CAGR Calculator (Compound Annual Growth Rate, Solve for Any Value)

Enter the three values you know out of starting value, ending value, period and CAGR (%), leave the one you want to find blank and press "Calculate". The blank value is worked out, and a year-by-year table and a growth curve are shown.

Starting value

Ending value

Period entered as

Period (years)

Period (months)

CAGR (%)

Enter exactly three values (it cannot calculate with all four filled in). Enter the CAGR as a percentage (for 10%, enter "10"; enter a negative number for a decline). Enter the period in years (decimals such as 2.5 are fine) or in months; months are converted with years = months ÷ 12. The values can be in any unit (dollars, people, items and so on).

Result and graph

Fill in three of the fields on the left and press "Calculate". The result, a year-by-year table and a graph will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and graphs
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

From a starting value, an ending value and the number of years, find the compound annual growth rate (CAGR). For example, "sales grew from $1 million to $1.61 million in 5 years" is about 10% growth a year. CAGR turns uneven growth into "the same percent every year" that would give the same result
Enter any three of the four fields (starting value, ending value, years, CAGR) and the fourth is worked out. "How many years to double sales at 10% growth a year?" (years), "What will it be in 5 years at 15% a year?" (ending value) and "How big do we need to be now to hit the target?" (starting value) all work
Enter the period as years, including decimals such as 2.5, or as months, such as 18. The years to reach a target are shown both as a decimal and rounded up (the year in which you reach it)
A year-by-year table and a graph that lays the growth curve (starting value × (1 + CAGR) to the power of the years) over the straight line for "the same amount every year" show the difference from the simple average growth rate at a glance

CAGR depends only on two values, the starting value and the ending value. The ups and downs in the years between do not enter the calculation (they are invisible to it). To smooth investment returns when you know every year's return, the related "Average Return Calculator" fits better. For a one-year change ("how many percent up from last year?"), the "Percentage Change Calculator" is quicker.

What is this calculation used for?

"X% CAGR" in business plans and investor presentations

Startup business plans and public companies' earnings presentations describe the growth of sales or users as, for example, "a 3-year CAGR of 25%". If sales grew from $2 million to $3.9 million in 3 years, the CAGR is $(3.9 \div 2)^{1/3} - 1 \approx 0.249$, or about 25%.
Even when growth is uneven from year to year, one number shows how fast something grew, so you can compare companies of different sizes over different periods. But CAGR looks only at two points, so it hides what happened in between, such as a jump in the last year only. When reading a report, check the year-by-year figures too.

Working back to "how many years to double sales?" (setting business goals)

To the question "if we keep growing 10% a year, when will sales double?", the years formula answers $\log 2 \div \log 1.1 \approx 7.27$ years, so sales pass double by the end of year 8. The other way around, if the goal is to double in 5 years, the CAGR needed is $2^{1/5} - 1 \approx 0.149$, or growth of about 14.9% a year.
Once "how many percent a year gets us to the goal" is a number, you can break it down into a sales target for each year (the year-by-year table).

Comparing the growth of app users or members (running a web service)

If an app's monthly users grew from 50,000 to 200,000 (4 times) in 2 years, the CAGR is $4^{1/2} - 1 = 1$: 100% a year (doubling every year). If another app grew 4 times in 4 years, it is $4^{1/4} - 1 \approx 0.414$, about 41% a year. The same "4 times" grew at very different speeds, and CAGR makes that visible.
The same formula works as is for things not measured in dollars, like numbers of users.

Forecasting market size (behind "the market will reach $X billion by 2030")

If a market is expected to grow 8% a year on average, a market worth $1 billion today can be estimated at $1 \times 1.08^5 \approx 1.469$, or about $1.47 billion, in 5 years (the ending value formula). Many "market size in X years" figures in industry reports are built with this kind of calculation.
Keep in mind that the number assumes the growth rate stays the same from now on. Check the assumptions of the forecast (what percent, for how many years) before you use it.

Looking at population change as a yearly rate (local statistics)

If a town's population fell from 50,000 to 40,000 in 10 years, the CAGR is $(4 \div 5)^{1/10} - 1 \approx -0.0221$, a decline of about 2.2% a year. Simply dividing "20% down in 10 years" into "2% down a year" is a little off, because decline also compounds (each year's drop is measured from the population after the last drop).
A decline uses exactly the same formula; the CAGR is just negative. With the years formula, you can also work back to how many years until the population falls below 30,000 if the same pace continues.

