Gordon Growth Model (GGM) Equation, Derivation & Rearrangements

Core idea

Overview

Developed by Myron J. Gordon and Eli Shapiro in the 1950s, the Gordon Growth Model (GGM) is a fundamental stock valuation method that calculates a stock's intrinsic value by discounting its expected future dividends. It assumes that dividends will grow at a constant rate indefinitely and that the required rate of return is greater than the dividend growth rate.

When to use: The GGM is best applied to mature, stable, dividend-paying companies with predictable and constant dividend growth, such as utilities or consumer staples. It is also frequently used to calculate the terminal value in multi-stage discounted cash flow (DCF) models.

Why it matters: This model is crucial for investors to estimate the intrinsic value of a stock, helping them determine if a stock is undervalued or overvalued relative to its market price. It provides a foundational understanding of how future dividends contribute to a stock's current valuation and serves as a benchmark for investment decisions.

Symbols

Variables

P = Current Stock Price / Intrinsic Value, D_1 = Expected Dividend Per Share Next Year, r = Required Rate of Return / Cost of Equity, g = Constant Dividend Growth Rate

P

Current Stock Price / Intrinsic Value

1/ t im e

D_{1}

Expected Dividend Per Share Next Year

u ni tl ess

r

Required Rate of Return / Cost of Equity

k J / m o l

g

Constant Dividend Growth Rate

s^{-} 1

Walkthrough

Derivation

Derivation of Gordon Growth Model (GGM)

The Gordon Growth Model (GGM) is a widely used dividend discount model that values a stock based on the present value of its future dividends, assuming a constant growth rate in perpetuity. This derivation shows how the GGM formula is obtained from the general Dividend Discount Model by applying the properties of an infinite geometric series.

Dividends are expected to grow at a constant rate (g) indefinitely.
The required rate of return (r) is constant.
The required rate of return (r) must be greater than the dividend growth rate (g) (i.e., r > g). This condition ensures that the sum of the infinite series converges to a finite value.

1

Start with the Dividend Discount Model (DDM)

The fundamental principle of stock valuation states that the intrinsic value of a stock today ($P_0$) is the present value of all its future expected dividends ($D_t$), discounted at the investor's required rate of return (r).

P_{0} = t = 1 \sum \infty \frac{D _{t}}{( 1 + r ) ^{t}}

Note: This is the general form of the Dividend Discount Model, which serves as the foundation for many dividend-based valuation models.

2

Define Future Dividends with Constant Growth

The Gordon Growth Model assumes that dividends grow at a constant rate 'g' indefinitely. If $D_1$ is the dividend expected at the end of the first period, then the dividend at any future period 't' can be expressed as $D_t = D_1(1+g)^{t-1}$.

D_{t} = D_{1} (1 + g)^{t - 1}

Note: $D_1$ represents the dividend expected one period from now. If $D_0$ (the current dividend) is given, then $D_1 = D_0(1+g)$.

3

Substitute Constant Growth Dividends into the DDM

By substituting the expression for $D_t$ from the previous step into the Dividend Discount Model, we expand the sum to show the present value of each future dividend growing at rate 'g'.

P_{0} = \frac{D _{1}}{( 1 + r ) ^{1}} + \frac{D _{1} ( 1 + g )}{( 1 + r ) ^{2}} + \frac{D _{1} ( 1 + g ) ^{2}}{( 1 + r ) ^{3}} + \dots

Note: This step clearly illustrates the infinite stream of growing dividends being discounted back to the present.

4

Rearrange and Identify as an Infinite Geometric Series

We factor out $D_1$ and rearrange the terms to reveal the structure of an infinite geometric series. The first term of this series is $a = \frac{1}{1+r}$ and the common ratio is $k = \frac{1+g}{1+r}$.

P_{0} = D_{1} [\frac{1}{1 + r} + \frac{1 + g}{( 1 + r ) ^{2}} + \frac{( 1 + g ) ^{2}}{( 1 + r ) ^{3}} + \dots] P_{0} = D_{1} [\frac{1}{1 + r} + \frac{1}{1 + r} (\frac{1 + g}{1 + r}) + \frac{1}{1 + r} (\frac{1 + g}{1 + r})^{2} + \dots]

Note: Recognizing this pattern is crucial for simplifying the infinite sum. The condition $r > g$ ensures that the common ratio $k < 1$, which is necessary for the series to converge.

5

Apply the Formula for the Sum of an Infinite Geometric Series

The sum (S) of an infinite geometric series with first term 'a' and common ratio 'k' (where $|k| < 1$) is given by the formula $S = \frac{a}{1-k}$. We substitute our identified 'a' and 'k' into this formula.

