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Compound annual growth rate
Compound annual growth rate
Compound annual growth rate
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CAGR calculator
Value Year
Initial value $100 1990
Final value $800 2005
CAGR of 14.9% over 15 years
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Compound annual growth rate (CAGR) is a business, economics and investing term representing the mean annualized growth rate for compounding values over a given time period.[1][2] CAGR smoothes the effect of volatility of periodic values that can render arithmetic means less meaningful. It is particularly useful to compare growth rates of various data values, such as revenue growth of companies, or of economic values, over time.[3]

Equation

[edit]

For annual values, CAGR is defined as:

where is the initial value, is the end value, and is the number of years.

CAGR can also be used to calculate mean annualized growth rates on quarterly or monthly values. The numerator of the exponent would be the value of 4 in the case of quarterly, and 12 in the case of monthly, with the denominator being the number of corresponding periods involved.[4]

In practice, CAGR calculations are often performed in Microsoft Excel. A convenient built-in function is , where represents the number of periods, denotes the present value (initial investment), and represents the future value (final value of the investment). The IRR function returns the equivalent constant interest rate per period, effectively matching the CAGR when applied over a specified period.[5]

Applications

[edit]

These are some of the common CAGR applications:

  • Calculating and communicating the mean returns of investment funds[6]
  • Demonstrating and comparing the performance of investment advisors[6]
  • Comparing the historical returns of stocks with bonds or with a savings account[6]
  • Forecasting future values based on the CAGR of a data series (you find future values by multiplying the last datum of the series by (1 + CAGR) as many times as years required). As with every forecasting method, this method has a calculation error associated.
  • Analyzing and communicating the behavior, over a series of years, of different business measures such as sales, market share, costs, customer satisfaction, and performance.
  • Calculating mean annualized growth rates of economic data, such as gross domestic product, over annual, quarterly or monthly time intervals.[7]

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Compound annual growth rate

View on Grokipedia
from Grokipedia
The compound annual growth rate (CAGR) is a financial metric that measures the mean annual growth rate of an , revenue, or other value over a specified multi-year period, assuming steady each year to smooth out fluctuations and provide a consistent annualized return figure. It is widely used by investors, analysts, and businesses to evaluate and compare the performance of assets, companies, or economies over time, as it normalizes irregular growth patterns into a single, hypothetical constant rate. CAGR is calculated using the formula:
CAGR=(Ending ValueBeginning Value)1n−1\text{CAGR} = \left( \frac{\text{Ending Value}}{\text{Beginning Value}} \right)^{\frac{1}{n}} - 1
where nn represents the number of years in the period. This geometric mean approach accounts for the effects of compounding, differing from simple arithmetic averages that can overstate growth by ignoring reinvested returns. For example, if an investment starts at $10,000 and ends at $19,487 after five years, the CAGR would be approximately 14.27%, indicating the equivalent steady annual growth rate. In practice, CAGR finds applications in portfolio analysis, where it helps assess long-term returns on , mutual funds, or ; in , for projecting or growth; and in economic reporting, such as tracking GDP expansion. It is particularly valuable for comparing disparate investments, as it eliminates the impact of timing differences in returns and focuses on overall . Despite its utility, CAGR has notable limitations: it assumes constant growth without volatility, potentially misleading users about exposure during market downturns; it ignores intermediate flows like dividends or additional investments; and it may overstate performance if the period includes irregular events like mergers or economic shocks. Therefore, it should be used alongside other metrics, such as standard deviation for volatility or (IRR) for flow considerations, to provide a fuller picture of viability.

Fundamentals

Definition

The compound annual growth rate (CAGR) represents the mean annual growth rate of an or metric over a specified period longer than one year, assuming that profits or gains are reinvested at the end of each period. It calculates the of the annual growth factors, providing a smoothed measure of growth that accounts for the effects of over time. This metric treats the growth as if it occurred at a steady rate each year, enabling consistent comparison across different time frames or s. In contrast to a simple arithmetic average of annual growth rates, which can overstate performance in cases of volatility, CAGR normalizes irregular returns into a single, constant annual rate. For instance, it reveals the true compounded impact of ups and downs, avoiding the distortion that might suggest zero net growth from equal gains and losses in successive periods. Essential prerequisites for understanding CAGR include the concept of compound growth, where returns build upon prior returns, and key terms such as the beginning value (the initial amount invested), the ending value (the final amount after the period), and the number of periods (n, typically measured in years).

