Growing Perpetuity (Gordon Growth)

Builds onPresent Value of a Perpetuity — if this page feels steep, start there.

PV = \frac{PMT_1}{r - g}

$PV$: the price today of the growing stream
$PMT_1$: the subscript 1 matters: this is NEXT period's payment, not today's
$r$: the return investors require per period
$g$: the constant rate the payment grows at, forever
$r - g$: the net margin: discounting pulls value down at r while growth pushes it up at g — only the difference does the pricing

Why it must be true

If the payment grows at $g$ forever, discounting at $r$ only "wins" by the margin $r-g$ . The stream is priced as if it were a level perpetuity at that net rate. This is the Gordon growth model behind every dividend-discount valuation.

Two first-principles checks: the numerator is next period's payment $PMT_1$ , not today's; and the formula only converges when $r > g$ — a company can't outgrow its discount rate forever, or its price would be infinite.

The derivation

Discount the growing stream term by term:

PV = \frac{PMT_1}{1+r} + \frac{PMT_1(1+g)}{(1+r)^2} + \frac{PMT_1(1+g)^2}{(1+r)^3} + \dots

This is geometric with ratio $\frac{1+g}{1+r} < 1$ (requires $r > g$ ). Summing:

PV = \frac{PMT_1}{1+r} \cdot \frac{1}{1 - \frac{1+g}{1+r}} = \frac{PMT_1}{r - g}

When to reach for it

A payment stream growing at a constant rate forever — dividend-discount valuations and the terminal value in nearly every DCF model.

Listen for

“growing at g% forever / constant growth”“next year's dividend”“Gordon growth”“terminal value”

Back-of-the-envelope

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

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Same reciprocal table as the flat perpetuity, applied to the margin: r − g of 4% → 25× next year's payment; 2% → 50×.
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Growth is powerful: halving the margin doubles the price. If two answers differ by roughly 2×, check whether you used r or r − g.
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Quick yield check: dividend yield = r − g. A stock at 25× next dividend implies a 4% margin — does that match the given r and g?

Traps in applying it

✗Using this year's payment PMT₀ instead of next period's PMT₁ = PMT₀(1+g).
✗Letting g ≥ r — the series diverges and the formula is meaningless.
✗Adding g to the discount rate instead of subtracting it from r.

Limits & criticisms

Its central assumption — constant growth forever — is impossible: any g above long-run GDP growth implies the firm eventually becomes the economy. And because the denominator is the thin margin r − g, tiny changes in either input swing the valuation enormously — the standard criticism of every DCF terminal value built on it.

Where it came from

John Burr Williams proposed in The Theory of Investment Value (1938) that a stock is worth its discounted dividends; Myron Gordon and Eli Shapiro (1956) closed the infinite sum for constant growth, giving the model Gordon's name. It was a deliberate answer to the question "what disciplined number can replace market froth?" after the 1929 crash.

Today it is the workhorse of equity terminal values, utility rate regulation, and quick sanity checks: given a price, it exposes the growth assumption the market is silently making.

One identity, 3 questions

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

Price growing income

PV = \frac{PMT_1}{r-g}

The Gordon growth valuation: growth thins the discounting margin to r − g, so faster growth means a higher price for the same payment.

Drill this face →

Implied total return

r = \frac{PMT_1}{PV} + g

Rearranged, return decomposes into income yield plus growth — the two ways a growing asset pays you.

Drill this face →

Growth priced into the market

g = r - \frac{PMT_1}{PV}

Given a price and required return, this face reveals the perpetual growth the market is implicitly assuming — a reality check on valuations.

Drill this face →

On the BA II Plus

Worked example: A share trades at $177.78 with next period's dividend of $8.00 growing at 3.5% forever. What required return does the price imply?

1.8 [÷] 177.7778 [=]income yield PMT₁/PV
2.[+] 0.035 [=]add back growth

→ 8%

Where it leads

Master this and the following come almost for free:

Cost of Equity (Dividend Discount)Justified Leading P/E

Drill this formula Back to the library