How many percent a year your pay or prices have risen (money basics)

For example, if someone earned $40,000 a year 20 years ago and earns $60,000 now, the CAGR is $(60 \div 40)^{1/20} - 1 \approx 0.0205$, growth of about 2% a year. The rise in prices (inflation) can also be turned into an average yearly percent with the same formula, so you can compare the growth of your pay and of prices with the same yardstick.
Turning a long-term change into a yearly rate is what CAGR does best. "50% more in 20 years" sounds big, but as a yearly rate it is about 2%, a number you can relate to.

Formulas and graphs

Formula for the compound annual growth rate (CAGR)

Graph

Standard notation (the usual math form)

$G$ $=$ $V_n$ $\div$ $V_0$

$r$ $=$ $G$ $1/n$ $-$ $1$

In words (symbols replaced with words)

③ $G$: growth multiple $=$ ① $V_n$: ending value $\div$ ② $V_0$: starting value

⑤ $r$: CAGR $=$ $G$: growth multiple ④ to the $1/n$ power ($n$ = years) $-$ $1$

The formula in words

① Divide the $V_n$: ending value

② by the $V_0$: starting value

③ to get the $G$: growth multiple (how many times larger it got)

④ raise it to the $1/n$ power ($n$ = years) to turn it into the multiple per year,

⑤ then subtract 1 (the original amount) and you get the $r$: CAGR (multiply by 100 for a percent)

Quick example

If sales grew from $1,000,000 to $1,210,000 in 2 years, the CAGR is (in thousands of dollars)

$G$: growth multiple $=$ ending value (1210) $\div$ starting value (1000)

$r$: CAGR $=$ growth multiple (1.21) to the $1/2$ power (2 years) $-$ $1$

$1210 \div 1000 = 1.21$

$1.21^{1/2} = \sqrt{1.21} = 1.1$

$1.1 - 1 = 0.1\ \ (10\%)$

Key idea

CAGR is the percent per year that, if it stayed the same every year, would take you from the starting value to the ending value. In the example above, $1,000,000 growing 10% every year (×1.1, then ×1.1) becomes exactly $1,210,000 after 2 years. Even if real sales jumped in year 1 and stayed flat in year 2, the CAGR is still 10%. CAGR looks only at the two end points, so the ups and downs in between do not enter the calculation. "To the $1/n$ power" is the number that gives the original number when raised to the $n$th power (the $n$th root). For 2 years it is the square root ($\sqrt{\ }$), and for 3 years the cube root. On a calculator or in Excel, type "^(1/n)".

Formula for the ending value (the value after n years)

Standard notation (the usual math form)

$V_n$ $=$ $V_0$ $\times$ $(1 + r)$ $n$

In words (symbols replaced with words)

④ $V_n$: ending value $=$ ① $V_0$: starting value $\times$ ② $(1 + r)$: multiple per year ③ $n$: years

The formula in words

① Take the $V_0$: starting value

② multiply it by the $(1 + r)$: multiple per year

③ over and over, once for each of the $n$: years

④ and you get the $V_n$: ending value

Quick example

If sales of $1,000,000 grow at a CAGR of 10%, sales after 3 years are (in thousands of dollars)

$V_3$: ending value $=$ starting value (1000) $\times$ $(1 + 0.1)$ 3 years

$1.1^{3} = 1.1 \times 1.1 \times 1.1 = 1.331$

$1000 \times 1.331 = 1331$

Key idea

This formula has exactly the same shape as compound interest. "10% growth every year" is "×1.1 every year", so after 3 years it is $1.1 \times 1.1 \times 1.1 = 1.331$ times as large. It is a 33.1% increase, not "10% × 3 years = 30%", because from year 2 on, the 10% is added to the value after it has already grown (the same way interest earns interest in compounding). The years $n$ can be a decimal such as 2.5. Then a calculation such as $1.1^{2.5}$ can be done with "^" on a calculator or in Excel.

