Delayed Perpetuity: Formula, Two-Step Calculation, and Sensitivity

The delayed perpetuity formula is $PV_0 = (C / r) / (1 + r)^N$, where $C$ is the recurring payment, $r$ is the discount rate per period, and $N$ is the number of periods before the value $C/r$ is measured. You compute it in two steps: first value the perpetuity as $C/r$ at the point one period before the first payment arrives, then discount that lump sum back to today.

What the Delay Changes

A standard perpetuity pays $C$ at the end of every period starting one period from now, and its present value is $C/r$.¹ A delayed perpetuity is the same infinite stream, except nothing arrives for the first $N$ periods. The first check lands at the end of Period $N+1$, and payments continue forever from there.

Three inputs drive the calculation:

The recurring payment $C$ once the stream begins.
The discount rate $r$, meaning your required return or cost of capital per period.
The delay $N$, the number of full periods that pass before any cash arrives.

The delay is what forces an extra round of discounting on top of the ordinary perpetuity value. Miscount it and the answer is off by a factor of $(1+r)$.

The Two-Step Calculation

The logic behind the formula is worth understanding, because it protects you from the timing errors that plague these problems.

Step 1: Value the Perpetuity at the End of the Delay

Stand at Time $N$, the moment just before payments begin. From that vantage point, the first payment arrives one period later at Time $N+1$, which is exactly the setup of an ordinary perpetuity. So the value at Time $N$ is:

$PV_N = C / r$

Suppose you expect $100 per year forever, the first payment arrives four years from now (a three-year delay), and your discount rate is 5%. Standing at the end of Year 3, the perpetuity is worth $100 / 0.05 = 2{,}000$.¹ That single number captures every $100 payment from Year 4 onward, but it’s a value sitting three years in the future.

Step 2: Discount Back to Today

Treat the $PV_N$ from Step 1 as a lump sum arriving at Time $N$ and bring it to Time 0:

$PV_0 = PV_N / (1 + r)^N$

Continuing the example: $PV_0 = 2{,}000 / (1.05)^3 = 2{,}000 / 1.157625 \approx 1{,}727.68$.²

Combined into one expression:

$PV_0 = (C / r) / (1 + r)^N$

Getting the Exponent Right

The single most common mistake with delayed perpetuities is miscounting the exponent by one period. Different textbooks define the delay variable differently, so pin down what your variable means before plugging in numbers.

One rule keeps things straight: $C/r$ always gives the value exactly one period before the first payment. If the first payment arrives at Time 4, then $C/r$ is the value at Time 3. If the first payment arrives at Time 10, then $C/r$ is the value at Time 9. Count the periods from that point back to today, and that number is your exponent.²

You may see the formula written as $PV_0 = (C / r) / (1 + r)^{n-1}$, where $n$ is the period in which the first payment arrives rather than the length of the wait. It produces the same answer. If the first payment is in Year 4, then $n = 4$ and the exponent is $4 – 1 = 3$, identical to using $N = 3$.

A Cross-Check

A delayed perpetuity is a regular perpetuity with the first $N$ payments removed. You can therefore compute it a second way: the value of a regular perpetuity minus the present value of an $N$-period ordinary annuity paying the same $C$. Both methods must produce the same number. If they don’t, there’s a timing error somewhere in your setup.

The Growing Delayed Perpetuity

Cash flows often grow rather than staying flat. If payments increase at a constant rate $g$ per period after the first one, Step 1 changes to:

$PV_N = C / (r – g)$

Step 2 is unchanged. The combined formula is:

$PV_0 = (C / (r – g)) / (1 + r)^N$

This requires $g < r$. If the growth rate equals or exceeds the discount rate, the value blows up to infinity and the model no longer applies.[mfn]Investopedia. Perpetuity: Financial Definition, Formula, and Examples[/mfn] In practice, analysts typically use a long-run growth rate around 2% to 3% and a discount rate materially above that.

This is the same structure behind the Gordon Growth Model and the terminal value component of a discounted cash flow analysis.¹

How Sensitive the Answer Is to the Discount Rate

Because $r$ sits alone in the denominator of $C/r$, small changes in the discount rate cause outsized swings in value. Take the earlier example of $100 per year starting after a three-year delay:

At 4%: $PV_0 = (100 / 0.04) / (1.04)^3 = 2{,}500 / 1.1249 \approx 2{,}222$
At 5%: $PV_0 = (100 / 0.05) / (1.05)^3 = 2{,}000 / 1.1576 \approx 1{,}728$
At 6%: $PV_0 = (100 / 0.06) / (1.06)^3 = 1{,}667 / 1.1910 \approx 1{,}399$

Moving from 5% to 4% raises the value by about 29%. Moving from 5% to 6% cuts it by about 19%. A two-percentage-point swing from 4% to 6% nearly halves the number. The sensitivity gets worse with longer delays: stretch the wait to ten years and the same 4%-to-6% swing spreads the value from roughly $1,689 to $935.

The practical implication is that your discount rate assumption matters more than precision in any other input. When presenting a valuation that rests on a delayed perpetuity, show the result at two or three discount rates so the range of reasonable outcomes is visible.

Real Versus Nominal Discount Rates

Because the stream stretches to infinity, inflation has plenty of time to erode purchasing power, and the choice of rate has to match the cash flow assumption. Nominal rates are the ones quoted in markets and include expected inflation. Real rates strip inflation out. The approximate relationship: real rate ≈ nominal rate − expected inflation.

If $C$ is stated in today’s dollars and won’t adjust for inflation, use the nominal discount rate. If $C$ represents real purchasing power that will grow with inflation, use the real discount rate. Mixing a nominal rate with real cash flows, or the reverse, produces a number that is quietly wrong. For a growing perpetuity where $g$ already reflects inflation, use the nominal rate; the inflation adjustment sits inside the $r – g$ spread.

Perpetuity Due Adjustment

Everything above assumes an ordinary perpetuity, with each payment at the end of its period. A perpetuity due pays at the beginning of each period, shifting every cash flow forward by one.

For a delayed perpetuity due where the first payment lands at Time $N$ rather than Time $N+1$, the formula $C/r$ now measures value at Time $N-1$. Discount back $N-1$ periods:

$PV_0 = (C / r) / (1 + r)^{N-1}$

The result is exactly $(1+r)$ times larger than the ordinary version, because every payment arrives one period sooner. This distinction matters for instruments like certain lease agreements or annuity contracts where payments are due at the start of each period.

Where the Formula Gets Used

Delayed perpetuities model any situation where you pay upfront, wait through a silent period, and then collect cash flows indefinitely. Common applications include infrastructure projects with multi-year construction phases, preferred stock with deferred dividends, endowments and charitable trusts with waiting periods before distributions begin, and the terminal value component of a discounted cash flow model, where projected cash flows past the explicit forecast horizon are treated as a growing perpetuity and discounted back through the projection period.¹

1
Investopedia. Perpetuity: Financial Definition, Formula, and Examples
2
Investopedia. How to Calculate the Present Value of a Delayed Perpetuity