Which annual investment return would you prefer to have: 9% or 10%?
All things being equal, of course, anyone would rather earn 10% than 9%. But 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 annualized returns can be calculated and how these calculations can skew investors' perceptions of their investment returns.
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A Look at Economic Reality
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. So which method will determine how much extra cash an investor will actually have in his or her pocket at the end of the period?
Among the alternatives, 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 threeyear period:
Year 1: 15% Year 2: 10% Year 3: 5% 
To calculate the compound average return, we first add 1 to each annual return, which gives us 1.15, 0.9, and 1.05, respectively. We then multiply those figures together and raise the product to the power of onethird to adjust for the fact that we have combined returns from three periods.
Numerically this gives us:
(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 threeyear 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 = $103.5 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 
Disadvantages of the Common Calculation
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. (For related reading, see What is the difference between arithmetic and geometric averages?)
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 threeyear period:
15% + 10% + 5% = 10% 10%/3 = 3.33% 
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. (For more, see Overcoming Compounding's Dark Side.)
Claiming that we earned 3.33% per year compare to 2.81% may not seem like a significant difference. In our threeyear 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.
The Volatility Factor
The difference between the simple and compound average returns is also impacted 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% 
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 get back to breakeven. (For related reading, see The Uses And Limits Of Volatility.)
The opposite is also true; if volatility declines, the gap between the simple and compound averages will decrease. And if we earned the exact same return each year for three years  say with two different certificates of deposit  the compound and simple average returns would be identical.
Compounding and Your Returns
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? (For more, see Achieving Better Returns In Your Portfolio.)
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.
Reality check: the two sets of investors may have indeed received the same simple average returns, but so what? They most assuredly did not receive the same compound average return  the economically relevant average.
Conclusion
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 managers.
Which annual investment return would you prefer to have: 9% or 10%? The answer is: It depends on which return really puts more money in your pocket.

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