Average Return - It’s Not What You Think

Compound Annual Growth Rate (CAGR) is the proper way to express investment returns over a period, not (arithmetic) average

If you are used to averaging the last few months/years of an investment to determine your “average return”, you are not getting the answer you want.

Below will explain why and show you how to calculate CAGR.

Let’s start with a simple example:

A stock you own goes from $100 to $50. Your gain was ($50-$100)/$100 = -50%.

Now the stock goes from $50 to $100. Your gain was ($100-$50)/$50 = 100%.

You average arithmetic gain was 25%: (-50% + 100%)/2 = 25%.

But the stock started at $100, and ended there. There was no gain.

Claiming a gain of 25% is very misleading.

Investments Compound

The correct way to calculate investment returns is by multiplying successive gains/losses. Using the prior example, the calculations are:

$50/$100 * $100/$50 = 1.0

Each periods gain/loss is calculated as the ratio of (ending value)/(starting value). To get the net gain/loss, multiply the values from all periods.

CAGR is the calculation to determine average growth rate, which is the value most people think of when they say “average return”. CAGR answers the question “If I earn X% for Y% periods, how much did my balance change”.

CAGR Calculation Explained

CAGR is the Compound Annual Growth Rate.
This calculation shows the true annualized growth rate.
It calculates the geometric mean of investment returns.
CAGR uses the formula (End Value/Start Value)^(1/Years) - 1.

CAGR Example

You bought a stock for $100 in 2010.

You sold the stock for $280 in 2020.

CAGR = ($280/$100)^(1/(2020-2010)) - 1 = .108 = 10.8%.

Or, on average, each year the balance increased by 10.8%.

The reverse of the calculation is: ( 1 + 0.108)^10 = 2.8

I.E. the value of the investment increased by a factor of 2.8 over the 10 years.

CAGR - Generally Lower Than Arithmetic Average

Below is histogram of the output of 10,000 runs of a simulation that generates 30 year long random sequences of returns with arithmetic mean 7.0% (stdev 17%).

Average Return - It’s Not What You Think - AlgorithmicFIRE_image_1.png

The arithmetic average of returns is 7.1%, yet the average CAGR is just 5.7%. Using the arithmetic average for this data overstates annual returns by 1.4%.

This demonstrates how arithmetic averages generally overstate returns.

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