When Gordon Growth Valuation Breaks Down

When a Simple Perpetuity Stops Being Credible

The Gordon growth model is compact, but its simplicity depends on a demanding assumption: dividends grow at one constant rate forever. A valuation can look precise while being mathematically invalid or economically inconsistent with the firm’s outlook. The key checks are whether growth is below the required return, whether a stable perpetual growth rate is defensible, and whether the dividend pattern actually matches the model.

Let $D_1$ be the dividend expected one period from now, $r$ the required return on equity per period, and $g$ the constant dividend growth rate per period. The model values the present value of all future dividends. Its rate inputs must use the same time unit, and $D_1$, rather than the dividend just paid, belongs in the numerator.

Definition: The Gordon growth model (also called the constant-growth dividend discount model) estimates a share’s value as $P_0=D_1/(r-g)$, provided dividends grow at a constant rate $g$ forever, the required return $r$ is constant, and $r>g$. The dividend forecast and the required return must refer to the same periods. If the model’s assumptions or convergence condition fail, the formula does not provide a valid finite value.

Why the Required Return Must Exceed Growth

The condition $r>g$ is not a technical footnote. It is what makes the infinite sum of discounted dividends converge. If expected dividends grow as $D_t=D_1(1+g)^{t-1}$, then the present value of the dividend at time $t$ is $D_t/(1+r)^t$. Successive present values form a geometric series.

P_0=\sum_{t=1}^{\infty}\frac{D_1(1+g)^{t-1}}{(1+r)^t} =\frac{D_1}{1+r}\sum_{t=0}^{\infty}\left(\frac{1+g}{1+r}\right)^t

For a geometric series with first term $a$ and ratio $q$, its infinite sum is $a/(1-q)$ only when $|q|<1$. Here, with nonnegative dividend growth and a positive required return, $q=(1+g)/(1+r)$. The convergence condition $q<1$ is equivalent to $g<r$. Substitution gives

P_0=\frac{D_1}{1+r}\frac{1}{1-(1+g)/(1+r)} =\frac{D_1}{r-g}.

At $g=r$, each discounted dividend is the same positive amount, so adding infinitely many of them does not produce a finite value. When $g>r$, discounted dividends increase rather than shrink, so the sum also diverges. A zero or negative result obtained by inserting $g\ge r$ into $D_1/(r-g)$ is not a meaningful stock value; it is a warning that the formula has been used outside its domain.

Worked Example: Growth Equal to or Above the Required Return

A hypothetical company has just paid a dividend of $1.85. Its next dividend is forecast at $2.00, and an analyst assumes an 8% required return. Test the proposed constant-growth rates of 8% and 10%. Explain why a direct Gordon calculation is not a valid valuation.

Method. Check the geometric-series ratio $q=(1+g)/(1+r)$ before using the Gordon formula. This avoids interpreting a zero denominator or a negative computed price as a valid value.

At $g=8\%$ and $r=8\%$, the ratio is $q=1.08/1.08=1$. The present value of the first dividend is $2.00/1.08=\$1.85185185$. Because each following dividend also grows at 8%, its discounted value is the same $1.85185185. The sum of infinitely many equal positive amounts diverges.

The Gordon denominator is $0.08-0.08=0$, so $2.00/0$ is undefined. It is not a finite price.

At $g=10\%$, $q=1.10/1.08=1.0185185$, which is greater than 1. The first two discounted dividends are $2.00/1.08=\$1.85185185$ and $2.20/(1.08)^2=2.20/1.1664=\$1.88683196$. The second present value already exceeds the first, and subsequent present values continue to grow by the factor 1.0185185.

Direct substitution would give $2.00/(0.08-0.10)=-\$100$. That negative number is not an estimate of the share’s value: it results from applying the closed-form formula to a divergent series. Check: both proposed growth rates fail $r>g$, so neither produces a finite Gordon value.

