A perpetuity, sometimes called a perpetual annuity, is a cash-flow stream assumed to continue at regular intervals indefinitely. A level perpetuity pays the same amount each period; a growing perpetuity changes at a constant rate. Perpetuity formulas are useful in finance, but their values are highly sensitive to payment timing, discount rates, growth assumptions, and whether the cash flow can be sustained.
Key Takeaways
- A level perpetuity paying (C) one period from now has present value (C/r).
- A growing perpetuity paying (C_1) next period has present value (C_1/(r-g)), provided (r>g).
- The numerator must be the next expected cash flow, not an unlabeled current or historical amount.
- Cash-flow and discount-rate periods, currencies, tax bases, and inflation treatment must match.
- A terminal value calculated at the end of a forecast period must still be discounted to the valuation date.
- Small changes in (r-g) can create large valuation changes, especially when the spread is narrow.
- A perpetuity model does not guarantee that a security will pay forever or that a business will survive indefinitely.
For equal end-of-period payments beginning one period after the valuation date:
$$
PV=\frac{C}{r}
$$
where:
- (PV) is the value one period before the first payment;
- (C) is the cash flow each period; and
- (r) is the required return or discount rate per matching period, expressed as a decimal.
The formula assumes the first payment is one full period away. If a payment is due immediately, add that payment separately because it is not discounted.
Worked Example: Level Perpetuity
Assume an instrument is expected to pay 100 at the end of every year indefinitely and the appropriate annual discount rate is 5%:
$$
PV=\frac{100}{0.05}=2{,}000
$$
The value is 2,000 immediately after the most recent payment, assuming the next 100 payment is one year away. At a price of 2,000, the annual cash yield is 5%:
$$
\frac{100}{2{,}000}=5\%
$$
This does not establish a market price or guarantee the payment. Credit risk, call provisions, payment deferral, liquidity, taxes, and changing required returns can materially change value.
Why an Infinite Stream Can Have Finite Value
A level perpetuity is an infinite geometric series:
$$
PV=\frac{C}{1+r}+\frac{C}{(1+r)^2}+\frac{C}{(1+r)^3}+\cdots
$$
When (r>0), each later cash flow receives a smaller present-value weight. The series converges to (C/r). If the discount rate is zero, the sum does not converge to a finite value under this formula.
The finite result is mathematical. It does not mean very distant forecasts are reliable; it means discounting causes their individual present values to approach zero.
If the next-period cash flow is (C_1) and cash flows then grow at constant rate (g):
$$
PV=\frac{C_1}{r-g}
$$
The standard formula requires:
$$
r>g
$$
The rate and growth assumptions must use the same period and basis. A nominal discount rate should be paired with nominal cash flows; a real discount rate should be paired with real cash flows.
If the most recent cash flow is (C_0), first calculate next period’s amount:
$$
C_1=C_0(1+g)
$$
Using (C_0) directly in the numerator understates value when (g>0).
Worked Example: Growing Perpetuity
Assume the next annual cash flow is 120, long-run growth is 2%, and the annual discount rate is 8%:
$$
PV=\frac{120}{0.08-0.02}=2{,}000
$$
The 120 must be the year-one cash flow. If 120 were the just-paid amount instead, the next cash flow would be 122.40 and the value would be 2,040 under the same assumptions.
Sensitivity to Discount Rate and Growth
Holding next year’s cash flow at 120 and the discount rate at 8%:
| Perpetual growth | Spread (r-g) | Value |
|---|
| 0% | 8% | 1,500.00 |
| 1% | 7% | 1,714.29 |
| 2% | 6% | 2,000.00 |
| 3% | 5% | 2,400.00 |
| 4% | 4% | 3,000.00 |
The value doubles as the denominator falls from 8% to 4%. This convex sensitivity makes unsupported terminal-growth or discount-rate changes especially dangerous. A narrow spread can produce a large number that appears precise but rests on fragile assumptions.
Terminal Value in a DCF Model
Analysts often use a growing perpetuity to estimate the value of cash flows after an explicit forecast period. Suppose year-five free cash flow is 10.0 million, stable growth is 2.5%, and the matching discount rate is 8.5%.
First estimate year-six cash flow:
$$
FCF_6=10.0(1.025)=10.25
$$
Then calculate terminal value at the end of year five:
$$
TV_5=\frac{10.25}{0.085-0.025}=170.83
$$
That is a year-five value, not a present value. Discount it five years to the valuation date:
$$
PV(TV)=\frac{170.83}{(1.085)^5}\approx113.61
$$
The analyst must then add the present value of years one through five. Omitting the explicit cash flows or failing to discount terminal value creates a material overstatement.
Cash Flow and Discount Rate Must Match
| Cash flow being valued | Matching rate concept |
|---|
| Cash flow available to all capital providers | Weighted average cost of capital or another enterprise discount rate |
| Cash flow available only to equity holders | Cost of equity or another equity discount rate |
| Fixed contractual payment | Yield or required return reflecting timing, credit, options, and liquidity |
| Real cash flow excluding inflation | Real discount rate |
| Nominal cash flow including inflation | Nominal discount rate |
This table states consistency principles, not a universal prescription. The instrument, accounting purpose, jurisdiction, and valuation standard can require more specific treatment.
