Compounded Annual Growth rate (CAGR) is a business and investing specific term for the smoothed annualized gain of an investment over a given time period.^[1] CAGR is not an accounting term, but remains widely used, particularly in growth industries or to compare the growth rates of two investments because CAGR dampens the effect of volatility of periodic returns that can render arithmetic means irrelevant. CAGR is often used to describe the growth over a period of time of some element of the business, for example revenue, units delivered, registered users, etc.

Formula[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{CAGR}(t_0,t_n) = \left( \frac{V(t_n)}{V(t_0)} \right)^\frac{1}{t_n-t_0} - 1

• Failed to parse (Missing texvc executable; please see math/README to configure.): V(t_0)

: start value, Failed to parse (Missing texvc executable; please see math/README to configure.): V(t_n)
: finish value, Failed to parse (Missing texvc executable; please see math/README to configure.): t_n - t_0
: number of years.

• Actual or normalized values may be used for calculation as long as they retain the same mathematical proportion.
• The CAGR can also be calculated as the geometric mean of 1 plus each year's return (i.e. +3% becomes 1.03 and -2% becomes 0.98), minus 1.^[2]

Example[link]

Suppose the revenues of a company for four years, V(t) in above formula, have been:

Failed to parse (Missing texvc executable; please see math/README to configure.): {t_n-t_0}

= 2007 - 2004 = 3

Then the CAGR of revenues over the three-year period from the end of 2004 to the end of 2007 is:

Failed to parse (Missing texvc executable; please see math/README to configure.): {\rm CAGR}(0,3) = \left( \frac{200}{100} \right)^\frac{1}{3} - 1 = 0.2599 = 25.99%

Verification:

If you multiply the initial value by (1 + CAGR) three times (because we calculated for 3 years) you will get exactly the final value again. This is:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(t_n) = V(t_0) \times (1 + {\rm CAGR})^n

For n = 3:

Failed to parse (Missing texvc executable; please see math/README to configure.): = V(t_0) \times (1 + {\rm CAGR}) \times (1 + {\rm CAGR}) \times (1 + {\rm CAGR})

Failed to parse (Missing texvc executable; please see math/README to configure.): = 100 \times 1.2599 \times 1.2599 \times 1.2599 = 200

For comparison:

• the Arithmetic Mean Return (AMR) would be the sum of annual revenue changes (compared with the previous year) divided by number of years, or:

Failed to parse (Missing texvc executable; please see math/README to configure.): \text{AMR}=\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i = \frac{1}{n} (x_1+\cdots+x_n) = \frac{ 15% + 30.4% + 33.3%}{3} = 26.26%.

In contrast to CAGR, you cannot obtain Failed to parse (Missing texvc executable; please see math/README to configure.): V(t_n)

by multiplying the initial value, Failed to parse (Missing texvc executable; please see math/README to configure.): V(t_0)

, three times by (1 + AMR) (unless all annual growth rates are the same).

• the Arithmetic Return (AR) or simple return would be the ending value minus beginning value divided by the beginning value:

Failed to parse (Missing texvc executable; please see math/README to configure.): \text{AR}=\frac{V_f - V_i}{V_i} = \frac{200-100}{100} = 100%.

Applications[link]

These are some of the common CAGR applications:

• Calculating and communicating the average returns of investment funds^[3]
• Demonstrating and comparing the performance of investment advisors^[3]
• Comparing the historical returns of stocks with bonds or with a savings account^[3]
• Forecasting future values based on the CAGR of a data series (you find future values by multiplying the last datum of the series by (1 + CAGR) as many times as years required). As every forecasting method, this method has a calculation error associated.
• Analyzing and communicating the behavior, over a series of years, of different business measures such as sales, market share, costs, customer satisfaction, and performance.

References[link]

See also[link]

• Geometric mean
• Arithmetic mean
• Average annual return