Growth rate answers a simple question: what constant per-period rate turns a starting number into an ending number? Enter any two values and the time between them and this calculator solves for r directly, useful for checking a claimed investment return, a population trend, or a company's revenue growth without building a full model. To project a rate forward instead of solving for one, see the investment growth calculator.
Growth rate equation
r = (x₁ / x₀)1/t − 1. This is the geometric mean rate that bridges the two observations, also known as compound annual growth rate (CAGR) when t is in years.
Worked example
A portfolio rises from $50,000 to $80,000 over 8 years. r = (1.6)^(1/8) − 1 = 6.05% per year. The simple total return of 60% does not annualise to 7.5%; that arithmetic mean ignores compounding.
Why use CAGR
CAGR strips volatility from the comparison and reports the steady equivalent rate. Two funds that both turned $10,000 into $20,000 over 10 years have identical CAGR (7.18%) even if one zigzagged and one grew smoothly. The same logic extends to any pair of measurements, not just money: bacterial counts, website traffic, or a country's GDP all get the same treatment when comparing performance across differing time spans.
Discrete rate vs continuous rate
This calculator reports two related numbers: the discrete per-period rate r, and the continuous rate k = ln(x₁/x₀) / t, the one used in calculus-based models built on e^(kt). For the same move from $1,000 to $1,500 over 5 years, r = 8.45% but k = ln(1.5) / 5, about 8.11%. The two differ because k assumes compounding happens continuously, while r assumes it happens once per period. Use k when the underlying process is modeled with the natural exponential function, covered on the continuous growth calculator. Over short periods or low rates the two numbers are nearly identical; the gap widens as the rate or the number of periods increases, which matters most for fast-growing startups or hyperinflationary economies where discrete and continuous compounding can diverge by several percentage points.
Checking a real growth claim
Suppose a company reports revenue grew from $2.4 million to $4.1 million over 3 years and calls it "70% growth." That 70% is the simple total return, not an annual rate. The annualised figure is r = (4.1/2.4)^(1/3) − 1, about 19.5% per year, the number that should be compared against a competitor's annual growth rate rather than against the 3-year total. The same 70% figure reported over just 1 year instead of 3 would annualise to a full 70% per year, a very different story. As a result, the time period t matters just as much as the percentage change itself.
How much a small rate difference is worth
Run forward instead of solved backward, the same formula shows how much a few percentage points matter. $10,000 growing at 5% for 20 years reaches about $26,533. The same $10,000 at 8% for 20 years reaches about $46,610, roughly 76% more from just a 3-point difference in the annual rate. Entering those two endpoints back into this calculator confirms it: $10,000 to $26,533 over 20 years returns r = 5.00%, and $10,000 to $46,610 over 20 years returns r = 8.00%.
Negative rates and decline
The formula handles shrinkage exactly as well as growth. A population or revenue base falling from 1,000,000 to 820,000 over 4 years has r = (0.82)^(1/4) − 1, about -4.84% per year. The same negative-rate logic drives the half-life calculator and the exponential decay calculator, both of which are really just this formula's negative branch given its own name and units.
FAQ
How do I calculate growth rate from two values?
Divide the final value by the initial value, raise the result to the power 1/t, and subtract 1: r = (x₁/x₀)^(1/t) - 1. Growing from $1,000 to $1,500 over 5 years gives r = (1.5)^(1/5) - 1, about 8.45% per year, the constant annual rate that connects the two points exactly.
What is CAGR?
CAGR stands for compound annual growth rate, the same r = (x₁/x₀)^(1/t) - 1 formula applied with t measured specifically in years. It answers one question: what single steady annual rate, compounding every year, turns the starting value into the ending value? A fund moving from $200,000 to $350,000 over 6 years has a CAGR of (1.75)^(1/6) - 1, about 9.78%.
Is CAGR the same as average growth rate?
No. The simple arithmetic average of yearly percent changes overstates true performance whenever returns are volatile, an effect known as volatility drag. CAGR is the geometric mean instead, the one rate that compounds cleanly from the first value to the last. A portfolio that gains 50% one year and loses 50% the next has a 0% arithmetic average, but its real two-year CAGR is about -13.4%, since $100 becomes $150 then $75.
Can growth rate be negative?
Yes, a negative result indicates decline rather than growth, and the same formula still applies without changes. If a value falls from $10,000 to $7,000 over 4 years, r = (0.7)^(1/4) - 1, about -8.53% per year, because x₁/x₀ is less than 1, which makes the fourth root less than 1 as well.
What time unit should t use?
Use whatever period matches the rate you want: years for an annual rate, months for a monthly rate, quarters for a quarterly rate. To convert a monthly rate to an annual one, compound it twelve times: r_year = (1 + r_month)^12 - 1. A 0.5% monthly rate compounds to about 6.17% per year, noticeably more than simply multiplying 0.5% by 12.
How does this differ from the simple return?
Simple return, (x₁ - x₀) / x₀, measures total percent change while ignoring how long it took to get there. Growth rate annualises that change so periods of different lengths become comparable. A 60% simple return over 8 years and a 60% simple return over 2 years are the same total gain, but they annualise to about 6.05% and about 26.5% per year, very different growth rates.