Formula for the starting value (what you need now to hit a target)

Standard notation (the usual math form)

$V_0$ $=$ $V_n$ $\div$ $(1 + r)$ $n$

In words (symbols replaced with words)

④ $V_0$: starting value $=$ ① $V_n$: ending value $\div$ ② $(1 + r)$: multiple per year ③ $n$: years

The formula in words

① Divide the $V_n$: ending value

② by the $(1 + r)$: multiple per year

③ raised to the power of the $n$: years

④ and you get the $V_0$: starting value

Quick example

To reach $1,331,000 in sales in 3 years at a CAGR of 10%, the sales you need now are (in thousands of dollars)

$V_0$: starting value $=$ ending value (1331) $\div$ $(1 + 0.1)$ 3 years

$1.1^{3} = 1.331$

$1331 \div 1.331 = 1000$

Key idea

This is just the ending value formula rearranged into "starting value = ...". Something grown by multiplying goes back to where it started when you divide by the same multiple. Use it to work back to how big you need to be now to hit a target in $n$ years.

Formula for the years to reach a target

Graph

Standard notation (the usual math form)

$G$ $=$ $V_n$ $\div$ $V_0$

$n$ $=$ $\log G$ $\div$ $\log(1 + r)$

In words (symbols replaced with words)

③ $G$: growth multiple $=$ ① $V_n$: ending value $\div$ ② $V_0$: starting value

⑤ $n$: years $=$ $\log G$: log of the growth multiple $\div$ ④ $\log(1 + r)$: log of the multiple per year

The formula in words

① Divide the $V_n$: ending value

② by the $V_0$: starting value

③ to get the $G$: growth multiple , then take its logarithm $\log G$,

④ divide it by the $\log(1 + r)$: log of the multiple per year

⑤ and you get the $n$: years (the base of the log can be 10 or $e$; as long as the top and bottom use the same base, the result is the same)

Quick example

The years it takes for sales of $1,000,000 to reach $2,000,000 (double) at a CAGR of 10% are (in thousands of dollars)

$n$: years $=$ $\log 2$ (growth multiple $2000 \div 1000 = 2$) $\div$ $\log(1 + 0.1)$

$\log 2 \approx 0.30103,\ \ \log 1.1 \approx 0.041393$

$0.30103 \div 0.041393 \approx 7.27$

Key idea

The logarithm ($\log$) answers "to what power must 1.1 be raised to get 2?". The answer is about 7.27 years, so at 10% growth a year, sales pass double by the end of year 8 (at the end of year 7 they are only $1.1^7 \approx 1.949$ times as large, not there yet). The calculator shows both this decimal number of years and the number rounded up. By the way, the Rule of 72, "years to double ≈ 72 ÷ growth rate (%)", gives $72 \div 10 = 7.2$ years in your head, very close to the calculation above.

For reference: the simple average growth rate (how it differs from CAGR)

Graph

Standard notation (the usual math form)

$g$ $=$ $\left(\dfrac{V_n}{V_0} - 1\right)$ $\div$ $n$

In words (symbols replaced with words)

③ $g$: simple average growth rate $=$ ① total growth $\left(\dfrac{\text{ending value } V_n}{\text{starting value } V_0} - 1\right)$ $\div$ ② $n$: years

The formula in words

① Divide the total growth (how many percent it grew over the whole period)

② by the $n$: years

③ and you get the $g$: simple average growth rate (multiply by 100 for a percent)

Quick example

If sales grew from $1,000,000 to $1,210,000 in 2 years (21% growth in total), the simple average growth rate is

$g$: simple average growth rate $=$ total growth (0.21) $\div$ 2 years

$1210 \div 1000 - 1 = 0.21$

$0.21 \div 2 = 0.105\ \ (10.5\%)$

Key idea

For the same "$1,000,000 → $1,210,000 in 2 years", the simple average growth rate is 10.5% but the CAGR is 10%. The simple average is the straight-line view, "it grew by the same amount ($105,000) every year". CAGR is the curved view, "it grew by the same percent (×1.1) every year". If you call it "10.5% growth a year" from the simple average, then really growing 10.5% every year would give $1.105^2 = 1.221$ times, which looks bigger than the actual 1.21 times. The longer the period and the higher the growth, the wider this gap. When you talk about growth as a yearly rate, CAGR is the standard.

CAGR (compound annual growth rate) is "ending value ÷ starting value" raised to the $1/n$ power, minus 1: the percent per year if growth stayed the same every year. To find the ending value, the starting value or the years instead, you only rearrange the formula, so if you know any three of the four, the fourth can always be calculated.

