AI Math
打开应用

CAGR Calculator

Back out the compound annual growth rate between any two values, step by step
CAGR from 10,000 to 18,500 over 6 years
Revenue grew from 2.4 million to 5.9 million in 5 years - what is the CAGR?
How many years at 12% CAGR to double 50,000?
A portfolio gains 50% then loses 50% - what is the CAGR?

The CAGR Formula

Compound annual growth rate is the single constant rate that would carry a beginning value to an ending value over a given number of years:

CAGR=(VendVbegin)1n−1\text{CAGR} = \left(\frac{V_{\text{end}}}{V_{\text{begin}}}\right)^{\frac{1}{n}} - 1

  • VbeginV_{\text{begin}} — the value at the start
  • VendV_{\text{end}} — the value at the end
  • nn — the number of years between the two observations, not the number of data points

It is the inverse of compounding: where Vend=Vbegin(1+r)nV_{\text{end}} = V_{\text{begin}}(1+r)^n solves forward for a value, CAGR solves backward for the rate. That makes it a smoothed figure — it describes the endpoints only and says nothing about the path between them. Two series with wildly different volatility can share a CAGR.

Rearrangements and the Average-Return Trap

The same relation answers three other questions:

Vend=Vbegin(1+r)n,Vbegin=Vend(1+r)n,n=ln⁡(Vend/Vbegin)ln⁡(1+r)V_{\text{end}} = V_{\text{begin}}(1+r)^n, \qquad V_{\text{begin}} = \frac{V_{\text{end}}}{(1+r)^n}, \qquad n = \frac{\ln(V_{\text{end}}/V_{\text{begin}})}{\ln(1+r)}

The last one uses logarithms because nn sits in the exponent; any log base works as long as both are the same.

For a non-whole horizon, put the years in as a decimal: 42 months is n=3.5n = 3.5.

CAGR is not the arithmetic mean of yearly returns. The arithmetic mean adds and divides; CAGR multiplies and takes a root, which is a geometric mean of the growth factors:

1+CAGR=(1+r1)(1+r2)⋯(1+rn)n1 + \text{CAGR} = \sqrt[n]{(1+r_1)(1+r_2)\cdots(1+r_n)}

Whenever the yearly returns vary, the geometric mean is strictly below the arithmetic mean, and the gap widens with volatility. Only CAGR reproduces the actual ending value.

Common Mistakes to Avoid

  • Counting data points instead of intervals: values for 2018 through 2024 span n=6n = 6 years, not 7.
  • Averaging the annual percentages: use the geometric mean, or the CAGR will not reproduce the ending value.
  • Applying CAGR to a series that crosses zero or goes negative: the ratio Vend/VbeginV_{\text{end}}/V_{\text{begin}} must be positive, or the root is undefined.
  • Forgetting to subtract 1: the root gives the growth factor; CAGR is that factor minus 1.
  • Comparing CAGRs over different horizons: a 3-year and a 10-year CAGR are not like-for-like.
  • Reading CAGR as a forecast: it is a description of two observed endpoints, nothing more.

示例题目

Step 1: Ratio: 18500/10000=1.8518500/10000 = 1.85
Step 2: Take the 6th root: 1.851/61.85^{1/6}. Using logs, ln⁡1.85≈0.6151856\ln 1.85 \approx 0.6151856, divided by 6 gives 0.10253090.1025309
Step 3: e0.1025309≈1.1079716e^{0.1025309} \approx 1.1079716
Step 4: CAGR≈1.1079716−1=0.1079716\text{CAGR} \approx 1.1079716 - 1 = 0.1079716
Step 5: Check: 10000×(1.1079716)6≈18,50010000 \times (1.1079716)^6 \approx 18{,}500
Answer: CAGR≈10.797%\text{CAGR} \approx 10.797\% per year

Step 1: Ratio: 5.9/2.4≈2.45833335.9/2.4 \approx 2.4583333
Step 2: ln⁡2.4583333≈0.8994653\ln 2.4583333 \approx 0.8994653, divided by 5 gives 0.17989310.1798931
Step 3: e0.1798931≈1.1970937e^{0.1798931} \approx 1.1970937
Step 4: CAGR≈0.1970937\text{CAGR} \approx 0.1970937
Answer: CAGR≈19.709%\text{CAGR} \approx 19.709\% per year

Step 1: Arithmetic mean: (50%−50%)/2=0%(50\% - 50\%)/2 = 0\%
Step 2: Actual growth factor: 1.50×0.50=0.751.50 \times 0.50 = 0.75, so \100endsatends at$75$
Step 3: CAGR=0.751/2−1\text{CAGR} = 0.75^{1/2} - 1; 0.75≈0.8660254\sqrt{0.75} \approx 0.8660254
Step 4: CAGR≈−0.1339746\text{CAGR} \approx -0.1339746
Answer: An average return of 0%0\% but a CAGR of ≈−13.397%\approx -13.397\% per year - the CAGR is the one that matches the \75$ ending value

常见问题

CAGR = (ending value / beginning value)^(1/n) − 1, where n is the number of years between the two values. Multiply by 100 to state it as a percentage.

CAGR is a geometric mean of growth factors, while the average return is an arithmetic mean of percentages. Whenever returns vary, the geometric mean is strictly smaller, because a loss removes more value than the same-sized gain adds back. Only CAGR reproduces the true ending value.

Yes. If the ending value is below the beginning value the ratio is less than 1, its root is less than 1, and subtracting 1 gives a negative rate. The formula breaks down only if a value is zero or negative, since the root of a non-positive ratio is undefined.

Express n in years as a decimal. Thirty months is n = 2.5, and 100 days is n = 100/365 ≈ 0.274. The formula is otherwise unchanged.

免费试用 AI-Math

任何数学问题都能获得分步解答。拍照上传或输入问题即可。

开始解题