Compound annual growth rate

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Compound annual growth rate (CAGR) is a business and investing specific 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 different data sets such as revenue growth of companies in the same industry. [3]

Formula[edit]

\mathrm{CAGR}(t_0,t_n) = \left( {V(t_n)/V(t_0)} \right)^\frac{1}{t_n-t_0} - 1

Example[edit]

In this example, we will compute the CAGR over three periods. Presume that the year-end revenues of a business for four years, V(t) in above formula, have been:

Year-End12/31/200412/31/2007
Year-End Revenue9,00013,000


{t_n-t_0} = 2007 - 2004 = 3


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

{\rm CAGR}(0,3) = \left( \frac{13000}{9000} \right)^\frac{1}{3} - 1 = 0.13 = 13% - it's 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. (In reality, growth is seldom constant.)


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:


V(t_n) = V(t_0) \times (1 + {\rm CAGR})^n


For n = 3:


= V(t_0) \times (1 + {\rm CAGR}) \times (1 + {\rm CAGR}) \times (1 + {\rm CAGR})
= 9000 \times 1.1304 \times 1.1304 \times 1.1304 = 13000



For comparison:


\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 V(t_n) by multiplying the initial value, V(t_0), three times by (1 + AMR) (unless all annual growth rates are the same).



\text{AR} = \frac{V_f - V_i}{V_i} = \frac{13000-9000}{9000} = 44.44%.

Applications[edit]

These are some of the common CAGR applications:

References[edit]

See also[edit]