Solving for an Unknown in a Perpetuity
A level perpetuity has no final payment, so its price is determined by the amount of each payment, the interest rate, and when the first payment is made. In many problems, two of these quantities are known and the third must be found. The algebra is short; choosing the formula that matches the payment dates is the important step.
The formulas in this tutorial assume a positive level payment once per year, a constant positive effective annual interest rate, and a price measured at time 0. The payments continue indefinitely. A perpetuity-immediate pays at the end of each year, beginning at time 1. A perpetuity-due pays at the beginning of each year, beginning immediately at time 0.
The same formulas can be rearranged to solve for a payment or an interest rate. It is useful to write all three versions for each payment pattern before substituting numbers.
| Payment pattern | Price | Payment from price and rate | Rate from price and payment |
|---|---|---|---|
| Perpetuity-immediate | \(P=R/i\) | \(R=Pi\) | \(i=R/P\) |
| Perpetuity-due | \(P=R(1+i)/i\) | \(R=Pi/(1+i)\) | \(i=R/(P-R)\) |
The due formulas can also be expressed using the annual discount rate \(d=i/(1+i)\): a perpetuity-due has \(P=R/d\) and \(R=Pd\). When solving for the interest rate from \(P\) and \(R\), however, the table’s expression \(i=R/(P-R)\) makes clear that the due price includes a payment made immediately.
Why the Formulas Can Be Rearranged This Way
For an immediate perpetuity, each payment is discounted one more year than the previous payment. With \(v=1/(1+i)\), its present value is the infinite geometric series \(Rv+Rv^2+Rv^3+\cdots\). Since \(i>0\), \(0<v<1\), so the series converges.
A due perpetuity has one additional payment at time 0. After that payment, its remaining payments are exactly an immediate perpetuity viewed at time 0. Therefore its price is the payment today plus the value of the remaining stream:
Solving for the unknowns now follows by ordinary algebra. For an immediate perpetuity, \(P=R/i\) gives \(R=Pi\) and \(i=R/P\). For a due perpetuity, \(P=R(1+i)/i=R+R/i\). Rearranging this last expression gives \(P-R=R/i\), so \(i=R/(P-R)\). With positive payments and a positive rate, a due price must be greater than the payment made at time 0: \(P>R\).
A payment at time 1 indicates an immediate perpetuity; a payment at time 0 indicates a due perpetuity.
Use \(P=R/i\) for an immediate perpetuity or \(P=R(1+i)/i\) for a due perpetuity.
Choose the version that isolates the requested unknown, then substitute the price, payment, and rate using consistent annual units.
Substitute the answer back into the price formula and confirm that the first payment is treated at the correct date.
Worked Examples
Worked Example: Finding Prices for Both Payment Timings
A hypothetical fund distributes $840 at the end of every year forever. A second contract distributes the same amount annually, with its first payment made immediately. At an effective annual interest rate of 7%, find each contract’s price at time 0 and the difference in prices.
Method and timing check. The first contract is a perpetuity-immediate; the second is a perpetuity-due. The interest rate is \(i=0.07\), and the level payment is \(R=\$840\). The immediate formula applies only to the end-of-year payments.
For the immediate perpetuity,
The price is $12,000. For the due perpetuity,
The due price exceeds the immediate price by
Check. The due contract pays $840 immediately, then has the same end-of-year payment stream as the immediate contract. Thus its value should be exactly one payment more: \(\$12{,}000+\$840=\$12{,}840\). This also avoids the common mistake of treating an immediate first payment as if it were one year away.
Worked Example: Finding the Payment of a Perpetuity-Due
A hypothetical endowment is priced at $21,000. It earns an effective annual interest rate of 5% and makes a level payment at the beginning of every year, including an immediate payment at purchase. Find the annual payment.
Method and common mistake avoided. Because the first payment is immediate, use the due rearrangement \(R=Pi/(1+i)\), not the immediate rearrangement \(R=Pi\). Here \(P=\$21{,}000\) and \(i=0.05\).
Substitute the known values:
The level annual payment is $1,000. To check, the due price calculated from this payment is
If the immediate formula had been used, it would give \(R=21{,}000(0.05)=\$1{,}050\). That is the payment for a perpetuity-immediate priced at $21,000, not for this due perpetuity. The timing assumption changes the payment by $50.
Worked Example: Finding the Interest Rate from a Due Price
A hypothetical perpetual grant costs $12,720 and pays $720 at the beginning of each year, with the first payment made immediately. Find the effective annual interest rate. Give the rate to four significant figures.
Method and timing check. This is a perpetuity-due, so solve \(P=R(1+i)/i\) for \(i\). The known values are \(P=\$12{,}720\) and \(R=\$720\). First isolate the value of the payments after the immediate one:
For a due perpetuity, \(P-R=R/i\). Therefore,
The effective annual interest rate is \(6.000\%\). Check. At this rate, the immediate payment is $720 and the remaining immediate perpetuity is worth \(720/0.06=\$12{,}000\). Their total is \(\$720+\$12{,}000=\$12{,}720\), as quoted.
A timing error would use the immediate formula \(i=R/P\), giving \(720/12{,}720=0.05660377\), or \(5.660\%\) to four significant figures. That calculation incorrectly treats the first $720 payment as occurring at the end of year 1. The price \(P\) already includes a payment at time 0, so the appropriate denominator for the due-rate formula is \(P-R\), not \(P\).
Interpreting the Answer and Avoiding Traps
When price and payment are given, the rate formula depends on the payment timing. For an immediate perpetuity, dividing the payment by the price gives the effective annual rate. For a due perpetuity, first subtract the payment made at time 0 from the price, then divide the payment by what remains. That remainder is the value of the end-of-year stream.
For positive \(R\) and \(i\), the price of an immediate perpetuity exceeds zero, and the price of a due perpetuity exceeds its immediate payment. If a problem gives a purported due price \(P\leq R\), the data cannot represent a positive-rate level due perpetuity under these assumptions: after paying \(R\) immediately, no positive value would remain for the future payments. Check for a misread timing condition, mismatched units, or inconsistent data rather than applying the rate formula mechanically.
The rate \(i\) must be effective for the interval between payments. In these examples payments are annual, so it is an effective annual rate. If payments occur at another frequency, an annual rate cannot simply be substituted as though it were effective per payment period; the rate and payment interval must be made consistent first. The formulas here also presume a level payment continuing forever, not a payment that changes over time or stops after a specified number of years.
Check Your Understanding
For each problem, identify the payment timing, solve for the requested quantity, and check that the result fits the corresponding perpetuity formula.
- A hypothetical fund pays $540 at the end of each year forever. At an effective annual interest rate of 4.5%, find its price at time 0. (Answer: $12,000)
- A perpetuity-due is priced at $25,200 when the effective annual interest rate is 5%. Find its level annual payment. (Answer: $1,200)
- A perpetuity-immediate costs $18,750 and pays $1,125 at each year-end forever. Find the effective annual interest rate. (Answer: 6.000%)
- A hypothetical perpetual award costs $18,900 and pays $900 immediately and at the beginning of every subsequent year. Find the effective annual interest rate. (Answer: 5.000%)
- At an effective annual interest rate of 6%, a hypothetical contract pays $660 each year forever. Find the prices of the immediate and due versions, then find how much more the due version costs. (Answer: immediate price $11,000; due price $11,660; difference $660)