Introduction
Growth rates describe the percentage change in a variable over a specific time period. They are used across biology, economics, corporate performance and investing to summarize how quickly something (revenue, GDP, population, dividends, portfolio value) is growing or shrinking. Growth rates may be reported as simple period-to-period changes, compounded annual rates, or continuous rates depending on the context and the desired interpretation.
Key concepts
– Simple growth rate (period-over-period): (Ending − Beginning) / Beginning.
– Compound Annual Growth Rate (CAGR): the constant annual rate that would take you from a beginning value to an ending value over n years.
– Continuous growth rate: assumes continuous compounding; use natural logarithms.
– Internal and sustainable growth rates: formulas that express how fast a company can grow without external financing given return and retention metrics.
– Dividend growth is key to valuation models such as the Gordon Growth Model (DDM).
1. How to calculate basic growth rates (period-to-period)
Practical steps
1. Decide the beginning (BV) and ending value (EV) and the period (e.g., 1 year, 1 quarter).
2. Use the basic formula:
Growth rate = (EV − BV) / BV
3. Convert to percentage: multiply by 100.
Example
– Revenue from $50M to $60M in one year:
Growth = (60 − 50) / 50 = 0.20 = 20%.
Notes
– For quarter-over-quarter or month-over-month, apply the same formula to sequential periods.
– Watch for base effects: very small denominators inflate percentages.
2. Compound Annual Growth Rate (CAGR)
Why use it
CAGR summarizes multi-period growth as a single average annual rate, smoothing year-to-year volatility. It is not the same as an actual year-by-year return if returns vary.
Formula
CAGR = (EV / BV)^(1/n) − 1
where n = number of years (or periods adjusted to years).
Practical steps (manual)
1. Divide ending value by beginning value.
2. Raise the result to the 1/n power.
3. Subtract 1 and convert to a percentage.
Example
– BV = $100, EV = $180, n = 3 years:
CAGR = (180/100)^(1/3) − 1 = 1.8^(0.3333) − 1 ≈ 0.203 = 20.3% per year.
Excel
– Formula: =(EV/BV)^(1/n)-1
– Or: =POWER(EV/BV,1/n)-1
– For irregular cash flows, use XIRR to find an annualized return.
Limitations
– Hides volatility and path dependency (ups and downs).
– Assumes reinvestment and smooth growth.
3. Continuous growth and instantaneous rate
When used
Population growth models or some economic/biological models assume continuous compounding.
Formula
r = ln(EV / BV) / t
where r is the continuous annual growth rate, t is years, ln is natural log.
Interpretation
If r = 0.05 (5%), then EV = BV * e^(0.05t).
4. Calculating GDP growth rate
Simple year-over-year GDP growth
Growth = (GDP_thisYear − GDP_lastYear) / GDP_lastYear
Quarterly rates and annualizing
– Quarter-over-quarter percentage change: (Q_t / Q_{t-1}) − 1
– To annualize a quarterly compounding rate: annualized = (Q_t / Q_{t-1})^4 − 1
Example
– GDP year 1 = 20,000; year 2 = 20,600:
Growth = (20,600 − 20,000) / 20,000 = 0.03 = 3% annual growth.
5. Growth rates in valuation — dividend growth & Gordon Growth Model
Gordon Growth Model (GGM)
Price = D1 / (k − g)
where:
– D1 = expected dividend next period,
– k = required rate of return (discount rate),
– g = constant expected dividend growth rate.
Practical steps
1. Estimate next year’s dividend (D1) and a sustainable growth rate (g).
2. Ensure g < k (otherwise formula breaks).
3. Plug into the model to solve for intrinsic stock price or for implied g given price.
Caveat
The GGM assumes perpetual and constant growth and is most appropriate for mature firms with stable dividends.
6. Internal Growth Rate (IGR) and Sustainable Growth Rate (SGR)
Definitions
– Internal Growth Rate (IGR): the maximum growth rate a firm can achieve without external financing, given its return on assets and retention ratio.
– Sustainable Growth Rate (SGR): the maximum growth rate a firm can achieve without changing its leverage, given return on equity and retention ratio.
Formulas
– IGR = (ROA × b) / (1 − ROA × b)
– SGR = (ROE × b) / (1 − ROE × b)
where b = retention ratio (1 − dividend payout ratio).
