Present Value of a Perpetuity with Growth

Core idea

Overview

The Present Value of a Perpetuity with Growth formula, often called the Gordon Growth Model, is a fundamental tool in finance for valuing assets that are expected to generate a stream of cash flows indefinitely, with each cash flow growing at a constant rate. It discounts these future growing cash flows back to their present value, providing a single figure that represents the current worth of that future income stream. This model is particularly useful for valuing stocks, real estate, or businesses that are assumed to have perpetual life and stable growth.

When to use: Apply this formula when valuing an asset that is expected to generate cash flows indefinitely, and these cash flows are projected to grow at a constant, stable rate. It's crucial that the discount rate (r) is greater than the growth rate (g) for the formula to yield a meaningful, finite present value. This model is commonly used in equity valuation, particularly for mature companies with predictable growth.

Why it matters: This equation is vital for investors and financial analysts as it provides a theoretical framework for determining the intrinsic value of income-generating assets. It helps in making investment decisions, assessing the fairness of asset prices, and understanding the impact of growth rates and discount rates on valuation. Its application extends to corporate finance for capital budgeting and strategic planning.

Symbols

Variables

$C_{1}$ = Cash Flow in Period 1, r = Discount Rate, g = Growth Rate, PV = Present Value

C_{1}

Cash Flow in Period 1

$

r

Discount Rate

%

g

Growth Rate

%

PV

Present Value

$

Walkthrough

Derivation

Formula: Present Value of a Perpetuity with Growth

Derives the formula for the present value of an infinite stream of cash flows growing at a constant rate.

Cash flows grow at a constant rate (g) indefinitely.
The discount rate (r) is constant and greater than the growth rate (g).
The first cash flow (C1) occurs at the end of the first period.

1

Define Present Value as Sum of Discounted Cash Flows:

The present value (PV) is the sum of all future cash flows, each discounted back to the present. C1 is the cash flow in the first period, and it grows by (1+g) each subsequent period.

P V = \frac{C _{1}}{( 1 + r ) ^{1}} + \frac{C _{1} ( 1 + g )}{( 1 + r ) ^{2}} + \frac{C _{1} ( 1 + g ) ^{2}}{( 1 + r ) ^{3}} + \dots

2

Factor and Recognize as Geometric Series:

Factor out C1. The expression in the brackets is an infinite geometric series where the first term is a = 1/(1+r) and the common ratio is x = (1+g)/(1+r).

P V = C_{1} [\frac{1}{1 + r} + \frac{1 + g}{( 1 + r ) ^{2}} + \frac{( 1 + g ) ^{2}}{( 1 + r ) ^{3}} + \dots]

3

Apply Infinite Geometric Series Sum Formula:

The sum of an infinite geometric series a + ax + ax^2 + ... is a / (1-x), provided |x| < 1. Here, the first term is C1/(1+r) and the common ratio is (1+g)/(1+r). The condition |x|<1 implies r > g.

P V = \frac{C _{1} \cdot \frac{1}{1 + r}}{1 - \frac{1 + g}{1 + r}}

4

Simplify the Expression:

Simplify the denominator by finding a common denominator. The (1+r) terms in the numerator and denominator of the main fraction cancel out.

P V = \frac{C _{1} / ( 1 + r )}{\frac{( 1 + r ) - ( 1 + g )}{1 + r}} = \frac{C _{1} / ( 1 + r )}{\frac{r - g}{1 + r}}

5

Final Formula:

This is the simplified formula for the present value of a perpetuity with growth, also known as the Gordon Growth Model.

P V = \frac{C _{1}}{r - g}

Note: This formula is only valid when the discount rate (r) is strictly greater than the growth rate (g).

Result

P V = \frac{C _{1}}{r - g}

Source: Brealey, Myers, and Allen, Principles of Corporate Finance, Chapter 2: Present Value and the Opportunity Cost of Capital

Free formulas

Rearrangements

Solve for $C_{1}$

Present Value of a Perpetuity with Growth: Make C1 the subject

C_{1} = P V \cdot (r - g)

To make $C_{1}$ (Cash Flow in Period 1) the subject, multiply both sides of the equation by $(r - g)$ .

Difficulty: 2/5

Solve for $r$

Present Value of a Perpetuity with Growth: Make r the subject

r = \frac{C _{1}}{P V} + g

To make $r$ (Discount Rate) the subject, first isolate the $(r - g)$ term, then add $g$ to both sides.

Difficulty: 3/5

Solve for $g$

Present Value of a Perpetuity with Growth: Make g the subject

g = r - \frac{C _{1}}{P V}

To make $g$ (Growth Rate) the subject, first isolate the $(r - g)$ term, then subtract $r$ and multiply by -1, or rearrange the terms.

Difficulty: 3/5

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

Visual intuition

Graph

The graph forms a hyperbola because the discount rate appears in the denominator, meaning the present value decreases as the discount rate increases. For an economics student, this shape illustrates that higher discount rates significantly reduce the current worth of future cash flows, while very small discount rates cause the present value to rise sharply. The most important feature of this curve is that the present value never reaches zero, reflecting that even with a high discount rate, an infinite stream of growing cash flows retains some positive value.

Graph type: hyperbolic

Why it behaves this way

Intuition

The formula sums an infinite series of future cash flows, each growing by 'g' but discounted by 'r', where the net effect (r-g) ensures the sum converges to a finite present value, like a diminishing but never-ending

PV

The current monetary worth of an infinite stream of future cash flows.

