Perpetuities and Their Present-Value Formula

What Makes a Payment Stream a Perpetuity?

A long-term agreement may describe payments that continue without a stated final payment. A level perpetuity is the idealized version of such a stream: equal payments continue indefinitely. Its value is not the sum of the undiscounted payments, which would be unbounded, but the present value of those payments after discounting each one to a specified valuation date.

Definition: A level perpetuity is a stream of equal payments of amount $C$ that continues indefinitely at regular intervals. An ordinary level perpetuity pays at the end of each interval. Its present value is measured immediately before the first payment is due.

The interest rate must match the payment interval. For example, with annual payments, use an effective annual rate $i$; with quarterly payments, use the effective rate per quarter. The ordinary-perpetuity formula below requires a fixed, positive rate per payment period. It also assumes that the payments are deterministic, level, and certain to be made: there is no mortality contingency, payment risk, or random payment amount in the model.

For annual payments of $C$, the first payment is at time 1, the second at time 2, and so on. If the annual effective rate is $i$, the present value of a payment at time $k$ is $Cv^k$, where $v=1/(1+i)$. This timing convention is essential: a payment made immediately at time 0 is not discounted and requires a timing adjustment.

Deriving the Present-Value Formula

Start with a finite ordinary annuity of $n$ equal payments. Its present value at time 0 is the sum of the discounted payments. For $i>0$, the finite geometric sum can be written as follows:

C(v+v^2+\cdots+v^n) =C\frac{v(1-v^n)}{1-v} =C\frac{1-v^n}{i}.

The last equality follows from $1-v=i/(1+i)=iv$. Now let the number of payments increase without bound. Because $i>0$, $v=1/(1+i)$ lies strictly between 0 and 1, so $v^n$ tends to 0 as $n$ tends to infinity. Therefore, the value of the finite annuity tends to the value of the perpetuity:

\lim_{n\to\infty}C\frac{1-v^n}{i} =C\frac{1-0}{i} =\frac{C}{i}.

Theorem: For a level ordinary perpetuity with payment $C$ at the end of every period and a fixed effective rate $i>0$ per payment period, the present value immediately before the first payment is $$ PV=\frac{C}{i}. $$

The limit explains why the finite-annuity formula becomes $C/i$: the term representing the discounted value of the distant final payment, $v^n$, disappears as the finite stream is extended indefinitely. The formula is not valid at $i=0$, because then there is no discounting and the sum of infinitely many positive payments does not converge. It also does not apply with a negative rate: for $-1<i<0$, $v>1$, so the discounted terms do not approach zero.

Applying the Formula Carefully

A reliable calculation begins with the timeline, not with a memorized expression. Confirm the payment amount, the interval between payments, the rate per interval, and whether the first payment is at the end or beginning of an interval. For an ordinary perpetuity, the first payment occurs one period after the valuation date, so the formula $C/i$ is directly applicable.

Set the valuation date.
For an ordinary perpetuity, value the stream immediately before its first end-of-period payment.

Match the rate to the payment period.
Use the effective rate for one payment interval, and verify that it is positive.

Apply the limit formula.
Divide the level payment per period by the positive rate per period.

Check the timing and units.
Confirm that the first payment is not immediate and that the result is expressed in the same currency units as the payment.

Worked Example: A Trust With Payments Continuing Indefinitely

A hypothetical trust makes a fixed payment of $8,400 at the end of every year, beginning one year from today, and continues indefinitely. The effective annual interest rate is 3.5%. Find the present value of the full stream and the present value of only the first two payments. Assume payments are certain and level; there is no mortality contingency or random payment distribution.

Method. Treat the full stream as an ordinary level perpetuity. The payment interval is one year, so the rate per period is $i=0.035$. Since $i>0$, the formula applies:

PV_{\text{full}}=\frac{C}{i} =\frac{\$8{,}400}{0.035} =\$240{,}000.00.

For the first two payments alone, discount the year-1 and year-2 amounts separately. The second payment is discounted for two years, not one:

PV_{\text{first two}} =\frac{\$8{,}400}{1.035} +\frac{\$8{,}400}{(1.035)^2} =\$8{,}115.9420\ldots+\$7{,}841.4899\ldots =\$15{,}957.43.

The present value of all payments from year 3 onward, measured at time 0, is the full value less the first-two-payment value:

PV_{\text{year 3 onward}} =\$240{,}000.00-\$15{,}957.43 =\$224{,}042.57.

As a check, the stream from year 3 onward is the same perpetual pattern shifted two years later, so its value at time 0 is $(\$240{,}000)/(1.035)^2=\$224{,}042.57$. All money amounts are rounded to the nearest cent only at the final step. A common mistake is to discount the year-2 payment by one year, or to misadd the two discounted amounts; the correct year-2 discount factor is $(1.035)^{-2}$.

Worked Example: Comparing a Perpetuity With Its First Three Payments

A hypothetical endowment pays $5,000 at the end of each year forever. The effective annual interest rate is 5%. Find the value of the full stream, the value of the first three payments, and the value of all payments from year 4 onward, each at time 0. Assume deterministic payments, no mortality contingency, and no payment uncertainty.

Method. First value the full ordinary perpetuity. The rate is positive and matches the annual payment interval:

PV_{\text{full}}=\frac{\$5{,}000}{0.05}=\$100{,}000.00.