Practical steps
1. Compute ROA or ROE from financials.
2. Calculate b from dividend policy.
3. Plug into the formula to find the maximum internal/sustainable growth.
7. What is a “normal” or “good” growth rate?
– Mature large companies: single-digit to low-teens revenue/earnings growth (e.g., 2–10%) is common.
– High-growth tech or earlier-stage firms: double-digit to 30%+ annual growth is common.
– Startups: early-stage benchmarks vary—VCs often look for rapid month-over-month (MoM) user or revenue growth; 10–20% MoM can be very strong in early traction phases. “Good” depends on business model, market size, margins, and capital intensity.
Guidance
Always compare to relevant peers/industry averages and consider absolute scale: 20% growth on $1B revenue is very different from 20% on $1M.
8. Calculating population growth
Discrete annual growth (CAGR-like)
Growth rate = (Population_end / Population_start)^(1/n) − 1
Continuous growth
r = ln(Pop_end / Pop_start) / t
Example
Population grows from 1,000 to 1,500 in 5 years:
Discrete rate = (1500/1000)^(1/5) − 1 ≈ 0.0845 = 8.45% per year.
Continuous r = ln(1.5) / 5 ≈ 0.0811 = 8.11% per year.
9. How to calculate growth rates in Excel — practical formulas
– Period-over-period: =(Current − Previous) / Previous
– CAGR: =(EV/BV)^(1/n)-1 or =POWER(EV/BV,1/n)-1
– Continuous annual rate: =LN(EV/BV)/n
– Annualize quarterly rate: =(Q_t/Q_{t-1})^4-1
– XIRR for irregular cash flows: =XIRR(values, dates)
Formatting: multiply results by 100 or format as Percentage.
10. Adjusting growth for inflation, taxes and fees (real and after-tax growth)
– Real growth rate ≈ (1 + nominal) / (1 + inflation) − 1.
– After-tax return depends on tax treatment; for simple approximation: after-tax nominal ≈ nominal × (1 − tax rate) for fully taxable flows, but more accurate approach applies taxes to actual cash flows and recomputes CAGR or XIRR.
– For investments, compute net-of-fees returns by using actual net cash flows into/out of the portfolio.
11. Example calculations (quick reference)
– Simple growth: BV $200 → EV $260 in 1 year: (260−200)/200 = 30%
– CAGR over 4 years: BV $200 → EV $400: CAGR = (400/200)^(1/4)-1 = (2)^(0.25)-1 ≈ 18.9%
– GDP quarterly annualized: Q1 = 5,000, Q2 = 5,100: quarterly growth = 5,100/5,000 − 1 = 2%; annualized = (1.02)^4 − 1 ≈ 8.24%.
12. Limitations and common pitfalls
– Smoothing hides volatility: CAGR masks intermediate declines and recoveries.
– Base effects and outliers can distort percentage changes.
– Assumption mismatches: constant growth assumptions (e.g., GGM) often don’t hold.
– Seasonality: compare year-over-year rather than sequential quarters without adjustment.
– Data quality and accounting changes can change reported values without real economic change.
– Comparing across industries without normalization can mislead—benchmarks matter.
13. Practical checklist for analysts and investors
1. Define the period and frequency (annual, quarterly, monthly).
2. Choose the right metric: simple growth for short periods, CAGR for multi-year averages, continuous rate for certain models.
3. Adjust for inflation, taxes and seasonality when relevant.
4. Compare to industry peers and macro benchmarks (industry CAGR, GDP growth).
5. Use appropriate Excel formulas or financial functions (XIRR for irregular flows).
6. Note assumptions (constant growth, reinvestment, payout ratio).
7. Run sensitivity tests: show outcomes for different growth assumptions.
Fast facts
– Growth rates can be positive (expansion) or negative (contraction).
– CAGR is descriptive, not predictive of volatility or path.
– For valuation, the assumed growth rate often drives the majority of modeled value.
– Industry norms vary widely; context is essential.
The bottom line
Growth rates are fundamental metrics for evaluating economies, companies, investments and populations. Choosing the correct growth measure (simple, compound, continuous), understanding its assumptions and limitations, and applying proper adjustments (inflation, taxes, seasonality) are critical to meaningful analysis and decision-making.
Reference
– Investopedia — Growth Rates
Editor’s note: The following topics are reserved for upcoming updates and will be expanded with detailed examples and datasets.