Two-Stage Dividend Discount Models

Why Use a Two-Stage Dividend Model?

A company’s dividends may grow rapidly for a limited period and then settle into a more sustainable long-run pattern. A two-stage dividend discount model represents those phases separately: it forecasts dividends individually during the first stage, then values the continuing stable-growth stream as a perpetuity. This makes the valuation more flexible than assuming one growth rate applies from the next dividend onward.

The model is an application of the dividend discount principle: a share’s value is the present value of the future dividends expected by its owner. The key is to keep the timing consistent. The stable-stage value is calculated at the end of the explicit forecast, not at the valuation date, and must then be discounted back.

Definition: Let $D_0$ be the dividend just paid, at time 0. Suppose dividends grow at specified rates $g_1,\ldots,g_N$ over the next $N$ years, followed by a constant annual growth rate $g_s$ from year $N$ onward. Let $r$ be the annual required return on the share. The two-stage dividend discount value at time 0 is the present value of dividends $D_1,\ldots,D_N$ plus the present value of the stable-growth dividends after year $N$.

Forecasting Dividends and Locating the Terminal Value

The first-stage growth rates need not be equal. Starting with the most recently paid dividend, calculate each forecast dividend using its own rate:

D_t=D_{t-1}(1+g_t), \qquad t=1,\ldots,N.

The stable growth rate applies after the explicit forecast. Therefore, the first dividend in the stable stage is $D_{N+1}=D_N(1+g_s)$. The stable-stage value is located at time $N$, immediately before that dividend is paid. The complete valuation equation is:

P_0=\sum_{t=1}^{N}\frac{D_t}{(1+r)^t}+\frac{P_N}{(1+r)^N}, \qquad P_N=\frac{D_{N+1}}{r-g_s}.

Here $P_N$ is the value, at time $N$, of dividends $D_{N+1},D_{N+2},\ldots$. It is not an additional dividend. The first term of $P_0$ discounts each explicitly forecast dividend to time 0; the second discounts the value of all later dividends from time $N$ to time 0.

Assumptions and convergence: The required return $r$ and growth rates are annual effective rates, dividends are paid annually at year-end, and the forecast dividends are paid to shareholders. The stable-stage perpetuity converges mathematically when $\left|(1+g_s)/(1+r)\right|<1$. In the usual financial setting, dividends remain nonnegative and $r>-1$, with $g_s\geq -1$; then this condition is equivalent to $r>g_s$. Under those usual assumptions, the Gordon formula $P_N=D_{N+1}/(r-g_s)$ applies. If the convergence condition fails, the stable-stage infinite sum does not have a finite value under this model.

Why the Stable-Stage Formula Holds

At time $N$, the first stable-stage dividend $D_{N+1}$ arrives one year later. Each subsequent dividend grows by $1+g_s$ and is discounted by one additional factor of $1+r$. Thus the value at time $N$ is a geometric series:

P_N =\frac{D_{N+1}}{1+r} \left[ 1+\frac{1+g_s}{1+r} +\left(\frac{1+g_s}{1+r}\right)^2+\cdots \right].

For a geometric series with first term 1 and common ratio $q$, the infinite sum is $1/(1-q)$ when $|q|<1$. Applying that identity here gives:

P_N =\frac{D_{N+1}}{1+r} \frac{1}{1-\frac{1+g_s}{1+r}} =\frac{D_{N+1}}{1+r} \frac{1+r}{r-g_s} =\frac{D_{N+1}}{r-g_s}.

This derivation explains both the formula and its condition. In the standard nonnegative-dividend setting with $r>-1$ and $g_s\geq-1$, the common ratio is nonnegative, so convergence requires it to be less than 1, which is equivalent to $r>g_s$. The more general mathematical condition is the absolute-value condition above; merely comparing $r$ and $g_s$ without considering the allowed rates is not a complete convergence test.

