Compound annual growth rate

Compound annual growth rate (CAGR) is a business, economics and investing term for the geometric progression ratio that provides a constant rate of return over the time period. ^{[1] [2]} CAGR is not an accounting term, but it is often used to describe some element of the business, for example revenue, units delivered, registered users, etc. CAGR dampens the effect of volatility of periodic returns that can render arithmetic means irrelevant. It is particularly useful to compare growth rates from various data sets of common domain such as revenue growth of companies in the same industry or sector. ^[3]

CAGR is equivalent to the more generic exponential growth rate when the exponential growth interval is one year.

Formula

CAGR is defined as:

${\displaystyle \mathrm {CAGR} (t_{0},t_{n})=\left({\frac {V(t_{n})}{V(t_{0})}}\right)^{\frac {1}{t_{n}-t_{0}}}-1}$

where ${\displaystyle V(t_{0})}$ is the initial value, ${\displaystyle V(t_{n})}$ is the end value, and ${\displaystyle t_{n}-t_{0}}$ is the number of years.

Actual or normalized values may be used for calculation as long as they retain the same mathematical proportion.

Example

In this example, we will compute the CAGR over a three-year period. Assume that the year-end revenues of a business over a three-year period, ${\displaystyle V(t)}$, have been:

Year-End	2004-12-31	2007-12-31
Year-End Revenue	9,000	13,000

Therefore, to calculate the CAGR of the revenues over the three-year period spanning the "end" of 2004 to the "end" of 2007 is:

${\displaystyle {\rm {CAGR}}(0,3)=\left({\frac {13000}{9000}}\right)^{\frac {1}{3}}-1=0.13=13\%}$

Note that this is a smoothed growth rate per year. This rate of growth would take you to the ending value, from the starting value, in the number of years given, if growth had been at the same rate every year.

Verification:

Multiply the initial value (2004 year-end revenue) by (1 + CAGR) three times (because we calculated for 3 years). The product will equal the year-end revenue for 2007. This shows the compound growth rate:

${\displaystyle V(t_{n})=V(t_{0})\times (1+{\rm {CAGR}})^{n}}$

For n = 3:

${\displaystyle =V(t_{0})\times (1+{\rm {CAGR}})\times (1+{\rm {CAGR}})\times (1+{\rm {CAGR}})}$

${\displaystyle =9000\times 1.1304\times 1.1304\times 1.1304=13000}$

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:

${\displaystyle {\text{AMR}}={\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {1}{n}}(x_{1}+\cdots +x_{n})={\frac {11.11\%+10\%+8.33\%}{3}}=9.81\%.}$

In contrast to CAGR, you cannot obtain ${\displaystyle V(t_{n})}$ by multiplying the initial value, ${\displaystyle V(t_{0})}$, 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:

${\displaystyle {\text{AR}}={\frac {V(t_{n})-V(t_{0})}{V(t_{0})}}={\frac {13000-9000}{9000}}=44.44\%.}$

CAGR can also be used to calculate average annualized growth rates on quarterly or monthly data. The numerator of the exponent would be the value of 4 in the case of quarterly, and 12 in the case of monthly, with the denominator being the number of observations involved. ^[4]

Applications

These are some of the common CAGR applications:

• Calculating and communicating the average returns of investment funds ^[5]
• Demonstrating and comparing the performance of investment advisors ^[5]
• Comparing the historical returns of stocks with bonds or with a savings account ^[5]
• 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 with 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.
• Calculating average annualized growth rates of economic data, such as gross domestic product, over annual, quarterly or monthly time intervals. ^[6]

See also

• Annual growth %
• Arithmetic mean
• Average annual return
• Continuous compounding
• Geometric mean
• Exponential growth

References

1. Mark J. P. Anson; Frank J. Fabozzi; Frank J. Jones (3 December 2010). The Handbook of Traditional and Alternative Investment Vehicles: Investment Characteristics and Strategies. John Wiley & Sons. pp. 489–. ISBN   978-1-118-00869-0.
2. root. "Compound Annual Growth Rate (CAGR) Definition | Investopedia". Investopedia. Retrieved 2016-03-04.
3. Emily Chan (27 November 2012). Harvard Business School Confidential: Secrets of Success. John Wiley & Sons. pp. 185–. ISBN   978-1-118-58344-9.
4. "How is average annual growth calculated?". Bureau of Economic Analysis. January 11, 2008.
5. "Compound Annual Growth Rate CAGR: Summary and Forum". www.12manage.com. Retrieved 2019-05-02.
6. "How is average annual growth calculated?". Bureau of Economic Analysis. January 11, 2008.