Finite Annuities as Truncated Perpetuities

Why a Finite Annuity Can Be Viewed as a Truncated Perpetuity

A level finite annuity and a level perpetuity have the same payment amount and timing at first. The difference is that the finite annuity stops, while the perpetuity continues. This gives a useful way to value the finite stream: start with the value of the perpetuity, then subtract the value of the payments that occur after the finite annuity ends.

This viewpoint is especially helpful for keeping track of payment dates. The omitted tail does not start on the date of the last included payment; it starts on the next payment date. The tail must also be valued at the same date as the original stream, or discounted back to that date.

Definition: An ordinary level annuity-immediate with $n$ payments of $C$ pays $C$ at the end of each period, at times $1,2,\ldots,n$. A level perpetuity-immediate pays $C$ at the end of every period, at times $1,2,3,\ldots$. The rate $i$ is the effective rate per payment period, with $i>0$ when using the perpetuity method.

If the periods are years, $i$ is the effective annual rate. If payments are made at a different regular interval, the rate used in the formulas must be the effective rate per such interval. We assume level payments, a constant rate, and no default, fees, or other adjustments to the cash flows. The finite annuity itself can be valued at $i=0$, but a perpetuity has no finite present value at zero interest.

Deriving the Ordinary-Annuity Formula

Let $v=1/(1+i)$. A perpetuity-immediate has present value $C/i$. The payments beyond the $n$th payment begin at time $n+1$. Their present value at time 0 is the value of a perpetuity-immediate at time $n$, discounted $n$ periods.

\text{PV at time 0 of tail} = \sum_{k=n+1}^{\infty} C v^k = \frac{C v^{n+1}}{1-v} = \frac{C}{i}v^n

The geometric series converges because $0<v<1$ when $i>0$. Also, $1-v=i/(1+i)$, so $v/(1-v)=1/i$. Subtracting the tail from the full perpetuity leaves exactly the payments at times $1$ through $n$:

\begin{aligned} PV_0 &= \frac{C}{i} - \frac{C}{i}v^n \\ &= \frac{C}{i}(1-v^n) \\ &= C a_{\overline{n}|} \end{aligned}

This derivation explains why the tail is discounted by $v^n$, not $v^{n+1}$. At time $n$, the tail is a perpetuity-immediate worth $C/i$: its first payment is one period later, at time $n+1$. Discounting that time-$n$ value back to time 0 requires $n$ periods.

Worked Example: Subtracting the Tail of an Ordinary Annuity

A hypothetical fund will pay $1,200 at the end of each year for 8 years. At an effective annual interest rate of 5%, find the present value at time 0 using the perpetuity-tail method. Name the method and identify the timing error it helps avoid.

Method and setup. Treat the 8 payments at times 1 through 8 as the first 8 payments of a $1,200 perpetuity-immediate. The perpetuity is valued at time 0, and the omitted tail consists of payments at times 9, 10, and later.

Perpetuity value. The full perpetuity is worth $C/i=1{,}200/0.05=\$24{,}000.00$ at time 0.

Tail value. At time 8, a perpetuity-immediate paying $1,200 per year is worth $1{,}200/0.05=\$24{,}000.00$. Discount it 8 years to time 0. Here $1.05^8=1.4774554438$, so $v^8=1/1.4774554438\approx0.6768393620$. Thus the omitted tail is worth $\$24{,}000(0.6768393620)=\$16{,}244.14$ at time 0.

Subtract and check. The finite annuity is worth $\$24{,}000.00-\$16{,}244.14=\boxed{\$7{,}755.86}$. Equivalently, $1{,}200(1-v^8)/0.05=\$7{,}755.86$. The result is positive and less than the perpetuity value, as it must be. The common timing error is to discount the tail by $v^9$; that would treat its value as though it were first located at time 9, rather than valuing the perpetuity at time 8.

Payment Timing: Annuities Due

An annuity-due pays at the beginning of each period. With $n$ payments of $C$, its payment dates are $0,1,\ldots,n-1$. To use a perpetuity comparison, begin with a perpetuity-due, whose payments occur at times $0,1,2,\ldots$. Its value is $C(1+i)/i$, since it is an ordinary perpetuity-immediate shifted one period earlier.

The payments omitted from the finite annuity are at times $n,n+1,\ldots$. At time $n$, that tail is itself a perpetuity-due worth $C(1+i)/i$; discounting it $n$ periods gives its value at time 0. Therefore the finite annuity-due value is the perpetuity-due value minus this discounted tail.

PV_0 = \frac{C(1+i)}{i} - \frac{C(1+i)}{i}v^n = C\ddot{a}_{\overline{n}|}

Worked Example: Removing the Tail of an Annuity-Due

A hypothetical service contract requires five payments of $900, with the first paid immediately and the remaining payments at the beginnings of the next four years. Find its present value at an effective annual rate of 6%, using a perpetuity-due and its tail. State the mistake avoided.

Method and dates. The payments are at times 0, 1, 2, 3, and 4. Start with a $900 perpetuity-due at time 0. Its omitted tail begins at time 5, so at time 5 it is another perpetuity-due.

Full perpetuity-due. Its time-0 value is $900(1.06)/0.06=\$15{,}900.00$.

Tail and subtraction. Since $1.06^5=1.3382255776$, $v^5\approx0.7472581729$. The omitted tail at time 0 is $\$15{,}900(0.7472581729)=\$11{,}881.40$. Subtracting gives $\$15{,}900.00-\$11{,}881.40=\boxed{\$4{,}018.60}$.

