Perpetuities, Gordon Growth, and Uneven NPV

Perpetuity = an annuity that never ends. PV-perpetuity = C / r, where C is the constant payment per period and r is the discount rate. Derivation: take the PV-annuity formula PMT × [(1 − (1+r)⁻ⁿ) / r] and let n → ∞. The (1+r)⁻ⁿ term drives to zero (any positive rate compounds future periods to negligible PV), leaving PMT/r. Worked example: a charitable endowment that promises $40,000/year forever, earning 5% on its investments, requires PV = $40,000 / 0.05 = $800,000. That is the lump-sum donation needed to fund the scholarship in perpetuity. The British government issued perpetuity bonds called "consols" from 1751 to 2015, and they really did pay forever — until the Treasury redeemed them.

Read the table sideways: distant cash flows at any reasonable discount rate contribute almost nothing to present value. This is why valuation analysts can lazily say "the discounted value of cash flows beyond year 30 is rounding error" without losing accuracy. It is also why the perpetuity formula works in practice — even though no real company truly pays forever, discounting at any rate above zero shrinks distant decades to negligible PV. The math forgives the assumption.

Three discipline checks for Gordon Growth. (1) C₁ is NEXT period's cash flow, NOT the trailing one. If a company's last dividend was $2.00 and you expect 4% perpetual growth, C₁ = $2.08, not $2.00. Skipping this step understates fair value by exactly the growth rate. (2) r > g is MANDATORY. If g ≥ r, the formula returns infinity or negative — the math is telling you the assumption is impossible. Real-world reading: you cannot have a company growing faster than your required return forever, because if it could, every investor on Earth would buy it and drive the price up until r > g again. (3) g is a LONG-RUN sustainable rate, not a near-term burst. A startup growing 50% per year cannot grow at 50% forever; eventually it converges to GDP-trend (~3-4% nominal). Use the long-run g, not the visible-period one.

The Dividend Discount Model (DDM) is just Gordon Growth applied to dividends. Stock fair value = D₁ / (r − g), where D₁ is next year's dividend, r is your required return on equity (typically 8-12% for established US stocks), and g is the long-run sustainable dividend growth rate (typically 2-6%). Worked example: Procter & Gamble pays a $4.00 trailing dividend, expected to grow at 4% forever, and you require 9%. Fair value = $4.00 × 1.04 ÷ (0.09 − 0.04) = $4.16 / 0.05 = $83.20. Compare to the actual stock price; if the stock trades below $83.20, the DDM says it is undervalued (subject to your assumptions about r and g). DDM works best for stable dividend payers (utilities, consumer staples, banks); it breaks down for growth stocks that don't pay dividends or where g approaches r.

When cash flows are uneven (project years 1-5 produce $200, $350, −$100, $500, $800 in irregular amounts) you cannot use the closed-form annuity or perpetuity formulas. You discount each cash flow individually using PV = CFₜ / (1+r)ᵗ, then sum the per-period PVs. Net Present Value (NPV) = the sum of all discounted future cash flows minus the initial investment. NPV > 0 means the project earns more than your discount rate; NPV < 0 means it destroys value. This per-period discount-and-sum mechanic IS the DCF method that dcf-1 introduces and dcf-7 builds IRR pitfalls on top of. The valuation industry is, mechanically, just NPV + Gordon Growth glued together.

Read the worked NPV. The first 5 years contribute roughly $740 to the value; the Gordon Terminal Value contributes the remaining $7,310 — about 91% of the total. This concentration is not a bug; it is structural. Most real businesses have most of their value in the long run, just like the perpetuity table you read above. This is why DCF analysts spend so much time arguing about the terminal value — get r or g wrong by 1% and the answer swings by 30%. Get the year-1 cash flow wrong by 10% and the answer barely moves.