Applying the Two-Stage Model

Worked Example: Unequal Growth Rates in the Explicit Stage

A hypothetical share has just paid a dividend of $2.00. Dividends are forecast to grow by 20% in year 1, 10% in year 2, and 0% in year 3. From year 4 onward, they are expected to grow by 4% annually. If the required annual return is 10%, find the share value at time 0. Use the two-stage method and avoid treating the stable-stage value as if it were at time 0.

Method and dividend forecast. The first-stage horizon is $N=3$, $D_0=\$2.00$, and the applicable growth rates are 20%, 10%, and 0%. Therefore:

D_1=2.00(1.20)=2.40,\qquad D_2=2.40(1.10)=2.64,\qquad D_3=2.64(1.00)=2.64.

The first stable-stage dividend is:

$$ D_4=2.64(1.04)=2.7456. $$

Stable-stage value at time 3. The required return exceeds stable growth, $0.10>0.04$, so the usual Gordon formula applies:

P_3=\frac{2.7456}{0.10-0.04}=\frac{2.7456}{0.06}=\$45.76.

Discount each component to time 0. The year-1 dividend has discount factor $1.1$, the year-2 dividend has factor $1.1^2=1.21$, and both the year-3 dividend and $P_3$ have factor $1.1^3=1.331$:

P_0 =\frac{2.40}{1.1} +\frac{2.64}{1.21} +\frac{2.64}{1.331} +\frac{45.76}{1.331} =\frac{54{,}208}{1{,}331} \approx \$40.73.

The valuation check is that the terminal value is dated at year 3 and uses $D_4$, the dividend one year after that date. The terminal value is discounted three years, just like $D_3$. Rounding the final value to the nearest cent gives $40.73. The common mistake avoided is discounting $P_3$ only two years because it is formed using dividends that begin in year 4.

Worked Example: A Strong First-Year Forecast Followed by Stable Growth

A hypothetical company has just paid a $1.50 dividend. Its forecast growth is 30% in year 1 and 15% in year 2; beginning in year 3, dividends are expected to grow at 5% annually. An investor’s required annual return is 11%. Calculate the share value today, using the stable-stage value at the correct date.

Method. Forecast the two explicit dividends first. Then calculate the value at time 2 of dividends from year 3 onward, and discount all three present-value components to time 0.

D_1=1.50(1.30)=\$1.95,\qquad D_2=1.95(1.15)=\$2.2425, $$ $$ D_3=2.2425(1.05)=\$2.354625.

Since $r=0.11$ exceeds $g_s=0.05$, the stable-stage value at time 2 is:

P_2=\frac{D_3}{r-g_s} =\frac{2.354625}{0.11-0.05} =\$39.24375.

The year-1 dividend is discounted by $1.11$. The year-2 dividend and terminal value are both discounted by $1.11^2=1.2321$:

P_0 =\frac{1.95}{1.11} +\frac{2.2425}{1.2321} +\frac{39.24375}{1.2321} \approx 1.756757+1.820063+31.851107 \approx \$35.43.

The final amount, rounded to the nearest cent, is $35.43. The calculation keeps the full values through the sum; the displayed component decimals are for checking. The sum is consistent with the time-2 value: $P_2$ includes dividends beginning in year 3, while the separately discounted dividends are only $D_1$ and $D_2$. A common mistake is to include $D_3$ separately as well as in $P_2$, which would count it twice.

Worked Example: A Declining Dividend Before Stable Growth

A hypothetical share has just paid $4.00. Its dividend is forecast to decline by 10% in year 1, grow by 5% in year 2, and then grow at a stable 3% annually from year 3 onward. If the required annual return is 9%, calculate the value today. Identify why a terminal value dated at year 3 would need a different discount period.

Forecast the explicit dividends. The first-stage rates are $-10\%$ and $5\%$, so:

D_1=4.00(0.90)=\$3.60,\qquad D_2=3.60(1.05)=\$3.78.

The first stable-stage dividend is:

D_3=3.78(1.03)=\$3.8934.

Value the stable tail at time 2. Since $0.09>0.03$, the stable-stage value is finite under the usual financial assumptions:

P_2=\frac{3.8934}{0.09-0.03} =\frac{3.8934}{0.06} =\$64.89.

