Valuing a Perpetuity After Its Payments Begin

Introduction

A perpetuity does not stop paying, but the value of its remaining payments depends on when you measure it. At a payment date, the next payment may still be due, or it may just have been made. That one timing detail changes which cash flows belong in the value.

This tutorial values a level perpetuity at a focal date after its payments have begun. The method is to identify the next unpaid payment, then value that payment and all later payments as a perpetuity. All examples assume a deterministic level payment stream, with no expenses, default, or other adjustments.

Definition: A level annual perpetuity pays a fixed amount $R$ at regular annual intervals without a final payment. Assume a constant positive effective annual interest rate $i$, with $v=1/(1+i)$. A perpetuity-immediate has payments at the ends of annual periods; a perpetuity-due has a payment at the start of each period. At a valuation date, only payments not yet made are included. A payment scheduled for that exact date is included if it is still due.

The Next Unpaid Payment Determines the Value

Use the payment timeline, not simply the number of payments that have already occurred. If the next payment is one year after the focal date, the remaining stream is an ordinary perpetuity-immediate. If the next payment is due immediately, include that payment and then value the subsequent annual payments. If the next payment is a fraction of a year away, discount it for that fraction of a year before accounting for the later payments.

For a level perpetuity-immediate, the value one period before its first payment is $R a_{\overline{\infty}|}=R/i$. If a payment is due immediately, the value also includes $R$ now; the remaining later payments have value $R/i$.

\text{Next payment in one year: } \frac{R}{i} \qquad \text{Next payment due now: } R+\frac{R}{i}=\frac{R}{d}, \quad d=\frac{i}{1+i}

The second value is $R/d$, where $d$ is the effective annual discount rate corresponding to $i$. Equivalently, it is $R\ddot{a}_{\overline{\infty}|}$. The distinction is not a change in the payment amount or interest assumption; it is whether the payment on the focal date remains unpaid.

Key result: At a payment date, the value immediately before the payment is $R/d$, because the payment due now is included. The value immediately after the payment is $R/i$, because that payment has already been made and the next one is a year away.

Why the Timing Formulas Hold

Let the next unpaid payment be due $h$ years after the focal date, where $0\leq h\leq 1$. Assume fractional-year accumulation is consistent with the annual effective rate, so the accumulation over $t$ years is $(1+i)^t$. Each later payment is one year after the previous one. Thus the payments occur at $h,h+1,h+2,\ldots$ years from the focal date.

Discounting those payments to the focal date gives a geometric series. Its first term is $R v^h$, and each later term is multiplied by $v$. Since $i>0$, $0<v<1$, so the series converges.

V=R v^h+R v^{h+1}+R v^{h+2}+\cdots =R v^h(1+v+v^2+\cdots) =\frac{R v^h}{1-v} =\frac{R v^h}{d}

The last equality follows from $1-v=1-1/(1+i)=i/(1+i)=d$. If $h=1$, the next payment is one year away and $Rv/d=R/i$. If $h=0$, a payment is due now and $R/d=R+R/i$. Immediately after a payment has been made, the next payment is one year away, so the value is $R/i$, not $R/d$.

Fix the focal date.
Write down the date at which the value is required. “At the payment date” is not precise enough if it is unclear whether the payment has been made.

Find the next unpaid payment.
Determine whether it is due now, in one year, or after some other interval. Do not count a payment that has already been paid.

Value that payment and all later ones.
Use $R v^h/d$ when the next payment is $h$ years away, under the stated fractional-year accumulation assumption.

Move the value only if needed.
If the question asks for a value at a different date, discount or accumulate the value using the specified interest model.

Worked Examples

Worked Example: Value Immediately Before and After a Payment

A fund pays $R=\$1{,}200$ at the end of every year forever. The effective annual interest rate is $i=5\%$. Find the value at the date of the fourth payment, first immediately before that payment and then immediately after it. State the method and identify the timing error to avoid.

Method. Value the remaining stream at the specified focal date. Immediately before the fourth payment, that payment is still unpaid. Immediately after it, the next payment is one year away.

The effective annual discount rate is $d=i/(1+i)=0.05/1.05=1/21$.

Immediately before the fourth payment: include the $\$1{,}200$ due now and all later payments.

V_{\text{before}}=\frac{R}{d} =\frac{1{,}200}{1/21} =\$25{,}200

Immediately after the fourth payment: the fourth payment has already been made, so only payments beginning one year later remain.

V_{\text{after}}=\frac{R}{i} =\frac{1{,}200}{0.05} =\$24{,}000

Check. The two values differ by $\$25{,}200-\$24{,}000=\$1{,}200$, exactly the payment due at that date. A common mistake is to use $R/d$ after the payment has been made; that incorrectly counts the fourth payment a second time.

Worked Example: Value at a Later Date and Its Value Today

A hypothetical trust makes annual payments of $\$1{,}000$, with the first payment at the end of year 1 and one payment each year forever after that. The effective annual interest rate is $25\%$. Find the value of the remaining payments at year 4 immediately before the year-4 payment. Then find the value today of that remaining stream alone, and verify that adding the earlier payments gives the original value of the full perpetuity today.

