Sensitivity Of Gordon Growth Cost To Inputs

Why Input Sensitivity Matters

A Gordon growth estimate can look precise even when its inputs are uncertain. A small change in a forecast dividend, market price, or long-run growth assumption can change the estimated cost of equity. To interpret the result, separate the arithmetic effect of each input from the reliability of the assumptions behind it.

The estimate uses the next expected annual dividend, $D_1$, the current share price, $P_0$, and the constant annual dividend growth rate, $g$. Rates and dividend timing must use consistent annual conventions. In particular, $D_1$ is the dividend expected one period from now, not the dividend just paid.

Definition: Under the constant-growth dividend model, dividends are expected to grow at a constant annual rate $g$ indefinitely, and the required return $r$ is greater than $g$. The current share price satisfies $P_0=D_1/(r-g)$. Rearranging gives the Gordon growth cost-of-equity estimate $r=D_1/P_0+g$.

The model assumes a continuing stream of dividends that grows at the same rate each period, with $r>g$, so the discounted dividend series converges. It is a model-based estimate, not a guarantee that the inputs or the model describe future outcomes. The earlier material on dividend and rate timing conventions is useful here: $D_1$, $P_0$, and $g$ must refer to compatible periods.

How Each Input Changes the Estimate

Write the estimate as the sum of a dividend yield and a growth rate. Holding the other two inputs fixed, a larger next dividend raises the estimate, a larger share price lowers it, and a larger growth rate raises it one-for-one.

r(D_1,P_0,g)=\frac{D_1}{P_0}+g

The local sensitivities are the partial derivatives. They describe the change in the estimate for a small change in one input while the other inputs remain fixed.

\frac{\partial r}{\partial D_1}=\frac{1}{P_0}, \qquad \frac{\partial r}{\partial P_0}=-\frac{D_1}{P_0^2}, \qquad \frac{\partial r}{\partial g}=1

These derivatives have different units. For example, $\partial r/\partial D_1$ is a change in annual rate per dollar of dividend per share, while $\partial r/\partial P_0$ is a change in annual rate per dollar of share price. The derivative with respect to $g$ means that a one-percentage-point increase in $g$ produces a one-percentage-point increase in $r$, if $D_1$ and $P_0$ do not change.

Why the Price Effect Is Negative

The estimate’s dividend-yield component is $D_1/P_0$. With $D_1$ fixed, increasing $P_0$ spreads the same dividend over a larger price, so the yield falls. Differentiating the quotient with respect to price gives the negative derivative.

\frac{\partial}{\partial P_0}\left(\frac{D_1}{P_0}\right) = D_1\frac{\partial}{\partial P_0}(P_0^{-1}) = -\frac{D_1}{P_0^2}

For finite changes, use the formula itself rather than treating a derivative as exact. If the inputs change by $\Delta D_1$, $\Delta P_0$, and $\Delta g$, the exact change is:

\Delta r= \frac{D_1+\Delta D_1}{P_0+\Delta P_0} -\frac{D_1}{P_0} +\Delta g

This expression also shows why the separate effects of simultaneous changes should be calculated carefully: the dividend and price appear together as a ratio. A derivative-based approximation is most useful for small changes; direct recalculation is preferable when changes are larger.

Worked Example: Separating Three Input Changes

A hypothetical share has a current price of $ \$40.00$, a next expected annual dividend of $ \$2.40$, and an estimated annual dividend growth rate of $4.0\%$. An analyst revises the dividend to $ \$2.52$, the price to $ \$42.00$, and growth to $4.5\%$. Calculate the original and revised cost estimates, and identify each input’s effect in sequence.

Method. Apply $r=D_1/P_0+g$, changing one input at a time from the original values. This makes the contributions interpretable and avoids confusing percentage-point changes with percentage changes.