Now discount $D_1$ for one year, and both $D_2$ and $P_2$ for two years:

P_0 =\frac{3.60}{1.09} +\frac{3.78}{1.09^2} +\frac{64.89}{1.09^2} \approx 3.302752+3.181550+54.616620 \approx \$61.10.

Thus the value rounded to the nearest cent is $61.10. The stable-stage value is at time 2 because $D_3$ is the first dividend it represents. If instead the stable-stage value were dated at time 3, its numerator would have to be $D_4$, and that value would be discounted for three years. Using a time-3 value with $D_3$ as its numerator would misalign the dividend timing.

Interpreting the Result and Avoiding Traps

The model separates two distinct sources of value: the present value of near-term dividends forecast individually, and the present value of the longer stable-growth tail. The terminal value can be a substantial part of the total, so an error in its date, growth rate, or discounting can materially affect the result. The model is useful only when the stable stage is a reasonable representation of dividends after the explicit forecast; it does not establish that the forecasts themselves are reliable.

Item	Correct interpretation	Frequent error
First stable dividend	$D_{N+1}=D_N(1+g_s)$	Using $D_N$ in the numerator
Terminal value date	$P_N=D_{N+1}/(r-g_s)$, at time $N$	Treating $P_N$ as a time-0 value
Present value of the tail	Discount $P_N$ by $(1+r)^N$	Discounting it for $N-1$ years or not discounting it
Stable-stage condition	Check $\left\|(1+g_s)/(1+r)\right\|<1$	Using the Gordon formula without checking convergence

A particularly important exam trap is the terminal-value timing. The value at time $N$ includes the stream starting with $D_{N+1}$, not the dividend $D_N$, which was already forecast and is included separately. Another is confusing the finite first-stage forecast with the stable-stage perpetuity: the first-stage dividends are discounted one by one, while only the later stream is summarized by $P_N$. These timing checks are aligned with the SOA VEE Accounting and Finance syllabus.

Key takeaway: Forecast $D_1,\ldots,D_N$ using the specified first-stage rates, form $D_{N+1}$ using stable growth, calculate $P_N=D_{N+1}/(r-g_s)$ only when the perpetuity converges, and discount $P_N$ by $(1+r)^N$.

Check Your Understanding

For each problem, keep the terminal value’s date explicit and check the stable-growth condition before using the Gordon formula.

A hypothetical share has just paid $1.20. Its dividend grows by 25% in year 1 and 0% in year 2, then grows at 4% forever from year 3. The required annual return is 9%. Find the share value at time 0. (Answer: $28.90)
A hypothetical company has just paid a $2.50 dividend. Dividends decline by 20% in year 1, grow by 10% in year 2, and then grow steadily at 2%. The required annual return is 8%. Find the share value today. (Answer: $35.80)
A share has just paid $1.00. Its dividend grows by 20% in year 1, after which it is forecast to grow at 8% forever. If the required annual return is 7%, can the standard stable-stage dividend model assign a finite value? State the convergence check. (Answer: No; the ratio is $1.08/1.07>1$, so the stable-stage series does not converge.)
A hypothetical share has just paid $3.00. Its dividend grows by 10% in year 1 and 0% in year 2, then grows at 5% forever. If the required annual return is 9%, calculate $P_2$ and $P_0$. (Answer: $P_2=\$86.625$; $P_0=\$78.72$)

Item	Correct interpretation	Frequent error
First stable dividend	\(D_{N+1}=D_N(1+g_s)\)	Using \(D_N\) in the numerator
Terminal value date	\(P_N=D_{N+1}/(r-g_s)\), at time \(N\)	Treating \(P_N\) as a time-0 value
Present value of the tail	Discount \(P_N\) by \((1+r)^N\)	Discounting it for \(N-1\) years or not discounting it
Stable-stage condition	Check \(\left\|(1+g_s)/(1+r)\right\|<1\)	Using the Gordon formula without checking convergence

What You'll Learn