Worked Gordon Model Valuations

Using the Gordon Model to Value a Share

A common-stock valuation can depend on a small number of inputs, but a timing error in any one of them can materially change the result. The Gordon growth model values a share from its expected future dividends when those dividends are expected to grow at a constant rate indefinitely. A central practical question is whether the dividend provided is the most recently paid amount or the next amount expected.

Use $D_0$ for the dividend just paid, $D_1$ for the dividend expected one period from now, $g$ for the constant dividend growth rate per period, and $r$ for the required return per period. For annual dividends, $g$ and $r$ must both be annual rates. The model’s value is a per-share value at the valuation date.

Definition: Under the constant-growth Gordon model, the value $P_0$ of a common share is the present value of dividends expected to grow at a constant rate $g$ forever. If $D_1$ is the next expected dividend and the required return is $r>g$, then $P_0=D_1/(r-g)$. When the known dividend is the most recently paid dividend $D_0$, first calculate $D_1=D_0(1+g)$.

The required return $r$ is the return investors require for the share’s risk. The growth rate $g$ represents the assumed constant growth in dividends—not necessarily growth in earnings, sales, or the market price. The model is useful only when its assumptions are plausible enough for the valuation question. In particular, the same constant growth rate must continue indefinitely, and the required return must exceed that growth rate. If $r\leq g$, the infinite series used by the model does not converge to a finite positive value.

Why the Next Dividend Goes in the Numerator

The value at time 0 is based on dividends that will be paid after time 0. The dividend just paid, $D_0$, is already in the past; it is not one of the future cash flows being discounted. If dividends grow at rate $g$, the first future dividend is $D_1=D_0(1+g)$, followed by $D_2=D_1(1+g)$, and so on.

P_0=\frac{D_1}{1+r}+\frac{D_1(1+g)}{(1+r)^2}+\frac{D_1(1+g)^2}{(1+r)^3}+\cdots

Factor out $D_1/(1+r)$. Each successive term is the previous term multiplied by $q=(1+g)/(1+r)$. For nonnegative rates with $r>g$, $0<q<1$, so the infinite geometric series converges. Its sum is its first term divided by $1-q$:

P_0=\frac{D_1}{1+r}\cdot\frac{1}{1-\frac{1+g}{1+r}} =\frac{D_1}{1+r}\cdot\frac{1+r}{r-g} =\frac{D_1}{r-g}.

This derivation explains both the formula and its conditions: the numerator is the first dividend after the valuation date, and a finite value requires the discounted dividend series to converge. If the input is already $D_1$, do not grow it again. If the input is $D_0$, convert it to $D_1$ before applying the formula.

Worked Valuations

Worked Example: Starting With the Most Recently Paid Dividend

A hypothetical utility share has just paid an annual dividend of $1.80. Analysts expect dividends to grow at 4% per year indefinitely, and investors require a 10% annual return. Calculate the Gordon model value per share. All rates and dividends are annual.

Method and timing check. The stated $1.80 dividend has just been paid, so it is $D_0$, not $D_1$. The next expected dividend is:

D_1=D_0(1+g)=\$1.80(1.04)=\$1.872.

The required return exceeds growth: $r-g=0.10-0.04=0.06$. Thus the model’s convergence condition is met. Apply the Gordon formula:

P_0=\frac{D_1}{r-g}=\frac{\$1.872}{0.06}=\$31.20.

Check. The units are dollars per share: a dollar dividend divided by a dimensionless annual rate spread gives a dollar value. Using $D_0$ directly would give $30.00, understating the model value by $1.20 because it omits the growth from the dividend just paid to the next dividend. The correct value is $31.20 per share.

Worked Example: When the Next Dividend Is Given Directly

A hypothetical manufacturer is expected to pay an annual dividend of $2.16 one year from now. This is the next expected dividend. Dividends are forecast to grow at 4% annually thereafter, and the required annual return is 10%. If the observed market price is $35.50, calculate the model value and compare it with the market price.

Method and timing check. The stated $2.16 is already $D_1$; no growth adjustment is needed. The annual rate spread is:

$$ r-g=0.10-0.04=0.06. $$

Therefore:

P_0=\frac{\$2.16}{0.06}=\$36.00.

Compare the unrounded model value with the market price:

\$36.00-\$35.50=\$0.50.

The model value is $0.50 per share above the market price. This is a comparison under the stated model inputs, not a guarantee that the market price will rise or that the assumptions will prove accurate. Applying the 4% growth rate to $2.16 first would wrongly treat the already-forward dividend as $D_0$, counting one period of growth twice.

