Deriving the Gordon Growth Formula

From Growing Dividends to a Stock Value

A constant-growth stock valuation can be understood as the sum of the present values of future dividends. The key step is recognizing that, when each dividend grows at the same rate and each is discounted at the same required return, the discounted dividends form a geometric sequence. This derivation makes both the Gordon formula and its convergence condition explicit.

Assume dividends are paid at the end of each year. Let $D_1$ be the dividend due one year from now, $g$ the constant annual dividend growth rate, and $r$ the required annual return. The rates must be measured over the same annual period. In this setting, $P_0$ denotes the value at time 0 of the future dividend stream.

Definition: Under the constant-growth dividend model, the dividend in year $t$ is $D_t=D_1(1+g)^{t-1}$, for $t=1,2,\ldots$. The value $P_0$ is the sum of the present values of all these dividends, discounted at the required return $r$.

The model used in the derivation requires an indefinitely continuing dividend stream, a constant growth rate, and a constant required return. Assume $1+r>0$ and $1+g>0$, so the discount and growth factors are positive. The derivation will establish the additional condition needed for that infinite stream to have a finite present value. These conditions concern the mathematical setup; a formula result is not meaningful if the stream or rates do not fit it.

The Geometric-Series Derivation

The first dividend is $D_1$. Each later dividend is the previous one multiplied by $1+g$, while its discount factor includes one additional year of required return. Therefore, the present value of each dividend is the preceding dividend’s present value multiplied by the same ratio, $q=(1+g)/(1+r)$.

Write the dividend stream.
The dividends in years 1, 2, 3, and onward are $D_1,\ D_1(1+g),\ D_1(1+g)^2,\ldots$.

Discount each dividend.
The year-$t$ dividend is received $t$ years from now, so its present value is $D_t/(1+r)^t$.

Identify the common ratio.
Each successive discounted dividend is multiplied by $q=(1+g)/(1+r)$.

Take the infinite-series limit.
Use the geometric-series limit only if the discounted terms shrink sufficiently for the series to converge.

P_0=\sum_{t=1}^{\infty}\frac{D_t}{(1+r)^t} =\frac{D_1}{1+r}\left[1+\frac{1+g}{1+r}+\left(\frac{1+g}{1+r}\right)^2+\cdots\right]

For a finite geometric series with first term $a$ and common ratio $q\ne1$, the sum of the first $n$ terms is $a(1-q^n)/(1-q)$. Here, $a=D_1/(1+r)$. If $0\le q<1$, then $q^n$ approaches 0 as $n$ grows. The infinite sum is consequently $a/(1-q)$. Substituting the values of $a$ and $q$ gives the result.

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}

Theorem: If dividends grow at a constant rate $g$, are paid annually beginning one year from now, and are discounted at a constant annual required return $r$, then their infinite present value is $P_0=D_1/(r-g)$ when $r>g$, under the positive-factor assumptions above.

The cancellation in the derivation is valid because the series converges. Under the stated positive-factor assumptions, $q<1$ exactly when $(1+g)/(1+r)<1$, which is equivalent to $g<r$. If $r=g$, then $q=1$: every discounted dividend has the same positive value, and adding infinitely many such terms does not produce a finite sum. If $r<g$, then $q>1$: discounted dividends do not shrink; they grow. The infinite series again has no finite sum. In either case, inserting the rates into $D_1/(r-g)$ does not provide a valid finite stock value.

Worked Example: Deriving the Value and Checking the First Terms

A hypothetical share is expected to pay $D_1=\$3.60$ at the end of year 1. Dividends are forecast to grow by 3% per year indefinitely, and the annual required return is 9%. Find the value from the infinite dividend stream, then calculate the present value of the first three dividends and relate it to the infinite sum.

Method. First check convergence, then apply the geometric-series formula. Here, $r=0.09$ and $g=0.03$, so $r-g=0.06>0$. The common ratio is $q=1.03/1.09=103/109$, approximately $0.9450$, which is less than 1.

The infinite value is

P_0=\frac{\$3.60}{0.09-0.03}=\frac{\$3.60}{0.06}=\$60.00

For the first three dividends, $D_1=\$3.60$, $D_2=\$3.60(1.03)=\$3.708$, and $D_3=\$3.708(1.03)=\$3.81924$. Their present values, before rounding, are

\frac{\$3.60}{1.09}=\$3.302752\ldots,\qquad \frac{\$3.708}{1.09^2}=\frac{\$3.708}{1.1881}=\$3.120949\ldots,\qquad \frac{\$3.81924}{1.09^3}=\frac{\$3.81924}{1.295029}=\$2.949154\ldots

The first three present values sum to $\$9.372855\ldots$. The finite-series formula gives the same check:

\frac{\$3.60}{1.09}\left(1+q+q^2\right) =\frac{\$3.60}{1.09}(1+q+q^2) =\$60.00(1-q^3) =\$9.372855\ldots

The remaining dividends have present value $\$60.00-\$9.372855\ldots=\$50.627145\ldots$, which is also $\$60.00q^3$. The two checks agree. Common mistake avoided: using only the first three dividends as though they were the entire stream would understate the infinite value; their present value is a finite partial sum, not the Gordon value.

Worked Example: Converting the Current Dividend to Next Year’s Dividend

A hypothetical company has just paid a dividend of $\$2.40$. Its dividends are expected to grow by 4% annually, and investors require a 10% annual return. Determine the value of the future dividend stream, taking care to identify which dividend belongs in the numerator. Also calculate the present value of the first three future dividends as a check.

Method. The reported dividend is $D_0$, the dividend just paid; the formula uses $D_1$, the dividend one year from now. Calculate $D_1=\$2.40(1.04)=\$2.496$. Since $r-g=0.10-0.04=0.06>0$, the series converges.

