4.4Lesson 4.4 · Module 4

Valuing a Company with Multiple Growth Stages

The Gordon Growth Model assumes one constant growth rate forever. Real companies grow rapidly, then transition, then mature. Multi-stage valuation separates temporary cash flows from stable cash flows.

Separate explicit forecast from terminal value
$T V_{N}$ = $D_{N + 1}$ /(r − $g_{S}$ )
Discount both sections to today
Compare one-stage vs multi-stage valuations

Start

Learning objectives

By the end of this lesson, you should be able to:

1Explain why one constant growth rate cannot value a real company.
2Identify the three growth stages: rapid growth, transition, and maturity.
3Value a stock as the sum of explicit-forecast dividends plus a terminal value.
4Compute the terminal value $T V_{N}$ = $D_{N + 1}$ / (r − $g_{S}$ ).
5Explain why the terminal value uses $D_{N + 1}$ , not $D_{N}$ .
6Compare no-growth, multi-stage, and perpetual-growth valuations.
7Recognize when terminal value dominates total value.

From Lesson 4.3

The Gordon Growth Model values a stock with a single, constant growth rate that lasts forever: $P_{0} = D_{1} / (r - g)$ . That model only works when $g < r$ and the growth rate is genuinely sustainable over the indefinite future. A company growing dividends at 15–18% per year cannot do so forever — eventually market saturation, competition, and sheer size drag growth back toward the economy-wide rate. This lesson shows how to value a company that grows fast for a while, then settles.

4.1Section 1

Why one growth rate is not enough

Real companies do not grow at a single rate forever. A successful young firm may grow earnings and dividends at 15–20% for several years as it captures market share. But that pace cannot last: competition enters, the addressable market saturates, and the company's own size makes percentage growth harder to sustain. Eventually growth slows toward the rate of the overall economy.

The Gordon model cannot describe this pattern. If you plug in the high early growth rate, you get an absurdly high value and a formula that may not even converge. If you plug in the low mature rate, you ignore years of rapid dividend growth that a buyer today would actually receive. The truth lies between — and that is exactly what multi-stage valuation captures.

4.2Section 2

Three stages of corporate life

Analysts usually split a company's life into three stages. The lengths and intensities differ by firm, but the shape is remarkably consistent.

Stage 1 — Growth

Rapid expansion. The firm reinvests most of its earnings, pays little or no dividend, and grows quickly. Margins may be rising.

Stage 2 — Transition

Competition catches up. Growth decelerates year by year, payout rises as attractive reinvestment opportunities fade, and the firm begins to behave more like a mature business.

Stage 3 — Mature

Sustainable growth at or near the economy-wide rate. Stable, predictable payout. This is the regime where the Gordon model finally applies.

4.3Section 3

The central formula

Multi-stage valuation splits the stock's value into two pieces: the present value of dividends you forecast explicitly (years 1 through N), plus the present value of a terminal value that captures everything after year N.

Multi-stage DDM

P_{0} = t = 1 \sum N \frac{D _{t}}{( 1 + r ) ^{t}} + \frac{T V _{N}}{( 1 + r ) ^{N}}

Stock value = present value of the explicit-forecast dividends + present value of the terminal value, each discounted back to today.

$P_{0}$: stock value today
$D_{t}$: dividend in year t (explicit forecast)
$T V_{N}$: terminal value at the end of year N
$r$: cost of equity (discount rate)
$N$: length of the explicit forecast

The first sum values each near-term dividend individually. The second term lumps every dividend from year N+1 onward into a single terminal value, then discounts that value back to today.

Terminal value (Gordon at time N)

T V_{N} = \frac{D _{N + 1}}{r - g _{S}}

Terminal value is the Gordon Growth Model computed at time N, using the next dividend $D_{N + 1}$ and the stable, mature growth rate $g_{S}$ .

$D_{N + 1}$: first dividend of the mature stage = $D_{N}$ × (1 + $g_{S}$ )
$g_{S}$: sustainable, stable growth rate ( $g_{S}$ < r)
$r$: cost of equity

Once the company has settled into its mature stage, the Gordon model applies. $T V_{N}$ is the value, at time N, of that growing perpetuity starting with $D_{N + 1}$ .

Definition · Terminal value is not liquidation

Terminal value is not the price for which the company is sold, and it is not a liquidation or scrap value. It is the present value, as of time N, of all cash flows the company will ever produce after year N. The business keeps operating indefinitely; the terminal value just packages that infinite tail into one number.

