arithmetic mean return is stable over time must mean that geometric mean return is not, and vice versa. [Note that there is no evidence that either is stable.]

2.40 The choice then, Smither's & Co (2003) argue, comes down to preference about which measure is assumed to be stable. They did not find consensus in the literature on the point. However, they did express a slight preference for assuming that returns follow a lognormal distribution, i.e. that the geometric mean return is stable, due to its ability to better describe the features of financial returns (as discussed above). They recommended that the relevant arithmetic mean returns are "best built up from a more data consistent framework in which returns are lognormally distributed, so means should be estimated with reference to mean log returns, or (virtually identically, geometric (compound) averages)."

2.41 Therefore, the Commission's assumption (based on the use of the geometric mean) would be in line with both a common assumption about the distribution of the equity returns, and the (tentative) recommendation of the Joint Regulators study. However, there no definitive consensus on the point. Further, justifying the use of geometric mean from the lognormality angle could leave the CAA open to questions of how skewness of returns (driving the lognormal characterisation of the distribution) is taken into account elsewhere in the determination. Previously we discussed using models based on skew, though this was rejected at an early stage (for reasons that remain valid). As we discussed at that time, once we begin to model skewness, we might have to take into account that investors could care about it. If investors do care about the skewness of returns, it should be taken into account in more comprehensive way than using it just as a basis of an argument when it seems convenient.

Whilst this is mostly right, there is one significant blot in this reasoning.

The blot is the meaningless distinction drawn between “normal returns” and “lognormal returns” in the phrase “arithmetic mean of normal returns or lognormal returns” in ¶2.39. The arithmetic mean is a simple calculation based on historical returns: the average of a list of numbers. That mean will be the same irrespective of what distribution is used to model the returns, and it makes no sense to draw a distinction between the “arithmetic mean of normal returns” and that of “lognormal returns”.

Perhaps the authors meant to draw a distinction between the expectation value of future returns depending on whether these returns were deemed to be drawn from a normal or a lognormal distribution? That would make more sense. But then there is a difference between the expectation value of a lognormal distribution and the mean of logarithmic returns, which the reasoning seems to ignore. The mean of logarithmic returns is a suitable statistical estimator of the centre (or median) of the lognormal distribution, but the arithmetic average of returns that follow that lognormal distribution is greater, and the gap can be calculated by reference to an estimate of volatility. This is essentially the method espoused by Smithers & Co's 2003 and 2006 reports: calculate the geometric average on historical data, and then add 1 or 2 percentage points to convert it into an arithmetic average. This method for building up an arithmetic average is noted by the CAA's consultants at the end of ¶2.40 but they seem to take no notice of the adjustment.

Does this blot matter? Let's get the blue pencil out and see.

Strike out the third sentence of ¶2.39 which contains the offending phrase.

The rest of ¶2.39 survives.

¶2.40 survives.

The first sentence of ¶2.41 dies: the use of a geometric mean (or arithmetic mean of log returns) is not in line with any assumption about distribution of returns, or with the Smithers & Co 2003 report.

The second sentence of ¶2.41 survives.

The third sentence of ¶2.41 dies: there is nothing to "justify" the use anymore.

The rest of ¶2.41 is not affected.

But the CAA's argument for ignoring BAA's submission is no longer supported by the corrected report.

That was with a fairly strict blue pencil, which deleted the warning to CAA in ¶2.41 as it was polluted by the earlier errorenous reasoning. But let's use a gentler and more generous blue pencil, which tries to preserve that idea. Instead of simply striking out the third sentence of ¶2.41, we would replace it with something like the following corrected version:

"Using the geometric average on the basis that it is an estimate of the median of a lognormal distribution of returns would leave the CAA open to questions of how skewness of returns around this median value is taken into account in the determination."

Given that the text as printed makes no sense because of the blot noted above, this is perhaps a more plausible view of how the CAA should have interpreted the report from its consultants.

But that view only moves the error around. If the CAA did understand the fact that using a geometric mean amounted to choosing the median of a skewed lognormal distribution instead of the average, then it needed to address the resulting bias as part of its analysis of BAA's submission.

Could the CAA say that the remarks at the end of ¶2.41 address the question of skew? The consultants refer to their previous work, which one must assume to be this December 2006 report (61 pages, PDF). The only discussion of skew in that report is clearly expressed as relating to “Distributions with the same mean and variance but differing skewness”. Here, we need to consider whether using the median of a skewed lognormal distribution instead of the mean is acceptable. No reasonable CAA official could have thought that the December 2006 paper had addressed the issue.

