# Compound Annual Growth Rate (CAGR) > CAGR says how fast something grew each year, as if it grew by the same share every year, from its starting value to its ending value. Page: https://formulabro.yunusemrevurgun.com/finance/compound-annual-growth-rate/ Category: Finance Published: 2026-10-23 Updated: 2026-10-23 ## Formula $$ \text{CAGR} = \left(\frac{V_{\text{end}}}{V_{\text{start}}}\right)^{1/n} - 1 $$ Read it out loud: "CAGR equals V end divided by V start, raised to the power one over n, then minus one. CAGR means compound annual growth rate. V start is the value at the start, V end is the value at the end, and n is the number of years between them." ## Before you start - **Exponents and fractional powers**: A small raised number, the **exponent**, says how many times to multiply a number by itself. A fraction as an exponent means a root instead: $x^{1/3}$ is the **cube root** of $x$, the number that gives $x$ when multiplied by itself three times. So $1.331^{1/3} = 1.1$, because $1.1 \times 1.1 \times 1.1 = 1.331$. - **Growth factor**: To make a number grow by a rate, multiply it by 1 plus the rate. A rise of 10% multiplies the amount by 1.1. A ratio such as $\frac{1331}{1000} = 1.331$ is a **growth factor**, or multiple: the whole result, including the original amount, compared with the start. - **Start value and end value**: $V_{\text{start}}$ is the value at the beginning of the period, and $V_{\text{end}}$ is the value at the end. The letter $V$ stands for value, measured in dollars here. The small words under the letter say which moment is meant. - **Number of years, $n$**: The letter $n$ counts the years between the start and the end. The time can be a whole number or a decimal, such as 1.5 years. It counts the periods of growth, not the number of year-end values you see on a chart. - **Percent as a decimal**: A percent means out of 100, so 10% is written as 0.10 in a formula. To turn a decimal into a percent, multiply by 100, so 0.1 becomes 10%. The result of a CAGR formula is a decimal per year until you convert it into a percent. - **Brackets and the root**: The large brackets in $\left(\ldots\right)^{1/n}$ group the whole ratio, so the root applies to the entire ratio and not only to its top. The minus 1 sits outside the brackets, so it is subtracted after the root has been taken. More notation help: https://formulabro.yunusemrevurgun.com/math-basics/ ## What each symbol means - $\text{CAGR}$: **Compound annual growth rate**, the steady yearly growth rate, written as a decimal per year. A CAGR of 0.10 is the same as 10% per year. It is the one rate that would take the start value to the end value. - $V_{\text{start}}$: The **value at the start**, such as the amount of money first put into a fund, in dollars. - $V_{\text{end}}$: The **value at the end** of the period, the amount the money has reached by then, in dollars. - $n$: The **number of years** between the start and the end. It can be a whole number such as 3, or a decimal such as 1.5 for a year and a half. ## The formula piece by piece - $V_{\text{end}}$: Start with the final value, measured in dollars. It is the amount the money has reached at the end of the period. - $\frac{V_{\text{end}}}{V_{\text{start}}}$: Divide the end value by the start value to get the total growth multiple. It says how many times the money multiplied over the whole period, with all the years taken together. - $\left(\frac{V_{\text{end}}}{V_{\text{start}}}\right)^{1/n}$: Take the nth root of the multiple by raising it to the power 1 over $n$. This spreads the total growth evenly across the years and gives the growth factor for a single year. - $\left(\frac{V_{\text{end}}}{V_{\text{start}}}\right)^{1/n} - 1$: Subtract 1 to remove the original amount from the yearly factor. What is left is the growth rate per year, written as a decimal. - $\text{CAGR} = \left(\frac{V_{\text{end}}}{V_{\text{start}}}\right)^{1/n} - 1$: The whole formula in one sentence: **the CAGR is the steady yearly rate that turns the start value into the end value over n years, found by taking the nth root of the total growth multiple and subtracting 1.** ## What the formula is doing, and why CAGR answers one question: **what single steady yearly rate would take the money from its start value to its end value?** Actual growth is uneven, with some good years and some poor ones. The CAGR smooths all of that into one yearly rate, so you can compare investments that ran for different lengths of time on the same scale. **Why divide the end value by the start value?** The ratio gives the total growth multiple, which is how many times the money multiplied over the whole period. If a fund grew from 1000 dollars to 1331 dollars, the ratio is 1.331, so each starting dollar became 1.331 dollars. This total covers every year at once. **Why take the nth root?** Suppose the money grew by the same factor $x$ in each of $n$ years. Then the total multiple would be $x$ multiplied by itself $n$ times. To go back from the total to the yearly factor, we take the nth root, which is the same as raising the total to the power $\frac{1}{n}$. The cube root of 1.331 is 1.1, so the yearly factor is 1.1. **Why subtract 1?** A yearly factor of 1.1 keeps the original amount and adds 0.1 of growth on top. Subtracting 1 removes the original amount and leaves only the growth rate, 0.1, which is 10% per year. A factor below 1 gives a negative rate, which means the value fell. **Why is it called compound?