Arithmetic Mean

Investment Returns

There are 2 types of average returns: arithmetic and geometric.

Ilmanen (2011) illustrates the concept simply: "The arithmetic average of two successive returns of +50% and -50% is 0%, but the geometric average is -13% or [(1 + 0.5) * (1 - 0.5)](1/2) = 0.87]"¹.

The classic O'Shaughnessey (1998) text defines the difference as: "Arithmetic average [is] the average return over the period, [while the] geometric average [is] the average annual compound return over the period"⁵.

Jakab (2016) advocates in favor of adjusting for the effect of compounding, stating that "instead of that arithmetic average, investment returns are calculated as a geometric average to get what is called a compound annual return"⁴.

In Excel

=RRI(#_Years, First_Value, Last_Value)

or

=RATE(#_Years, , -First_Value, Last_Value)

or program your own macro:

Public Function CAGR(First, Last, Periods)

CAGR = ((Last / First) ^ (1 / Periods)) - 1

End Function

e.g. 1, 2, 3, 4

should be entered as: =CAGR(1,4,3)

giving the same result as =RRI(3, 1, 4) or =RATE(3, , -1, 4) or 58.74% once solved.

Back To The World Of Investing

There are a few properties to note:

• "The geometric mean of a set of numbers is always smaller than the arithmetic mean"² (i.e. AM > GM). Ilmanen (2011) gives an example: "For the S&P GSCI the gap between arithmetic and geometric means is 2% (12% minus 10%), reflecting 20% volatility"¹. Hence the reason is volatility: "Lower volatility (smoother returns) has no impact on the arithmetic mean but does boost the geometric mean"¹.

• The Compound Annual Growth Rate (CAGR) assumes gains are reinvested automatically. Non-geometric averages do not.

The geometric average corresponds to the experience of a buy-and-hold investor who neither adds capital nor takes capital out of a hedge fund. The arithmetic average is the optimal estimator from a statistical point of view, and it corresponds more closely to the experience of an investor who adds and redeems capital in order to keep a constant dollar exposure to the hedge fund.

Source: Lasse Heje Pedersen. Efficiently inefficient: how smart money invests and market prices are determined. Princeton University Press, 2015. [B114]

More about variance (aka volatility):

Basic finance theory, including single-period Markowitz or mean variance optimization and the single-period CAPM, requires that one use arithmetic means (AM) as inputs. However, most investors are interested in wealth compounding and this is better captured by geometric means (GM, also called compound average returns or compound annual returns). When I present historical evidence I focus on GMs, which show realized buy-and-hold returns. An asset’s GM is always lower than its AM, and the difference is larger for high-volatility assets. An excellent approximation of the relation between GM and AM is: GM≈AM - Variance/2.

The debate about which method is better (geometric or arithmetic) still rages. For instance, Gray and Carlisle (2012) favor geometric – as does Ilmanen – because it "penalizes volatility"³.

An asset’s or a portfolio’s geometric mean (GM) (i.e., its compound return) depends both on its arithmetic mean (AM) and its variance. Higher volatility erodes compound returns and thus wealth accumulation over time. [...] There has been an ongoing debate in financial circles for decades as to which measure is more appropriate for assessing long-term expected returns and terminal wealth. Textbooks have traditionally sided with AM, while practitioners often prefer GM, perhaps because the latter is correct for measuring realized compound return. The AM–GM distinction is related to the difference between mean and median. Long-horizon cumulative asset returns are highly skewed: expected returns are unbounded from above—asset prices can rise hundreds of percent or more—but have a bounded downside as they can ‘‘only’’ lose -100%. Due to positive skew ness caused by a few extremely positive outcomes, the mathematical expected return (AM) can be a very atypical outcome. While AMs overstate the long-run future return investors are likely to realize, GMs are more like medians that give more realistic forecasts for long-horizon investors. In the presence of estimation errors, an unbiased estimate of future portfolio value depends more on the GM the longer the time horizon (for a given historical sample length).

Source: Antti Ilmanen. Expected returns: An investor's guide to harvesting market rewards. John Wiley & Sons, 2011. [B115]

See Also: academiccriticism.com

Sources

¹: Antti Ilmanen. Expected returns: An investor's guide to harvesting market rewards. John Wiley & Sons, 2011. [B115]

²: Ernie Chan. Quantitative trading: how to build your own algorithmic trading business. Vol. 430. John Wiley & Sons, 2009. [B094]

³: Wesley R. Gray and Tobias E. Carlisle. Quantitative Value: A Practitioner's Guide to Automating Intelligent Investment and Eliminating Behavioral Errors. John Wiley & Sons, 2012. [B102]

⁴: Spencer Jakab. Heads I Win, Tails I Win: Why Smart Investors Fail and How to Tilt the Odds in Your Favour. Penguin, 2016. [B171]

⁵: James O'Shaughnessey. What works on wall street. McGraw-Hill Companies, 1998. [B200]