S = \frac{a}{1 - k} for ∣ k ∣ < 1 P_{0} = \frac{\frac{D _{1}}{1 + r}}{1 - \frac{1 + g}{1 + r}}

Note: The condition $r > g$ is vital here, as it ensures that $k = \frac{1+g}{1+r} < 1$, allowing the series to converge to a finite value.

6

Simplify to the Gordon Growth Model Formula

By simplifying the complex fraction, we cancel out the $(1+r)$ terms in the numerator and denominator, arriving at the final, concise formula for the Gordon Growth Model.

P_{0} = \frac{\frac{D _{1}}{1 + r}}{\frac{( 1 + r ) - ( 1 + g )}{1 + r}} P_{0} = \frac{\frac{D _{1}}{1 + r}}{\frac{1 + r - 1 - g}{1 + r}} P_{0} = \frac{\frac{D _{1}}{1 + r}}{\frac{r - g}{1 + r}} P_{0} = \frac{D _{1}}{1 + r} \cdot \frac{1 + r}{r - g} P_{0} = \frac{D _{1}}{r - g}

Note: This final formula is elegant and widely used for valuing companies with stable, perpetually growing dividends.

Result

P_{0} = \frac{\frac{D _{1}}{1 + r}}{\frac{( 1 + r ) - ( 1 + g )}{1 + r}} P_{0} = \frac{\frac{D _{1}}{1 + r}}{\frac{1 + r - 1 - g}{1 + r}} P_{0} = \frac{\frac{D _{1}}{1 + r}}{\frac{r - g}{1 + r}} P_{0} = \frac{D _{1}}{1 + r} \cdot \frac{1 + r}{r - g} P_{0} = \frac{D _{1}}{r - g}

Source: Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance. McGraw-Hill Education.

Free formulas

Rearrangements

Solve for P

Make P the subject

P = \frac{D _{1}}{r - g}

This is the standard form of the Gordon Growth Model used to calculate the intrinsic value of a stock.

Difficulty: 1/5

Solve for D_1

Make D_1 the subject

D_{1} = P (r - g)

Solve for the expected dividend in the next period using the stock price and discount factors.

Difficulty: 2/5

Solve for r

Make r the subject

r = \frac{D _{1}}{P} + g

Solve for the required rate of return using the stock price and growth parameters.

Difficulty: 3/5

Solve for g

Make g the subject

g = r - \frac{D _{1}}{P}

Solve for the constant dividend growth rate using the stock price and required return.

Difficulty: 3/5

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

Visual intuition

Graph

Graph preview unavailable for this formula.

Why it behaves this way

Intuition

Imagine a 'magic money tree' that continuously produces fruit (dividends). The first fruit (D₁) will appear next year. Each year after that, the tree produces more fruit, and the amount grows at a constant rate (g). You want to buy this tree today. The price (P) you are willing to pay is the present value of all the fruit it will ever produce. Your required rate of return (r) is how much you value future fruit less than current fruit. The difference (r - g) in the denominator represents the 'net' discount rate, where the growth of the fruit partially offsets the effect of discounting.

P

Current Stock Price

This is the fair market value of the stock today, representing what an investor should be willing to pay for the future stream of growing dividends.

D_{1}

Expected Dividend in the Next Period

This is the dividend payment an investor expects to receive one period (e.g., one year) from now. It's the starting point for the perpetually growing dividend stream.

r

Required Rate of Return

This is the minimum annual return an investor expects to earn from the stock, considering its risk. It's the discount rate used to bring future dividends back to their present value.

g

Constant Dividend Growth Rate

This is the constant annual rate at which the company's dividends are expected to grow indefinitely into the future. It reflects the company's long-term earnings growth and payout policy.

Signs and relationships

r - g: The difference between the required rate of return (r) and the dividend growth rate (g) is crucial. For the model to yield a finite and positive stock price, 'r' must be strictly greater than 'g' (r > g). If 'r' were equal to 'g', the denominator would be zero, implying an infinite stock price, which is economically illogical. If 'r' were less than 'g', the denominator would be negative, implying a negative stock price, which is also nonsensical. This condition ensures that the present value of future dividends converges to a finite, positive value. The subtraction itself shows that the growth in dividends (g) effectively reduces the net discount rate applied to the future cash flows, making the stock more valuable.

One free problem

Practice Problem

A company is expected to pay a dividend of $1.50 next year (D1). The required rate of return for the stock is 12%, and dividends are expected to grow at a constant rate of 5% indefinitely. What is the intrinsic value of the stock according to the Gordon Growth Model?