(Added on 20 March 2008: the same consultants' March 2007 paper for CAA (52 pages, PDF) does not address the point either.)

And the CAA's March 2008 decision contains no such analysis of any bias or skew issues.

In conclusion, irrespective of how one interprets its arguments or its consultants' report, the conclusion seems to be invalidated by an error.

The relative stability of geometric and arithmetic averages is all very interesting, but it is not relevant to the point that BAA was making, which relates to the bias between the two averaging methods.

Whilst the CAA and its consultants might well be right to think that the geometric mean provides a more stable or reliable statistical analysis of historical data, this is irrelevant to the question of whether an adjustment needs to be made to the geometric average in order to produce a number that is relevant to setting the allowed rate of return in price controls.

## Does it matter?

Does the CAA's error matter to anyone in the real world, away from the intricacies of financial theory?

The natural answer is yes: the CAA seems to have neglected a rather good argument for increasing the allowed return on capital by the best part of 1 per cent (i.e. dozens of millions of pounds of income each year), on the basis of a concern about stability of estimates which is irrelevant to the estimation of the expectation value of the required return.

The judge over its shoulder says that the CAA “must not have exercised its discretion on the basis of irrelevant factors” and “must have taken into account factors which it is under a duty to consider”, and that “failure to follow either rule will usually lead to a decision being held to be invalid”.

If I am right, then there has been a clear failure of this kind.

The question that BAA would need to consider before suing is what their remedy would be if they are successful in proving that the CAA has wrongly disregarded some of the arguments in their consultation response on the cost of capital.

In this context, it is important to remember that the return on capital that the CAA has used was the number recommended by the Competition Commission. The flawed reasoning discussed above was only used to support the following conclusion:

[10.40] The CAA concludes that no compelling case has been put forward for adjusting the 2.5-4.5 per cent range proposed by the Commission for the ERP.

If the CAA had not relied on irrelevant concerns about stability to ignore BAA's submissions, or if the CAA's decision was quashed, it is hard to imagine that the CAA would do anything other than adopt the Competition Commission's reasoning, and tell BAA that if they don't like it they'll have to prove that the Competition Commission was wrong too.

Now the Competition Commission's reasoning leading to that number is not exactly convincing (analysing it would need a separate article, to address the weird and probably mistaken reliance on "ranges" which are narrower than both the spread of evidence and any plausible statistical confidence interval). ~~But I do not think that the Competition Commission relied on blatantly irrelevant considerations like the CAA's latest decision. Furthermore, there probably was a proper and well-managed consultation in the course of the Competition Commission's inquiry. I would guess that BAA would have a hard time to prove that the Competition Commission's approach was so plainly wrong that the CAA should not have relied on the Competition Commission's final report.~~

~~So I am not really expecting a challenge from BAA on this point.~~

What would be rather funny (*) is if the airlines that are not happy with the additional forecast capital expenditure taken into account in the CAA's price limits would use this point — although seemingly an error in their favour — as part of a wider attack on the CAA's decision.

(*) I am only a neutral spectator in all this — I am allowed to find this sort of thing amusing.

## Correction, 19 March 2008

I had rashly assumed in writing the original version of this article that there would have been reasonable consultation on draft findings by the Competition Commission. But the inquiry webpage seems to indicate that it fact there was not — no draft findings, no BAA submission on price control issues other than the initial one in May 2007.

Furthermore, there seems to be some doubt about whether the evidence on cost of capital issues relied upon by the Competition Commission was properly exposed to, and tested by, BAA.

Contrary to my initial uninformed prejudice, it seems quite arguable that its submission to the CAA in January 2008 was the only way that BAA had to challenge the approach and findings of the Competition Commission. As a result, it could turn out to be important that the CAA has disregarded that submission on erroneous grounds.

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## Links

- CAA decision on Heathrow and Gatwick price controls, March 2008
- Consultancy report (51 pages, PDF) published by the CAA, March 2008
- BAA submissions on cost of capital (135 pages, PDF), January 2008
- Competition Commission report on Heathrow and Gatwick price controls, October 2007
- Smithers & Co report to UK economic regulators, February 2003
- Smithers & Co report to Ofgem, September 2006

Article added to Viewpoint: Franck by Franck Latrémolière on 16 March 2008. Full list of articles.

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Last changed by Franck at 6:59 PM on Wednesday 28 May 2008.