Growth rates and compounding
Calculate compounded growth and distinguish a steady annual equivalent from an arithmetic average.
Calculate multi-period growth and a compound annual growth rate, distinguishing a compounded result from an arithmetic average.
Overview
The owner of Alder Garden Supplies wants to describe three years of sales growth. Annual revenue rose from $100,000 to $133,100. She calls that “33.1% growth per year.” A prospective buyer asks how she calculated it.
Both businesses and all figures in this lesson are fictional teaching assumptions.
Recall Lesson 023: If revenue rises from $100 to $120, then falls back to $100, why are the percentage increase and decrease different?
Apply growth to the changing base
A growth rate measures a change relative to its starting value. One period's rate is (ending value − starting value) ÷ starting value. Multiply by 100 to express it as a percentage.
To grow $100,000 by 10%, add $10,000, giving $110,000. Equivalently, multiply $100,000 by 1.10. This growth factor includes the original amount, represented by 1, plus the decimal growth rate, 0.10.
Compounding applies the next period's change to the updated value. Another 10% increase adds $11,000 to $110,000, producing $121,000. The same percentage now acts on a larger base.
For a decline, use a factor below one. A 20% decline means retaining 80% of the starting amount, so multiply by 0.80. Across several periods, multiply the factors in sequence.
Worked example
Alder's completed annual revenues are below. Each row covers a full year, using the same revenue definition and business scope. “Year 0” names the baseline year, not a year with no sales.
| Annual period | Revenue | Change from preceding year |
|---|---|---|
| Year 0 | $100,000 | Baseline |
| Year 1 | $110,000 | 10% |
| Year 2 | $121,000 | 10% |
| Year 3 | $133,100 | 10% |
There are three growth intervals between four annual observations. Revenue in Year 3 is $100,000 × 1.10 × 1.10 × 1.10 = $133,100.
Total growth across those intervals is ($133,100 − $100,000) ÷ $100,000 × 100 = 33.1%. Adding 10% + 10% + 10% would give 30%, overlooking the changing base.
For a constant rate, the compact formula is:
Ending value = starting value × (1 + r)n.
Here r is the rate as a decimal and n is the number of equal-length growth intervals. The exponent n means multiply the factor by itself that many times. For Alder, r = 0.10 and n = 3.
The owner's correct description is “Revenue grew 33.1% in total over three annual intervals, equivalent to 10% per year compounded.” The $133,100 is Year 3 revenue, not the sum of revenue earned across all four years.
Find the equivalent annual rate
The compound annual growth rate, or CAGR, is the constant yearly rate that would connect the starting and ending values. It summarizes the endpoints even when actual yearly rates differ.
CAGR = (ending value ÷ starting value)1/n − 1.
Here n counts years between the observations. Raising the ratio to 1/n finds its nth root: the factor that, multiplied by itself n times, reproduces the ratio. On a calculator, enter the ratio, use the power key with (1 ÷ n), then subtract 1.
For Alder, $133,100 ÷ $100,000 = 1.331. The cube root is 1.10 because 1.10 × 1.10 × 1.10 = 1.331. Subtract 1 to obtain 0.10, then multiply by 100 to report 10%.
This calculation uses the geometric mean of growth factors, which preserves their compounded product. OpenStax explains why growth calculations use decimal multipliers before finding this mean and subtracting one. Source: OpenStax.
Distinguish the arithmetic average
An arithmetic average adds the annual rates and divides by their count. It describes the average of those observations, but generally does not reproduce the ending value when compounded.
Consider $100 growing 25% and then falling 20%. The calculation is $100 × 1.25 × 0.80 = $100. Total growth is zero, and the two-year CAGR is zero.
The arithmetic average is (25% + (−20%)) ÷ 2 = 2.5%. Compounding 2.5% twice would produce $100 × 1.025 × 1.025 = $105.0625, which does not match the actual $100 endpoint.
Always label the measure. CAGR also hides the path: a business can reach the same endpoint through steady growth or large swings. Neither historical rate establishes next year's demand, profit, or cash.
Use comparable definitions and equal-length periods. A monthly compound rate is monthly until converted to an annual rate. Today's CAGR formula assumes positive starting and ending values; zero or negative endpoints require a different explanation rather than forcing this calculation.
Exercise
Allow 8–10 minutes. You lead established manufacturer Summit Fixtures. Its completed annual revenues were $800,000 in Year 0, $1,000,000 in Year 1, and $968,000 in Year 2. All years have the same length and revenue definition. There were no acquisitions or changes in business scope.
A manager reports, “Average growth was 10.9% annually, so use 10.9% compound growth to describe our results.” Costs and cash collections have not been supplied.
- Calculate each annual growth rate and their arithmetic average.
- Calculate total growth from Year 0 to Year 2. Verify it by multiplying the two annual growth factors.
- Calculate CAGR. Use two growth intervals; the square root of 1.21 is 1.10. Explain how it differs from 10.9%.
- If next year's revenue grew at the historical CAGR, what would it be? Write a two-sentence correction to the manager that separates the historical result from this future assumption.
Try the calculations before reading the answer.
Answer
Year 1 growth is ($1,000,000 − $800,000) ÷ $800,000 × 100 = 25%. Year 2 growth is ($968,000 − $1,000,000) ÷ $1,000,000 × 100 = −3.2%.
The arithmetic average is (25% − 3.2%) ÷ 2 = 10.9%. That part of the manager's calculation is correct.
Total growth is ($968,000 − $800,000) ÷ $800,000 × 100 = 21%. The annual factors are 1.25 and 0.968. Their product is 1.21, so $800,000 × 1.21 = $968,000. Adding the rates would give 21.8%, which does not reproduce the result.
CAGR is ($968,000 ÷ $800,000)1/2 − 1 = 1.10 − 1 = 0.10, or 10%. Check: $800,000 × 1.10 × 1.10 = $968,000. Using 10.9% twice would instead give $800,000 × 1.109 × 1.109 = $983,904.80.
A further 10% increase would produce $968,000 × 1.10 = $1,064,800 next year. This is a conditional calculation. Summit's recent decline makes it especially useful to investigate orders, prices, and customer demand before adopting that assumption.
A correction could read: “Revenue grew 21% over two annual intervals, equivalent to a 10% CAGR; 10.9% is the arithmetic average of the two observed rates. Next year's $1,064,800 scenario assumes another 10% increase and needs demand evidence before becoming a budget.”
The revenue figures alone cannot establish profit or available cash. Review the underlying yearly results alongside CAGR.
Recall answer: The increase uses $100 as its base, giving 20%. The decrease uses $120, giving $20 ÷ $120 × 100 ≈ 16.7%. Compounding likewise respects each period's updated base.
Applying this lesson
Keep the starting value, ending value, number of intervals, and growth measure together. Add a factor-by-factor calculation and its CAGR check to your calculation sheet. Tomorrow you will audit a growth report that also includes sales mix and margin.
Sources
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