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.
What you will learn
- 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
The formula
- C
- payment each period, first one at the end of period 1
- r
- discount rate per period
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.
Common 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.