Perpetuity vs. Annuity
| Feature | Perpetuity | Annuity |
|---|
| Number of payments | Indefinite | Finite |
| Level-payment present value | (C/r) | (C[1-(1+r)^{-n}]/r) |
| Main additional input | Sustainability | Number and timing of payments |
| Typical use | Terminal value, long-lived claims | Loans, leases, retirement payments |
An Annuity ending after 30 payments is not a perpetuity, even if the term is long. Conversely, a legal instrument with no stated maturity may still fail to behave like a reliable perpetuity if payments can be suspended or the issuer can call it.
Where Perpetuity Logic Appears
- Terminal value: A DCF model may treat post-forecast free cash flow as growing at a stable rate.
- Preferred shares: A fixed, indefinite dividend can sometimes be approximated as a level perpetuity, subject to issuer and contract risks.
- Consols and perpetual debt: No stated maturity can make fixed coupons resemble a perpetuity, but call, deferral, and credit terms matter.
- Endowments: A spending policy may seek an indefinite distribution stream, although returns and distributions vary.
- Infrastructure and real assets: Long-duration cash flows may use a continuing-value assumption rather than a literal infinite forecast.
The formula is a model, not evidence that the underlying cash flows satisfy its assumptions.
How to Evaluate a Perpetuity Assumption
- Identify the valuation date and first payment date.
- Confirm whether the numerator is current cash flow or next-period cash flow.
- Define the cash flow before choosing the discount rate.
- Match enterprise cash flow with an enterprise rate and equity cash flow with an equity rate.
- Keep periods, currency, inflation, and tax treatment consistent.
- Test whether long-run growth is economically sustainable.
- Link growth to reinvestment and return on capital when valuing a business.
- Review credit, call, deferral, cancellation, and finite-life risks.
- Discount a future terminal value back to the valuation date.
- Run sensitivity analysis across plausible (r) and (g) values.
Common Mistakes and Limitations
- Using the current cash flow instead of (C_1): Growing perpetuity value uses the next expected payment.
- Allowing (g\ge r): The standard formula no longer produces a finite convergent value.
- Mixing cash-flow definitions: Equity and enterprise cash flows require different discount-rate concepts.
- Mixing real and nominal inputs: Inflation treatment must be consistent.
- Forgetting terminal-value timing: Value at the forecast horizon is not value today.
- Assuming growth requires no reinvestment: Sustainable business growth generally consumes capital or depends on operating economics.
- Treating a legal perpetuity as a certain payment stream: Default, deferral, call, regulation, and restructuring can interrupt payments.
- Relying on one point estimate: A small denominator makes value highly sensitive to assumptions.
Public Source Checks
- OpenStax’s Perpetuities chapter derives the level and growing perpetuity formulas and explains their use in preferred-stock and corporate-finance valuation.
- New York University professor Aswath Damodaran’s present-value primer sets out level and growing perpetuity formulas, payment timing, and the requirement that perpetual growth remain below the discount rate.
- Damodaran’s terminal-value guidance explains why DCF models use a continuing-value assumption after an explicit forecast period.
- The Federal Reserve Bank of St. Louis Time Value of Money module covers present value, future value, interest, inflation, and opportunity cost.
- Present Value: The value at the valuation date of future cash flows discounted at a stated rate.
- Discount Rate: The rate used to translate future cash flow into present value.
- Annuity: A finite series of regular payments.
- Discounted Cash Flow: A valuation method that discounts forecast cash flows and a supported continuing value.
- Preferred Stock: An equity security whose stated dividends may sometimes be modeled using perpetuity logic.
- Time Value of Money: The broader framework for comparing cash flows at different dates.
FAQs
Is a perpetual annuity the same as a perpetuity?
Usually yes. Both terms describe a payment stream modeled as continuing indefinitely. Confirm the context because an insurance annuity contract may have life-contingent, contractual, or product-specific terms that a simple perpetuity formula does not capture.
Can a perpetuity have a finite present value?
Yes. When a positive discount rate causes later payments to receive progressively smaller present-value weights, the infinite level-payment series converges to a finite value.
Why must the discount rate exceed the perpetual growth rate?
The standard growing-perpetuity series converges only when the discount rate exceeds growth. A smaller or equal discount rate also implies an unstable long-run valuation assumption under this model.
Is terminal value the same as present value?
No. Terminal value is normally measured at the end of the explicit forecast period. It must be discounted back to the valuation date before being combined with earlier cash flows.
Does perpetual growth mean a company grows faster forever?
No. A perpetuity model assumes a stable long-run growth rate. The assumption should be economically sustainable and consistent with reinvestment, returns on capital, inflation, and the valuation currency.
This article is educational only and does not provide individualized investment, valuation, accounting, tax, or legal advice.