Original estimate. The dividend yield is $2.40/40.00=0.06=6.0\%$. Therefore,

r_{\text{original}}=6.0\%+4.0\%=10.0\%

Dividend revision. At the original price, the revised yield is $2.52/40.00=0.063=6.3\%$. This raises the estimate by $0.3$ percentage points, to $10.3\%$.

Price revision. With the revised dividend and growth still at $4.0\%$, the revised yield is $2.52/42.00=0.06=6.0\%$. The price increase lowers the estimate by $0.3$ percentage points, back to $10.0\%$.

Growth revision. Raising growth from $4.0\%$ to $4.5\%$ adds $0.5$ percentage points. The final estimate is $6.0\%+4.5\%=10.5\%$.

Check. Substituting all revised inputs directly gives $2.52/42.00+0.045=0.06+0.045=0.105$, or $10.5\%$. The sequential effects, $+0.3-0.3+0.5=+0.5$ percentage points, agree with the change from $10.0\%$ to $10.5\%$. The calculation avoids the mistake of applying the revised dividend yield to the old price in the final estimate.

Local Sensitivity and Exact Price Changes

The price derivative gives a first-order approximation. For a small price change $\Delta P_0$, with the dividend and growth held fixed, the approximate rate change is $-D_1\Delta P_0/P_0^2$. Because the ratio is curved in price, this approximation need not equal the exact change.

Worked Example: Comparing Approximate and Exact Price Effects

A hypothetical company’s next annual dividend is $ \$3.00$, its current share price is $ \$50.00$, and its annual growth estimate is $5.0\%$. Calculate the cost estimate if the price rises to $ \$55.00$, holding the dividend and growth fixed. Compare the exact change with the local derivative approximation.

Method. First calculate the original estimate, then recalculate using the new price. Use the derivative at the original price only as a local approximation.

Original estimate. The dividend yield is $3.00/50.00=0.06=6.0\%$, so $r=6.0\%+5.0\%=11.0\%$.

Exact revised estimate. The new yield is $3.00/55.00=0.05454545\ldots$, or $5.454545\ldots\%$. Thus the revised estimate is $5.454545\ldots\%+5.0\%=10.454545\ldots\%$. Rounded to four decimal places, it is $10.4545\%$, a decrease of $0.5455$ percentage points.

Local approximation. At the original inputs,

\frac{\partial r}{\partial P_0} =-\frac{3.00}{50.00^2} =-0.0012

This is a decrease of $0.12$ percentage points per dollar of price. For a $ \$5.00$ increase, the approximation is $-0.12\times5=-0.60$ percentage points, giving an approximate revised estimate of $10.40\%$.

Check. The exact revised estimate is $10.4545\%$, not $10.40\%$; the derivative is an approximation for a small change. The difference is $0.0545$ percentage points. Using the exact quotient avoids treating the local slope as exact over a $ \$5.00$ price change.

Growth Near the Required Return

There is an important distinction between sensitivity of the estimated cost and sensitivity of the model-implied price. In the cost estimate $r=D_1/P_0+g$, changing $g$ by one percentage point changes $r$ by one percentage point, with the observed price and next dividend held fixed. This formula does not become singular as $g$ approaches the resulting estimate.

The valuation form $P_0=D_1/(r-g)$, however, divides by the spread between required return and growth. If $r$ is fixed and $g$ approaches $r$ from below, that spread becomes small and the implied price rises sharply. At $g=r$, the constant-growth perpetuity does not converge, so the valuation formula does not apply.

Worked Example: A Narrow Return-Growth Spread

A hypothetical share is expected to pay $D_1=\$1.80$ next year. For illustration, hold the required annual return fixed at $11.0\%$. Compare the model-implied prices when annual growth is $8.0\%$, $10.5\%$, and $10.9\%$. State why the values become unstable as growth nears the required return.

Method. Use $P_0=D_1/(r-g)$, which requires $r>g$. The dividend and required return remain fixed in all three calculations.