Symbols and terms

Symbols

CAGR	C-A-G-R, or "kay-ger"	Compound annual growth rate - the percent per year if growth stayed the same every year. On this page it is written with the symbol $r$.
$V_0$	V sub zero	The starting value (the value at the start of the period). $V$ stands for value, and the small 0 at the lower right stands for year 0. (Example - sales of $1,000,000 five years ago)
$V_n$	V sub n	The ending value (the value after $n$ years). The small $n$ at the lower right stands for year $n$. (Example - sales of $1,610,510 this year)
$G$	G	The growth multiple: ending value ÷ starting value ($V_n \div V_0$), how many times larger it got over the whole period. It stands for growth. (Example - $1,000,000 → $1,210,000 gives $G = 1.21$)
$n$	n	The number of years from the starting value to the ending value. It stands for number. It can be a decimal such as 2.5 (18 months is $18 \div 12 = 1.5$ years).
$r$	r	The CAGR written as a decimal. It stands for rate. For 10%, $r = 0.1$; multiply by 100 to show it as a percent.
$1 + r$	one plus r	The multiple per year. For 10% growth, it is $1 + 0.1 = 1.1$. It is the original 1 plus the growth $r$, and over $n$ years it multiplies to $(1 + r)^n$.
$x^{1/n}$	x to the one over n	$x$ to the $1/n$ power: the number that gives $x$ when raised to the $n$th power (the $n$th root $\sqrt[n]{x}$). For 2 years it is the square root, and for 3 years the cube root. (Example - $1.21^{1/2} = \sqrt{1.21} = 1.1$)
$\log$	log	Logarithm: the symbol for "to what power must a number be raised to get this value?". This page uses it to work back to the years, as in "to what power must 1.1 be raised to get 2?". The base can be 10 (common log) or $e$ (natural log, $\ln$); as long as the top and bottom use the same base, the answer is the same.
$g$	g	The simple average growth rate: just the total growth divided by the years, for reference. It has its own letter to keep it apart from the CAGR $r$ (and it is a different quantity from the capital $G$, the growth multiple).

Terms

CAGR (compound annual growth rate)	Growth over several years turned into the percent per year that, kept the same every year, gives the same result. It depends only on the starting value, the ending value and the number of years; the ups and downs in between do not enter the calculation. It is the standard measure for comparing the growth of sales, users, market size, population and so on as a yearly rate.
compounding	Growth where the same percent is also added to what has already been added. Two years of 10% growth give $1.1 \times 1.1 = 1.21$ times (21% more), not "10% + 10% = 20% more". CAGR smooths growth using this compound view.
geometric mean	The average you get by multiplying $n$ numbers together and taking the $1/n$ power. CAGR is the geometric mean of the yearly multiples minus 1, which is different from the usual average (add up and divide, the arithmetic mean). For quantities linked by multiplication, like multiples, use the geometric mean.
simple average growth rate (arithmetic mean)	Just the total growth over the period divided by the number of years. It is the straight-line view, "the same amount every year", and it ignores compounding, so for growth over more than a year it comes out larger than the CAGR (for a decline, the yearly drop comes out smaller than the CAGR). When you talk about a yearly rate, CAGR is the standard.
growth multiple	Ending value ÷ starting value, how many times larger it got over the whole period. $1,000,000 growing to $1,610,510 is about 1.61 times. Subtract 1 and multiply by 100 to get the total growth (about 61% more).
nth root	The number that gives a number when raised to the $n$th power, written $\sqrt[n]{x}$ or $x^{1/n}$. If something grew 1.21 times in 2 years, the multiple per year is $\sqrt{1.21} = 1.1$, and this is the heart of the CAGR calculation.
logarithm (log)	The number for the power in "to what power must $a$ be raised to get $b$?", written $\log_{a} b$. It finds an exponent (a number of times multiplied), so it is used to work back to years, as in "how many years to double at 10% growth a year?".
Rule of 72	A rule of thumb for mental math - "years to double ≈ 72 ÷ yearly rate (%)". At 10% a year it takes about 7.2 years to double, and at 5% about 14.4 years. It comes very close to the exact value from logarithms.
year-over-year (YoY) growth	The change from last year to this year, as in "up 10% year over year". CAGR is the value this yearly change would have if it were the same every year.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.

