Present Value of a Perpetuity in 6 Minutes
A perpetuity is a stream of equal payments that goes on forever. That sounds like it should be worth infinity, but because far-off payments are discounted to almost nothing, the present value is simply the payment divided by the interest rate.
Perpetuity present value explained simply
A perpetuity pays the same amount every period forever, and its present value is the payment divided by the rate: PV = C ÷ r. In the example, $500 a year forever at 5% is worth $10,000 today.
What this perpetuity lesson covers
- What a perpetuity is and where you see one (preferred stock, consols)
- The present value of perpetuity formula: PV = C ÷ r
- Why an infinite stream has a finite value
- When the first payment timing changes the answer
Perpetuity formula, explained
- C
- payment each period, first one at the end of period 1
- r
- discount rate per period
How to value a perpetuity step by step (worked example)
An investment pays $500 at the end of every year, forever. The discount rate is 5%. What is it worth today?
- Inputs: C = $500, r = 0.05.
- Apply the formula: PV = $500 ÷ 0.05.
- PV = $10,000.
Answer: $10,000 today.
Perpetuity vs annuity
| Perpetuity | Annuity | |
|---|---|---|
| How long it pays | Forever | A fixed number of periods |
| Formula | C ÷ r | C × [1 − (1 + r)⁻ⁿ] ÷ r |
| Examples | Some preferred shares, endowment payouts | Loans, leases, fixed pensions |
Does anything really pay forever? David's take
A perpetuity is a lovely piece of maths: $500 a year forever is worth $10,000. In business, 'forever' hides inside many valuations, from terminal values to stock prices. I'm not thinking like a founder looking for a long-term moat. With AI, product lifecycles are compressing, from 50 years to 50 months to 50 weeks. That's a catchy line, not a statistic, but the direction is real. Use the perpetuity for things that really are long-lived, and be suspicious when a business plan quietly assumes one.
From David's concept: Guerrilla Producting (product lifecycle compression). This is David's opinion, alongside the standard textbook answer above.
Harder perpetuity questions
What if the first payment is today?
Add today's payment to the ordinary perpetuity: $500 + $500 ÷ 0.05 = $10,500.
What happens to the value if rates fall to 4%?
$500 ÷ 0.04 = $12,500. Perpetuities are very sensitive to the rate, because so much of their value sits far in the future.
Common perpetuity questions
What is the formula for the present value of a perpetuity?
PV = C ÷ r, where C is the fixed payment each period and r is the discount rate per period. It assumes the first payment comes one period from now and the payments never stop.
How can something that pays forever have a finite value?
Each payment further in the future is worth less today. Payments 50 or 100 years out are discounted to almost nothing, so the sum of all the present values converges to a finite number: C ÷ r.
What is an example of a perpetuity?
Preferred stock that pays a fixed dividend with no maturity is the classic example. British consols, government bonds with no end date, were another until the UK redeemed them in 2015. Endowments that pay out a fixed amount every year work the same way.
What if the first perpetuity payment is made today?
Then add today's payment to the standard formula: PV = C + C ÷ r. With $500 at 5%, that is $500 + $10,000 = $10,500.