Importance

The compound annual growth rate (CAGR) offers significant practical value by providing a standardized, single metric that captures the smoothed annual growth of an , metric, or over irregular periods, enabling straightforward comparisons across volatile or uneven trajectories. Unlike simple arithmetic averages, which can be skewed by short-term fluctuations, CAGR inherently accounts for effects, delivering a geometrically averaged rate that reflects the true annualized without distortion from interim volatility. This makes it particularly useful for long-term and , such as evaluating portfolios or economic indicators spanning multiple years, where it simplifies by presenting growth in an intuitive, annual form. CAGR's standardization has been widely adopted by financial regulators and professional bodies to ensure consistent and transparent reporting. For instance, the U.S. Securities and Exchange Commission (SEC) mandates the disclosure of average annual total returns—calculated as CAGR—for in shareholder reports, covering 1-, 5-, and 10-year periods, reinforced in modern tailored shareholder reports. Similarly, the CFA Institute's Global Investment Performance Standards (GIPS) require firms to present time-weighted annualized returns, effectively utilizing CAGR methodologies, for composite performance to facilitate fair comparisons across investment managers. This regulatory and standards-based adoption promotes uniformity in disclosures and economic data reporting, reducing ambiguity and enhancing investor confidence. Compared to alternatives like simple year-over-year growth rates or arithmetic means, CAGR excels by embedding the power of , which avoids understating long-term returns in scenarios with variable performance, and allows for apples-to-apples evaluations across disparate time frames, , or economic contexts. Its interpretability as an equivalent steady annual rate further distinguishes it, making complex growth patterns accessible without needing detailed volatility breakdowns. In practice, CAGR's simplicity drives its broad real-world use, including in corporate annual reports for or trends, and population projections by the , where it provides a clear, comparable measure of sustained expansion.

Calculation

Formula

The compound annual growth rate (CAGR) is calculated using the formula CAGR=(Ending ValueBeginning Value)1n−1,\text{CAGR} = \left( \frac{\text{Ending Value}}{\text{Beginning Value}} \right)^{\frac{1}{n}} - 1, where Ending Value represents the value of the investment or metric at the conclusion of the period, Beginning Value is the initial value at the start, and nn denotes the number of years or compounding periods over which the growth occurs. To compute the CAGR step by step:
  1. Compute the ratio: Ending Value ÷ Beginning Value.
  2. Take the nth root of that ratio, where n is the number of years.
  3. Subtract 1 from the result.
  4. Multiply by 100 to express as a percentage.
This calculation can be performed using a calculator, a spreadsheet formula such as =(Ending Value/Beginning Value)^(1/n)-1 in Excel or Google Sheets, or programming languages like Python with libraries such as NumPy. This formula expresses the result as a decimal, which can be multiplied by 100 to obtain a if desired, though the decimal form is standard for further computations. A common notation variation uses abbreviations such as EV for Ending Value and BV for Beginning Value, yielding CAGR=(EVBV)1n−1\text{CAGR} = \left( \frac{\text{EV}}{\text{BV}} \right)^{\frac{1}{n}} - 1. The parameter nn is typically an integer for annual to reflect yearly periods, but it can be adjusted to fractional values for sub-annual or multi-period analyses, provided the supports consistent intervals. Input values for Beginning Value and Ending Value must be positive to ensure mathematical validity, as negative bases in the exponentiation could lead to complex numbers; however, the formula accommodates negative growth scenarios where Ending Value is less than Beginning Value, resulting in a CAGR less than zero. The embedded assumptions include annual compounding, meaning growth is calculated as if it occurs uniformly each year, and reinvestment of returns at the same compound rate throughout the entire period without interruptions or variable rates.

Derivation

The compound annual growth rate (CAGR) is derived from the fundamental principle of compound growth, where an initial value grows exponentially over multiple periods at a constant annual rate. Consider an initial value V0V_0 that grows to a final value VnV_n over nn periods through annual at rate rr. The relationship is expressed as Vn=V0(1+r)nV_n = V_0 (1 + r)^n, which captures the reinvestment of growth each period, leading to exponential accumulation. To solve for the constant annual rate rr, rearrange the equation by dividing both sides by V0V_0: VnV0=(1+r)n\frac{V_n}{V_0} = (1 + r)^n. Taking the nnth root of both sides yields 1+r=(VnV0)1/n1 + r = \left( \frac{V_n}{V_0} \right)^{1/n}, so r=(VnV0)1/n−1r = \left( \frac{V_n}{V_0} \right)^{1/n} - 1. This form annualizes the total growth factor by extracting the geometric mean of the annual multipliers, assuming constant growth; specifically, it represents the rate that, when compounded annually, equates the initial and final values over nn periods. This derivation connects directly to exponential growth models, where the exponent nn in (1+r)n(1 + r)^n reflects compounding's multiplicative nature, and the power of 1/n1/n effectively "annualizes" the cumulative effect by distributing it evenly across periods. In contrast, simple interest follows a linear derivation from S=P(1+rt)S = P(1 + rt), solving to r=S−PPtr = \frac{S - P}{Pt}, without reinvestment and thus understating multi-period growth. For edge cases, when n=1n = 1, the formula simplifies to r=V1V0−1r = \frac{V_1}{V_0} - 1, reducing to the simple periodic return. For continuous compounding, an approximation arises by taking the natural logarithm: r≈eln⁡(Vn/V0)n−1r \approx e^{\frac{\ln(V_n / V_0)}{n}} - 1, though this is not the standard discrete CAGR used in practice.