Percents (Grade 6)	Being able to switch between percents and decimals ($10\% = 0.1$; $1.21$ times $= 21\%$ more) Knowing that "10% more" is the same as "the original value × 1.1" (the idea of percentage change)
Exponents (Grades 6–8)	Knowing that the small number at the upper right (the exponent) is how many times to multiply, as in $1.1^3 = 1.1 \times 1.1 \times 1.1$ Knowing that a square root ($\sqrt{1.21} = 1.1$) is the number that gives the original number when squared (Grade 8)
Compound interest (Algebra 1 and personal finance)	Knowing that something growing 10% every year becomes $1.1 \times 1.1 = 1.21$ times as large (21% more), not "10% + 10% = 20%" Knowing that this kind of growth, where the same percent is added to what has already been added, is called compounding
nth roots and rational exponents (Algebra 2)	Knowing that $x^{1/n}$ is the number that gives $x$ when raised to the $n$th power (the $n$th root) (example - $8^{1/3} = 2$) Knowing that an exponent that is a decimal (such as $1.1^{2.5}$) still makes sense
Logarithms (Algebra 2 and Precalculus)	Knowing that $\log_{a} b$ tells you to what power $a$ must be raised to get $b$ (example - $\log_{2} 8 = 3$) Using logarithms to find $n$ in $1.1^{n} = 2$ (with the change-of-base formula, $n = \log 2 \div \log 1.1$)
Expressions and rearranging equations (Grades 7–8)	Understanding a formula with letters such as $V_n = V_0 (1 + r)^n$, and being able to rearrange it to find another quantity, as in $V_0 = V_n \div (1 + r)^n$

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the CAGR

Starting value V0	1000
Ending value Vn	1610.51
Years n	5
CAGR r (%)	=((B2/B1)^(1/B3)-1)*100

Table to find the ending value

Starting value V0	1000
CAGR r (%)	10
Years n	3
Ending value Vn	=B1*(1+B2/100)^B3

Table to find the starting value

Ending value Vn	1331
CAGR r (%)	10
Years n	3
Starting value V0	=B1/(1+B2/100)^B3

Table to find the years to reach a target

Starting value V0	1000
Ending value Vn (target)	2000
CAGR r (%)	10
Years n	=LN(B2/B1)/LN(1+B3/100)
Years rounded up	=ROUNDUP(B4,0)

Table to find the simple average growth rate (for reference)

Starting value V0	1000
Ending value Vn	1210
Years n	2
Simple average growth rate g (%)	=(B2/B1-1)/B3*100

After pasting, B1 to B3 are your inputs and B4 is calculated automatically.
For example, the first table shows 10 in B4 (a CAGR of 10%), the second shows 1331 in B4 (the value after 3 years), the third shows 1000 in B4 (the starting value needed), the fourth shows about 7.27 in B4 (years) and 8 in B5 (years rounded up), and the fifth shows 10.5 in B4 (the simple average growth rate).
"^(1/B3)" is "to the 1/n power", and "LN" is the natural logarithm (log base $e$). The years formula gives the same answer with "LOG10" or "LOG", as long as the top and bottom use the same function.
CAGR can also be found with Excel's function "=RRI(years, starting value, ending value)" (in the first table, "=RRI(B3,B1,B2)*100").

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the CAGR

Starting value V0	1000
Ending value Vn	1610.51
Years n	5
CAGR r (%)	=((B2/B1)^(1/B3)-1)*100

Table to find the ending value

Starting value V0	1000
CAGR r (%)	10
Years n	3
Ending value Vn	=B1*(1+B2/100)^B3

Table to find the starting value

Ending value Vn	1331
CAGR r (%)	10
Years n	3
Starting value V0	=B1/(1+B2/100)^B3

Table to find the years to reach a target

Starting value V0	1000
Ending value Vn (target)	2000
CAGR r (%)	10
Years n	=LN(B2/B1)/LN(1+B3/100)
Years rounded up	=ROUNDUP(B4,0)

Table to find the simple average growth rate (for reference)

Starting value V0	1000
Ending value Vn	1210
Years n	2
Simple average growth rate g (%)	=(B2/B1-1)/B3*100

The same formulas as in Excel work as is (Google Sheets also has the RRI function). Copy the whole table, paste it into cell A1, and replace B1 to B3 with your own numbers.

