Deriving the Perpetuity-Immediate Value

What Makes a Perpetuity-Immediate Different?

A finite annuity has a last payment; a perpetuity has no last payment. Although a perpetuity contains infinitely many payments, its present value can be finite: each later payment is discounted more heavily, and the resulting sequence of present values can converge. The key is to identify the payment timing and the interest-rate condition that makes this convergence possible.

This tutorial derives the value of a level perpetuity-immediate in two connected ways: as the limit of finite annuity-immediate values, and as the sum of an infinite geometric series. The derivation and its assumptions are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: A level perpetuity-immediate pays a fixed amount $R$ at the end of every period, with the first payment one period from now and no final payment date. For annual payments at effective annual interest rate $i$, the payment dates are $t=1,2,3,\ldots$. The present value is measured at time $0$.

The word immediate describes the payment timing: the first payment is at the end of the first period, not at time $0$. The word level means that the payment amount stays constant. In the derivation below, periods are annual, payments are annual, and $i$ is a constant effective annual rate. These assumptions matter: a different payment frequency or changing interest rates requires a different valuation setup.

Start with a Finite Annuity-Immediate

First consider $n$ annual payments of 1, with the first at time $1$ and the last at time $n$. At effective annual rate $i$, the discount factor is $v=1/(1+i)$. The present value is the sum of the discounted payments:

a_{\overline{n}|}=v+v^2+\cdots+v^n

This is a finite geometric series. For $i>0$, its ratio $v$ is between 0 and 1, and $1-v=i/(1+i)=iv$. Applying the finite geometric-series identity gives:

a_{\overline{n}|} =\frac{v(1-v^n)}{1-v} =\frac{1-v^n}{i}

The factor $v$ in the numerator of the geometric-series expression is important. The first term of the series is $v$, because the first payment occurs at time $1$. A series that starts with 1 would describe a payment at time $0$, not this annuity-immediate.

Derivation: If $S_n=v+v^2+\cdots+v^n$, then factoring out $v$ gives $S_n=v(1+v+\cdots+v^{n-1})$. The finite geometric-series identity makes the parenthesized sum $(1-v^n)/(1-v)$, so $S_n=v(1-v^n)/(1-v)$. Since $1-v=iv$, cancellation of $v$ gives $S_n=(1-v^n)/i$. This proves the finite annuity-immediate formula from its discounted payments.

Take the Limit as the Number of Payments Grows

A perpetuity-immediate can be viewed as a sequence of finite annuities whose payment counts increase without bound. For $i>0$, $v=1/(1+i)$ satisfies $0<v<1$. Therefore $v^n$ approaches 0 as $n$ grows. Taking the limit in the finite annuity formula gives the present value of a perpetuity paying 1 per year:

\lim_{n\to\infty}a_{\overline{n}|} =\lim_{n\to\infty}\frac{1-v^n}{i} =\frac{1}{i}

If each payment is $R$ rather than 1, every discounted payment and every finite annuity value is multiplied by $R$. Thus the limit is $R/i$. This result requires a fixed positive effective annual rate and level annual payments made in arrears. If $i=0$, no payment is discounted and the sum of infinitely many positive payments diverges. If $-1<i<0$, then $v>1$, so the discounted terms do not approach zero and the series does not converge to a finite value.

Theorem: For annual level payments $R$ at times $1,2,3,\ldots$, with a constant effective annual interest rate $i>0$, the present value at time $0$ is $$ \text{PV}=\frac{R}{i}. $$ The formula is not a general value for any infinite payment stream. Its assumptions include level payments, annual periods, payment at each period-end beginning one period from now, and a positive constant effective annual rate.

The Same Result as an Infinite Geometric Series

The limiting argument and the infinite-series argument describe the same discounted cash flows. A level payment $R$ at time $k$ has present value $Rv^k$. Consequently, the perpetuity’s present value is the infinite sum:

R v+R v^2+R v^3+\cdots =R(v+v^2+v^3+\cdots)

Since $0<v<1$, this geometric series converges. Its first term is $v$, and its common ratio is $v$; therefore its sum is $v/(1-v)$. Using $1-v=iv$ yields:

R\frac{v}{1-v} =R\frac{v}{iv} =\frac{R}{i}

This second derivation makes the payment timing especially visible. The series starts with $Rv$, not $R$, because the first payment is discounted for one full period. The infinite series is not treated as an unexamined sum: convergence follows from $0<v<1$, which in turn follows from the assumption $i>0$.

Worked Example: Comparing Four Payments with the Infinite Stream

A hypothetical fund pays \$120 at the end of each year. Assume a constant effective annual interest rate of 5%, with valuation at time 0. Find the present value of the first four payments, the perpetuity value, and the value of the payments after year 4. Use the finite annuity formula and the perpetuity limit, and round dollar amounts to the nearest cent only at the end.

Method. Calculate the four-payment annuity-immediate first. Then subtract it from the limiting value to find the value contributed by later payments. This avoids treating the finite value as though it already includes all perpetuity payments.

The discount factor is:

v=\frac{1}{1.05}=0.9523809524

Its fourth power is $v^4=1/(1.05)^4=0.8227024748$. The present-value factor for four payments is:

a_{\overline{4}|} =\frac{1-v^4}{0.05} =\frac{1-0.8227024748}{0.05} =3.545950504

Thus the first four payments have present value $120(3.545950504)=\$425.5140605$, or \$425.51 to the nearest cent. The perpetuity limit is $120/0.05=\$2,400.00$. Therefore, the present value of payments after year 4 is $2,400-425.5140605=\$1,974.4859395$, or \$1,974.49.

