CAGR Equation, Derivation & Rearrangements

Core idea

Overview

The Compound Annual Growth Rate (CAGR) represents the mean annual growth rate of an investment over a specified period of time longer than one year. It acts as a smoothing mechanism that describes the rate at which an investment would have grown if it had grown at a steady rate on a compounded basis.

When to use: Use CAGR when you need to compare the performance of various investments over the same time horizon, especially when those investments have volatile year-to-year returns. It is the standard metric for reporting the growth of portfolios, business revenues, or market shares over multi-year periods.

Why it matters: CAGR provides a single, consistent figure that eliminates the 'noise' of market volatility, allowing for a direct 'apples-to-apples' comparison between different asset classes. It helps investors understand the geometric progression of their wealth, which is more accurate for long-term planning than a simple arithmetic average.

Symbols

Variables

FV = Final Value, PV = Initial Value, n = Periods (Years), CAGR = CAGR

F V

Final Value

£

P V

Initial Value

£

n

Periods (Years)

y e a r s

C A GR

CAGR

Walkthrough

Derivation

Formula: Compound Annual Growth Rate (CAGR)

CAGR is the constant annual rate that would grow a present value to a future value over n years, smoothing out year-to-year fluctuations.

Growth compounds at the same rate every year.
No withdrawals or additions are made during the period.

1

Find the total growth factor:

Divide the final value by the starting value to see how many times the investment has grown overall.

Total factor = \frac{F V}{P V}

2

Take the nth root and subtract 1:

The nth root distributes total growth evenly across n years. Multiply by 100 to express as a percentage. Example: £1,000 → £2,000 in 5 years gives (2)^(1/5) − 1 ≈ 14.87%.

C A GR = (\frac{F V}{P V})^{\frac{1}{n}} - 1

Result

C A GR = (\frac{F V}{P V})^{\frac{1}{n}} - 1

Source: GCSE Finance / Business — Growth & Investment

Free formulas

Rearrangements

Solve for $C A GR$

Make CAGR the subject

([\frac{F V}{P V}]^{\frac{1}{n}} - 1) \times 100

The Compound Annual Growth Rate (CAGR) formula already expresses CAGR as the subject. This process simply identifies the expression for CAGR.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph of CAGR (cagr) against the independent variable Future Value (FV) follows a power law shape. As FV increases, the CAGR rises at a decreasing rate, creating a concave curve. The graph crosses the x-axis when FV equals the Present Value (PV) and has a y-intercept at -100.

Graph type: power_law

Why it behaves this way

Intuition

Imagine a smooth, continuous curve representing the steady growth of an investment from its initial to its final value, even if its actual path was volatile

FV

The final value of the investment or asset at the end of the specified period.

Represents the accumulated wealth or size after growth.

PV

The initial value of the investment or asset at the beginning of the specified period.

Represents the starting capital or size.

n

The number of years over which the investment has grown.

Defines the duration for which the growth is being averaged.

Signs and relationships

(1/n) exponent: This exponent takes the nth root of the total growth factor (FV/PV), effectively annualizing the total growth to find the average annual multiplicative factor.
- 1: Subtracting 1 converts the annual growth factor into a pure annual growth rate, isolating the percentage increase beyond the original principal.
x 100: Multiplies the decimal growth rate by 100 to express it as a percentage, which is the standard convention for reporting growth rates.

Free study cues

Insight

Canonical usage

The CAGR is calculated using a dimensionless ratio of currency values over a time period (typically years) and is expressed as a percentage.

Common confusion

Using the number of data points instead of the number of intervals for 'n'. For example, from the start of 2018 to the end of 2022 is 5 years (n=5), even though there are 6 years of annual data points listed.

Dimension note

The ratio FV/PV is a dimensionless quantity. The resulting CAGR is a rate expressed as a percentage, representing the geometric mean growth per year.

Unit systems

$F V$ Currency (e.g., £, $, €) · Future Value. Must be in the same currency unit as PV.

$P V$ Currency (e.g., £, $, €) · Present Value. Must be in the same currency unit as FV.

$n$ years · The number of years over which the growth occurred.

One free problem

Practice Problem

An investor buys shares in a tech company for 1,000 and sells them exactly 5 years later for 1,610.51. What is the Compound Annual Growth Rate (CAGR) for this investment?

Initial Value1000 £

Final Value1610.51 £

Periods (Years)5 years

Solve for: $c a g r$

Hint: Divide the future value by the present value before calculating the 5th root.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

A fund growing from £1000 to £2000 in 5 years.

Study smarter

Tips

Always ensure the time period 'n' is calculated as the total number of years between the start and end dates.
Remember that CAGR assumes all returns are reinvested throughout the period.
Note that CAGR does not reflect the risk or volatility experienced during the investment term.

Avoid these traps

Common Mistakes

Using percentage instead of decimal for calculation.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

CAGR is the constant annual rate that would grow a present value to a future value over n years, smoothing out year-to-year fluctuations.

Use CAGR when you need to compare the performance of various investments over the same time horizon, especially when those investments have volatile year-to-year returns. It is the standard metric for reporting the growth of portfolios, business revenues, or market shares over multi-year periods.

CAGR provides a single, consistent figure that eliminates the 'noise' of market volatility, allowing for a direct 'apples-to-apples' comparison between different asset classes. It helps investors understand the geometric progression of their wealth, which is more accurate for long-term planning than a simple arithmetic average.

Using percentage instead of decimal for calculation.