Arithmetic vs. Geometric Return: Understanding the Key Differences

Imagine you’re planning a road trip across several states. You want to know the average speed you traveled each day. Let’s say on day one, you drove at an average of 60 miles per hour, on day two, 70 miles per hour, and on day three, 50 miles per hour. To find your arithmetic average speed, you would simply add up those speeds: 60 plus 70 plus 50, which equals 180. Then, you divide that sum by the number of days, which is three. So, 180 divided by 3 gives you an arithmetic average speed of 60 miles per hour. This is the arithmetic average return in the world of finance – it’s the sum of the returns in each period divided by the number of periods. It provides a simple, straightforward average.

Now, let’s shift gears and think about your savings account instead of a road trip. Suppose you invest $100. In the first year, your investment grows by 10 percent. In the second year, it grows by 20 percent. And in the third year, it unfortunately declines by 10 percent. If you calculate the arithmetic average return, it would be 10 percent plus 20 percent minus 10 percent, which totals 20 percent. Dividing 20 percent by three years gives you an arithmetic average return of approximately 6.67 percent per year.

However, this arithmetic average doesn’t quite tell the full story of how your initial $100 actually grew. Let’s track the actual growth. Starting with $100, a 10 percent gain in the first year brings it to $110. In the second year, a 20 percent gain on $110 brings it to $132. Finally, in the third year, a 10 percent loss on $132 reduces it to $118.80. So, after three years, your initial $100 has grown to $118.80.

The geometric average return, also known as the compound annual growth rate, reflects this actual growth more accurately. It essentially asks: “What constant annual return, compounded over three years, would have turned my $100 into $118.80?” Instead of simply averaging the percentages, it considers the compounding effect, meaning that returns are earned not just on the initial investment but also on the accumulated returns from previous periods.

To calculate the geometric average return conceptually, you would find the overall growth factor. In our example, your investment grew from $100 to $118.80 over three years. This is a total growth of 1.188 times the original investment. To find the equivalent annual growth rate, you take the cube root of 1.188, since it’s over three years. The cube root of 1.188 is approximately 1.0595. Subtracting 1 from this gives you 0.0595, or approximately 5.95 percent. This 5.95 percent is the geometric average return.

Notice that the geometric average return of 5.95 percent is lower than the arithmetic average return of 6.67 percent. This is almost always the case, and the difference becomes more pronounced when there is greater volatility in returns from year to year. The geometric average is a more accurate representation of the actual year-over-year growth you experienced in your investment portfolio because it accounts for the effect of compounding.

Think of it like this: the arithmetic average is like describing the average height of the steps you climbed in a staircase, whereas the geometric average is like describing the overall steepness of the staircase, reflecting the actual vertical gain you made. For understanding long-term investment performance and projecting future growth, the geometric average return, or compound annual return, is generally the more useful and realistic metric. It provides a clearer picture of the actual rate at which your money has grown over time, factoring in the ups and downs and the power of compounding. While the arithmetic average can be a simpler calculation and may be relevant in certain contexts, for long-term investment analysis, especially when returns fluctuate, the geometric average provides a more insightful measure of performance.