# Growth rate

The growth rate is the relative increase in a variable over a period of time or, if several periods are considered, the mean relative increase in a variable over a period of time.

An exponential process is often assumed here. Instead of using the growth rate is then usually with the growth factor expected. If the growth factor remains the same, one also speaks of the growth constant . This corresponds to a growth rate of 23% . ${\ displaystyle g}$ ${\ displaystyle \ lambda = g + 1}$${\ displaystyle g = 0 {,} 23 / {\ text {year}}}$${\ displaystyle \ lambda = 1 {,} 23 / {\ text {year}}}$

## definition

The discrete growth rate is the change in a time- dependent variable between two points in time and relative to its starting value : ${\ displaystyle g}$${\ displaystyle t}$${\ displaystyle A (t)}$${\ displaystyle t_ {0}}$${\ displaystyle t}$${\ displaystyle A (t_ {0})}$

${\ displaystyle g = {\ frac {A (t) -A (t_ {0})} {A (t_ {0})}}}$.

If you shorten the period more and more towards its starting point, i.e. if you form the limit value , then you get the steady growth rate at this point in time. It is the current change in size at a specific point in time relative to its value at that point in time. ${\ displaystyle w}$${\ displaystyle A (t)}$${\ displaystyle t_ {0}}$${\ displaystyle A (t_ {0})}$

${\ displaystyle w = {\ frac {1} {A (t_ {0})}} \ cdot {\ frac {dA} {dt}} (t_ {0})}$

The mean discrete growth rate over several periods of time is given by the general equation

${\ displaystyle \ operatorname {Growth rate} (t_ {0}, t) = \ left ({\ frac {A (t)} {A (t_ {0})}} \ right) ^ {\ frac {1} { n}} - 1}$

expressed, where the number of time spans between and and represents the size considered at the respective point in time . This is the growth rate from the geometric mean of the growth factors for the individual periods. ${\ displaystyle n = t-t_ {0}}$${\ displaystyle t_ {0}}$${\ displaystyle t}$${\ displaystyle A (t)}$${\ displaystyle t}$

## Annual growth rate ( Compound Annual Growth Rate )

A special growth rate is the compound annual growth rate (abbreviated CAGR ), a key figure for looking at investments, market developments , sales, etc. in business and economics. The CAGR represents the average annual growth of a quantity to be considered.

For the calculation, the current value is divided by the initial value. The -th root is taken from the result , where is the number of years that are considered. The Compound Annual Growth Rate therefore represents the mean percentage by which the starting value of a time series increases to hypothetical subsequent values ​​for the reporting years until the actual final value is reached at the end of the reporting period. Actual fluctuations in the following years in the meantime have no effect, the growth rate is constant. ${\ displaystyle n}$${\ displaystyle n}$

The formula for the CAGR is the same as that of the growth rate, with CAGR expressing size as the number of years. ${\ displaystyle n}$

Example: In 2004 a company had a turnover of € 1 million. In 2006 sales were € 1.21 million. The number of time units is 2006–2004 = 2. ${\ displaystyle n}$

${\ displaystyle \ operatorname {CAGR} (2004,2006) = \ left ({\ frac {1,210,000} {1,000,000}} \ right) ^ {\ frac {1} {2}} - 1 = 0 { ,} 1 = 10 \ \%}$

The annual growth rate is 10%. Therefore, if you multiply the starting value twice by the corresponding growth factor 1.1, you get the final value:

${\ displaystyle 1,000,000 \ cdot 1 {,} 1 \ cdot 1 {,} 1 = 1,210,000}$

## Specific growth rate in biotechnology

With exponential growth, the rate at which the cell mass changes ( ) at any point in time is proportional to the cell mass . The constant of proportionality is called the specific growth rate: ${\ displaystyle {\ tfrac {dX} {dt}}}$${\ displaystyle X}$${\ displaystyle \ mu}$

${\ displaystyle {\ frac {dX} {dt}} = \ mu X}$

Other parameters used to describe fermentation processes are the specific product formation rate and the specific substrate consumption .