The delayed perpetuity formula is $PV_0 = (C / r) / (1 + r)^N$, where $C$ is the recurring payment, $r$ is the discount rate, and $N$ is the number of periods before the stream begins. You calculate it in two steps: value the perpetuity as $C / r$ at the point one period before the first payment, then discount that lump sum back to today. Getting the answer right depends almost entirely on counting the delay correctly and picking a discount rate that matches your cash flow.
What Makes a Perpetuity Delayed
A standard perpetuity pays a fixed amount $C$ at the end of every period, starting one period from now, forever. Its present value is $C / r$.1Investopedia. Perpetuity: Financial Definition, Formula, and Examples A delayed perpetuity is the same infinite stream, except nothing arrives for the first $N$ periods. Payments kick in after that waiting phase and continue forever.
You need three inputs:
- The payment amount $C$, meaning the fixed cash flow received each period once the stream begins.
- The discount rate $r$, meaning your required rate of return or cost of capital.
- The delay period $N$, meaning the number of full periods that pass before any cash hits your account.
The delay period is what separates this from a plain perpetuity. With a regular perpetuity, the first payment arrives at the end of Period 1. With a delayed perpetuity, the first payment arrives at the end of Period $N + 1$, after the entire idle phase has elapsed. That gap demands an extra round of discounting that many people either forget or miscount.
The Two-Step Calculation
Every delayed perpetuity valuation follows the same logic. Pretend you’re standing at the moment just before payments begin, value the infinite stream from that vantage point, then ask how much that future lump sum is worth today.
Step 1: Value the Perpetuity at the End of the Delay
Imagine the delay period has just ended. You’re now at Time $N$, and the first payment of $C$ is about to arrive one period later at Time $N + 1$. From this vantage point, you’re looking at a plain ordinary perpetuity, so its value is the standard formula:
$PV_N = C / r$
Suppose you expect $100 per year forever, starting after a three-year delay, and your discount rate is 5%. Standing at the end of Year 3, the perpetuity is worth $100 / 0.05 = $2{,}000$.1Investopedia. Perpetuity: Financial Definition, Formula, and Examples That $2,000 is a single number capturing every future $100 payment from Year 4 onward. But it’s a future value sitting three years away, not today’s price.
Step 2: Discount That Value Back to Today
Treat the $2,000 from Step 1 as a lump sum arriving at Time $N$ and bring it back to Time 0:
$PV_0 = PV_N / (1 + r)^N$
In the example: $PV_0 = 2{,}000 / (1.05)^3 = 2{,}000 / 1.157625 \approx 1{,}727.68$.
So $1,727.68 is what you should pay today for the right to collect $100 every year forever, with the first check arriving in four years.2Investopedia. How to Calculate the Present Value of a Delayed Perpetuity Combining both steps 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, and if you don’t pin down what your variable means, you’ll be off by a factor of $(1 + r)$.
Here’s the rule that keeps everything straight: the standard perpetuity formula $C / r$ always gives you 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. From there, count how many periods separate that point from today, and that’s your exponent.2Investopedia. How to Calculate the Present Value of a Delayed Perpetuity
You might see the formula written as $PV_0 = (C / r) / (1 + r)^{n-1}$, where $n$ is the period of the first payment rather than the length of the delay. That looks different but gives the same answer. If the first payment is in Year 4, then $n = 4$ and the exponent is $4 – 1 = 3$, identical to using $N = 3$ as the delay. Confirm which convention you’re working with before plugging in numbers.
A Quick Cross-Check
A delayed perpetuity is just a regular perpetuity with the first $N$ payments lopped off. That means you can also calculate it as the value of a regular perpetuity minus the present value of an $N$-period annuity paying the same $C$. Both methods must produce the same number. If they don’t, there’s a timing error somewhere.
The Growing Version
Fixed payments are a simplification. Cash flows from infrastructure projects, real estate, or company dividends tend to grow over time. A growing delayed perpetuity assumes payments increase at a constant rate $g$ each period after the first one.
The only change is in Step 1. Instead of $C / r$, the value at the end of the delay becomes:
$PV_N = C / (r – g)$
Step 2 stays the same. 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 of the stream blows up to infinity, which means the model no longer applies and you need a different framework.[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.
Perpetuity Due Adjustment
Everything above assumes an ordinary perpetuity, where each payment arrives at the end of its period. A perpetuity due flips the timing so payments arrive at the beginning of each period. The distinction sounds minor, but it shifts every cash flow forward by one period, which changes the math.
For a delayed perpetuity due where the first payment falls at Time $N$ rather than Time $N + 1$, the standard formula $C / r$ gives the value one period before the first payment, which is now Time $N – 1$. You discount back $N – 1$ periods rather than $N$:
$PV_0 = (C / r) / (1 + r)^{N-1}$
The result is exactly $(1 + r)$ times larger than the ordinary version, because every payment arrives one period sooner. Certain lease agreements and annuity contracts pay at the start of each period, so the due variant applies there. Forgetting the distinction in a large valuation means overcharging or undercharging by a full period’s worth of time value on every payment into infinity.
Why the Discount Rate Dominates the Result
The perpetuity formula has an uncomfortable property. Small changes in the discount rate cause enormous swings in value. Because $r$ sits in the denominator by itself, the relationship is hyperbolic rather than linear.
Take the earlier example of $100 per year starting after a three-year delay. Nudging the discount rate by one percentage point in either direction produces the following:
- 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% increases the value by about 29%. Moving from 5% to 6% decreases it by 19%. A two-percentage-point swing from 4% to 6% nearly cuts the value in half. The sensitivity only gets worse with longer delays: stretch the delay to ten years and the same rate swing from 4% to 6% creates a spread of roughly $1,689 versus $935.
The practical takeaway is that your discount rate assumption matters far more than precision in any other input. If you’re presenting a valuation that rests on a delayed perpetuity, show the result at two or three different discount rates so the audience understands the range of reasonable outcomes.
Real Versus Nominal Discount Rates
Because a perpetuity stretches to infinity, inflation has plenty of time to erode purchasing power. Whether you use a real or nominal discount rate changes the result significantly, and the choice has to match your cash flow assumption.
The nominal interest rate is the rate quoted in markets, and it includes a built-in cushion for expected inflation. The real interest rate strips that inflation component out, reflecting only the true return on capital. The approximate relationship is: real rate ≈ nominal rate − expected inflation.
The rule is straightforward. If your cash flow $C$ is stated in today’s dollars and will not 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, will quietly produce a number that’s way off. For a growing perpetuity where the growth rate $g$ already accounts for inflation, the nominal discount rate is appropriate and the inflation adjustment is effectively embedded in the $r – g$ spread.