The Compound Annual Growth Rate as a Means to Measure Accurate Returns
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 yearoveryear 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 fiveyear 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 DefinedCAGR 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 twoyear example:
[(1 + 50%) x (1 + 100%) ^ (1/2)] 1 =
[(1.50) x (2.00) ^ (1/2)[ 1 = 0%
This makes much more sense. Let’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%)*(122%)*(1+45%)*(118%)*(1+44%))^(1/5))1
Below is an overview of why the difference between the arithmetic and geometric/CAGR returns vary so widely.