How to calculate it in Python

import math

start_value = 1000      # starting value (e.g., sales 5 years ago, in thousands of dollars)
end_value = 1610.51     # ending value (e.g., this year's sales)
years = 5               # years

# CAGR = (ending value / starting value)^(1/years) - 1
cagr = (end_value / start_value) ** (1 / years) - 1
print(f"CAGR: {cagr * 100:.2f}%")

# For reference: simple average growth rate = total growth / years
simple_avg = (end_value / start_value - 1) / years
print(f"Simple average growth rate: {simple_avg * 100:.2f}%")

# Working back 1: the value after n years from the starting value and CAGR
rate = 0.10             # CAGR (10%)
future_value = start_value * (1 + rate) ** 3
print(f"Value after 3 years: {future_value:.2f}")

# Working back 2: the years to reach a target from the starting value, target and CAGR
target_value = 2000
years_needed = math.log(target_value / start_value) / math.log(1 + rate)
print(f"Years to double: {years_needed:.2f} (rounded up: {math.ceil(years_needed)})")

Runs with the standard library only. In this example, the CAGR is 10.00%, the simple average growth rate is 12.21%, the value after 3 years is 1331.00, and the years to double are about 7.27 (8 rounded up). "** (1 / years)" is "to the 1/n power", and math.log is the natural logarithm. Change the starting value, ending value and years at the top and run it.

How to write it in LaTeX and other math languages (copy and paste)

Formula for the compound annual growth rate (CAGR)

r = (Vₙ ÷ V₀)^(1/n) − 1

r = \left(\dfrac{V_n}{V_0}\right)^{1/n} - 1

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>r</mi>
    <mo>=</mo>
    <msup>
      <mrow>
        <mo>(</mo>
        <mfrac><msub><mi>V</mi><mi>n</mi></msub><msub><mi>V</mi><mn>0</mn></msub></mfrac>
        <mo>)</mo>
      </mrow>
      <mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow>
    </msup>
    <mo>&#x2212;</mo>
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</math>

r = (V_n/V_0)^(1/n) - 1

(vn/v0)^(1/n) - 1

r := (V_n/V_0)^(1/n) - 1;

r = (V_n/V_0)^(1/n) - 1;

r = (V_n/V_0)^(1/n) - 1

Formula for the ending value (the value after n years)

Vₙ = V₀ × (1 + r)ⁿ

V_n = V_0 (1 + r)^{n}

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  <mrow>
    <msub><mi>V</mi><mi>n</mi></msub>
    <mo>=</mo>
    <msub><mi>V</mi><mn>0</mn></msub>
    <msup>
      <mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mo>)</mo></mrow>
      <mi>n</mi>
    </msup>
  </mrow>
</math>

V_n = V_0 (1 + r)^n

v0 (1 + r)^n

V_n := V_0*(1 + r)^n;

V_n = V_0*(1 + r)^n;

V_n = V_0 (1 + r)^n

Formula for the starting value (what you need now to hit a target)

V₀ = Vₙ ÷ (1 + r)ⁿ

V_0 = \dfrac{V_n}{(1 + r)^{n}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>V</mi><mn>0</mn></msub>
    <mo>=</mo>
    <mfrac>
      <msub><mi>V</mi><mi>n</mi></msub>
      <msup>
        <mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mo>)</mo></mrow>
        <mi>n</mi>
      </msup>
    </mfrac>
  </mrow>
</math>

V_0 = V_n/(1 + r)^n

vn/(1 + r)^n

V_0 := V_n/(1 + r)^n;

V_0 = V_n/(1 + r)^n;

V_0 = V_n/(1 + r)^n

Formula for the years to reach a target

n = log(Vₙ ÷ V₀) ÷ log(1 + r)

n = \dfrac{\log\left(V_n / V_0\right)}{\log(1 + r)}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>n</mi>
    <mo>=</mo>
    <mfrac>
      <mrow><mi>log</mi><mo>(</mo><msub><mi>V</mi><mi>n</mi></msub><mo>/</mo><msub><mi>V</mi><mn>0</mn></msub><mo>)</mo></mrow>
      <mrow><mi>log</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mo>)</mo></mrow>
    </mfrac>
  </mrow>
</math>

n = log(V_n/V_0)/log(1 + r)

