Perpetuities with Nonannual Payments

Why the Payment Interval Matters

A perpetuity may pay monthly, quarterly, or semiannually rather than annually. Its value still comes from discounting the payments, but the rate used for discounting must match the time between payments. For example, a quarterly payment stream must be valued using an effective quarterly rate, not by directly dividing the payment by an annual rate.

The second essential detail is payment timing. A perpetuity-immediate makes its first payment at the end of one payment interval. A perpetuity-due makes its first payment at the valuation date. If payments are deferred, the value depends on the date of the first payment and the number of payment intervals from the valuation date to the date just before that payment.

Definition: Suppose a level perpetuity pays $R$ at regular intervals, $m$ times per year. Let $j$ be the effective interest rate for one payment interval. A perpetuity-immediate pays at the end of each interval; a perpetuity-due pays at the beginning of each interval. The rate $j$, the payment interval, and the payment timing must all describe the same cash-flow schedule.

Converting the Interest Rate to the Payment Interval

If the given rate is an effective annual rate $i$, the effective rate $j$ for a payment interval of $1/m$ years is found by equating accumulation over one year. There are $m$ such intervals in a year, so accumulating at $j$ for $m$ intervals must give the same annual accumulation as $i$.

(1+j)^m=1+i \qquad\Longrightarrow\qquad j=(1+i)^{1/m}-1

This conversion preserves the effective annual rate. In general, $j$ is not $i/m$: dividing an effective annual rate by $m$ does not account for interest being earned on interest within the year.

If instead the stated rate is a nominal annual rate $i^{(m)}$ convertible $m$ times per year, its definition gives the rate per conversion period directly:

j=\frac{i^{(m)}}{m}

These conversion rules apply when the stated rate corresponds to the relevant annual or nominal convention and remains constant over the periods being valued. Do not use the nominal-rate rule on an effective annual rate, or the effective-rate conversion on a nominal rate that is already specified as convertible at the payment frequency.

Immediate and Due Perpetuity Values

For a perpetuity-immediate, the first payment $R$ is made one interval from the valuation date, the next payment is made two intervals from that date, and so on. Each payment is discounted using the interval rate $j$. The resulting value is:

Theorem: For a level perpetuity-immediate paying $R$ once per interval at effective interval rate $j>0$, the value one interval before the first payment is $R/j$. For a perpetuity-due with the same payment amount and interval rate, the value at the date of the first payment is $R(1+j)/j$.

The due value is larger because its first payment is received immediately rather than one interval later. Equivalently, a perpetuity-due consists of an immediate payment $R$ plus a perpetuity-immediate whose first payment is one interval from the due date.

\text{Perpetuity-immediate value}=\frac{R}{j} \qquad \text{Perpetuity-due value}=R+\frac{R}{j} =\frac{R(1+j)}{j}

Why the Perpetuity Formula Holds

The perpetuity-immediate value follows from a geometric series. The first payment is discounted by one interval, the second by two intervals, and each later payment by one additional interval. Thus the present value $V$ is:

V=\frac{R}{1+j}+\frac{R}{(1+j)^2}+\frac{R}{(1+j)^3}+\cdots

Set $v_j=1/(1+j)$ for this derivation only. Since $j>0$, $0<v_j<1$, and the infinite geometric series converges. Its sum is $R v_j/(1-v_j)$. Substituting $v_j=1/(1+j)$ gives:

V=\frac{R v_j}{1-v_j} =\frac{R/(1+j)}{1-1/(1+j)} =\frac{R/(1+j)}{j/(1+j)} =\frac{R}{j}

The condition $j>0$ matters: with a nonpositive interval rate, this infinite series does not have a finite value under the usual perpetuity model. The formula also assumes fixed payments, a constant rate, and payment dates that continue at the stated interval without interruption.

Worked Examples

Worked Example: Quarterly Payments at an Effective Annual Rate

A fund is to make level payments of $2,000 at the end of every three months forever. The effective annual interest rate is 21%, and it is constant. Find the value today. State the payment timing and the interval rate.

Method. This is a perpetuity-immediate because the first payment is made at the end of the first quarter. Convert the effective annual rate to an effective quarterly rate, then use $R/j$. The common mistake to avoid is dividing 21% by four.

There are four quarterly intervals in a year, so:

j=(1.21)^{1/4}-1=1.05-1=0.05

The interval rate is 5%. The value, one quarter before the first payment, is:

V=\frac{2{,}000}{0.05}=40{,}000

The value today is $40,000.00. As a timing check, the first $2,000 is discounted one quarter, not received today. Dividing by the annual rate instead would give $2{,}000/0.21$, which does not match the quarterly cash-flow spacing. The answer is checked by the perpetuity-immediate formula with the correctly converted 5% quarterly rate.

Worked Example: A Quarterly Perpetuity-Due at a Nominal Rate

A hypothetical trust pays $1,200 every three months, with the first payment made immediately and payments continuing forever. The annual nominal interest rate is 8% convertible quarterly. Find the value at the date of the first payment.

