Average Annual Return: Calculation & Example

How do you calculate the average annual, or annualized, return from playing poker or any investment? The principle remains the same. This article also serves as an example of both the arithmetic and geometric averages, explaining when it is appropriate—or even necessary—to use the geometric average instead of the more common arithmetic one.

Introduction

Calculating the average annual return has broad applicability and can be used for any investment, such as stocks. However, since our website primarily focuses on gaming, let's start with poker. Using a simple example, we'll highlight key considerations when applying averages. We will then also present an example related to the average annual (or annualized, derived from the Latin annum = year) return on an investment.

Figure 1: Average Annual Return Illustration (source: Craiyon)

Average Annual Return for a Poker Player

Imagine a poker player who starts playing with a certain bankroll (the term for a player’s funds in poker, managed through bankroll management). He performs well, earning a 100% return in his first year (in other words, his bankroll doubles). However, in the second year, he loses 50% of his entire balance.

Simple math shows that the player is back to where he started. It's important to note that the 50% loss in the second year is calculated based on the increased balance of 200% (the initial 100% plus the additional 100% from the first year). Since 50% of 200% is 100%, the player ends up back at his starting balance.

If this is making your head spin, it’s easier to calculate in dollars (the currency most commonly used in poker). Suppose the player starts with $100. In the first year, he earns a 100% return, gaining another $100, bringing his total to $200. In the second year, he loses 50% of his balance (50% of $200 = $100), leaving him with his original $100.

For this reason, when calculating the player’s average return, we cannot simply use:

Year 1: +100%
Year 2: -50%

If we applied a basic arithmetic average, we would get (100% - 50%) ÷ 2 = 25%, which is clearly incorrect.

The correct result—a zero average annual return—is obtained using the geometric average, which is designed specifically for compounding growth. To compute the average return, we multiply all individual values (growth rates), take the n-th root (where n is the number of periods), and subtract 1.

Since negative numbers cannot be used under a root, losses are expressed as proportions. For example, a 10% loss (0.10) means the new balance is 90% of the previous period’s (0.90). In our example, the player’s bankroll grew by 100% to 200% (2.00) and then dropped by 50% to 50% (0.50).

The player's average annual return is calculated as √(2 × 0.5) = 1 (100%), and after subtracting the initial value of 1 (or 100%), the correct result is 0 (0%). The average annual return is zero, meaning the balance remains unchanged over the two-year period.

Average Annualized Investment Return: Example

In the previous example, we highlighted the pitfalls of using a simple arithmetic average. In the next example, there is no such logical trap, and the difference between the arithmetic and geometric averages is smaller.

The table below shows the annual percentage returns of the broader U.S. stock market index S&P 500 from 2003 to 2013 (note: this spans 11 periods). The index reflects the movement of stock prices for the 500 most significant U.S. companies, including dividends.

We can observe, for example, the year 2008, when the financial and economic crisis hit hard, causing the S&P 500 index to drop by 37.22% (in reality, the decline was much steeper because stocks had been rising for five years prior, increasing the base from which they fell). Conversely, gold prices soared.

• Gold Surpassed Its All-Time High from August 2020, Selling for a Record $2,135 per Ounce on December 4, 2023.

To compute annualized returns using the geometric average, we must first convert percentage changes into growth factors (see the corresponding column). For example, a 37.22% decline in 2008 means the index dropped to 62.78% of its previous value (or 0.6278).

Source: Return Data – moneychimp.com [cited 2014-02-12], calculations by the author.

We multiply all growth factors together (in Excel, use =PRODUCT()), take the 11th root, or equivalently, raise the product to the power of 1/11 (^ for exponentiation in Excel), yielding the geometric mean. Excel also provides the function =GEOMEAN() for direct calculation.

Subtracting 1 (or 100%) from the geometric mean gives us the constant annual growth rate, or the annualized average return, which in this case is 9.15%. This means U.S. stocks in the S&P 500 grew at an average annual rate of 9.15%.

By contrast, the simple arithmetic average of the returns would be 10.97%, demonstrating that the geometric mean is always lower than the arithmetic mean, making it a more accurate measure for investment returns.

You Might Be Also Interested

• Investment Triangle Golden Rule;
• Stocks: Basics You Need to Know;
• Bonds: A Complete Guide to Understanding Bonds;
• Bitcoin: A Comprehensive Guide;
• Is Buying Stocks a Game of Chance?
• Commodities: A Comprehensive Guide;
• Gold as a Safe Haven: Navigating Turbulence;
• Gold Price Surpasses $2800 per Ounce: $3000 is Within Reach (2025-01-30);
• Gold Price Breaks December 2023 Record: What's Next (2024-03-05);
• Price of Gold Surpassed the Historic Record from August 2020 (2023-12-04);
• Silver in Modern Society;
• Copper as a Barometer of the State of the Economy;
• Dividend: What It Is and How to Get Them;
• Inflation: How to Protect Against It.

Based on the original Czech article: Průměrný roční výnos – výpočet u investice, pokeru, příklady.