Perpetuity-Due Values and First-Payment Timing

Why the First Payment Date Matters

A perpetuity is a stream of level payments that continues indefinitely. Its value depends not only on the payment amount and interest rate, but also on when the first payment is made. For a perpetuity-due, the first payment is made at the valuation date itself. That payment is not discounted: it is already payable now.

This is the key distinction from a perpetuity-immediate, whose first payment is one period after the valuation date. The later payments in the two streams occur at the same dates if both are annual, but the due perpetuity includes an additional payment at time zero. The earlier material on perpetuity-immediate values gives the value of the stream starting one year from now; here, we derive and use the value when a payment is due immediately.

Definition: A level annual perpetuity-due pays an amount $R$ at times $0,1,2,\ldots$, where time is measured in years and the valuation date is time 0. Assume a constant effective annual interest rate $i>0$, with $v=1/(1+i)$. Its present value is the sum of the time-0 payment and the present values of all subsequent payments.

The assumption $i>0$ matters: it makes the present-value terms shrink as payment dates recede, so the infinite sum converges. The formula below does not give a finite value when $i=0$, and a nonpositive rate does not produce a convergent series of discounted positive payments. Unless a problem states otherwise, the rate and payment frequency must match: an annual payment stream is valued using an effective annual rate.

Deriving the Perpetuity-Due Formula

At time 0, the first payment $R$ is worth $R$. The payment at time 1 is discounted one year, the payment at time 2 is discounted two years, and the same pattern continues. Thus the present value $P$ is an infinite geometric series.

P=R+Rv+Rv^2+Rv^3+\cdots

For $i>0$, $0<v<1$. The sum of a convergent geometric series with first term $R$ and common ratio $v$ is $R/(1-v)$. Since $v=1/(1+i)$, we have $1-v=i/(1+i)$. Therefore:

P=\frac{R}{1-v} =\frac{R}{1-\frac{1}{1+i}} =R\frac{1+i}{i} =\frac{R}{d}, \qquad d=\frac{i}{1+i}.

Here $d$ is the effective annual discount rate corresponding to $i$. In standard annuity notation, the value factor is $\ddot{a}_{\overline{\infty}|}=(1+i)/i=1/d$. The formula is also easy to justify by comparison with a perpetuity-immediate: that stream has payments at times $1,2,3,\ldots$, so its value is $R/i$. The due stream adds exactly one payment $R$ at time 0.

\ddot{a}_{\overline{\infty}|} =1+a_{\overline{\infty}|} =1+\frac{1}{i} =\frac{1+i}{i} =\frac{1}{d}

This relationship provides a useful timing check. At the same payment amount and positive rate, the perpetuity-due must be worth exactly one payment more than the perpetuity-immediate. It is not merely a perpetuity-immediate value multiplied by an arbitrary adjustment: the extra time-0 payment explains the difference.

Theorem: For a level annual payment $R$ made at times $0,1,2,\ldots$, with a constant effective annual rate $i>0$, the present value at time 0 is $R\ddot{a}_{\overline{\infty}|}=R(1+i)/i=R/d$. For the corresponding perpetuity-immediate, with payments at times $1,2,3,\ldots$, the value is $Ra_{\overline{\infty}|}=R/i$; hence the due value exceeds the immediate value by $R$.

Worked Examples

Worked Example: Valuing a Perpetuity-Due and Checking Its Timing

A scholarship fund will pay \$6,000 immediately and then \$6,000 at the end of every year forever. The effective annual interest rate is 5%. Find the value at the payment date. Also compare it with the value of a perpetuity-immediate paying \$6,000 annually at the same rate.

Method. The first payment is made at time 0, so value the stream as a perpetuity-due. The rate is $i=0.05$ per year, and the payment is $R=\$6{,}000$.

Due perpetuity. Its value factor is $(1+i)/i=1.05/0.05=21$. Therefore:

P_{\text{due}}=\$6{,}000(21)=\$126{,}000.

Immediate perpetuity comparison. If the same payment stream instead began at the end of year 1, its value would be $R/i$:

P_{\text{immediate}} =\frac{\$6{,}000}{0.05} =\$120{,}000.

The difference is $\$126{,}000-\$120{,}000=\$6{,}000$, exactly one payment. As a direct-sum check, $v=1/1.05$, so the due series is $\$6{,}000(1+v+v^2+\cdots)$; its sum is $\$6{,}000/(1-v)=\$6{,}000/ (1-1/1.05)=\$126{,}000$. The common mistake avoided here is treating the immediate payment as though it were one year away and reporting only \$120,000 as the due value.

Worked Example: Finding the Level Payment from a Fund Value

A fund has \$18,900 available at time 0 to support equal annual grants forever. The first grant is made immediately, and later grants are made at the end of each year. If the effective annual interest rate is 8%, find the grant amount and verify the result by separating the immediate grant from the later grants.

Method. Use the due perpetuity value equation $P=R(1+i)/i$, then solve for $R$. Here $P=\$18{,}900$ and $i=0.08$.

