The annuity that never ends
What if a payment stream goes on forever? Each year ₩100, every year, no expiration date. That's a perpetuity. The British government issued bonds like this in the 18th century — called "consols," they paid forever (until the UK retired them in 2015, but that's a different story). Some preferred stocks pay forever as long as the company exists. Some endowment funds operate this way.
The math is surprising in its simplicity. An annuity needs a finite n; for a perpetuity, n goes to infinity. You'd think infinite payments would be worth infinite money. They aren't — because the further-out payments are discounted so heavily, their contribution shrinks to (essentially) zero.
The closed-form: P = C / r
For a perpetuity paying C per period at discount rate r, the present value is:
Three letters. The simplest valuation equation in finance. ₩100 forever at 5%? P = 100 / 0.05 = ₩2,000. That's the price someone should pay today for an infinite stream of ₩100 annual payments.
If you double the payment to ₩200, the price doubles to ₩4,000. If the discount rate halves to 2.5%, the price doubles too (to ₩4,000). Makes sense: cheaper money (lower r) means future payments are penalized less, so the total is worth more today.
This is numerator–denominator play in its purest form
Stop and look at P = C / r. It's exactly the P = A / B shape from Track 1, lesson 3. C is the numerator. r is the denominator. P goes up if C goes up, or if r goes down. Two drivers, period. Same picture from the thesis lesson, with concrete finance variables now.
And every other valuation equation we'll see — Gordon Growth, DCF, even multiples like P/E — is just a perpetuity with extra trim. P = C/r is the bare frame. Adding growth gives Gordon. Adding finite years gives an annuity. Splitting the cash flow stream into different periods gives a multi-stage DCF. But the bones are this one equation.
Why it works — the math intuition
Why doesn't an infinite stream of payments add up to infinity? Because each payment further out is discounted by another factor of (1+r). After enough years, the discount factor is so big that the contribution of that payment is essentially zero.
Mathematically, the annuity formula PV = C × [1 − (1+r)^(-n)] / r takes n → ∞ and the term (1+r)^(-n) goes to zero. What's left? PV = C × 1 / r = C/r. The infinite series converges to a finite number. That's the limit concept from Track 1 lesson 8 quietly doing work in the background.
Real examples
Pure perpetuities are rare in practice (most things eventually end). But the model is useful as an approximation:
- Land that produces stable rent year after year — close to perpetuity (the land doesn't expire).
- Companies operating "forever" — the terminal value in a DCF model uses a perpetuity formula for everything beyond the explicit forecast horizon.
- Endowment funds that aim to spend only the income each year, never the principal.
- Old British consols (mentioned above) and a few preferred stocks.
The takeaway
A perpetuity is the simplest possible valuation. P = C / r. Two letters connected by a fraction. Numerator–denominator play in its purest form. It's also the seed of everything that follows — Gordon Growth in the next lesson is literally one tweak away from this. So burn this in: P = C / r is the prototype. Everything else is variation.