Examples

To illustrate the practical computation of the compound annual growth rate (CAGR), consider a basic example of an investment starting at $1,000 and growing to $1,500 over 3 years. The calculation uses the endpoints only: first, divide the ending value by the beginning value to get $1,500 / $1,000 = 1.5. Next, raise this ratio to the power of 1 divided by the number of years, or 1.51/31.5^{1/3}, which approximates 1.1447 (using a calculator or spreadsheet for the cube root). Subtract 1 from this result and multiply by 100 to express as a percentage: (1.1447 - 1) × 100 = 14.47%. This indicates an average annual growth rate of 14.47% over the period. For cases involving negative growth, suppose an asset's value declines from $2,000 to $1,200 over 5 years. Apply the same steps: $1,200 / $2,000 = 0.6, then 0.61/5≈0.90240.6^{1/5} \approx 0.9024. Subtracting 1 yields -0.0976, or -9.76% when expressed as a . This negative CAGR reflects an average annual contraction, useful for evaluating underperforming investments, though it smooths out yearly variations. In multi-period scenarios with intermediate data points, CAGR still relies solely on the starting and ending values, disregarding fluctuations in between. For instance, with annual values of $100 (year 0), $120 (year 1), $144 (year 2), and $172.8 (year 3), compute using the endpoints: $172.8 / $100 = 1.728, then 1.7281/31.728^{1/3} = 1.2 exactly. Subtract 1 to get 0.2, or 20%. The intermediate values, which show consistent 20% yearly growth here, are ignored in the formula, highlighting CAGR's focus on overall return. CAGR can also be used to project future values of an investment, assuming the historical growth rate continues. The formula for the projected value is Projected Value=Initial Investment×(1+CAGR)n,\text{Projected Value} = \text{Initial Investment} \times (1 + \text{CAGR})^n, where nn is the number of periods into the future. For example, for an initial investment of $100,000 with a projected CAGR of 40% over 5 years, the calculation is $100,000 \times (1.4)^5 \approx $100,000 \times 5.37824 = $537,824. To demonstrate the effects of long-term compounding, consider an initial investment of $1,000 growing at an 8.6% CAGR over 50 years, which yields approximately $62,000. This approach is commonly applied in finance to estimate total portfolio returns by annualizing the compounded growth, providing a smoothed measure of performance over time. CAGR can also be used to determine the required annual return to grow an investment to a target amount over a specific period. The formula for the required return rr is r=(target amountinitial amount)1/n−1,r = \left( \frac{\text{target amount}}{\text{initial amount}} \right)^{1/n} - 1, where nn is the number of years. For example, growing $1.3 million to $4.1 million in 20 years requires approximately 5.9% average annual return. Similarly, for instance, doubling a value, such as revenue, over approximately 4 years implies a CAGR of approximately 19%, calculated as (2)^{1/4} - 1 ≈ 0.1892 or 18.92%. To illustrate the application of compound growth calculations in revenue projections, consider a business starting with a present value of ₹27,548 crore and assuming an annual growth rate of 9% over 10 years. The step-by-step process involves multiplying the previous year's value by (1 + rate) for each year, which can be computed directly using the formula: Future Value = Present Value × (1 + rate)^n. Here, (1.09)^10 ≈ 2.3676, so ₹27,548 crore × 2.3676 ≈ ₹65,240 crore, representing approximately 2.4× growth. This method allows businesses to forecast future revenue assuming constant compounded growth. For computational efficiency, spreadsheets like simplify CAGR calculations with the formula =(ending_value / beginning_value)^(1 / number_of_years) - 1, entered directly into a cell (e.g., =(1500/1000)^(1/3)-1 for the basic example, yielding 0.1447 or 14.47%). This handles automatically via the POWER function if preferred: =POWER(ending_value / beginning_value, 1 / number_of_years) - 1. For non-integer periods, such as 2.5 years, substitute the fractional value directly (e.g., ^(1/2.5)), ensuring the time input accurately reflects completed periods from start to end. Always verify inputs for precision, as small errors in exponents can affect results.

