Which annual investment return would you prefer to earn: 9% or 10%?

All things being equal, of course, anyone would rather earn 10% than 9%. However, when it comes to calculating annualized investment returns, all things are not equal, and differences between calculation methods can produce striking dissimilarities over time. In this article, we'll show you how annualized returns can be calculated and how these calculations can skew investors' perceptions of their investment returns.

## The Compound Average

Just by noting that there are dissimilarities among methods of calculating annualized returns, we raise an important question: Which option best reflects reality? By reality, we mean economic reality. In other words, which method will show how much extra cash an investor will have in his or her pocket at the end of the period?

Among the choices, the geometric average (also known as the "compound average") does the best job of describing investment return reality. To illustrate, imagine that you have an investment that provides the following total returns over a three-year period:

• Year 1: 15%
• Year 2: -10%
• Year 3: 5%

To calculate the compound average return, we first add 1.00 to each annual return, which gives us values of 1.15, 0.9, and 1.05, respectively.

We then multiply those figures together and raise the product to the power of one-third to adjust for the fact that we have combined returns from three periods.

(1.15)*(0.9)*(1.05)^1/3 = 1.0281

Finally, to convert to a percentage, we subtract the 1 and multiply by 100. In doing so, we find that we earned 2.81% annually over the three-year period.

Does this return reflect reality? To check, we use a simple example in dollar terms:

• Beginning of Period Value = \$100
• Year 1 Return (15%) = \$15
• Year 1 Ending Value = \$115
• Year 2 Beginning Value = \$115
• Year 2 Return (-10%) = -\$11.50
• Year 2 Ending Value = \$103.50
• Year 3 Beginning Value = \$103.50
• Year 3 Return (5%) = \$5.18
• End of Period Value = \$108.67

If we simply earned 2.81% each year, we would likewise have:

• Year 1: \$100 + 2.81% = \$102.81
• Year 2: \$102.81 + 2.81% = \$105.70
• Year 3: \$105.7 + 2.81% = \$108.67

## The Simple Average

The more common method of calculating averages is known as the arithmetic mean, or simple average. For many measurements, the simple average is both accurate and easy to use. If we want to calculate the average daily rainfall for a particular month, a baseball player's batting average, or the average daily balance of your checking account, the simple average is a very appropriate tool.

However, when we want to know the average of annual returns that are compounded, the simple average is not accurate. Returning to our earlier example, let's now find the simple average return for our three-year period:

15% + -10% + 5% = 10%
10%/3 = 3.33%

Claiming that we earned 3.33% per year compared to 2.81% may not seem like a significant difference. In our three-year example, the difference would overstate our returns by \$1.66, or 1.5%. Over 10 years, however, the difference becomes larger: \$6.83, or a 5.2% overstatement. As we saw above, the investor does not actually keep the dollar equivalent of 3.33% compounded annually. This shows that the simple average method does not capture economic reality

## The Volatility Factor

The difference between the simple and compound average returns is also affected by volatility. Let's imagine that we instead have the following returns for our portfolio over three years:

• Year 1: 25%
• Year 2: -25%
• Year 3: 10%

If volatility declines, the gap between the simple and compound averages will decrease. Additionally, if we earned the same return each year for three years, for example, with two different certificates of deposit, the simple and compound average returns would be identical. In this case, the simple average return will still be 3.33%. However, the compound average return actually decreases to 1.03%.

The increase in the spread between the simple and compound averages is explained by the mathematical principle known as Jensen's inequality; for a given simple average return, the actual economic return—the compound average return—will decline as volatility increases. Another way of thinking about this is to say that, if we lose 50% of our investment, we need a 100% return to break even.

## Practical Application for Investments

What is the practical application of something as nebulous as Jensen's inequality? Well, what have your investments' average returns been over the past three years? Do you know how they have been calculated?

Let's consider the example of a marketing piece from an investment manager that illustrates one way in which the differences between simple and compound averages get twisted. In one particular slide, the manager claimed that because his fund offered lower volatility than the S&P 500, investors who chose his fund would end the measurement period with more wealth than if they invested in the index, despite the fact that they would have received the same hypothetical return. The manager even included an impressive graph to help prospective investors visualize the difference in terminal wealth.

In reality, the two sets of investors may have indeed received the same simple average returns, but that doesn't matter. They most assuredly did not receive the same compound average return—the economically relevant average.

## The Bottom Line

Compound average returns reflect the actual economic reality of an investment decision. Understanding the details of your investment performance measurement is a key piece of personal financial stewardship and will allow you to better assess the skill of your broker, money manager, or mutual fund manager.