How much a perpetual, growing income stream is worth today. A higher PV means the asset is more valuable now.

C_{1}

The expected cash flow received at the end of the first period.

The initial payment in the infinite series. A larger

C_{1}

directly increases the present value.

r

The discount rate, representing the required rate of return or opportunity cost of capital.

The rate at which future money is devalued to its present equivalent. A higher 'r' reduces the present value, reflecting higher risk or better alternative investments.

g

The constant rate at which future cash flows are expected to grow.

How fast the income stream increases each period. A higher 'g' increases the present value, as future payments are larger.

Signs and relationships

r - g: The difference 'r - g' represents the net effective discount rate. The growth rate 'g' reduces the impact of the discount rate 'r', making future cash flows relatively more valuable.

Free study cues

Insight

Canonical usage

This equation requires consistent monetary units for cash flows and present value, and consistent dimensionless units (decimals) for discount and growth rates, all over the same time period.

Common confusion

The most common confusion involves using percentage values directly in the formula for 'r' and 'g' instead of converting them to decimals. Another frequent mistake is not ensuring that the discount rate (r)

Dimension note

The discount rate (r) and growth rate (g) are dimensionless ratios, typically expressed as decimals in calculations. The present value (PV) and cash flow ( $C_{1}$ ) are expressed in monetary units.

Unit systems

PVCurrency (e.g., USD, EUR) - Represents the present value of the future cash flow stream. Must be in the same currency as C_1.

$C_{1}$ Currency (e.g., USD, EUR) - The expected cash flow to be received at the end of the first period. Must be in the same currency as PV.

$r$ Dimensionless (decimal) - The required rate of return or discount rate. Must be expressed as a decimal (e.g., 0.05 for 5%) for calculation and correspond to the same time period as C_1 and g.

$g$ Dimensionless (decimal) - The constant growth rate of the cash flows. Must be expressed as a decimal (e.g., 0.02 for 2%) for calculation and correspond to the same time period as C_1 and r.

One free problem

Practice Problem

A company is expected to pay a dividend of $100 next year, and these dividends are projected to grow at a constant rate of 5% indefinitely. If the required rate of return for this stock is 10%, what is the present value of this perpetuity?

Cash Flow in Period 1100 $

Discount Rate0.1 %

Growth Rate0.05 %

Solve for: PV

Hint: Ensure the discount rate is greater than the growth rate.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In valuing a dividend-paying stock with a stable, Present Value of a Perpetuity with Growth is used to calculate Present Value from Cash Flow in Period 1, Discount Rate, and Growth Rate. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Study smarter

Tips

Ensure r > g; otherwise, the formula results in an infinite or negative value, indicating the model is not applicable.
C1 represents the cash flow at the end of the first period, not the current period (C0).
Both r and g should be expressed as decimals (e.g., 5% as 0.05).
The model assumes constant growth and an infinite life, which are strong assumptions; use with caution and consider other valuation methods.

Avoid these traps

Common Mistakes

Using C0 instead of C1 for the initial cash flow.
Applying the formula when r is less than or equal to g.
Not converting percentages to decimals for r and g before calculation.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Derives the formula for the present value of an infinite stream of cash flows growing at a constant rate.

Apply this formula when valuing an asset that is expected to generate cash flows indefinitely, and these cash flows are projected to grow at a constant, stable rate. It's crucial that the discount rate (r) is greater than the growth rate (g) for the formula to yield a meaningful, finite present value. This model is commonly used in equity valuation, particularly for mature companies with predictable growth.

This equation is vital for investors and financial analysts as it provides a theoretical framework for determining the intrinsic value of income-generating assets. It helps in making investment decisions, assessing the fairness of asset prices, and understanding the impact of growth rates and discount rates on valuation. Its application extends to corporate finance for capital budgeting and strategic planning.

Using C0 instead of C1 for the initial cash flow. Applying the formula when r is less than or equal to g. Not converting percentages to decimals for r and g before calculation.

In valuing a dividend-paying stock with a stable, Present Value of a Perpetuity with Growth is used to calculate Present Value from Cash Flow in Period 1, Discount Rate, and Growth Rate. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Ensure r > g; otherwise, the formula results in an infinite or negative value, indicating the model is not applicable. C1 represents the cash flow at the end of the first period, not the current period (C0). Both r and g should be expressed as decimals (e.g., 5% as 0.05). The model assumes constant growth and an infinite life, which are strong assumptions; use with caution and consider other valuation methods.

References

Sources

Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
Wikipedia: Gordon growth model
Principles of Corporate Finance by Brealey, Myers, Allen
Investments by Bodie, Kane, Marcus
Gordon growth model (Wikipedia article)
Bodie, Zvi, Alex Kane, and Alan J. Marcus. Investments. McGraw-Hill Education.
Brealey, Richard A., Stewart C. Myers, and Franklin Allen. Principles of Corporate Finance. McGraw-Hill Education.
Ross, Stephen A., Randolph W. Westerfield, and Jeffrey Jaffe. Corporate Finance. McGraw-Hill Education.

Overview

Variables

Derivation

Define Present Value as Sum of Discounted Cash Flows:

Factor and Recognize as Geometric Series:

Apply Infinite Geometric Series Sum Formula:

Simplify the Expression:

Final Formula:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Present Value of a Perpetuity

Dividend Discount Model

Future Value of an Annuity

Frequently Asked Questions

Sources