P_0=\frac{D_1}{r-g} =\frac{\$2.496}{0.06} =\$41.60

The first three future dividends are $D_1=\$2.496$, $D_2=\$2.496(1.04)=\$2.59584$, and $D_3=\$2.59584(1.04)=\$2.6996736$. Discounting each at 10% gives

\frac{\$2.496}{1.10}=\$2.269091\ldots,\qquad \frac{\$2.59584}{1.10^2}=\frac{\$2.59584}{1.21}=\$2.145322\ldots,\qquad \frac{\$2.6996736}{1.10^3}=\frac{\$2.6996736}{1.331}=\$2.028303\ldots

The three present values total $\$6.442716\ldots$. With $q=1.04/1.10=52/55$, the finite-series check is $(\$2.496/1.10)(1-q^3)=\$41.60(1-q^3)=\$6.442716\ldots$. This agrees with the direct calculation. Common mistake avoided: using $D_0=\$2.40$ in the numerator gives $\$40.00$, which is not the value of the stated future dividend stream. The numerator must be the next dividend, $D_1$.

Worked Example: Testing Whether an Infinite Value Exists

A hypothetical share will pay $D_1=\$2.00$ one year from now. Evaluate whether its infinite dividend stream has a finite value under each of two forecasts: (a) $r=8\%$, $g=8\%$; and (b) $r=5\%$, $g=7\%$. In each case, show what happens to the discounted terms.

Method. Check the common ratio $q=(1+g)/(1+r)$ before using a Gordon value. In case (a), $q=1.08/1.08=1$. Every discounted dividend is $\$2.00/1.08=\$1.851852\ldots$. Thus, the first $n$ terms total $n(\$1.851852\ldots)$, which grows without bound as $n$ increases. There is no finite infinite-series value.

In case (b), $q=1.07/1.05=107/105$, approximately $1.019048$, which exceeds 1. The first discounted dividend is $\$2.00/1.05=\$1.904762\ldots$. The second dividend is $\$2.00(1.07)=\$2.14$, and its present value is $\$2.14/1.05^2=\$1.941043\ldots$, larger than the first. Successive discounted terms keep increasing, so the infinite sum does not converge to a finite value.

Common mistake avoided: substituting the rates mechanically in $D_1/(r-g)$ gives a zero denominator in case (a) and a negative denominator in case (b). Neither is a valid finite value. The convergence test shows why the formula cannot be applied in either case.

What the Formula Does—and Does Not—Establish

The derivation is a mathematical statement about a particular infinite stream. It does not establish that any given company’s dividends will actually grow at a constant rate forever, or that a selected required return is appropriate. Those judgments must be made before interpreting a calculated value. The formula is useful only after its cash-flow timing, growth pattern, and discounting period have been specified consistently.

A practical exam sequence is to identify the next dividend, put growth and required return on the same time basis, and test whether $r>g$ before calculating. If those checks pass, the denominator $r-g$ is the gap between the discount rate and dividend growth rate. If they fail, the issue is not a rounding adjustment: the infinite-series derivation does not yield a finite value.

Relationship between rates	Geometric ratio	Result for the infinite stream
$r>g$	$0<q<1$	Discounted dividends shrink; the series converges to $D_1/(r-g)$.
$r=g$	$q=1$	Discounted dividends remain constant and positive; the series diverges.
$r<g$	$q>1$	Discounted dividends grow; the series diverges.

The most important exam trap is treating $D_0$ as the first cash flow in the sum. With end-of-year payments, the first cash flow is $D_1$, received one year from now. Another is overlooking convergence: an algebraically familiar expression does not make a divergent series finite. These checks are central to using the derivation correctly and are aligned with the SOA VEE Accounting and Finance syllabus.

Key takeaway: The Gordon growth formula follows from summing an infinite geometric sequence of discounted dividends. Its value is $P_0=D_1/(r-g)$ only when the next dividend is correctly identified and the required return exceeds the growth rate.

Check Your Understanding

For each problem, first check dividend timing and whether the infinite series converges.

A hypothetical share will pay $D_1=\$4.00$ next year. Dividends grow by 5% annually, and the required annual return is 12%. Find the infinite-stream value. (Answer: $\$57.14$)
A hypothetical company has just paid $D_0=\$1.80$. Dividends grow by 3% annually, and the annual required return is 8%. Find $D_1$ and the value of the infinite stream. (Answer: $D_1=\$1.854$; value $=\$37.08$)
A share has $D_1=\$2.50$, with $r=7\%$ and $g=7\%$. State the present value of one discounted dividend and determine whether the infinite stream has a finite value. (Answer: Each discounted dividend is $\$2.50/1.07=\$2.3364\ldots$; there is no finite value.)
A share has $D_1=\$1.25$, with $r=9\%$ and $g=11\%$. Calculate the common ratio and state whether the infinite series converges. (Answer: $q=1.11/1.09\approx1.0183$; it diverges, so there is no finite value.)
A hypothetical share will pay $D_1=\$3.00$ next year; dividends then grow by 2% annually. With an 8% annual required return, find the present value of only the first four dividends. (Answer: $\frac{\$3.00}{1.08}\left[1+\frac{17}{18}+(\frac{17}{18})^2+(\frac{17}{18})^3\right]=\$10.22$, rounded to the nearest cent.)

Relationship between rates	Geometric ratio	Result for the infinite stream
\(r>g\)	\(0<q<1\)	Discounted dividends shrink; the series converges to \(D_1/(r-g)\).
\(r=g\)	\(q=1\)	Discounted dividends remain constant and positive; the series diverges.
\(r<g\)	\(q>1\)	Discounted dividends grow; the series diverges.

What You'll Learn