Method. Because the nominal rate is convertible quarterly, the effective rate per quarter is the nominal rate divided by four. The first payment is immediate, so this is a perpetuity-due. The common mistake to avoid is valuing it as an immediate perpetuity and omitting the payment at time zero.

The quarterly interval rate is:

j=\frac{0.08}{4}=0.02

A perpetuity-immediate with these payments would be worth $1{,}200/0.02=60{,}000$, one quarter before its first payment. For the due stream, include the $1,200 payable now:

V=1{,}200+\frac{1{,}200}{0.02} =1{,}200+60{,}000 =61{,}200

The value at the date of the first payment is $61,200.00. Equivalently, $1{,}200(1.02)/0.02=61{,}200$. The answer is checked by seeing that the due value is exactly one immediate payment more than the immediate perpetuity value.

Worked Example: A Deferred Semiannual Perpetuity

A hypothetical endowment will pay $5,250 every six months forever. The first payment is due 2.5 years from today, and the effective annual interest rate is 10.25%. Find the value today. Assume the rate is constant and payments occur exactly on the stated dates.

Method. First convert the effective annual rate to an effective semiannual rate. Then value the perpetuity one payment interval before its first payment and discount that value to today. The common mistake to avoid is counting years rather than semiannual intervals when discounting.

The effective semiannual rate $j$ satisfies $(1+j)^2=1.1025$. Since $1.05^2=1.1025$:

$$ j=0.05 $$

The first payment is at $t=2.5$ years. One semiannual interval before that payment is $t=2$ years. The value at $t=2$ is the value of a perpetuity-immediate:

V_{2}=\frac{5{,}250}{0.05}=105{,}000

There are four half-year intervals from today to $t=2$, so discount by four periods, not five:

V_0=\frac{105{,}000}{(1.05)^4} =\frac{105{,}000}{1.21550625} =86{,}383.7599\ldots

Rounded to the nearest cent, the value today is $86,383.76. As a timing check, the first payment at 2.5 years is five half-year intervals from today; the perpetuity value $105{,}000$ is dated one interval before it, at 2 years, which is four intervals from today. The same value can be checked by discounting the first payment and all later payments directly: $5{,}250(1.05)^{-5}/(1-(1.05)^{-1})=105{,}000(1.05)^{-4}$.

Deferred Payments and Interval Counting

For a deferred perpetuity-immediate, place the perpetuity value one payment interval before the first payment. If the first payment is at time $k/m$ years from today, this value is dated at $(k-1)/m$ years. Discount it by $k-1$ payment intervals to obtain the value today:

V_0=\frac{R}{j}(1+j)^{-(k-1)}

This expression assumes the first payment is at $k/m$, that all later payments occur at intervals of $1/m$ years, and that today is the valuation date. A payment at $t=0$ instead indicates a perpetuity-due; a payment at $t=1/m$ indicates an immediate perpetuity valued directly today, with no additional deferral discount.

The interval-counting step is a frequent source of errors. Mark the first payment date, move back exactly one payment interval to date the perpetuity value, and then count intervals from that date to the valuation date. Discounting by the number of intervals to the first payment rather than the number to the perpetuity value shifts the answer by one period.

Common Mistakes and Practical Checks

Using the annual rate in a nonannual formula. Convert an effective annual rate using $j=(1+i)^{1/m}-1$. Do not substitute $i$ for $j$.
Confusing effective and nominal rates. Divide $i^{(m)}$ by $m$ only when the stated rate is nominal and convertible $m$ times per year.
Using the immediate formula for a due perpetuity. A payment at the valuation date is not discounted. Add it, or use $R(1+j)/j$.
Discounting a deferred value by one interval too many. The value $R/j$ is dated one payment interval before the first payment, not at the first payment date.
Mixing payment frequencies. A quarterly payment stream needs a quarterly interval rate; a semiannual stream needs a semiannual interval rate.

These checks matter because even a correctly calculated perpetuity factor can produce the wrong value if its rate or valuation date does not match the cash flows. Before calculating, write down the payment interval, the first payment date, the rate convention, and the valuation date.

Check Your Understanding

For each problem, identify the matching interval rate and payment timing before calculating. Assume a constant rate and level payments continuing forever unless stated otherwise.

A perpetuity pays $300 at the end of each month. Its effective annual rate is $(1.01)^{12}-1$. Find its value today. (Answer: $30,000.00)
A perpetuity pays $900 every six months, with the first payment made immediately. The annual nominal rate is 6% convertible semiannually. Find its value at the first payment date. (Answer: $30,900.00)
A perpetuity pays $4,000 at the end of every six months, beginning six months from today. The effective annual rate is 10.25%. Find its value today. (Answer: $80,000.00)
A perpetuity pays $5,000 quarterly, with the first payment at $t=1.5$ years. The annual nominal rate is 4% convertible quarterly. Find its value today. (Answer: $475,732.84)
A perpetuity pays $700 every six months, with the first payment made today. The effective annual rate is 4.04%. Find its value at the date of the first payment. (Answer: $35,700.00)

What You'll Learn