Perpetuity Formula: Present Value and Growing Perpetuity

The perpetuity formula values a stream of payments that continues forever. Divide the next constant payment by the periodic discount rate, or divide the next growing payment by the discount rate minus the constant growth rate.

What each input means

C1 is the next payment, not a payment from the past. The subscript matters most in a growing perpetuity. If the most recent cash flow was $100 and it will grow 3% before the next payment, the numerator is $103. For a level perpetuity, every payment is the same, so the distinction is less visible but the first payment still normally occurs one period after the valuation date.

r is the return required for the cash flow's risk and timing. It is not automatically a bank rate. In an endowment example it might be a sustainable expected return. In equity valuation it may be the cost of equity. In a terminal-value calculation for free cash flow to the firm it is commonly a weighted average cost of capital. The rate must use the same period as the cash flow: an annual payment needs an annual rate.

g is a long-run constant growth assumption. It belongs only in the growing formula. The model requires r to exceed g; otherwise the infinite discounted series does not converge to a finite positive value. A long-run growth rate should also make economic sense. A company cannot outgrow the economy forever without eventually becoming implausibly large.

Why the timing is one period back

A perpetuity formula returns the value immediately before its first payment. If the first annual payment is at the end of next year, the result is a value today. If a terminal-value forecast has its first continuing cash flow in year 6, the formula produces a value at the end of year 5. That year-5 value must still be discounted to today.

This rule prevents a frequent one-year error. Do not treat a year-5 terminal value as though it were already a time-0 value. Put the first perpetuity cash flow on a timeline, step back one period, and label that point as the formula's valuation date.

For the endowment to distribute $84,600 at each year-end while preserving principal under the simplified constant-return assumption, it needs $1,781,052.63 at the start. At 4.75%, that balance earns $84,600 over the year; paying out those earnings returns the fund to its opening balance. Real endowments face variable returns, fees, and inflation, so this is a valuation illustration rather than a spending policy.

The growing case is worth $1,976,562.50 one period before the $126,500 cash flow. Notice that the denominator is only 6.4%, the difference between required return and growth. That makes the estimate very sensitive to either assumption. If r and g move close together, a small input change can produce a large value change. This sensitivity is a reason to show a range of terminal values rather than treating one result as exact.

Constant versus growing perpetuity

Use the constant formula when the payment amount stays fixed forever. Traditional examples include preferred shares with a fixed dividend and an endowment distributing only a fixed dollar amount. Because a fixed payment loses purchasing power when prices rise, a constant perpetuity is not the same as constant real spending.

Use the growing formula when each payment is expected to increase at the same percentage indefinitely. It often appears in the Gordon growth model and in discounted cash flow terminal value. The first cash flow in the numerator must be the one immediately after the valuation date. If you only have the cash flow at the valuation date, grow it once before inserting it.

Common mistakes

Using the current cash flow instead of the next cash flow. In a growing perpetuity, the numerator is C1. If C0 is the latest observed cash flow, apply one growth step first.

Forgetting where the value sits. A perpetuity starting in year 6 is valued at the end of year 5. Discount that result back five years to obtain time-0 value.

Allowing g to equal or exceed r. The closed-form growing-perpetuity result requires r greater than g. When the denominator is zero or negative, the assumptions do not describe a convergent present value.

Mixing periods. A monthly payment divided by an annual rate is not a valid match. Convert both inputs to monthly terms or both to annual terms.

Using a perpetuity for a finite stream. If payments end after 12 years, use an annuity or discount the 12 cash flows. “Very long” and “forever” are different assumptions.

Hiding sensitivity. A terminal value may be a large portion of a DCF estimate. Recalculate with reasonable combinations of r and g so the reader can see how much of the result depends on the forecast.

A quick reasonableness check

For a level perpetuity, multiplying the calculated value by r should reproduce the annual payment. For a growing perpetuity, multiplying the value by the r-minus-g spread should reproduce the next cash flow. This check catches percentage-entry mistakes, especially entering 4.75 rather than 0.0475. It does not prove that the chosen rates are reasonable; it proves only that the arithmetic matches the model.

Immediate and delayed first payments

A perpetuity due, with its first level payment made immediately, needs separate timing treatment. One payment sits at the valuation date and the remaining level stream starts one period later. Add the immediate payment to the value of the remaining ordinary perpetuity. Do not shift the first payment earlier while silently retaining the standard timing assumption.

A delayed perpetuity has the opposite issue. First value the stream one period before its first payment, then discount that single value back to the required date. For example, if the first payment occurs at the end of year 8, the standard formula gives a value at the end of year 7. Seven additional years of discounting are needed to reach time 0.

By the FinanceBrain Team · Last verified July 12, 2026 · How we produce and verify articles