Log[vn/v0]/Log[1 + r]

n := log(V_n/V_0)/log(1 + r);

n = log(V_n/V_0)/log(1 + r);

n = log(V_n/V_0)/log(1 + r)

For reference: the simple average growth rate (how it differs from CAGR)

g = (Vₙ ÷ V₀ − 1) ÷ n

g = \dfrac{V_n / V_0 - 1}{n}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>g</mi>
    <mo>=</mo>
    <mfrac>
      <mrow><msub><mi>V</mi><mi>n</mi></msub><mo>/</mo><msub><mi>V</mi><mn>0</mn></msub><mo>&#x2212;</mo><mn>1</mn></mrow>
      <mi>n</mi>
    </mfrac>
  </mrow>
</math>

g = (V_n/V_0 - 1)/n

(vn/v0 - 1)/n

g := (V_n/V_0 - 1)/n;

g = (V_n/V_0 - 1)/n;

g = (V_n/V_0 - 1)/n

How to have ChatGPT do the calculation

You are an assistant for business planning. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

A company's annual sales were $1,000,000 five years ago and $1,610,510 this year.
Find each of the following:
1. The compound annual growth rate, CAGR ((ending value ÷ starting value)^(1/years) − 1, as a percent to 2 decimal places)
2. For reference, the simple average growth rate ((ending value ÷ starting value − 1) ÷ years, as a percent), and in 1 or 2 sentences why it differs from the CAGR
3. Sales 3 years from the start if this CAGR continues (starting value × (1 + CAGR)^3)
4. Starting from this year's sales, the years until sales double at the same CAGR (log(2) ÷ log(1 + CAGR), both as a decimal and rounded up)

Show the formulas you used and the numbers from the execution result.

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Press the "Calculate" button
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CAGR	C-A-G-R, or "kay-ger"	Compound annual growth rate - the percent per year if growth stayed the same every year. On this page it is written with the symbol \(r\).
\(V_0\)	V sub zero	The starting value (the value at the start of the period). \(V\) stands for value, and the small 0 at the lower right stands for year 0. (Example - sales of $1,000,000 five years ago)
\(V_n\)	V sub n	The ending value (the value after \(n\) years). The small \(n\) at the lower right stands for year \(n\). (Example - sales of $1,610,510 this year)
\(G\)	G	The growth multiple: ending value ÷ starting value (\(V_n \div V_0\)), how many times larger it got over the whole period. It stands for growth. (Example - $1,000,000 → $1,210,000 gives \(G = 1.21\))
\(n\)	n	The number of years from the starting value to the ending value. It stands for number. It can be a decimal such as 2.5 (18 months is \(18 \div 12 = 1.5\) years).
\(r\)	r	The CAGR written as a decimal. It stands for rate. For 10%, \(r = 0.1\); multiply by 100 to show it as a percent.
\(1 + r\)	one plus r	The multiple per year. For 10% growth, it is \(1 + 0.1 = 1.1\). It is the original 1 plus the growth \(r\), and over \(n\) years it multiplies to \((1 + r)^n\).
\(x^{1/n}\)	x to the one over n	\(x\) to the \(1/n\) power: the number that gives \(x\) when raised to the \(n\)th power (the \(n\)th root \(\sqrt[n]{x}\)). For 2 years it is the square root, and for 3 years the cube root. (Example - \(1.21^{1/2} = \sqrt{1.21} = 1.1\))
\(\log\)	log	Logarithm: the symbol for "to what power must a number be raised to get this value?". This page uses it to work back to the years, as in "to what power must 1.1 be raised to get 2?". The base can be 10 (common log) or \(e\) (natural log, \(\ln\)); as long as the top and bottom use the same base, the answer is the same.
\(g\)	g	The simple average growth rate: just the total growth divided by the years, for reference. It has its own letter to keep it apart from the CAGR \(r\) (and it is a different quantity from the capital \(G\), the growth multiple).

CAGR Calculator (Compound Annual Growth Rate, Solve for Any Value)

Table of Contents

What you can do on this page

What is this calculation used for?

Formulas and graphs

Symbols and terms

Symbols

Terms

Good to know before you start

How to calculate it in Excel

How to calculate it in Google Sheets

How to calculate it in Python

How to write it in LaTeX and other math languages (copy and paste)

How to have ChatGPT do the calculation

How to Use

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