# Which mean do you mean?

November 28, 2017

“Average” is a synonym for the arithmetic mean. But for investors, the geometric mean is more relevant.

When talking investments, the word “average” (aka. mean) gets tossed around a lot. Investors use “average” when describing an investment’s long-term return. But which is more important? The simple average (the arithmetic mean) or the compound average (the geometric mean)?

## What is the arithmetic mean?

The arithmetic mean is the one we all know from grade school. (Unfortunately for their clients, it’s also the one that some fund managers use to advertise performance.)

Consider the following investment with two returns over two years:

Investment Year 1 Year 2
Capital $100$200 $100 Return +100% -50% To get the arithmetic mean, sum the returns and then divide by the number of returns: $\frac{100-50}{2}=25$ You started with$100. Two years later, you still have $100. Your arithmetic mean return was 25% per year. Hunh? In this case, the arithmetic mean describes the path your investment took to get nowhere. But it doesn’t tell us anything useful about investment performance. ## What is the geometric mean? To get the geometric mean, first divide the ending capital by the starting capital to get the total growth factor. From the example above: $GrowthFactor=\frac{EndingCapital}{StartingCapital}=\frac{100}{100}=1$ Next, put the total growth factor to the power of one over the number of periods. $GrowthFactor^{1/Periods}=1^{1/2}=1$ Next, subtract 1 from the result to get the decimal growth rate. (You need to subtract one to remove the initial investment from the ending amount. Otherwise, you’re counting your initial investment as part of the return.) $1-1=0$ Finally, multiply the result by 100 to convert the decimal return into a percentage. $0(100)=0%$ The geometric mean return is also called the compound average growth rate. As is obvious, an investment that starts at$100 and ends at $100 has a growth rate of 0%. ## Why the arithmetic mean is useless for investors Now consider the following three investments, each with two returns over two years: Investment Year 1 Year 2 Arithmetic Mean #1 capital$100 $200$100
#1 return 100% -50% 25.00% per year
#2 capital $100$120 $100 #2 return 20% -16.7% 1.65% per year #3 capital$100 $90$100
#3 return -10% 11.1% 0.55% per year

All three investments start and end the same, but their arithmetic mean returns differ.

Investors care about growth. But the arithmetic mean doesn’t measure growth; it measures change (aka. volatility). Volatility is irrelevant when calculating returns (or risk for that matter).

## The geometric mean is the compound growth rate

In contrast to the arithmetic mean, the geometric mean is the compound average growth rate. It answers the question:

“If applied every year, what growth rate would have my investment end where it ended, considering where it started?”

The difference is that the geometric mean takes compounding into effect. Compound growth is what investing is all about.

Let’s go back to our three no-growth examples:

Investment Year 1 Year 2 Geometric Mean
#1 capital $100$200 $100 #1 return 100% -50% 0% per year #2 capital$100 $120$100
#2 return 20% -16.7% 0% per year
#3 capital $100$90 \$100
#3 return -10% 11.1% 0% per year

We don’t even need to run the calculations. It’s obvious that these investments all performed the same. The degree of change varied in each, but the two-year performance was identical.

## Why use the arithmetic mean if it’s not what investors care about?

Because fund managers, economists, and short-term investors prefer it.

The arithmetic mean will always be greater than the geometric mean. For fund managers advertising their services, the arithmetic mean paints a better picture.

For economists, volatility equates with risk (rather than opportunity). The arithmetic mean, variance, and standard deviation are convenient ways to measure volatility. Similar to cell phones and MacDonald’s, regardless of their side effects, things that are convenient become popular.

The geometric mean is only relevant over a series of investments, and the longer, the better. If an investor is only considering one investment, or has a very short-term focus, then the arithmetic mean is a better predictor of what return the investor can expect.