Deferred Perpetuities and Their First Payment

Why the First Payment Date Matters

A perpetuity is a stream of equal payments that continues indefinitely. When the first payment is delayed, the perpetuity formula still provides a useful value—but not necessarily at the date you are asked to value. The key is to place that value at the right time: one payment period before the first payment. Then discount it back to the requested valuation date.

The main source of error is counting the delay incorrectly. If the first payment is at the end of year 8, a level annual perpetuity is valued at year 7, not at year 8. The amount at year 7 is then discounted seven years to find its value at time 0.

Definition: Suppose level payments of $C$ are made at the end of each period, beginning at time $n$, where $n$ is a positive integer. If the effective interest rate per payment period is $i>0$, the value at time 0 of this deferred perpetuity is $$ PV_0=\frac{C}{i}v^{n-1}, \qquad v=\frac{1}{1+i}. $$ The amount $C/i$ is the perpetuity value at time $n-1$, one period before the first payment. The factor $v^{n-1}$ discounts that amount to time 0.

Here, “time $n$” means that the first payment occurs at the end of the $n$th period. Thus, a perpetuity beginning at the end of year 1 is an immediate perpetuity valued at time 0, with no extra deferral. A perpetuity beginning at the end of year 6 is valued first at time 5, then discounted five periods to time 0.

Deriving the Deferred-Perpetuity Formula

Assume payments are certain, level, and made at the end of every period from time $n$ onward. Assume also that the rate $i$ is constant and positive for every period. At time $n-1$, the first payment is one period away, the second is two periods away, and so on. The value at that date is therefore the infinite sum

C v+C v^2+C v^3+\cdots

This is a geometric series with first term $Cv$ and common ratio $v$. Since $i>0$, $v=1/(1+i)$ satisfies $0<v<1$, so the infinite series converges. Its sum is

\frac{Cv}{1-v} =\frac{Cv}{1-\frac{1}{1+i}} =\frac{Cv}{\frac{i}{1+i}} =\frac{C}{i}.

This establishes why $C/i$ is located at time $n-1$: it is the value, at that date, of all payments beginning one period later. To move the value to time 0, discount it $n-1$ periods:

PV_0=\frac{C}{i}v^{n-1}.

The formula requires a positive rate, a constant rate per payment period, and an indefinitely continuing level stream of end-of-period payments. At $i=0$, a positive level perpetuity has no finite present value. If the first payment is at time 0, the payment timing differs from the assumptions above, so this formula does not apply as written.

A Reliable Timing Method

Before calculating, put the valuation date and first payment date on a timeline. Then find the date exactly one period before the first payment. This prevents the common off-by-one error: discounting from the first payment date, or counting the first payment itself as a discount period.

Fix the period and rate.
Identify the payment interval and use the effective interest rate per that interval. For annual payments, use an effective annual rate.

Locate the first payment.
Record its time relative to the requested valuation date. If the first payment is at time $n$, it is the end of period $n$.

Place $C/i$ one period earlier.
The perpetuity value is at time $n-1$, immediately before the first payment date.

Discount to the requested date.
Multiply by $v^{n-1}$ when the valuation date is time 0. For another valuation date, count the periods between that date and time $n-1$.

Worked Examples

Worked Example: Valuing a Perpetuity Beginning in Year 8

A hypothetical trust will pay $ \$1{,}800 $ at the end of each year forever, with its first payment at the end of year 8. The effective annual interest rate is 5%. Find the value at time 0, to the nearest cent.

Method. First locate the perpetuity value immediately before the first payment. Since the first payment is at time 8, that value is at time 7. The amount at time 7 is $C/i=\$1{,}800/0.05=\$36{,}000$.

There are seven years from time 0 to time 7, so discount by $v^7$, not $v^8$:

v^7=\frac{1}{(1.05)^7} =\frac{1}{1.40710042265625}, \qquad PV_0=\frac{36{,}000}{1.40710042265625} =\$25{,}584.527\ldots

Rounding only the final result to two decimal places gives $\$25{,}584.53$. As a check, this is less than the $ \$36{,}000 $ value at year 7, as it should be when discounting at a positive rate. The common mistake is to discount eight periods because the first payment is in year 8; that incorrectly places the perpetuity value at year 8 instead of year 7.

