This calculator gives estimates for education and planning. It is not investment, tax or legal advice; for decisions about your own situation, talk to a qualified financial professional.
Valuing payments that never end
A perpetuity is a series of payments that continues forever. It sounds infinite, yet its value today is finite, because each payment is discounted: a dollar received in 50 or 100 years is worth only cents now. The formula is one of the building blocks of finance, used to value preferred stock, real estate income, endowments and the terminal value of a company.
The formulas
With C the first payment, r the discount rate and g the growth rate of the payments, all per period:
- Level perpetuity: PV = C ÷ r
- Growing perpetuity: PV = C ÷ (r − g), valid only when r > g
- Perpetuity due, with the first payment today: multiply by (1 + r)
- Share of value from the first n periods: 1 − ((1 + g) ÷ (1 + r))ⁿ
Example
A payment of $1,000 a year, growing by 2% a year, discounted at 5%:
- Present value: 1,000 ÷ (0.05 − 0.02) = $33,333
- Without growth, the same payment is worth 1,000 ÷ 0.05 = $20,000
- The first 10 years account for only 25.2% of the growing perpetuity’s value
- Paying $33,333 for it gives a starting yield of 3%
The Gordon growth model
Myron Gordon applied the growing perpetuity to stocks: price = D₁ ÷ (r − g), where D₁ is next year’s dividend, r the return investors require and g the long-term dividend growth. A stock paying $2 next year, with a required return of 8% and 4% growth, is worth 2 ÷ 0.04 = $50. The same formula gives the terminal value in a discounted cash flow model, usually with a growth rate close to long-run inflation or GDP growth.
Sensitivity
Because r − g sits in the denominator, the value is very sensitive to both rates when they are close. In the example above, raising g from 2% to 3% increases the value by half, to $50,000. Small changes in assumptions can swing a valuation widely, which is why analysts test several scenarios.
Real-world perpetuities
British consols, issued from the 18th century, paid interest forever until the government redeemed the last of them in 2015. Today, perpetual bonds issued by banks, preferred shares with fixed dividends and ground rents behave like perpetuities. A university endowment that spends a fixed share of its value each year aims to support its programs in perpetuity, which requires its return to exceed its spending plus inflation.