Applications

Finance and Investments

In finance and investments, the compound annual growth rate (CAGR) serves as a key metric for evaluating and reporting the performance of various assets, including mutual funds, , and exchange-traded funds (ETFs). It provides a smoothed, annualized measure of returns that accounts for , enabling investors to assess long-term growth without the distortion of interim volatility. For instance, the historical CAGR of the index, representing large-cap U.S. equities, has averaged approximately 10.11% over the past century, illustrating its utility in benchmarking performance. Similarly, mutual funds and ETFs routinely disclose CAGR in prospectuses and performance summaries to highlight sustained growth, such as the annualized returns of equity funds over multi-year horizons. For example, for ETFs like the Vanguard S&P 500 ETF (VOO), average annual returns refer to the compound annual growth rate (CAGR), also known as annualized total returns, which include reinvested dividends and capital appreciation. Regulatory frameworks mandate the use of CAGR-like annualized returns in performance disclosures to ensure transparency and comparability for investors. Under the U.S. Securities and Exchange Commission's (SEC) Marketing Rule (Rule 206(4)-1), adopted in 2020 and effective from 2022, investment advisers must present performance results in advertisements over standardized 1-, 5-, and 10-year periods, with net performance shown alongside gross to reflect fees and expenses; these periods typically employ annualized s akin to CAGR for consistency. This requirement builds on earlier guidelines from the 1990s, where Form ADV Part 2A has required advisers to describe their performance calculation methods, often incorporating time-weighted annualized returns to avoid misleading claims. Additionally, rating agencies like Morningstar integrate CAGR into their fund evaluations; the firm's star ratings relies on compound annual total returns, calculated monthly and annualized over 3-, 5-, and 10-year periods, to rank funds against peers based on risk-adjusted performance. In portfolio analysis, CAGR facilitates comparisons across by providing a uniform growth metric, aiding decisions in diversification and allocation. For example, historical data from 1928 to 2023 shows U.S. equities achieving a return (equivalent to CAGR) of about 9.8%, compared to roughly 4.9% for long-term government bonds, underscoring equities' higher growth potential versus bonds' stability. This comparison is essential in constructing balanced portfolios, where CAGR helps quantify trade-offs in expected returns and risk. In , advisers project future portfolio values using conservative CAGR assumptions, such as 5% for balanced stock-bond mixes or 7% for equity-heavy allocations, to model sustainable withdrawal rates and longevity of savings. The projected future value is calculated as Future Value = Initial Value × (1 + CAGR)^n, where n is the number of years. For example, an initial investment of $100,000 growing at a 40% CAGR over 5 years yields $100,000 × (1.4)^5 ≈ $537,824. A specialized application is the time-weighted CAGR, which isolates investment manager performance by excluding the effects of client flows, such as deposits or withdrawals. This metric calculates the compounded growth rate across sub-periods defined by events, ensuring evaluations reflect market-driven returns rather than timing decisions. In client portfolio reporting, it is the standard under Global Investment Performance Standards (GIPS) and SEC guidelines, allowing advisers to demonstrate skill in asset selection and timing independent of external influences.

Business and Economics

In business contexts, the compound annual growth rate (CAGR) serves as a key metric for evaluating operational expansion, such as , , or customer base growth over multi-year periods. Software-as-a-Service (SaaS) companies frequently highlight ARR CAGR in earnings reports to illustrate recurring stability and scalability; for instance, as of 2025, reports a 25% CAGR in SaaS since 2019, reflecting . CAGR also plays a vital role in macroeconomic analysis by smoothing volatility in indicators like GDP, (where it represents the compound annual rate of price increase), or to reveal underlying long-term trends. For inflation, CAGR provides the smoothed annual growth rate of prices over a period; for example, a cumulative 65-66% rise over 20 years equates to an average annual rate of about 2.5%. The World Bank employs CAGR extensively in its development reports to quantify sustained economic progress; for example, it has tracked GDP CAGR since the 1970s to compare performance across countries, such as Georgia's 4.7% CAGR in GDP from 2010 to 2019 driven by services expansion. This approach aids policymakers in identifying structural shifts. Within , CAGR informs by quantifying a target company's historical growth trajectory, enabling acquirers to project synergies and value creation. EY research indicates that frequent M&A participants achieved higher enterprise value growth between 2015 and 2019, outperforming non-transactors with relatively stagnant growth. In sales forecasting and budgeting, firms use projected sales CAGR to allocate resources and set multi-year targets; McKinsey emphasizes integrating such projections into three- to seven-year financial plans to align with achievable growth amid uncertainties like market demand. A notable case is the tech industry boom from 1995 to 2005, where the sector's rapid adoption fueled significant revenue expansion, transforming global business models and contributing to broader economic surges as documented in sector analyses.

