Constant Growth Rate Calculator (CAGR)
Calculate the average annual (or per-period) growth rate required for a value to grow from a "Beginning Value" to an "Ending Value" over a specific "Number of Periods", assuming a constant rate of growth each period. This is often referred to as the Compound Annual Growth Rate (CAGR) when periods are years.
Enter the starting value, the ending value, and the number of time periods that elapsed between them. The result will be the constant percentage growth rate per period.
Enter Values
Understanding Constant Growth Rate (CAGR)
What is Constant Growth Rate / CAGR?
The Constant Growth Rate, often called Compound Annual Growth Rate (CAGR) when the periods are years, represents the hypothetical rate at which an investment, revenue stream, population, or other metric would have grown if it had grown at the same rate every period over a specified time frame. It smooths out volatile growth rates to provide a single, representative figure.
Formula for Constant Growth Rate
The formula used is derived from compound growth:
Ending Value = Beginning Value * (1 + Growth Rate)Number of Periods
Rearranging to solve for the Growth Rate:
Growth Rate = (Ending Value / Beginning Value)(1 / Number of Periods) - 1
Formula Breakdown:
Ending Value / Beginning Value: This calculates the total growth factor over the entire period.(...)(1 / Number of Periods): Taking this to the power of (1 / Number of Periods) finds the average *per-period* growth factor.... - 1: Subtracting 1 converts the growth factor into a growth *rate* (as a decimal), which is then usually multiplied by 100 to get a percentage.
This formula requires the Beginning Value and Number of Periods to be greater than zero.
Constant Growth Rate Examples
Explore how constant growth rate applies to different scenarios:
Example 1: Investment Growth (CAGR)
Scenario: An investment grew from $10,000 to $15,000 over 5 years.
1. Known Values: Beginning Value = 10000, Ending Value = 15000, Number of Periods = 5 (Years).
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (15000 / 10000)(1 / 5) - 1 = (1.5)0.2 - 1 ≈ 1.0845 - 1 = 0.0845
4. Result: 0.0845 * 100 = 8.45%
Conclusion: The Compound Annual Growth Rate (CAGR) was approximately 8.45% per year.
Example 2: Population Decline
Scenario: A town's population decreased from 50,000 to 40,000 over 10 years.
1. Known Values: Beginning Value = 50000, Ending Value = 40000, Number of Periods = 10 (Years).
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (40000 / 50000)(1 / 10) - 1 = (0.8)0.1 - 1 ≈ 0.9779 - 1 = -0.0221
4. Result: -0.0221 * 100 = -2.21%
Conclusion: The population had a constant annual decay rate of approximately -2.21%.
Example 3: Revenue Growth
Scenario: A company's quarterly revenue went from $250,000 to $350,000 over 6 quarters.
1. Known Values: Beginning Value = 250000, Ending Value = 350000, Number of Periods = 6 (Quarters).
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (350000 / 250000)(1 / 6) - 1 = (1.4)(1/6) - 1 ≈ 1.0577 - 1 = 0.0577
4. Result: 0.0577 * 100 = 5.77%
Conclusion: The constant quarterly growth rate was approximately 5.77%.
Example 4: Website Traffic Increase
Scenario: Monthly website visitors increased from 5,000 to 12,000 over 8 months.
1. Known Values: Beginning Value = 5000, Ending Value = 12000, Number of Periods = 8 (Months).
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (12000 / 5000)(1 / 8) - 1 = (2.4)0.125 - 1 ≈ 1.1196 - 1 = 0.1196
4. Result: 0.1196 * 100 = 11.96%
Conclusion: The constant monthly growth rate for website visitors was approximately 11.96%.
Example 5: Product Price Change
Scenario: The price of a product increased from $20 to $25 over 3 years.
1. Known Values: Beginning Value = 20, Ending Value = 25, Number of Periods = 3 (Years).
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (25 / 20)(1 / 3) - 1 = (1.25)(1/3) - 1 ≈ 1.0772 - 1 = 0.0772
4. Result: 0.0772 * 100 = 7.72%
Conclusion: The product price increased at a constant annual rate of approximately 7.72%.
Example 6: Inventory Reduction
Scenario: Inventory stock decreased from 1,000 units to 800 units over 4 months.
1. Known Values: Beginning Value = 1000, Ending Value = 800, Number of Periods = 4 (Months).
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (800 / 1000)(1 / 4) - 1 = (0.8)0.25 - 1 ≈ 0.9457 - 1 = -0.0543
4. Result: -0.0543 * 100 = -5.43%
Conclusion: Inventory decreased at a constant monthly rate of approximately -5.43%.