Check. The finite value and the omitted tail add to $\$425.5140605+\$1,974.4859395=\$2,400$, the perpetuity value. The tail is also $120v^4/0.05=\$1,974.4859395$, confirming the subtraction.

Worked Example: Building the Value from a Geometric Series

A hypothetical trust pays \$75 at each year-end, indefinitely, at a constant effective annual rate of 8%. Show the present value of the first three payments by discounting each separately, then use the convergent geometric series to obtain the value of all payments. Round final dollar amounts to the nearest cent.

Method. Write the discounted terms from their actual payment dates, then sum the geometric series. This makes it possible to check that the first term is discounted once.

Here $v=1/1.08=0.9259259259$, $v^2=0.8573388203$, and $v^3=0.7938322410$. The first three discounted payments are:

75v=69.44444444,\qquad 75v^2=64.30041152,\qquad 75v^3=59.53741808

Their sum is $69.44444444+64.30041152+59.53741808=\$193.28227404$, or \$193.28. For the entire stream, the discounted terms are $75(v+v^2+v^3+\cdots)$. Since $0<v<1$, the geometric series has sum $v/(1-v)$. The perpetuity value is therefore:

75\frac{v}{1-v} =75\frac{1}{0.08} =\$937.50

Check. The value of payments after the third is $75v^3/0.08=\$744.21772596$. Adding this to the first-three-payment value gives $\$193.28227404+\$744.21772596=\$937.50$. The total is consistent with the geometric-series result.

Worked Example: Seeing the Finite Values Approach the Limit

A hypothetical scholarship fund makes a level payment of \$200 at each year-end. Assume a constant effective annual rate of 10%. Calculate the present values of the first two payments and the first five payments, compare each with the infinite-stream limit, and round final dollar amounts to the nearest cent.

Method. Evaluate the finite annuity formula at two payment counts, then use the limit to identify how much value remains beyond each finite term. The common mistake this avoids is assuming that a long finite annuity has already reached the perpetuity value.

The discount factor is $v=1/1.10=0.9090909091$. Then $v^2=0.8264462810$ and $v^5=0.6209213231$. For two payments:

a_{\overline{2}|} =\frac{1-0.8264462810}{0.10} =1.735537190

The two-payment present value is $200(1.735537190)=\$347.1074380$, or \$347.11. Its difference from the perpetuity limit is $200(0.8264462810)/0.10=\$1,652.892562$, or \$1,652.89.

For five payments:

a_{\overline{5}|} =\frac{1-0.6209213231}{0.10} =3.790786769

The five-payment present value is $200(3.790786769)=\$758.1573538$, or \$758.16. The difference from the limit is $200(0.6209213231)/0.10=\$1,241.842646$, or \$1,241.84. The limiting value itself is $200/0.10=\$2,000.00$.

Check. For each payment count, the finite value plus the remaining tail equals \$2,000 before rounding. The five-payment value exceeds the two-payment value, as expected, because it includes three additional positive discounted payments.

Common Timing Error: Starting the Series at Time Zero

A frequent error is to write $R+Rv+Rv^2+\cdots$ for a perpetuity-immediate. That series assigns a payment of $R$ at time $0$, followed by payments at times $1,2,\ldots$. It therefore describes a different timing pattern. Its sum is $R/(1-v)=R(1+i)/i$, which is not the value of the immediate perpetuity.

For example, at $i=5\%$, the incorrect calculation for payments of \$100 would be $100(1.05)/0.05=\$2,100$. The immediate perpetuity’s first payment is at time 1, so its series starts with $100v$; the correct value is $100/0.05=\$2,000$. The extra \$100 in the incorrect calculation is exactly the additional time-zero payment that was mistakenly included.

Set the timeline.
For an immediate perpetuity, put the first payment one period from the valuation date and each later payment one period after the previous one.

Write the discounted payments.
For annual payments $R$, the terms are $Rv,Rv^2,Rv^3,\ldots$, not $R,Rv,Rv^2,\ldots$.

Check convergence.
A constant positive effective annual rate gives $0<v<1$, so the terms form a convergent geometric series.

Use the derived value.
The present value is $R/i$ under the stated annual, level-payment, immediate-timing assumptions.

The infinite-series derivation also explains when the shortcut cannot be used. If payments are not level, their amounts do not produce this geometric series with a single common factor $R$. If payments occur more than once per year, the period and rate must be made consistent with that payment frequency. If the first payment is at time $0$, the timing is different. These are not minor notation changes: they change the cash flows being valued or the terms of the series.

Key takeaway: A level perpetuity-immediate has present value $R/i$ because its discounted payments are $Rv+Rv^2+\cdots$, a convergent geometric series when $i>0$. The same value is the limit of $R a_{\overline{n}|}$ as $n$ tends to infinity.

Check Your Understanding

Use annual payments, a constant effective annual rate, and valuation at time 0 unless a question says otherwise.

A perpetuity-immediate pays \$1 at each year-end at an effective annual rate of 4%. What is its present value? (Answer: \$25.00)
A hypothetical fund pays \$180 at each year-end indefinitely. The effective annual rate is 6%. Find its present value. (Answer: \$3,000.00)
A three-payment annuity-immediate pays \$50 at the end of each year at an effective annual rate of 5%. Find its present value, rounding to the nearest cent. (Answer: \$136.16)
A perpetuity-immediate pays \$100 per year at an effective annual rate of 8%. What is the present value of the payments after the first six payments, measured at time 0? Round to the nearest cent. (Answer: \$787.71)
A level perpetuity-immediate pays \$40 per year, but the effective annual rate is 0%. Does the infinite stream have a finite present value? (Answer: No; the discounted payment sum diverges.)

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