How to use the perpetuity value calculator
- Enter the payment each period, which is the first payment.
- Enter the discount rate per period, the return you require, as a percent. Use the same period as the payments.
- Enter a growth rate per period if payments increase at a steady rate. Use 0 for level payments. Growth must be lower than the discount rate.
- Choose whether the first payment comes one period from now or today. Read the present value and the table of the first five payments. Copy, Print or export the CSV.
Formula
level, first payment in one period: PV = payment / rate
growing, first payment in one period: PV = payment / (rate - growth)
first payment today: PV x (1 + rate)
The rate and growth are per period, written as decimals in the formulas (8% is 0.08). The formula needs growth below the rate, because otherwise the value is unbounded. Each payment in the table is discounted by (1 + rate) for every period before it arrives.
$5,000 a period at 8%
With level payments of $5,000 starting in one period and a rate of 8%, the value is 5,000 / 0.08 = $62,500. If the first payment is paid today, multiply by 1.08 to get $67,500.
If payments grow 3% each period, the value is 5,000 / (0.08 - 0.03) = $100,000, or $108,000 with the first payment today. A check by adding thousands of discounted payments converges to the same number.
Why a forever stream has a finite value
Each payment is worth less today the further away it is, so the discounted payments shrink and add up to a finite total, even though there are infinitely many. The formula is a shortcut for that infinite sum. The first few payments hold most of the value, which the page shows as a share.
The result is very sensitive to the rate. A discount rate close to the growth rate makes the value large, and a small change in either number moves it a lot.
Using it sensibly
Perpetuities are a teaching tool and a way to model very long-lived payments, such as a stream whose end is far away. Real payments seldom last forever, so treat the answer as an upper limit when payments end eventually. The annuity calculator values payments that stop, and the present value calculator discounts a single amount.
For longer-term models see the discounted cash flow calculator and the IRR and NPV calculator. This is not investment advice.
Assumptions and limits
- The first payment is the amount you enter. Later payments grow at the rate you enter.
- The discount rate must be more than zero, and the growth rate must be below it. Negative growth is allowed.
- Payments, rates and periods share the same time unit, such as years or months.
- Taxes, inflation, risk and the chance that payments stop are not modeled. Nothing is stored or uploaded.
Frequently asked questions
What is the formula for a perpetuity?
For level payments that start in one period, the present value is the payment divided by the discount rate. For payments that grow at a steady rate, divide the first payment by the rate minus the growth rate.
Why must growth be below the discount rate?
If payments grew as fast as or faster than the discount rate, each discounted payment would not shrink and the total would be infinite. The formula only works when growth is lower than the rate.
What changes if the first payment is paid today?
The whole stream arrives one period sooner, so its value is higher by a factor of one plus the rate. The page offers this as an option and shows the effect.
Is a perpetuity realistic?
Rarely, because few payments last forever. It is still a useful approximation for payments that last a very long time, or as an upper limit on the value of a long stream.