Next, discount the first three payments to time 0. Their payment dates are years 1, 2, and 3:

\begin{aligned} PV_{\text{first three}} &=\frac{\$5{,}000}{1.05} +\frac{\$5{,}000}{(1.05)^2} +\frac{\$5{,}000}{(1.05)^3}\\ &=\$4{,}761.9048\ldots +\$4{,}535.1474\ldots +\$4{,}319.1880\ldots\\ &=\$13{,}616.24. \end{aligned}

Subtracting the finite portion from the full-stream value gives the value, at time 0, of payments from year 4 onward:

PV_{\text{year 4 onward}} =\$100{,}000.00-\$13{,}616.24 =\$86{,}383.76.

The check is that the three-payment value plus the remaining-stream value equals the full perpetuity value: $\$13{,}616.24+\$86{,}383.76=\$100{,}000.00$. Money is rounded to cents at the end. The main trap is treating the value of the remaining payments as though it were still at year 3; the subtraction above expresses both parts at the same valuation date, time 0.

When the First Payment Is Immediate

A level stream may instead pay at the beginning of each period. In that case, its first payment is at time 0, so the ordinary-perpetuity formula alone omits a payment. The remaining payments, beginning at time 1, form an ordinary perpetuity. Thus the value immediately before the payment at time 0 is the immediate payment plus the value of that ordinary stream.

PV_{\text{immediate-first}} =C+\frac{C}{i} =\frac{C(1+i)}{i}.

This adjustment follows from timing: the time-0 payment has discount factor 1, while the payments at times 1, 2, and later have the same pattern as an ordinary perpetuity. It is not a different discount rate or a change in the payment amount.

Worked Example: A Perpetual Payment Beginning Today

A hypothetical facility receives $1,200 immediately and then receives $1,200 at the beginning of each succeeding year forever. The effective annual rate is 4%. Find the value just before the immediate payment and check the value just after that payment. Assume all payments are certain and fixed; there is no mortality contingency or random payment distribution.

Method. The payment at time 0 is immediate. The remaining stream begins at time 1 and is an ordinary perpetuity. Its value at time 0 is:

PV_{\text{remaining stream}} =\frac{\$1{,}200}{0.04} =\$30{,}000.00.

Add the payment received immediately to obtain the value just before that payment:

PV_{\text{just before time 0 payment}} =\$1{,}200+\$30{,}000 =\$31{,}200.00.

Immediately after the time-0 payment is made, that payment is no longer part of the remaining stream. The value of the payments still due is therefore $30,000.00 at that instant. The check is that the value immediately before the payment exceeds the value immediately after it by exactly $1,200.00. Using $C/i$ alone for the value just before the payment would incorrectly omit the immediate cash flow.

Common Traps and Limits of the Model

The formula is compact, but its assumptions carry important information. In particular, “per period” must have the same meaning for the payment and the rate. An annual payment cannot be divided by a rate quoted for a different interval without first expressing both on a consistent period basis. The valuation date also matters: $C/i$ values an ordinary perpetuity immediately before its first payment, not a stream whose first payment is received immediately.

Using a zero rate. The expression $C/i$ is undefined at $i=0$, and an infinite stream of positive undiscounted payments has no finite present value.
Using a negative rate. For $-1<i<0$, $v>1$; later payments do not become negligible when discounted, so the infinite sum does not converge.
Mixing payment and rate periods. The rate must correspond to the interval between successive payments.
Ignoring the first-payment date. An immediate payment adds $C$ to the value of the ordinary stream.
Applying a deterministic formula to uncertain payments without adjustment. If payments may stop, vary, or fail, $C/i$ does not by itself model those risks.

A perpetuity is therefore a useful benchmark for a constant, indefinitely continuing cash-flow pattern, but it is an idealization. A finite contract, changing payments, uncertain survival, or uncertain payment performance requires a model that reflects those features rather than an unadjusted level-perpetuity formula.

Key takeaway: A level ordinary perpetuity with payment $C$ each period and a fixed positive rate $i$ per period has present value $C/i$, derived by taking the limit of a finite ordinary annuity. Confirm the rate period and payment timing before applying the formula.

Check Your Understanding

For each problem, assume fixed, certain payments with no mortality contingency or random payment distribution. Use a positive effective rate that matches the payment interval.

A fund pays $1,800 at each year-end forever. The effective annual rate is 6%. Find the present value immediately before the first payment. (Answer: $30,000.00)
A hypothetical scholarship trust pays $3,150 annually at year-end forever at an effective annual rate of 7%. Find the full-stream value and the present value of its first two payments. (Answer: Full stream $45,000.00; first two payments $5,695.26)
A facility receives $2,000 at each year-end forever. At an effective annual rate of 5%, find the time-0 present value of the first four payments and of payments from year 5 onward. (Answer: First four $7,091.90; year 5 onward $32,908.10)
A grant pays $900 immediately and then $900 at the beginning of every year forever. At an effective annual rate of 3%, find the value just before the immediate payment and the value just after it. (Answer: Before $30,900.00; after $30,000.00)
A hypothetical perpetual year-end payment stream pays $3,600 annually. Compare its present values at effective annual rates of 4% and 6%, and find the dollar and percentage decrease when the rate rises from 4% to 6%. (Answer: Values $90,000.00 and $60,000.00; decrease $30,000.00, or 33.33% of the 4% value)

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