4.4Section 4

The timeline: explicit forecast + terminal value

Picture the dividend stream laid out along a timeline. Years 1 through N are forecast individually — each dividend is discounted on its own. At year N the terminal value attaches, representing everything from N+1 onward, and that single lump is discounted back N periods to today.

Time 0 (today)D₁D₂D₇TV₇ at year 7D₈, D₉, … forever

Each green dividend is discounted back to time 0 on its own. The amber terminal value, which itself represents the entire infinite tail, is also discounted back to time 0 — but only once, using $(1 + r)^{7}$ .

4.5Section 5

A full worked example

Suppose $D_{0} = $1$ , dividends grow at $g_{H} = 6%$ for $N = 7$ years, then stabilize at $g_{S} = 0%$ thereafter, and the cost of equity is $r = 20%$ . We build the dividend table, compute the terminal value, and sum the present values.

t	Dₜ	(1.20)ᵗ	PV(Dₜ)
1	$1.0600	1.2000	$0.8833
2	$1.1236	1.4400	$0.7803
3	$1.1910	1.7280	$0.6893
4	$1.2625	2.0736	$0.6089
5	$1.3382	2.4883	$0.5379
6	$1.4185	2.9860	$0.4751
7	$1.5036	3.5832	$0.4196
PV of dividends (years 1–7)			≈ $4.39

Terminal value and total

T V_{7} = \frac{D _{8}}{r - g _{S}} = \frac{D _{7} ( 1 + g _{S} )}{r - g _{S}}

With $g_{S}$ = 0%, D₈ = D₇ = $1.5036, so TV₇ = 1.5036 / 0.20 = $7.518.

P_{0} = PV of D_{1} \dots D_{7} $4.39 + PV of T V_{7} \frac{$7.518}{( 1.20 ) ^{7}} = $4.39 + $2.10

P₀ ≈ $6.49

The seven explicitly-forecast dividends contribute about $4.39. The terminal value, once discounted back seven years, contributes another $2.10. Together they give roughly $6.49.

4.6Section 6

One-stage vs multi-stage vs perpetual growth

Holding $D_{0} = $1$ and $r = 20%$ fixed, compare three assumptions about growth. The multi-stage answer sits between the two extremes.

Assumption	Formula	P₀
(a) No growth, 0% forever	D₁ / r = 1.06 / 0.20	$5.30
(b) 6% for 7 yr, then 0% (multi-stage)	Σ PV(Dₜ) + PV(TV₇)	$6.49
(c) 6% forever (Gordon)	D₁ / (r − g) = 1.06 / 0.14	$7.57

Treating the company as if it never grows understates value — it ignores real near-term dividend growth. Treating 6% as perpetual overstates value — no company grows above the economy forever. Multi-stage lands in between, capturing the temporary growth honestly without projecting it out to infinity.

4.7Section 7

Adding stable growth after year N

The previous example set stable growth to zero for simplicity. In practice mature companies do grow, slowly, with inflation and the economy. Suppose instead that after year 7 the company grows at $g_{S} = 3%$ .

Terminal value with positive stable growth

T V_{7} = \frac{D _{8}}{r - g _{S}} = \frac{D _{7} ( 1 + g _{S} )}{r - g _{S}}

D₈ = D₇ × (1 + 3%) = 1.5036 × 1.03 = $1.5487. Then TV₇ = 1.5487 / (0.20 − 0.03) = 1.5487 / 0.17 ≈ $9.11.

P V (T V_{7}) = \frac{$9.11}{( 1.20 ) ^{7}} \approx $2.54, P_{0} \approx $4.39 + $2.54 \approx $6.93

P₀ ≈ $6.93

Positive stable growth raises both the numerator (D₈ is larger) and shrinks the denominator (r − $g_{S}$ is smaller), so the terminal value — and the total price — is higher than the zero-growth-stable case.

4.8Section 8

Two-stage vs three-stage models

The multi-stage idea can be implemented with two stages or three. The trade-off is realism against simplicity.

Two-stage

High growth for N years, then an immediate jump to the stable rate. Simple to compute, but the cliff at year N is rarely realistic.

Three-stage

High growth, a gradual transition, then stable growth. More believable, at the cost of more assumptions.

15%12%10%8%6%4% (stable)

A three-stage model lets growth decelerate gradually — 15% down to a stable 4% — rather than dropping off a cliff. Each year of the transition still gets its own dividend in the explicit sum.

4.9Section 9

Why the terminal uses $D_{N + 1}$ , not $D_{N}$

The explicit forecast already discounts $D_{N}$ as the last dividend in the sum. The terminal value starts one period after the explicit forecast, at $D_{N + 1}$ . Starting at $D_{N}$ would count the year-N dividend twice — once in the explicit sum and once inside the terminal perpetuity.