D11.5

Required Rate of Return / Cost of Equity0.12 kJ/mol

Constant Dividend Growth Rate0.05 s^-1

Solve for: P

Hint: Apply the formula P = D1 / (r - g) directly.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

An investor is evaluating a mature utility company, 'Stable Power Co.', known for its consistent dividend payments. Stable Power Co. just paid a dividend of $2.00 per share (D0), and analysts expect its dividends to grow at a steady rate of 3% annually. If the investor requires a 10% rate of return on their investment, the GGM can be used to estimate the intrinsic value of Stable Power Co.'s stock.

Study smarter

Tips

Always ensure that the required rate of return (r) is strictly greater than the dividend growth rate (g) to avoid meaningless results.
Remember that D1 represents the *next year's* expected dividend, not the current or last paid dividend (D0). If D0 is given, calculate D1 as D0 * (1 + g).
The GGM is highly sensitive to changes in its inputs; even small adjustments to 'r' or 'g' can significantly alter the valuation.
Use the GGM in conjunction with other valuation methods, especially for companies that don't perfectly fit its strict assumptions.

Avoid these traps

Common Mistakes

Assuming a truly constant dividend growth rate in perpetuity, which is unrealistic for most companies over very long periods.
Applying the model to companies that do not pay dividends or have erratic dividend growth patterns.
Using a dividend growth rate (g) that is equal to or greater than the required rate of return (r), which leads to a negative or infinite stock price, rendering the model invalid.
Confusing the current dividend (D0) with the next period's expected dividend (D1) in the numerator of the formula.
Over-relying on the model's output without considering its inherent assumptions and limitations, particularly its sensitivity to input variables.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Gordon Growth Model (GGM) is a widely used dividend discount model that values a stock based on the present value of its future dividends, assuming a constant growth rate in perpetuity. This derivation shows how the GGM formula is obtained from the general Dividend Discount Model by applying the properties of an infinite geometric series.

The GGM is best applied to mature, stable, dividend-paying companies with predictable and constant dividend growth, such as utilities or consumer staples. It is also frequently used to calculate the terminal value in multi-stage discounted cash flow (DCF) models.

This model is crucial for investors to estimate the intrinsic value of a stock, helping them determine if a stock is undervalued or overvalued relative to its market price. It provides a foundational understanding of how future dividends contribute to a stock's current valuation and serves as a benchmark for investment decisions.

Assuming a truly constant dividend growth rate in perpetuity, which is unrealistic for most companies over very long periods. Applying the model to companies that do not pay dividends or have erratic dividend growth patterns. Using a dividend growth rate (g) that is equal to or greater than the required rate of return (r), which leads to a negative or infinite stock price, rendering the model invalid. Confusing the current dividend (D0) with the next period's expected dividend (D1) in the numerator of the formula. Over-relying on the model's output without considering its inherent assumptions and limitations, particularly its sensitivity to input variables.

An investor is evaluating a mature utility company, 'Stable Power Co.', known for its consistent dividend payments. Stable Power Co. just paid a dividend of $2.00 per share (D0), and analysts expect its dividends to grow at a steady rate of 3% annually. If the investor requires a 10% rate of return on their investment, the GGM can be used to estimate the intrinsic value of Stable Power Co.'s stock.

Always ensure that the required rate of return (r) is strictly greater than the dividend growth rate (g) to avoid meaningless results. Remember that D1 represents the *next year's* expected dividend, not the current or last paid dividend (D0). If D0 is given, calculate D1 as D0 * (1 + g). The GGM is highly sensitive to changes in its inputs; even small adjustments to 'r' or 'g' can significantly alter the valuation. Use the GGM in conjunction with other valuation methods, especially for companies that don't perfectly fit its strict assumptions.

References

Sources

Gordon, M.J., & Shapiro, E. (1956). Capital Equipment Analysis: The Required Rate of Profit. Management Science, 3(1), 102-110.
Ross, S., Westerfield, R., & Jordan, B. (2010). Essentials of Corporate Finance (7th ed.). New York, NY: McGraw-Hill, Irwin.
Brealey, R., Myers, S., & Allen, F. (2010). Principles of Corporate Finance. Maidenhead, Berkshire: McGraw-Hill.
Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.). John Wiley & Sons.
Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance. McGraw-Hill Education.

Gordon Growth Model (GGM)

Overview

Variables

Derivation

Start with the Dividend Discount Model (DDM)

Define Future Dividends with Constant Growth

Substitute Constant Growth Dividends into the DDM

Rearrange and Identify as an Infinite Geometric Series

Apply the Formula for the Sum of an Infinite Geometric Series

Simplify to the Gordon Growth Model Formula

Rearrangements

Graph

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Two-Stage Dividend Discount Model

Capital Asset Pricing Model (CAPM)

Frequently Asked Questions

Sources