The growing perpetuity: derive it properly, because everything downstream stands on it. Now let the cash flow grow at a constant rate g forever. You get CF₁ at t=1, CF₁(1+g) at t=2, CF₁(1+g)² at t=3, … Present value:
PV = CF₁/(1+r) + CF₁(1+g)/(1+r)² + CF₁(1+g)²/(1+r)³ + …
It's a geometric series with first term a = CF₁/(1+r) and common ratio x = (1+g)/(1+r). An infinite geometric series converges if and only if |x| < 1. Here that means g < r. When it converges, S = a/(1−x):
PV = [CF₁/(1+r)] / [1 − (1+g)/(1+r)]
Multiply numerator and denominator by (1+r), and the denominator becomes (1+r) − (1+g) = r − g. So:
PV = CF₁ / (r − g): the Gordon growth formula.
Common mistakes with the Gordon growth model
That derivation just taught you three things. People who only memorize the formula get them wrong for their whole careers.
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(1) The numerator is next period's cash flow. Remember that the series began at CF₁, one period out. If you're handed today's cash flow, CF₀, the formula becomes PV = CF₀(1+g)/(r−g). Forget the (1+g) and you understate value by g%. That looks small. On a ₹9 stream growing at 4% with r = 11%, though, it's the difference between ₹128.57 and ₹133.71, and on a terminal value it moves the whole valuation.
(2) g < r is the convergence condition, not a convention. If g ≥ r, each term of the series is as large as the last, or larger, in present-value terms. The sum diverges, and the "value" is infinite. The algebra CF₁/(r−g) will still spit out a number if you feed it g > r. It'll be a negative number, and that's how you know someone let the spreadsheet do their thinking. Economics enforces the math too. No cash-flow stream can grow faster than the economy forever, because it would eventually become the economy and then exceed it. So perpetual g is capped. The terminal-value rules you'll formalize later name the number that does the capping: g ≤ the risk-free rate in your own discount-rate stack, the same figure you put in the denominator, in the same currency. On a US-dollar stack built off a ~4.2% ten-year Treasury the cap is 4.2%. A mature company typically deserves 2.5 to 3%. On the credit-cleaned rupee stack you'll build for the cost of capital, a 6.5% G-sec less a 2.0-point sovereign default spread, the risk-free rate is 4.5% and so is the cap. Build the rupee stack on the raw 6.5% G-sec instead and the cap is 6.5%. Both stacks are defensible. Taking the rate from one and the cap from the other isn't. These caps are nominal. They differ across currencies only because rupee inflation runs higher, which the real-versus-nominal section explains.
A company can grow 25% for years. That belongs in the growing annuity of the explicit forecast. It never goes in the perpetuity.
(3) PV explodes as g approaches r, so tiny assumption changes swing valuations violently. Take CF₁ = ₹100 and r = 12%:
| g | 0% | 4% | 6% | 8% | 10% | 11% | 11.5% |
|---|---|---|---|---|---|---|---|
| PV = 100/(0.12−g) | 833 | 1,250 | 1,667 | 2,500 | 5,000 | 10,000 | 20,000 |
Going from g = 10% to 11%, one percentage point, doubles the value. That non-linearity is why terminal-value assumptions dominate DCF outputs, and you'll measure by how much later on. It's also why "just bump growth half a point" is never innocent, and why experienced analysts present valuation as a range over (r, g) pairs instead of a single point.
How do you find the growth rate a share price implies?
First live application: reading what a price implies. Run Gordon backward. Given price P and the trailing, just-paid dividend D₀, the market-implied growth is g = (r·P − D₀)/(P + D₀), which you get by solving P = D₀(1+g)/(r−g) for g. Here are two illustrations. The figures are illustrative, modeled on ~FY24/FY25 scale. Pull actual prices and dividends from screener.in or stockanalysis.com when you rework them:
- Hindustan Unilever at ~₹2,400 with a ~₹42 dividend. At r = 12%, the price implies g ≈ 10.1% forever. That's a whisker under the discount rate, and above any defensible perpetual cap for even a great franchise. You can reach a conclusion in 30 seconds. Either the market expects dividends to grow far faster than GDP for a very long explicit period, or it's applying a lower r, or the price embeds optimism. You're looking at India's "premium multiple regime" through a perpetuity.
- Coca-Cola at ~$60 with a ~$1.94 dividend. At r = 8%, implied g ≈ 4.6% forever, near US nominal GDP growth. That's demanding, but coherent for a century-old moat.
One formula, and you're already doing expectations investing. It's crude and dividend-only, but it has the same structure as the reverse DCF you'll build later. Notice one more thing in the comparison. The Indian implied g and the US implied g are in different currencies with different inflation. A 10% rupee growth and a 4.6% dollar growth are closer in real terms than they look, and the real-versus-nominal section makes that precise.