Worked Example: Valuing From a Later Valuation Date

A hypothetical foundation pays $ \$2{,}160 $ annually forever, with the first payment at the end of year 5. The effective annual interest rate is 5%. Find the value at time 2, to the nearest cent.

Method. The first payment is at time 5, so the perpetuity value is at time 4. Its value there is $ \$2{,}160/0.05=\$43{,}200 $.

The requested date, time 2, is two years before time 4. Therefore, discount two periods:

PV_2=\frac{43{,}200}{(1.05)^2} =\frac{43{,}200}{1.1025} =\$39{,}183.673469\ldots

Rounded to two decimal places, the value at time 2 is $\$39{,}183.67$. The time difference is $4-2=2$ periods. Discounting from time 5 would be wrong because the $ \$43{,}200 $ value is located at time 4, not at the date of the first payment.

Worked Example: Inferring the First Payment Date

A hypothetical endowment pays $ \$10{,}000 $ at the end of each year forever. The effective annual interest rate is 25%, and its value at time 0 is $ \$20{,}480 $. Assuming the first payment occurs at the end of an integer year, find the year of the first payment.

Method. If the first payment occurs at time $n$, the perpetuity value is $C/i=\$10{,}000/0.25=\$40{,}000$ at time $n-1$. The given present value therefore satisfies

20{,}480=40{,}000v^{n-1}, \qquad v=\frac{1}{1.25}=0.8.

Divide by $ \$40{,}000 $: $20{,}480/40{,}000=0.512$. Since $0.8^3=0.512$, it follows that $n-1=3$, hence $n=4$. The first payment is at the end of year 4.

Check. The value at time 3 is $ \$40{,}000 $; discounting three periods gives $ \$40{,}000(0.8)^3=\$20{,}480 $, exactly the stated value. A common mistake is to interpret the three discount periods as meaning the first payment is at time 3. They lead to the value date $n-1$; the first payment is one period later, at time 4.

Common Timing Errors and Practical Checks

The most useful check is to ask where the amount $C/i$ is located before applying any discount factor. It is always one full payment period before the first payment under the end-of-period assumptions. The number of discount periods is the distance from the requested valuation date to that location—not automatically the year number of the first payment.

First payment date	Location of $C/i$	Discount periods from time 0
End of year 1	Time 0	0
End of year 4	Time 3	3
End of year 8	Time 7	7

The rate must also match the period. If payments are annual, the formula uses an effective annual rate; if payments are made at another frequency, the rate must be appropriate to that payment period. Do not combine an annual payment interval with a rate quoted for a different interval without first establishing an equivalent periodic rate.

Finally, the formula values a level perpetuity. If payment amounts change over time, they are not all represented by the same $C$; if payments stop after a finite number of periods, the stream is not a perpetuity. Those differences change the valuation problem and must be handled using assumptions appropriate to the cash flows.

Key takeaway: For level payments $C$ at the end of each period, with the first payment at time $n$, locate $C/i$ at time $n-1$. Discount from that date to the requested valuation date, counting periods from the value date—not from the first payment date.

Check Your Understanding

For each problem, identify the date of the first payment and the date at which $C/i$ is located before calculating.

A hypothetical fund pays $ \$3{,}000 $ annually forever, beginning at the end of year 4. The effective annual rate is 25%. Find its value at time 0. (Answer: $ \$6{,}144.00 $)
A hypothetical grant pays $ \$9{,}600 $ annually forever, beginning at the end of year 7. The effective annual rate is 25%. Find its value at time 3. (Answer: $ \$19{,}660.80 $)
A level annual perpetuity is worth $ \$18{,}000 $ at time 0. Its first payment is at the end of year 4, and the effective annual rate is 20%. Find the annual payment. (Answer: $ \$6{,}220.80 $)
A hypothetical perpetual payment of $ \$7{,}500 $ is made annually at an effective annual rate of 25%. Its value at time 0 is $ \$15{,}360 $. If the first payment is at the end of an integer year, identify that year. (Answer: End of year 4)
A hypothetical perpetuity pays $ \$5{,}000 $ annually, with its first payment at the end of year 6. The effective annual rate is 25%. Find its value at time 2 and at time 4. (Answer: $ \$10{,}240.00 $ at time 2; $ \$16{,}000.00 $ at time 4)

What You'll Learn