Limitations

Key Shortcomings

One significant shortcoming of the compound annual growth rate (CAGR) is its failure to account for volatility in returns, as it assumes a steady, compounded growth path over the period. By out fluctuations, CAGR can present a misleadingly optimistic view of performance, potentially masking substantial risks associated with erratic ups and downs in value. For instance, an experiencing extreme gains followed by sharp losses might yield a respectable CAGR while exposing investors to high volatility that standard deviation measures would highlight. This oversight implies consistent growth that did not occur, leading to underestimation of risk in . Another limitation arises from CAGR's reliance solely on endpoint values—the initial and final amounts—ignoring intermediate cash flows, timing of investments, or withdrawals. This endpoint bias distorts the true , as it treats all growth as if it compounded uniformly without considering opportunities for reinvestment or the impact of interim transactions. For example, additional contributions during the period would be erroneously attributed to the asset's growth rather than external inputs, inflating the perceived . Such an approach fails to reflect the real-world dynamics of portfolio management where timing and flows significantly influence outcomes. CAGR is also ill-suited for short time periods, typically fewer than three years, because the compounding assumption becomes unreliable with limited data points, amplifying the effects of outliers or irregular patterns. In these cases, the metric can produce inaccurate projections, as historical short-term results are less predictive of future annualized growth. Moreover, applying CAGR to non-annual data without proper adjustment further misleads, as it presupposes even intervals that may not align with the underlying periodicity. Finally, CAGR often overstates the consistency of growth by implying a stable trajectory where volatility or uneven progress prevailed. A classic illustration is an asset that doubles in value (100% gain) in one year followed by a 50% loss the next, resulting in a 0% CAGR over two years despite the evident and . This smoothing effect can deceive users into viewing irregular performance as reliably steady, particularly in volatile sectors like equities or startups.

Comparisons to Alternatives

The compound annual growth rate (CAGR) differs from the return in its treatment of compounding effects over multiple periods. While the arithmetic mean simply averages periodic returns, CAGR employs the to reflect the compounded growth rate, providing a more accurate measure of long-term performance. For instance, consider an starting at $1 that experiences a +50% return in the first period (reaching $1.50) followed by a -50% return (ending at $0.75), resulting in a total loss of 25%. The arithmetic mean return is 0% [(+50% + (-50%))/2], which misleadingly suggests no net change, whereas the CAGR is approximately -13.4% [($0.75/$1)^{1/2} - 1], correctly capturing the diminished value due to the asymmetric impact of changes on varying bases. This overstatement by the arithmetic mean arises because it ignores the multiplicative nature of returns, making CAGR preferable for assessing compounded growth in investments or economic metrics. In contrast to the (IRR), CAGR assumes steady growth between initial and final values without considering intermediate s. IRR, solved iteratively as the discount rate that sets the of all cash inflows and outflows to zero, incorporates the timing and magnitude of multiple cash flows, making it suitable for evaluating projects or investments with uneven contributions, such as or . For example, in a scenario with an initial outflow of $1,000 followed by inflows of $400, $500, $600, and $700 over four years, the IRR might be 36.4%, while CAGR based solely on endpoints ($2,200 final value) yields 21.7%, highlighting IRR's sensitivity to patterns. Thus, IRR is the better choice for irregular investment streams, whereas CAGR simplifies analysis for endpoint-focused evaluations like performance over a fixed horizon. CAGR also contrasts with the time-weighted (TWR), which both eliminate the distorting effects of external s but differ in . TWR calculates compounded returns by geometrically linking subperiod returns (e.g., quarterly) around dates, offering a detailed view of manager performance independent of timing decisions. In practice, TWR might show 15.3% annualized for a fund with varying quarterly gains, while CAGR, using only overall start and end values, provides a smoothed equivalent but lacks subperiod insights. CAGR's simplicity suits quick, holistic assessments of total growth, but TWR is preferred for precise in portfolios with frequent flows. Alternatives like the are chosen over CAGR when risk adjustment is essential, as CAGR measures raw growth without penalizing volatility. The , defined as (portfolio return - )/standard deviation of returns, evaluates excess return per unit of risk, aiding comparisons of similar assets like mutual funds. For non-compounding scenarios, such as linear trends in sales or population data without reinvestment effects, a simple arithmetic average suffices, avoiding CAGR's assumption of .

References

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