Worked Example: Comparing Two Shares With Different Inputs

An analyst evaluates two hypothetical shares. Share A just paid an annual dividend of $1.20; its expected constant growth rate is 4%, its required return is 10%, and its market price is $19.50. Share B just paid an annual dividend of $1.80; its expected constant growth rate is 2%, its required return is 8%, and its market price is $32.00. Calculate each model value and state how it compares with its market price. All rates and dividends are annual.

Method. For each share, first convert its most recently paid dividend to the next expected dividend, then use that share’s own required return and growth rate.

Share A. Its next expected dividend is:

D_{1,A}=\$1.20(1.04)=\$1.248.

Its rate spread is $0.10-0.04=0.06$, so:

P_{0,A}=\frac{\$1.248}{0.06}=\$20.80.

The model value exceeds Share A’s market price by $\$20.80-\$19.50=\$1.30$ per share.

Share B. Its next expected dividend is:

D_{1,B}=\$1.80(1.02)=\$1.836.

Its rate spread is $0.08-0.02=0.06$, so:

P_{0,B}=\frac{\$1.836}{0.06}=\$30.60.

The model value is below Share B’s market price by $\$32.00-\$30.60=\$1.40$ per share. The two comparisons use the same calculation method, but each share’s own dividend, growth rate, and required return. The correct model values are $20.80 for A and $30.60 for B.

Interpreting Inputs and Results

For annual dividends, use annual rates in decimal form in the denominator. For example, 8% is entered as 0.08, not 8. If a problem supplies a dividend and describes it as “just paid,” it is $D_0$. If it describes the dividend as “expected next year,” it is $D_1$. Pay attention to the valuation date: the numerator must be the first dividend that will be received after that date.

The denominator $r-g$ is the spread between the required return and dividend growth rate. It is not a discount factor, and it is not the required return by itself. A smaller positive spread produces a larger model value for a fixed $D_1$; this follows directly from dividing the same positive dividend by a smaller denominator. That mathematical sensitivity does not establish that either input is known with certainty.

When comparing a model value with a quoted market price, calculate the difference from the unrounded model result and round the difference only at the end. If displayed figures have already been rounded, their difference may not equal the difference calculated from the unrounded values. State whether a reported comparison is based on unrounded calculations or on displayed rounded prices.

Common Mistakes and Exam Traps

Using $D_0$ as though it were $D_1$. The formula needs the next dividend. If the most recently paid dividend is given, calculate $D_1=D_0(1+g)$ first.
Growing a dividend twice. If the problem explicitly provides the next expected dividend, use it directly as $D_1$. Multiplying it by $1+g$ again shifts the cash flow one period too far.
Ignoring the condition $r>g$. Check the spread before calculating. If $r\leq g$, the constant-growth infinite series does not produce a finite positive Gordon value.
Mixing periods. An annual growth rate cannot be subtracted from a monthly required return without converting them to consistent periods. This tutorial’s calculations use annual dividends and annual rates.
Rounding too early. Retain full precision in the dividend and valuation calculations, then round the final monetary amounts to the nearest cent. Do not use the difference between rounded display values as though it were an unrounded difference.

Key takeaway: Identify whether the dividend supplied is $D_0$ or $D_1$, use matching per-period rates, verify $r>g$, and then calculate $P_0=D_1/(r-g)$. The result is a model-based present value under the constant-growth assumptions, not a certain future market price.

Check Your Understanding

For each problem, use annual dividends and annual rates unless the question states otherwise. Round monetary answers to the nearest cent.

A hypothetical retailer has just paid a dividend of $2.00 per share. Dividends are expected to grow at 3% annually, and the required annual return is 8%. Find the Gordon model value. (Answer: $41.20 per share)
A hypothetical technology firm is expected to pay a dividend of $1.75 one year from now. Its constant annual dividend growth rate is 2.5%, and its required annual return is 7.5%. Find the value; do not treat the stated dividend as the one just paid. (Answer: $35.00 per share)
A hypothetical transport company just paid an annual dividend of $0.90. Expected constant growth is 6%, and the required return is 12%. If its market price is $16.20, find the model value and the model value minus market price. (Answer: $15.90 per share; model value minus market price is -$0.30 per share)
A hypothetical insurer just paid an annual dividend of $3.00. It is expected to grow at 4% annually forever, and the required annual return is 10%. Find the next expected dividend and the Gordon value. (Answer: $D_1=\$3.12$; $52.00 per share)
A hypothetical share just paid an annual dividend of $1.60. Growth is expected to be 5% annually, and the required return is 9%. A candidate uses $1.60/(0.09-0.05)=\$40.00$. Identify the timing error and calculate the correct value. (Answer: The candidate used $D_0$ instead of $D_1$; $D_1=\$1.68$ and the correct value is $42.00 per share)

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