Calculating investment performance is one of the first things finance students must learn in business school. Along with risk, return is a fundamental concept that is clearly important when dealing with wealth and how to grow it over time. The compound annual growth rate, or CAGR for short, represents one of the most accurate ways to calculate and determine returns for individual assets, investment portfolios and anything that can rise or fall in value over time.

The CAGR represents the year-over-year growth rate of an investment over a specified time period. And as the name implies, it uses compounding to determine the return on the investment, which we will see below is a more accurate measure when those returns are more volatile.

Average Returns

Frequently, investment returns are stated in terms of an average. For instance, a mutual fund may report an average annual return of 15% over the past five years made up of the following annualized returns:

 Year 1 26% Year 2 -22% Year 3 45% Year 4 -18% Year 5 44%

This type of return is known as the arithmetic average return and is mathematically correct. It represents the average mutual fund return over a five-year period.

 Average return 15.00%

But is this the best way to report the investment returns? Perhaps not. Take the example of a fund that reported a negative return of 50% during its first year but doubled in price for a return of 100% in the second year. The arithmetic average return is 25%, or the average of -50% and 100%. However, the investor ended the period with the same amount of money as he started. \$100 that falls 50% equals \$50 at the end of the first year. If that \$50 doubles in the second year, it returns to the original \$100.

CAGR Defined

CAGR helps fix the limitations of the arithmetic average return. As we know intuitively, the return in the above example was 0% as the \$100 investment at the beginning of year one was the same \$100 at the end of year two. This means the CAGR is 0%.

To calculate the CAGR, you take the nth root of the total return, where "n" is the number of years you held the investment, and subtract one. This also consists of adding one to each percentage return and multiplying each year together. In the two-year example:

[(1 + 50%) x (1 + 100%) ^ (1/2)] -1 =

[(1.50) x (2.00) ^ (1/2)[ -1 = 0%

This makes much more sense. Letâ€™s return to the mutual fund example above with five years of performance data:

 Year 1 26% Year 2 -22% Year 3 45% Year 4 -18% Year 5 44%

Here, the arithmetic average return was 15% but the CAGR/geometric return is only 11%. It is calculated as follows:

=(((1+26%)*(1-22%)*(1+45%)*(1-18%)*(1+44%))^(1/5))-1

Below is an overview of why the difference between the arithmetic and geometric/CAGR returns vary so widely.

Differences Between Average Returns

Mathematically, the geometric return is equal to the arithmetic return minus half the variance. Variance starts to get into the investment risk discussion and is calculated along with an investmentâ€™s standard deviation, both of which deal with volatility. As you can see, the more volatile the returns become, the bigger the difference between arithmetic and CAGR returns. Below is a way to get to the CAGR if you have the arithmetic average and standard deviation:

(1 + rave)2 - StdDev2 = (1 + CAGR)2

As you can see, the larger the standard deviation, the larger the differences between the arithmetic return and CAGR.

To more clearly define the differences between the two, it is accurate to describe the CAGR as what was actually earned per year on average, compounded annually. The arithmetic return represents what was earned during a typical, or average, year. Both are right, but the CAGR is arguably more accurate. However, most average returns are likely to be based on arithmetic calculations, so be sure to find out which return is being referred to.

Additionally, arithmetic returns do not account for compounding. The CAGR and geometric returns take compounding into consideration.

The discussion above pertains to a portfolio that does not see any cash flows. When money is either added or subtracted from a portfolio, it is important to calculate dollar-weighted average returns.

The Bottom Line

There are different types of average investment returns. The arithmetic average is the one most investors are familiar with and represents adding up investment returns and dividing it by the number of investment periods. It is simply an average return. The CAGR, or geometric return, is more complicated to calculate but is at the end of the day a more accurate measure of compounded average returns. It is more useful to extrapolate returns into the future, and these will usually be smaller than the arithmetic average, especially when returns are more volatile. Investors need to be aware of the difference between each, and then they can take into consideration the risk, or volatility, of investment returns to help explain any differences that arise.

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