At $g=8.0\%$. The spread is $11.0\%-8.0\%=3.0\%=0.03$. The price is $1.80/0.03=\$60.00$.

At $g=10.5\%$. The spread is $11.0\%-10.5\%=0.5\%=0.005$. The price is $1.80/0.005=\$360.00$.

At $g=10.9\%$. The spread is $11.0\%-10.9\%=0.1\%=0.001$. The price is $1.80/0.001=\$1,800.00$.

Check. Each price satisfies $P_0(r-g)=\$1.80$: for example, $\$1,800.00\times0.001=\$1.80$. The growth rate remains below the required return in every case, but the implied price increases rapidly as the spread narrows. This illustrates valuation instability: small errors in estimating the spread can cause large changes in price.

Common Traps and Practical Interpretation

A sensitivity calculation is only as useful as its timing and units. When inputs are revised, state which are held fixed, work in rates or percentage points consistently, and recalculate the full expression to confirm the result.

Using the last dividend instead of the next dividend. The formula uses $D_1$. If the last paid dividend is $D_0$, it must first be forecast under the stated growth assumption: $D_1=D_0(1+g)$.
Reversing the price effect. A higher price, with the dividend unchanged, lowers $D_1/P_0$ and therefore lowers the estimated cost. For example, with a fixed $D_1=\$3.00$, moving from a price of $\$50.00$ to $\$55.00$ reduces the yield from $6.0\%$ to about $5.4545\%$.
Confusing percentage points and relative percentages. Moving from $4.0\%$ growth to $4.5\%$ is an increase of $0.5$ percentage points, not $0.5\%$ of the original rate.
Applying a derivative to a large change as if it were exact. The price derivative is local. For a material price change, calculate the new dividend yield directly.
Ignoring the return-growth condition in valuation. $P_0=D_1/(r-g)$ requires $r>g$. A small positive spread can produce a very large price; a zero or negative spread does not yield a finite positive value under this model.

For an analyst, the derivatives help identify which inputs drive the estimate, while the finite-change calculation gives the actual result for a specified revision. The Gordon growth method is most informative when its long-run growth assumption is credible and the return-growth spread is not so small that modest input error overwhelms the estimate. This distinction is a useful part of cost-of-equity analysis aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

For each hypothetical case, calculate the requested estimate or sensitivity. Keep annual rates and dividend timing consistent.

A share has $D_1=\$1.50$, $P_0=\$30.00$, and $g=4.0\%$. Calculate the cost estimate, then recalculate it if only $D_1$ rises by $ \$0.15$. (Answer: $9.0\%$ initially; $9.5\%$ after the dividend revision, an increase of $0.5$ percentage points.)
A share has $D_1=\$2.10$, $P_0=\$42.00$, and $g=3.5\%$. The price changes to $ \$45.50$ and growth changes to $4.0\%$, with the dividend unchanged. Calculate the revised estimate. (Answer: $8.6154\%$, or $8.62\%$ to two decimal places.)
A share has $D_1=\$2.40$, $P_0=\$48.00$, and $g=5.0\%$. The dividend rises to $ \$2.52$, price falls by $ \$6.00$, and growth falls by $0.25$ percentage points. Calculate the original and revised estimates and the net change. (Answer: $10.0\%$ initially; $10.75\%$ revised; increase of $0.75$ percentage points.)
For $D_1=\$4.00$, $P_0=\$80.00$, and $g=4.0\%$, calculate the original estimate, the derivative approximation to a $ \$4.00$ price increase, and the exact revised estimate. (Answer: $9.0\%$ initially; approximate decrease $0.25$ percentage points, giving $8.75\%$; exact estimate $8.7619\%$.)
With $D_1=\$2.40$ and fixed required return $r=12.0\%$, calculate the model-implied prices at $g=10.0\%$ and $g=11.0\%$. What is the percentage increase in price? (Answer: $\$120.00$ and $\$240.00$, respectively; the price increases by $100\%$.)

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