Time Value of Money
corePresent Value of a Perpetuity
Builds onPresent Value of an Ordinary Annuity — if this page feels steep, start there.
- the price today of receiving the payment forever
- the fixed payment received every period, without end
- income divided by the rate: the size of pot whose interest alone funds the payment forever
- the required return per period, as a decimal
Reading the notation
Why it must be true
A perpetuity pays forever, yet its price is finite — because distant payments are discounted into insignificance. The formula is the cleanest statement of "price = income ÷ yield" in all of finance.
Think of it from the owner's side: if you hold and it earns each period, you can withdraw exactly forever without touching the principal. So an income of forever needs a pot of . Preferred stock and UK consols are priced exactly this way.
The derivation
Take the ordinary annuity factor and let :
The term — the value of the tail — vanishes: payments beyond a few decades contribute almost nothing to price.
When to reach for it
A level payment with no end date — preferred dividends, perpetual bonds, endowment spending, or the terminal value shortcut in a DCF.
Listen for
Back-of-the-envelope
Estimate it in your head first — then the calculator only confirms.
- ≈
Divide by the rate = multiply by its reciprocal. Memorize the yield table: 4% → 25×, 5% → 20×, 8% → 12.5×, 10% → 10× the payment.
- ≈
Backwards is just as fast: a $2,000 perpetuity priced at $25,000 yields 2,000 ÷ 25,000 = 8%.
- ≈
Answers that are the payment times a single-digit number are almost always the 'multiplied instead of divided' trap — a perpetuity is worth many multiples of one payment.
Traps in applying it
- ✗PMT/r assumes the first payment arrives one period from now — a payment today adds PMT on top.
- ✗Mixing an annual payment with a quarterly rate.
- ✗Confusing the coupon rate (sets the payment) with the required return (does the discounting).
Limits & criticisms
Nothing actually pays forever — issuers call bonds, firms cut dividends, inflation erodes level payments. The formula is also extremely rate-sensitive at low yields: at r = 2%, a 1-point rate change moves the price by a third. It values a frozen payment; growth needs the Gordon version.
Where it came from
Perpetuities are real securities, not a textbook fiction. Dutch water boards issued perpetual bonds in the 1600s — at least one, from 1648, still pays interest today — and Britain's famous consols (1751) consolidated the national debt into perpetual annuities that traded for over 250 years. Traders could read the formula off the market: price equals coupon over yield.
Today it prices preferred stock, underpins the terminal value in every DCF model, and supplies endowment spending rules — the questions "what is income forever worth?" and "what can I spend forever?" are the same equation.
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 = income ÷ yield
The pricing face: a finite pot whose interest exactly funds the payment forever. Preferred stock and consols trade on this line.
Yield = income ÷ price
Read from market prices: the return a perpetual payment stream offers at today's price — the dividend yield in its purest form.
Sustainable income
The endowment face: what a pot can pay out each period forever without touching principal. Spending rules for foundations start here.
On the BA II Plus
Worked example: An endowment must fund a grant of $780.00 every period in perpetuity, earning 3.5% per period. How large must the endowment be?
- 1.780 [÷] 0.035 [=]PMT ÷ r (rate as a decimal)
→ $22,285.71
Where it leads
Master this and the following come almost for free: