Perpetuity: Definition, Formulas, and Examples
What is a perpetuity?
A perpetuity is a financial instrument or cash-flow stream that pays a fixed amount at regular intervals forever — it has no maturity date. In practice, perpetuities are rare as real securities, but the concept is central to valuation models (for example, the dividend discount model) and to calculating terminal value in discounted cash flow (DCF) analysis.
Why an infinite stream can have a finite value
Even though payments continue indefinitely, the present value (PV) of those payments can be finite because each future payment is discounted. As time passes, each payment is worth less in today’s dollars due to the time value of money; the sum of an infinite sequence of discounted payments converges to a finite number when an appropriate discount rate is used.
Explore More Resources
Basic formulas
- Level (fixed) perpetuity
- PV = C / r
- C = cash payment received each period (first payment one period from now)
-
r = discount rate per period
-
Growing perpetuity (payments grow at constant rate g each period)
- PV = C1 / (r − g)
- C1 = cash payment in the next period (i.e., first payment)
-
g = constant growth rate (must satisfy r > g for the formula to converge)
-
Terminal value (used in valuation at the end of an explicit forecast period)
- If CFn is the cash flow in year n, the terminal value at year n is:
TVn = CFn × (1 + g) / (r − g) = Cn+1 / (r − g) - To get today’s value, discount TVn back to present: TV0 = TVn / (1 + r)^n
Example: growing perpetuity as terminal value
A company is projected to have cash flow of $100,000 in year 10. Assume a long-term growth rate g = 3% and cost of capital r = 8%. The terminal value at year 10 equals the value of a growing perpetuity starting in year 11:
TV10 = $100,000 × (1 + 0.03) / (0.08 − 0.03) = $103,000 / 0.05 = $2,060,000
Explore More Resources
To include that in a DCF, you would discount the $2.06 million back to present value using (1 + r)^10.
Growing vs. fixed perpetuity
- Fixed perpetuity: identical payments forever; PV = C / r.
- Growing perpetuity: payments increase at a constant rate g; PV = C1 / (r − g). Because payments grow, a growing perpetuity has a higher present value than a fixed one (assuming r > g).
Practical use and limitations
- Perpetuities are uncommon as actual securities today (historical example: British consols, phased out in 2015), but the perpetuity concept is widely used:
- Valuing stocks via dividend discount models.
- Estimating terminal value in corporate valuations.
- Important assumptions and cautions:
- For a growing perpetuity, the discount rate must exceed the growth rate (r > g); otherwise the PV is undefined or infinite.
- Real-world growth rates are unlikely to remain constant forever; the model is an approximation.
- Results are sensitive to the chosen discount rate and growth rate.
Perpetuity versus annuity
- Annuity: fixed stream of payments for a predetermined number of periods (has a maturity).
- Perpetuity: payments continue without end (no maturity).
- Both are valued by discounting future cash flows, but the formulas differ because of the duration.
Key takeaways
- A perpetuity pays a constant (or systematically growing) cash flow forever.
- PV of a level perpetuity: PV = C / r.
- PV of a growing perpetuity: PV = C1 / (r − g), with r > g required.
- Perpetuity concepts are essential in DCF valuation and terminal value calculations, despite few true perpetual securities existing today.