Example 7: Value Doubles
Scenario: How long does it take for something to double if it grows at a constant rate of 10% per period? (This calculator solves for rate, but we can use it to check doubling time based on rate).
1. Known Values: Beginning Value = 100, Ending Value = 200 (doubled), Number of Periods = 7.27 (approx periods to double at 10%). Let's use 7 periods.
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (200 / 100)(1 / 7) - 1 = (2)(1/7) - 1 ≈ 1.1041 - 1 = 0.1041
4. Result: 0.1041 * 100 = 10.41%
Conclusion: If it doubles in exactly 7 periods, the constant growth rate was about 10.41%. (Note: This calculator finds rate given values and periods, not the other way around, but this shows the relationship).
Example 8: Using Decimals
Scenario: A metric changes from 0.5 to 0.8 over 3 periods.
1. Known Values: Beginning Value = 0.5, Ending Value = 0.8, Number of Periods = 3.
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (0.8 / 0.5)(1 / 3) - 1 = (1.6)(1/3) - 1 ≈ 1.1696 - 1 = 0.1696
4. Result: 0.1696 * 100 = 16.96%
Conclusion: The constant growth rate was approximately 16.96% per period.
Example 9: Short Term Growth
Scenario: A value increased from 500 to 520 in just 1 period.
1. Known Values: Beginning Value = 500, Ending Value = 520, Number of Periods = 1.
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (520 / 500)(1 / 1) - 1 = (1.04)1 - 1 = 1.04 - 1 = 0.04
4. Result: 0.04 * 100 = 4.00%
Conclusion: The constant growth rate for that single period was 4.00%.
Example 10: Value Stagnation
Scenario: A value remained unchanged from 750 to 750 over 6 periods.
1. Known Values: Beginning Value = 750, Ending Value = 750, Number of Periods = 6.
2. Formula: Growth Rate = (Ending Value / Beginning Value)(1 / Periods) - 1
3. Calculation: Growth Rate = (750 / 750)(1 / 6) - 1 = (1)(1/6) - 1 = 1 - 1 = 0
4. Result: 0 * 100 = 0.00%
Conclusion: The constant growth rate was 0.00% per period, indicating no overall change.
Understanding Percentage and Rates
A percentage is a number expressed as a fraction of 100... The constant growth rate is typically expressed as a percentage for ease of understanding.
Interpreting the Growth Rate
A positive growth rate means the value increased over the period. A negative growth rate means it decreased. A 0% rate means it stayed the same. A -100% rate means the value dropped to zero.
Frequently Asked Questions about Constant Growth Rate (CAGR)
1. What is the Constant Growth Rate?
It's the hypothetical rate at which something would need to grow each period (e.g., year, month) at a steady, compounded pace to get from a starting value to an ending value over a specific number of periods.
2. What does CAGR stand for?
CAGR stands for Compound Annual Growth Rate. It's the same concept as the constant growth rate, specifically applied when the periods are years.
3. What inputs do I need for this calculator?
You need three things: the value you started with (Beginning Value), the value you ended with (Ending Value), and how many periods passed between those two values (Number of Periods).
4. Can the growth rate be negative?
Yes. If the Ending Value is less than the Beginning Value, the calculator will correctly show a negative growth rate, representing a constant rate of decay or decrease per period.
5. Why can't the Beginning Value or Number of Periods be zero or negative?
The formula involves division by the Beginning Value and taking a root based on the Number of Periods. Mathematically, these operations are undefined or result in complex numbers/errors if the inputs are zero or negative in this context.
6. What happens if the Ending Value is zero?
If the Beginning Value is positive and the Ending Value is exactly zero, the calculated growth rate will be -100%, meaning the value decreased completely over the given periods.
7. Is Constant Growth Rate (CAGR) an actual rate?
Not necessarily. Unless the growth was *actually* constant each period, the calculated CAGR is a smoothed, hypothetical rate that represents the average compounding growth over the entire period. Actual period-to-period growth rates might vary significantly.
8. How is this different from a simple average growth rate?
Simple average growth typically just sums the total change and divides by the number of periods. Constant growth rate is a *compounding* rate. It assumes that the growth each period is applied to the *new* value from the previous period, which is a more accurate representation of many real-world growth processes (like investments).
9. What kind of "Periods" can I use?
The periods must be consistent units of time. If you measure your values yearly, the periods are years, and the result is annual. If you measure them monthly, the periods are months, and the result is monthly. Ensure your periods match the time intervals between your Beginning and Ending Values.
10. Can I use this to project future growth?
The calculated constant growth rate can be used as a basis for *projection*, but it assumes that the *past* constant rate will continue into the future, which may not be realistic. It's a tool for historical analysis and hypothetical future modeling.