Growth Must Also Be Stable and Economically Plausible

Passing the mathematical test $r>g$ does not, by itself, justify the model. Gordon growth assumes that the same growth rate continues indefinitely and that the required return remains constant. It is therefore unsuitable when dividends are expected to grow rapidly for a limited period, vary materially from year to year, or change because of a planned shift in payout policy. A rate inferred from a short, unusual period is not automatically a credible perpetual rate.

A common solution to a temporary growth phase is to forecast the transition explicitly. Estimate dividends during the nonconstant period, then use a constant-growth value only for the later, stable phase. The terminal value must be calculated at the date just before the stable-phase dividends begin to be received, using the dividend expected one period after that date. Discount both the forecast dividends and terminal value back to today.

Worked Example: Rapid Growth That Settles to a Stable Rate

A hypothetical share has just paid a $2.00 dividend. The next two annual dividends are expected to grow by 12% per year. From year 3 onward, dividends are expected to grow by 4% per year forever. The required return is 10% per year throughout. Estimate the share value today.

Method. Forecast the two transition dividends, calculate a Gordon terminal value at the end of year 2 using the year-3 dividend, and discount all amounts to time 0. This avoids incorrectly extending the temporary 12% growth rate forever.

The first dividend is $D_1=\$2.00(1.12)=\$2.24$. The second is $D_2=\$2.24(1.12)=\$2.5088$. The first dividend under stable growth is $D_3=\$2.5088(1.04)=\$2.609152$.

The stable phase satisfies the Gordon condition because $r=10\%$ is greater than $g=4\%$. At time 2, the value of dividends from year 3 onward is $P_2=D_3/(r-g)=2.609152/(0.10-0.04)=\$43.4858667$.

The time-0 value includes $D_1$ at time 1 and both $D_2$ and $P_2$ at time 2:

P_0=\frac{2.24}{1.10}+\frac{2.5088+43.4858667}{(1.10)^2} =2.0363636+38.0121212 =40.0484848.

Rounding only the final value to the nearest cent gives $P_0=\$40.05$. Check: the terminal value is at time 2, and the first dividend used in that terminal value is $D_3$, the dividend one year later.

Even with stable growth, a small gap between $r$ and $g$ can make the estimate highly sensitive to input choices. Since the denominator is $r-g$, a small change in either rate can change the value substantially. An output with several decimal places does not imply that the inputs are known with comparable precision.

Worked Example: A Valid but Fragile Estimate

A hypothetical company is expected to pay $D_1=\$4.00$ next year. An analyst uses a required return of 9% and compares perpetual growth assumptions of 7%, 8%, and 8.5%. Calculate the three values and interpret their sensitivity.

Method. Apply the Gordon formula separately to each rate, after checking that each is below 9%. The mistake to avoid is treating a large result as robust merely because the denominator remains positive.

At $g=7\%$, $r-g=0.09-0.07=0.02$, so $P_0=4.00/0.02=\$200.00$.

At $g=8\%$, $r-g=0.09-0.08=0.01$, so $P_0=4.00/0.01=\$400.00$.

At $g=8.5\%$, $r-g=0.09-0.085=0.005$, so $P_0=4.00/0.005=\$800.00$.

All three calculations satisfy $r>g$, but raising growth from 7% to 8.5% quadruples the estimate, from $200.00 to $800.00, with $D_1$ and $r$ unchanged. Check: the formula is mathematically applicable in each case, but the estimates are highly exposed to uncertainty in the perpetual growth assumption.

When the Perpetual Assumption Conflicts with the Outlook

A constant growth rate can be mathematically admissible yet still misrepresent the business. For example, if a company is expected to stop operating, make a one-time liquidation distribution, or change its payout materially, a perpetually growing dividend stream does not describe the forecast. Likewise, dividends that fluctuate or decline at different rates cannot be summarized faithfully by one constant $g$ unless the simplifying assumption is explicitly justified.

A constant decline is not automatically a model failure. If positive dividends are genuinely expected to decline at the same rate forever, $g$ is constant and negative; the model can still apply when $r>g$. The issue is not whether growth is positive, but whether the constant rate and perpetual dividend stream match the outlook. When they do not, forecast the relevant cash distributions explicitly and value any later phase using assumptions that fit that phase.

