Hong Life CalculatorsUnderstanding the perpetuity calculator — concept and principle
A perpetuity is a stream of payments that continues forever, such as an endowment payout, a preferred share dividend or a consol bond. This calculator finds its present value, with or without growth, and shows how much of that value comes from early and distant payments.
How it is calculated
- Level perpetuity: PV = C ÷ r, where C is the payment per period and r the discount rate per period.
- Growing perpetuity: PV = C ÷ (r − g), valid only when r is above g.
- Both assume the first payment arrives one period from now. For a perpetuity due (first payment today), multiply by (1 + r).
- Share of value from the first N payments = 1 − (1 + g)^N ÷ (1 + r)^N for a growing stream (1 − (1 + r)^−N when g = 0).
The principle
The present value of a payment far in the future is tiny, so an infinite stream of payments has a finite value. Summing C ÷ (1 + r)^k for k = 1 to infinity is a geometric series that collapses to C ÷ r. Intuitively, if you invested C ÷ r at rate r, it would pay exactly C each period without ever touching the principal. Growth raises the value because later payments are bigger, but if g reached r the payments would grow as fast as they are discounted and the value would be infinite.
Worked example
Default inputs: payment 75,000 USD a year, discount rate 3.5%, growth 0%, first payment one period from now.
- PV = 75,000 ÷ 0.035 ≈ 2,142,857 USD; with no growth the level value is the same.
- The payment in period 10 is still 75,000 USD.
- Share from the first 10 payments = 1 − 1.035^−10 ≈ 29.1%.
- Share from payments after the 30th = 1.035^−30 ≈ 35.6%.
- Payment ÷ value = 75,000 ÷ 2,142,857 = 3.5%, the discount rate.
| Discount rate per period (%) | Present value of the perpetuity |
|---|---|
| 0.875 | $8,571,429 |
| 1.75 | $4,285,714 |
| 2.62 | $2,857,143 |
| 3.5 | $2,142,857 |
| 4.38 | $1,714,286 |
| 5.25 | $1,428,571 |
| 7 | $1,071,429 |
How to read the result
The value is very sensitive to r: at 3% the same payment is worth 2.5 million, at 4% only 1.875 million. That sensitivity is why endowments and analysts debate the right discount rate. In valuation, the growing perpetuity is the basis of the Gordon growth model and of terminal values in discounted cash flow models.
Common mistakes
- Using a growth rate equal to or above the discount rate.
- Mixing an annual payment with a monthly rate.
- Forgetting the (1 + r) adjustment when the first payment is today.
- Treating a long but finite annuity as a perpetuity without checking the difference.
Frequently asked questions
What is the formula for a perpetuity?
PV = C ÷ r for level payments, or C ÷ (r − g) for payments growing at g.
How much must an endowment hold to pay 75,000 a year forever?
At a 3.5% return, about 2.14 million, if returns and payouts stay steady.
What is the difference between a perpetuity and an annuity?
An annuity stops after a set number of payments; a perpetuity never stops.
Sources: Brealey, Myers & Allen, Principles of Corporate Finance (perpetuities and growing perpetuities) · Gordon, M. J. (1959). Dividends, Earnings, and Stock Prices. Review of Economics and Statistics, 41(2).