Check and trap. Directly, the value is $900(1.06/0.06)(1-v^5)=\$4{,}018.60$. The answer is below the $15,900 perpetuity-due value. Treating these payments as an ordinary annuity would shift every payment one year later and understate the value. The tail in this due-annuity calculation also starts at time 5, not at time 4, because time 4 is the final included payment date.

Deferred Annuities and the Tail’s Valuation Date

A deferred finite annuity uses the same subtraction idea, but both pieces must be valued at the same date. One convenient approach is to value the finite payments at the date immediately before the first payment, then discount that value to the requested date. Another is to start with a deferred perpetuity and subtract its deferred tail. Either method depends on counting the payment dates precisely.

For example, if the first payment is at time $m+1$, the payment dates are $m+1$ through $m+n$. At time $m$, those payments form an ordinary annuity-immediate. The omitted perpetuity tail begins at $m+n+1$, and at time $m+n$ it is a perpetuity-immediate. These dates determine the appropriate discounting.

Worked Example: A Deferred Annuity Valued Before Its First Payment

A hypothetical trust pays $700 at times 4, 5, 6, 7, 8, and 9. At an effective annual interest rate of 4%, find its value at time 3, immediately before the first payment. Use the truncated-perpetuity method, and explain the timing check.

Identify the six payments. The first payment is one year after time 3, so from time 3 the stream is an ordinary annuity-immediate with $n=6$, not a perpetuity with its first payment at time 3.

Perpetuity at time 3. A $700 perpetuity-immediate valued at time 3 is worth $700/0.04=\$17{,}500.00$.

Tail at time 3. The sixth included payment is at time 9. The omitted payments begin at time 10. At time 9, the omitted tail is worth $\$17{,}500.00$, so at time 3 its value is $\$17{,}500v^6$. Since $1.04^6=1.2653190188$, $v^6\approx0.7903145250$, and the tail is worth $\$17{,}500(0.7903145250)=\$13{,}830.50$ at time 3.

Subtract and check. The value is $\$17{,}500.00-\$13{,}830.50=\boxed{\$3{,}669.50}$. The direct check is $700(1-v^6)/0.04=\$3{,}669.50$. The tail is discounted six periods from time 9 to time 3; using seven periods would incorrectly place the tail’s perpetuity value at time 10.

A Reliable Valuation Procedure

The perpetuity method is most reliable when the dates are written down before any formula is used. It can be organized as a short sequence of decisions:

Fix the valuation date.
State the date at which the answer is required. If the first payment is deferred, do not assume the value is already at time 0 or at the first payment date.

List the included payment dates.
Count the payments and identify whether they occur at period ends or period beginnings. The final included payment date determines where the tail starts.

Value the matching perpetuity.
Use a perpetuity-immediate for end-of-period payments or a perpetuity-due for beginning-of-period payments, with the same payment amount, rate, and valuation-date convention.

Value the omitted tail at the same date.
Locate the tail’s first payment, value the tail as a perpetuity at the date appropriate to its timing, and discount it to the annuity’s valuation date.

Subtract and check.
Subtract the tail from the full perpetuity. For positive payments and positive interest, the finite value should be positive and less than the matching perpetuity value.

What the Method Does—and Does Not—Assume

The truncation identity is a cash-flow identity: the full perpetuity consists of the finite annuity payments plus the later tail. It does not require that an actual issuer intend to make an infinite number of payments. The perpetuity is a mathematical comparison stream that makes the finite stream easier to express.

The formula $C(1-v^n)/i$ assumes level payments at equally spaced period ends and a constant rate per period. If payments vary, the finite value is still the sum of each payment discounted to the valuation date, but one level perpetuity cannot represent the whole stream. If the payment interval and the rate period do not match, first express the rate per payment period; otherwise the exponents and discount factors do not correspond to the cash-flow dates.

Key takeaway: For a finite level annuity, value the matching perpetuity and subtract the perpetuity tail that begins after the final included payment. Payment timing determines which perpetuity to use, and the time gap between the tail’s valuation date and the requested valuation date determines the discount exponent.

Check Your Understanding

For each problem, identify the payment timing and valuation date before calculating.

A hypothetical fund pays $300 at the end of each year for 6 years. At an effective annual rate of 5%, find its value at time 0 by subtracting the tail of a perpetuity-immediate. (Answer: $1,522.71)
A hypothetical equipment lease requires four annual end-of-year payments of $1,000. At an effective annual rate of 8%, find the time-0 value and the present value of the omitted tail separately. (Answer: Annuity $3,312.13; tail $9,187.87)
A hypothetical grant pays $2,500 immediately and at the beginning of each of the next two years. At an effective annual rate of 4%, find its value at time 0 using a perpetuity-due comparison. (Answer: $7,215.24)
A hypothetical account pays $900 at times 3, 4, 5, 6, and 7. The effective annual interest rate is 100%. Find the value at time 2, immediately before the first payment. (Answer: $871.88)
A hypothetical project pays $600 at each year-end for 10 years. At an effective annual rate of 5%, find its present value and verify that it is less than the value of the corresponding perpetuity-immediate. (Answer: $4,633.04; corresponding perpetuity $12,000.00)

These techniques are aligned with the SOA VEE Accounting and Finance syllabus. In a written valuation, showing the payment dates and the location of the omitted tail makes the result easier to verify and exposes the most common source of error: discounting the tail by one period too many or too few.

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