Two-Stage Dividend Discount Model

Builds onGrowing Perpetuity (Gordon Growth) — if this page feels steep, start there.

V_0 = \sum_{t=1}^{n} \frac{D_0(1+g_1)^t}{(1+r)^t} + \frac{1}{(1+r)^n}\left[\frac{D_0(1+g_1)^n(1+g_2)}{r-g_2}\right]

$D_0$: the dividend just paid (today), the starting point for stage one's growth
$g_1$: the high-growth rate assumed for the first n years
$n$: the number of years the high-growth stage lasts
$g_2$: the stable, permanent growth rate assumed forever after year n
$r$: the required return on equity, discounting every stage
$\frac{D_0(1+g_1)^n(1+g_2)}{r-g_2}$: the terminal value at year n: the whole stable-growth tail, collapsed into one lump sum

Why it must be true

No company grows at one rate forever — a young firm's blistering growth eventually cools into something closer to the overall economy's pace. The two-stage model prices that honestly: value the dividends explicitly for $n$ years of high growth $g_1$ , then hand everything after that off to the Gordon growth model at a sustainable, permanent rate $g_2$ .

This is simply the growing perpetuity you already know, used TWICE: once implicitly (each high-growth dividend is just a growing cash flow, discounted one at a time) and once explicitly, to collapse the entire infinite tail of stable-growth dividends into a single terminal value at the end of year $n$ — which then itself gets discounted back to today.

The derivation

Stage one: discount each of the $n$ high-growth dividends individually, since they don't yet fit a single closed-form perpetuity:

\text{Stage 1} = \sum_{t=1}^{n} \frac{D_0(1+g_1)^t}{(1+r)^t}

Stage two: once growth settles into the permanent rate $g_2$ , the entire remaining stream — starting with year $n+1$ 's dividend — IS a growing perpetuity, valued as of year $n$ :

TV_n = \frac{D_n(1+g_2)}{r - g_2}, \qquad D_n = D_0(1+g_1)^n

That terminal value is a lump sum sitting at year $n$ — discount it back to today like any other single future amount, and add it to stage one:

V_0 = \text{Stage 1} + \frac{TV_n}{(1+r)^n}

When to reach for it

Valuing a company whose current growth is clearly temporary (high-growth or cyclical), by explicitly modeling a finite high-growth phase before it settles into perpetual, sustainable growth.

Listen for

“two-stage dividend discount model”“grows at … for the next n years, then settles at …”“high-growth phase followed by stable/terminal growth”“terminal value at the end of the explicit forecast period”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Two separate perpetuities, not one blended rate: never average g1 and g2 into a single growth assumption — the model's whole point is keeping the two phases distinct.
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The terminal value formula needs D_{n+1} = D_n(1+g_2), NOT D_n itself — a classic off-by-one-period slip.
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Sanity check the terminal value's weight: for a long high-growth phase or a high discount rate, TV's present value shrinks; for a short phase or low r, TV usually dominates V0 — most of the price is a bet on the STABLE phase, not the exciting one.

Traps in applying it

✗Forgetting to discount the terminal value back n periods — it's a lump sum sitting at year n, not at year 0.
✗Using D_n instead of D_n(1+g_2) in the terminal value's numerator.
✗Applying g2 ≥ r — the terminal perpetuity only converges when the required return exceeds the stable growth rate.

Limits & criticisms

The model's honesty is also its fragility: value is extremely sensitive to r − g2 in the terminal value, so a tiny change in either assumption swings the whole valuation. It also assumes growth switches ABRUPTLY from g1 to g2 at year n, when real companies decelerate gradually — the three-stage model and the H-model exist specifically to smooth that transition. And like any DDM, it only works for dividend-paying companies with a policy an analyst can actually model.

Where it came from

The two-stage extension followed naturally from Gordon and Shapiro's constant-growth model (1956) once analysts needed to value young or cyclical companies whose current growth obviously could not persist forever — a straight Gordon model applied to a 30%-growth startup would imply an ever-expanding share of the entire economy. Multi-stage DDMs (two-stage, three-stage, H-model) became standard equity-research tools through the 1960s–80s specifically to make growth assumptions explicit and falsifiable rather than buried in a single blended rate.

One identity, 1 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Value today, two stages priced separately

V_0 = \sum_{t=1}^{n} \frac{D_0(1+g_1)^t}{(1+r)^t} + \frac{1}{(1+r)^n}\left[\frac{D_0(1+g_1)^n(1+g_2)}{r-g_2}\right]

The only face this model has: explicit high-growth dividends, plus a collapsed, discounted stable-growth tail.

Drill this face →

On the BA II Plus

Worked example: Current dividend $2.75; growth of 11% for 3 periods years, then 4% forever after; required return 11.5%. Value the stock using a two-stage dividend discount model.

1.\$2.75 [×] 1.11 [=] \$3.05year 1 dividend, discounted to \$2.74
2.\$2.75 [×] 1.2321 [=] \$3.39year 2 dividend, discounted to \$2.73
3.\$2.75 [×] 1.3676 [=] \$3.76year 3 dividend, discounted to \$2.71
4.\$3.76 [×] (1 [+] 0.04) [÷] (0.115 [−] 0.04) [=]terminal value at year n
5.[÷] 1.3862 [=]discount terminal value back to today

→ $45.80

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