Method. At year 4 immediately before the scheduled payment, the next payment is due now. First value that stream at year 4; then discount that value four years to today. Keep the earlier year-1, year-2, and year-3 payments separate.

The effective discount rate is $d=0.25/1.25=0.20$.

At year 4 immediately before the payment, the remaining stream includes the payment due now:

V_4=\frac{1{,}000}{0.20}=\$5{,}000

The accumulation factor over four years is $(1.25)^4=2.44140625$. Therefore, the year-4 remaining-stream value discounted to today is

V_{0,\text{remaining}} =\frac{5{,}000}{2.44140625} =\$2{,}048

Now discount the earlier payments to today:

1{,}000v+1{,}000v^2+1{,}000v^3 =800+640+512 =\$1{,}952

Adding the earlier payments to the remaining stream gives $\$1{,}952+\$2{,}048=\$4{,}000$. Independently, the full perpetuity-immediate value today is $R/i=\$1{,}000/0.25=\$4{,}000$, so the check agrees.

Timing trap. The $\$5{,}000$ at year 4 is the value immediately before the year-4 payment. Discounting it to today values only the year-4-and-later stream; it does not include payments from years 1 through 3. Adding those earlier payments is necessary to recover the value of the full perpetuity.

Worked Example: Next Payment Between Annual Dates

A hypothetical endowment pays $\$660$ once each year forever. The effective annual interest rate is $44\%$, and fractional-year accumulation follows $(1+i)^t$. At a focal date after payments have begun, the next unpaid payment is due in half a year. Find the value at the focal date. Explain why it is not an immediate perpetuity-due value.

Method. Discount each annual payment from its actual time to the focal date. The payments occur in $0.5,1.5,2.5,\ldots$ years.

Since $(1.44)^{1/2}=1.2$, the discount factor for half a year is $v^{0.5}=1/1.2=5/6$. Also,

d=\frac{0.44}{1.44}=\frac{11}{36}

The next payment is not due now, so discount it for half a year and value the remaining annual pattern:

V=\frac{660v^{0.5}}{d} =\frac{660(5/6)}{11/36} =550\left(\frac{36}{11}\right) =\$1{,}800

Check. The first payment contributes $\$660(5/6)=\$550$. The next payment is one year later, so its value is $\$660/(1.2)^3$; each successive term is smaller by a factor of $1/1.44$. The geometric series therefore has sum $\$550/[1-(1/1.44)]=\$1{,}800$, as above.

A common mistake is to use $R/d=\$2{,}160$, which assumes a payment is due immediately. Here the next payment is half a year away, so the entire due-now perpetuity value would be too large.

Common Timing Errors

The formulas are short; most errors come from describing the payment status imprecisely. A reliable solution states both the focal date and whether a payment scheduled for that date has already been made.

Counting a payment twice. Immediately after a payment, exclude it. The next payment is one year away, so the value is $R/i$.
Leaving out a payment due now. Immediately before a scheduled payment, include it. The value is $R/d$, not $R/i$.
Using the right formula at the wrong date. $R/i$ and $R/d$ describe different payment timings. They are not interchangeable labels for the same value.
Discounting a tail as if it were the whole stream. A value found at a later date covers only the payments remaining from that date. Payments already made before the focal date are not part of that value.
Ignoring a fractional interval. If the next payment is not one year away, account for the time to that payment under the stated interest model rather than treating it as due now or in a full year.

On problems aligned with the SOA Exam FM (Financial Mathematics) syllabus, a small timeline is often the fastest way to avoid these errors. Mark the focal date, mark the next unpaid payment, and then write the cash flows being valued. Only after that should you select a perpetuity formula.

Key takeaway: Value the unpaid cash flows from the focal date. A payment due immediately gives $R/d$; after that payment has been made, the remaining annual perpetuity is worth $R/i$. For a next payment $h$ years away, under consistent fractional-year accumulation, the value is $Rv^h/d$.

Check Your Understanding

For each problem, identify the next unpaid payment before calculating. Assume level annual payments, a constant effective annual rate, and no other cash flows. Where a half-year interval is specified, use fractional-year accumulation $(1+i)^t$.

A perpetuity pays $\$600$ at each year-end, and $i=6\%$. Find its value immediately after a payment has been made. (Answer: $\$10{,}000$)
A perpetuity pays $\$1{,}500$ annually, and $i=5\%$. Find its value immediately before a scheduled payment. (Answer: $\$31{,}500$)
A perpetuity pays $\$720$ annually, and $i=8\%$. At a focal date, its next payment is one year away. Find its value at that date. (Answer: $\$9{,}000$)
A perpetuity pays $\$400$ annually, and $i=25\%$. Find the values immediately before and immediately after a scheduled payment, and state their difference. (Answer: $\$2{,}000$ before, $\$1{,}600$ after, difference $\$400$)
A perpetuity pays $\$1{,}000$ annually, and $i=21\%$. Its next payment is due in half a year; fractional-year accumulation is consistent with the annual rate. Find its value at the focal date. (Answer: $\$5{,}238.10$)

What You'll Learn