R=P\frac{i}{1+i} =\$18{,}900\frac{0.08}{1.08} =\$18{,}900\frac{8}{108} =\$1{,}400.

Check using the payment split. The immediate grant costs \$1,400 at time 0. The remaining grants, beginning one year from now, form a perpetuity-immediate with value:

\frac{\$1{,}400}{0.08}=\$17{,}500.

Together, the immediate grant and the later stream have value $\$1{,}400+\$17{,}500=\$18{,}900$, equal to the fund value. The calculation avoids the common error of using $R=iP$, which would be appropriate for a perpetuity-immediate, not for a perpetuity-due. The due payment is $i/(1+i)$ times the fund value.

Worked Example: Testing a Quoted Price

A seller offers a contract that pays \$1,200 immediately and \$1,200 at the end of every year forever. The effective annual interest rate for valuation is 5%, and the quoted price is \$26,000. Determine the value of the payment stream and whether the quote is above or below that value.

Method. Since the payment at time 0 is included, value the contract as a perpetuity-due with $R=\$1{,}200$ and $i=0.05$.

P_{\text{due}} =\$1{,}200\frac{1.05}{0.05} =\$1{,}200(21) =\$25{,}200.

The quoted price exceeds the calculated value by $\$26{,}000-\$25{,}200=\$800$. Thus, at the stated rate, the contract’s payment stream is worth \$25,200, and the quote is \$800 above that value.

Independent timing check. The payments after time 0 are a perpetuity-immediate worth $\$1{,}200/0.05=\$24{,}000$. Adding the immediate payment gives $\$24{,}000+\$1{,}200=\$25{,}200$. This check prevents discounting the time-0 payment or accidentally adding an extra payment to a formula that already includes it.

Common Timing Mistakes and Practical Checks

Perpetuity-due calculations are compact, but a small timing misread changes the value by a full payment. Before substituting into a formula, write down the valuation date and identify the first payment date. A payment described as “immediate,” “today,” or “at the date of purchase” is generally at time 0, unless the problem explicitly defines another valuation convention.

Using $R/i$ for a due stream. That is the value of payments at times $1,2,3,\ldots$. For payments at times $0,1,2,\ldots$, use $R(1+i)/i$, or add the time-0 payment to the immediate-perpetuity value.
Discounting the first payment. A time-0 payment is not discounted. Applying a factor $v$ to it moves it to a different payment date.
Counting the first payment twice. If using $R(1+i)/i$, the immediate payment is already included. Do not add another $R$ after using that formula.
Mixing annual rates and payment periods. The displayed formulas assume annual payments and an effective annual rate. A different payment frequency requires a rate and period that match; do not apply an annual formula unchanged to monthly payments.
Ignoring the convergence condition. The infinite present-value sum relies on a positive effective annual rate. A finite-term annuity formula or a different model is needed if the assumptions do not support a convergent sum.

Mark the valuation date.
Set it to time 0 and record the interest rate per payment period.

Locate the first payment.
If it occurs at time 0 and equal payments continue every period, the stream is due.

Apply the due value factor.
For annual payments and $i>0$, calculate $R(1+i)/i$, or $R/d$.

Check the result.
Value the later payments as an immediate perpetuity and add one payment. The two methods must agree.

This topic is aligned with the SOA Exam FM (Financial Mathematics) syllabus because it tests the interaction between cash-flow timing, discounting, and equations of value. A reliable solution makes the payment dates explicit before choosing the formula. The relationship $P_{\text{due}}=P_{\text{immediate}}+R$ is especially useful as a reasonableness check: for a positive payment, the due value must be greater than the immediate value by exactly $R$, not by an amount that depends on a second discounting of the time-0 cash flow.

Check Your Understanding

For each problem, identify the first payment date before calculating. Assume annual payments and a constant effective annual interest rate.

A community fund pays \$900 immediately and \$900 at each year-end forever. The annual effective rate is 6%. Find the value at time 0. (Answer: \$15,900)
A perpetuity-immediate pays \$1,800 at each year-end forever at an effective annual rate of 9%. A matching perpetuity-due adds the same payment at time 0. Find the value of the due perpetuity and the difference between the two values. (Answer: Due value \$21,800; difference \$1,800)
A fund worth \$42,000 must pay an equal annual amount forever, with the first payment immediately. The effective annual rate is 7%. Find the theoretical annual payment, rounded to the nearest cent. (Answer: \$2,747.66)
A trust pays \$2,500 at time 0 and \$2,500 at the end of every year thereafter. At an effective annual rate of 4%, calculate the value by separating the immediate payment from the later stream. (Answer: \$65,000)
A contract requires a purchase price of \$34,000 and pays \$1,600 immediately plus \$1,600 at each year-end forever. At an effective annual rate of 5%, calculate the value of the stream and state how far the quoted price is above or below that value. (Answer: Value \$33,600; quoted price is \$400 above value)

What You'll Learn