Avoiding double-counting

D_{N + 1} = D_{N} \times (1 + g_{S})

Grow the last explicit dividend $D_{N}$ by one period at the stable rate to obtain $D_{N + 1}$ , the first dividend of the mature stage.

This single-step growth bridges the explicit forecast into the stable perpetuity without overlapping any cash flow.

4.10Section 10

A zero-dividend growth stage

Many young companies pay no dividend at all during their growth stage. Suppose $D_{1} = D_{2} = \dots = D_{5} = 0$ , and only starting in year 6 does the firm begin distributing cash. The stock can still have substantial positive value — because the terminal value captures the dividends that begin once the company matures.

Value is entirely terminal

When every explicit dividend is zero, the first sum vanishes and $P_{0} = P V (T V_{N})$ . The entire value comes from discounting the terminal value. This is common for early-stage firms that reinvest everything today in order to pay out far in the future.

4.11Section 11

A note on the discount rate

In the most general DDM, each dividend can be discounted at a different rate that reflects its own risk — early cash flows might be discounted at one rate, mature cash flows at another. This lesson uses a single, constant $r$ for every period. That is a simplification chosen for clarity; the key ideas — explicit forecast, terminal value, and $D_{N + 1}$ — are identical whichever discount-rate schedule you choose.

Try itMulti-stage valuation builder

P₀ = Σ Dₜ/(1+r)ᵗ +

T V_{N}

/(1+r)ᴺ

Value a company through growth and maturity

Set the most recent dividend D₀, the number of high-growth years N, the high-growth rate $g_{H}$ , the stable growth rate $g_{S}$ , and the cost of equity r. The builder forecasts each dividend, attaches a Gordon terminal value at year N, and discounts everything back to today.

Most recent dividend D₀High-growth years N

High-growth rate

g_{H}

6.0%

Stable growth rate

g_{S}

0.0%

Cost of equity r20.0%

Year t	Dividend Dₜ	Discount (1+r)ᵗ	PV of Dₜ
1	$1.0600	1.2000	$0.8833
2	$1.1236	1.4400	$0.7803
3	$1.1910	1.7280	$0.6892
4	$1.2625	2.0736	$0.6088
5	$1.3382	2.4883	$0.5378
6	$1.4185	2.9860	$0.4751
7	$1.5036	3.5832	$0.4196
PV of explicit dividends (years 1–7)			$4.3942

D_{N + 1}

D_{N}

× (1 +

g_{S}

)

$1.5036

$1.5036 × (1 + 0.0%)

T V_{N}

D_{N + 1}

/ (r −

g_{S}

)

$7.52

$1.5036 ÷ (20.0% − 0.0%)

PV of TV =

T V_{N}

/ (1+r)ᴺ

$2.10

$7.52 ÷ 1.20^7

Total value P₀

$6.49

$4.3942 + $2.10

Share of value from terminal value

32.3%

PV of dividends 67.7%PV of terminal 32.3%

Terminal value often represents a large share of total value because it bundles every dividend from year 8 onward into a single perpetuity. The further out the horizon, the larger this share tends to be.

4.12Concept check

Worked check: high growth, then stable

Try itWorked check

D₀ = $2, high growth 10% for 3 years, stable growth 4%, r = 12%

Find the dividends, the terminal value, and the total value.

4.13Common questions

Terminal value, timing, and the stable stage

04Mastery

Summary and mastery check

Try itLesson 4.4 mastery check

Pass with 3 of 5 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
The Gordon model assumes:
02
Terminal value $T V_{N}$ represents:
03
Why does the terminal formula use $D_{N + 1}$ ?
04
D₀ = $1, 6% growth for 7 years then 0%, r = 20%. Approximate value?
05
If stable growth rises from 0% to 3% (r = 20%), TV:

Lesson summary

1The Gordon model assumes one perpetual growth rate and cannot handle temporary high growth.
2Multi-stage valuation separates explicit-forecast dividends from terminal value.
3Terminal value $T V_{N}$ = $D_{N + 1}$ /(r − $g_{S}$ ) captures all cash flows after the forecast period.
4The terminal formula uses $D_{N + 1}$ to avoid double-counting $D_{N}$ .
5Terminal value often represents a large share of total value.
6Multi-stage valuations lie between no-growth and perpetual-growth extremes.

Replay this lesson Continue to From Earnings to Dividend Growth