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Perpetuity

A perpetuity is an annuity in which the constant periodic payments begin on a fixed date and continue indefinitely. It is sometimes referred to as a perpetual annuity. Examples include:

Perpetual (irredeemable) bonds that pay interest (coupons) forever.
Rights to earnings from rental properties or royalties in perpetuity.
Dividends from holding stock in a company indefinitely.
Annuity incomes intended to last a lifetime.

An infinite stream of payments has, surprisingly, a finite price, because receipts far in the future are worth almost nothing today. There is no principal to repay — the stream is the whole asset. Assuming payments begin at the end of the current period, the price is simply the periodic payment over the appropriate discount rate:

PV = \frac{M}{r}

The intuition takes one sentence: a pot of $PV$ invested at rate $r$ throws off $PV \times r$ of income every year forever without shrinking, so for that income to equal $M$ , the pot must be $M ∕ r$ .

The growing perpetuity — the version you will actually use. A fixed payment forever is rare. A payment that grows forever is everywhere: rents that rise with inflation, dividends that increase annually, a business whose earnings creep up with the economy. If the payment grows at a constant rate $g < r$ , the sum still converges, to:

PV = \frac{M_{1}}{r - g}

where $M_{1}$ is the payment one period from now. This is the Gordon growth model, and it underpins critical formulas across the rest of this book:

Stock valuation: With $M_{1}$ the next dividend, this is the dividend discount model. A stock paying $3 next year, growing 4% annually, at a 9% required return, is worth $$ 3 ∕ (0.09 - 0.04) = $ 60$ .
Real estate cap rates: The capitalization rate is this formula inverted. $Value = NOI ∕ cap rate$ — where NOI is net operating income, the property’s rents minus operating expenses, before financing — is exactly a perpetuity, which means a cap rate represents $r - g$ (a required return net of expected growth), not a simple dividend yield. That is why identical buildings trade at different cap rates in different markets: the market is pricing different $g$ (section “Capitalization Rate (CAP rate)”).
DCF terminal value: Any discounted cash flow model must stop projecting at some point and assume a steady state. The terminal value at year $N$ is $\frac{C_{N + 1}}{r - g}$ , discounted back — and in most models it accounts for most of the total valuation.

Handle $g$ with suspicion. The denominator $r - g$ is a small difference between two uncertain numbers, which makes the result violently sensitive to both. With $r = 9 %$ : at $g = 4 %$ the multiple is $20 \times$ ; at $g = 6 %$ it is $33 \times$ ; at $g = 7 %$ it is $50 \times$ . A two-point change in an assumption nobody can verify nearly doubled the valuation. And if you ever find yourself with $g \geq r$ , the formula returns a negative or infinite value — which is the math telling you the assumption is mathematically impossible, not signaling an infinitely valuable asset. No business grows faster than its discount rate forever; if it did, it would eventually become the entire economy. Cap long-run $g$ at something near nominal GDP growth and treat any model that needs more as a model built to reach a predetermined answer.

One application deserves the last word, because it governs the retirement chapters: a portfolio meant to last forever is a growing perpetuity in reverse. Sustainable spending is the real return net of the growth you must retain to keep pace with inflation — the structural reason safe withdrawal rates cluster in the 3–4% range instead of the 7% headline return (section “Safe Withdrawal Rate — Why 4%?”).

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