** Each year's growth is added to the amount that has already grown, so growth builds on growth. That is compounding. The CAGR is the steady yearly rate that gives exactly the same result as that compounding over the whole period. It is not the simple average of the yearly rates. ## An everyday picture Think of a snowball rolled down a hill. Each roll picks up snow in proportion to the size the snowball already has, so it grows by a steady share each time it rolls. If you know its size at the top, its size at the bottom and how many rolls it took, you can find the one steady share per roll that connects the start to the end. The CAGR does the same thing for money: it is the steady yearly share that links the starting value to the ending value. ## Worked example: How fast did a savings fund grow? A savings fund was worth 1000 dollars at the start. Three years later it was worth 1331 dollars, with no money added or taken out in between. What compound annual growth rate did the fund achieve? 1. **Write down the start value, the end value and the number of years.** The fund started at 1000 dollars, so $V_{\text{start}} = 1000$ dollars. It ended at 1331 dollars, so $V_{\text{end}} = 1331$ dollars. The time between them is three years, so $n = 3$. 2. **Find the total growth multiple.** Dividing the end value by the start value tells us how many times the money multiplied over the whole three years. We have $\frac{1331}{1000} = 1.331$, so each starting dollar became 1.331 dollars. 3. **Take the cube root.** Raising a number to the power $\frac{1}{3}$ is the same as taking its cube root, the number that gives the original when multiplied by itself three times. This spreads the total growth evenly across the years: $1.331^{1/3} = 1.1$, because $1.1 \times 1.1 \times 1.1 = 1.331$. 4. **Subtract 1 to get the rate.** The number 1.1 is the yearly growth factor, and it includes the original dollar. Subtracting 1 keeps only the growth: $1.1 - 1 = 0.1$ per year. Written as a percent, the fund grew at 10% per year. 5. **Check by growing the money year by year.** At 10% per year, 1000 dollars becomes $1000 \times 1.1 = 1100$ dollars after year one, then $1100 \times 1.1 = 1210$ dollars after year two, and $1210 \times 1.1 = 1331$ dollars after year three. That matches the end value, so the answer is right. 6. **Read the answer.** The fund's CAGR is 10% per year, the one steady yearly rate that takes 1000 dollars to 1331 dollars in three years. Actual yearly returns may have been higher in some years and lower in others, so the CAGR describes the whole path with one number. **Answer:** The savings fund grew at a compound annual growth rate of **10% per year**. Its value rose from 1000 dollars to 1331 dollars over three years, which is the same as growing by 10% in each of those years. ## Common mistakes - Averaging the yearly percentages instead of using the start and end values. The average of a rise of 50% and a fall of 50% is 0%, but 1000 dollars that rises 50% and then falls 50% ends at 750 dollars, which is a CAGR of about minus 13.4% per year. - Putting the number of years in the exponent instead of its reciprocal. The exponent is $\frac{1}{n}$, so for three years the power is $\frac{1}{3}$, not 3. Using 3 would give a number far too large. - Forgetting to subtract 1. The ratio 1.1 is the growth factor, the amount the money is multiplied by. The growth rate is 0.1, or 10% per year, which is the number left after subtracting 1. - Counting the wrong number of years. The number $n$ is the time between the start and the end. Three years of growth means $n = 3$, even though a chart may show four year-end values, one for the start and one for each year. ## Where you will meet it - Comparing investments, funds or savings accounts that ran for different lengths of time, on one yearly scale. - Company reports on revenue or customer numbers over several years, quoted as one growth rate per year. - Market size and sales forecasts, where a yearly growth rate summarizes how fast a market grew between two dates. - Comparing a fund's growth with a benchmark over the same period, using the same yearly rate for both. ## Questions people ask ### Why is it called compound? Each year's growth is added to an amount that has already grown, so growth builds on growth. The CAGR is the steady yearly rate that gives the same result as this compounding over the whole period. ### Does CAGR show the ups and downs? No. It uses only the start and end values, so it hides the path. A fund that rose a lot and then fell can have the same CAGR as a steady fund. Look at the yearly values as well before drawing conclusions. ### Can a CAGR be negative? Yes. If the end value is smaller than the start value, the ratio is less than 1, its root is less than 1, and the answer is negative. For example, going from 1000 dollars to 750 dollars over two years gives a CAGR of about minus 13.4% per year. ### Does CAGR work when the period is not a whole number of years? Yes. Use the length of the period in years, which can be a decimal. For example, a period of 1.5 years gives the exponent $\frac{1}{1.5}$, which is the same as $\frac{2}{3}$. ## Related formulas - [Return on Investment (ROI)](https://formulabro.yunusemrevurgun.com/finance/return-on-investment/) - [Internal Rate of Return (IRR)](https://formulabro.yunusemrevurgun.com/finance/internal-rate-of-return/) - [Geometric Series](https://formulabro.yunusemrevurgun.com/computer-science/geometric-series/) ## Sources - [Compound annual growth rate, Wikipedia](https://en.wikipedia.org/wiki/Compound_annual_growth_rate) - [Compound interest, Wikipedia](https://en.wikipedia.org/wiki/Compound_interest) - [Rate of return, Wikipedia](https://en.wikipedia.org/wiki/Rate_of_return) From Formula Bro (https://formulabro.yunusemrevurgun.com/), a side project developed by yunusemrevurgun.com.