Worked Example: A Declining Dividend and a Finite Business Life

A hypothetical firm has just paid a $1.50 dividend. It is expected to pay $1.20 at the end of year 1 and $0.90 at the end of year 2, then cease operations. Shareholders are also expected to receive a $8.00 liquidation distribution at the end of year 2. The required return is 10% per year. Find the value today and explain why a perpetuity based on the first decline is misleading.

Method. Discount the actual expected distributions at their payment dates. This avoids extending a temporary dividend decline forever and omitting the liquidation cash flow.

The year-1 dividend has present value $1.20/1.10=\$1.0909091$. At year 2, the dividend and liquidation distribution total $\$0.90+\$8.00=\$8.90$, with present value $8.90/(1.10)^2=8.90/1.21=\$7.3553719$.

P_0=\frac{1.20}{1.10}+\frac{0.90+8.00}{(1.10)^2} =1.0909091+7.3553719 =8.4462810.

Rounding only the final value to the nearest cent gives $P_0=\$8.45$. If an analyst instead treats the 20% fall from $1.50 to $1.20 as a permanent growth rate, a Gordon calculation would be $1.20/(0.10-(-0.20))=\$4.00$. That calculation assumes the dividend declines by 20% every year forever and includes no liquidation payment; neither assumption matches the stated outlook. Check: the direct present-value calculation includes every stated distribution at its correct date.

A Practical Diagnostic

Before relying on a Gordon estimate, separate mathematical validity from forecast credibility. The following sequence makes those checks explicit:

Identify the dividend and period.
Confirm that the numerator is the next expected dividend $D_1$, and that the dividend and rate periods match.

Test convergence.
Check that the required return exceeds perpetual growth, $r>g$. If it does not, the infinite discounted-dividend series has no finite sum.

Test the long-run story.
Ask whether constant growth, a constant required return, and continuing dividends are consistent with the firm’s expected payout and business outlook.

Test sensitivity.
Recalculate using plausible nearby values of $r$ and $g$. A very small $r-g$ signals that the estimate may be dominated by uncertain assumptions.

Change the model if needed.
Forecast dividends over periods with distinct growth patterns, and use a perpetual terminal value only if a stable phase is defensible. For a finite outlook, value the specified distributions directly.

A common exam trap is to treat the Gordon formula as an algebraic instruction rather than a result derived under particular assumptions. Substituting a growth rate at or above the required return cannot rescue a nonconvergent series. A second trap is to force one perpetual rate onto an unstable forecast simply because it produces a single number. State the model, test its conditions, and explain why the long-run assumptions fit before interpreting the estimate. This diagnostic approach is aligned with the SOA VEE Accounting and Finance syllabus.

Key takeaway: Use $P_0=D_1/(r-g)$ only when dividends grow at a defensible constant rate forever and $r>g$. If growth is unstable, the business outlook is finite, or the estimate is extremely sensitive, model the distinct forecast periods and distributions rather than presenting an unsupported Gordon value.

Check Your Understanding

For each problem, check both the arithmetic and whether the assumptions permit the valuation method.

A hypothetical share is expected to pay $D_1=\$2.50$. The required return and proposed perpetual growth rate are both 7%. Is a finite Gordon value available? (Answer: No; $r=g$, so the discounted-dividend series diverges.)
A company has just paid $1.80. Its next dividend grows by 8%, after which dividends grow by 3% forever. The required return is 9%. Find the value today. (Answer: $32.40.)
A share is expected to pay $D_1=\$5.00$, with a constant perpetual growth rate of 9% and required return of 11%. Find its Gordon value. (Answer: $250.00.)
A firm is expected to pay dividends of $2.00, $2.40, and $2.70 at the ends of years 1, 2, and 3. Beginning in year 4, dividends grow by 4% forever. If the required return is 10%, calculate its value today. (Answer: $40.99.)
A firm will pay $3.00 at the end of year 1, then $2.40 and a separate $12.00 liquidation distribution at the end of year 2. It will make no later payments. At a 10% required return, what is the value today? (Answer: $14.63.)

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