Valuing a Finite Annuity Followed by a Perpetuity

Introduction

Some cash flows have two distinct stages: a specified number of level payments, followed by level payments that continue indefinitely. The first stage is a finite annuity; the second is a perpetuity. The valuation is straightforward once the payment dates are clear, but a misplaced transition date can omit a payment or count one twice.

This tutorial considers annual payments under a constant effective annual interest rate. The central method is to value both segments at a common focal date, then discount that combined value if the requested value is earlier. The payment timeline—not just the words “followed by”—determines which annuity and perpetuity factors apply.

Definition: Suppose a cash flow pays $C$ at the ends of years $1,\ldots,n$, then pays $R$ at the ends of years $n+1,n+2,\ldots$. The finite segment is an $n$-payment annuity-immediate, and the continuation is a perpetuity-immediate whose first payment is one year after time $n$. Assume a constant effective annual rate $i>0$, with $v=1/(1+i)$. The value at time 0 is $$ C a_{\overline{n}|}+v^n\frac{R}{i}. $$

Value Both Segments at the Transition Date

At time $n$, the finite payments have already been made, so their present value at time 0 can be calculated directly using $a_{\overline{n}|}$. The perpetuity has not yet made its first payment: it begins at time $n+1$. Therefore, it is worth $R/i$ at time $n$, and its time-0 value is $v^n(R/i)$.

The phrase “at the transition date” needs care. Here, time $n$ is the date of the last finite payment, while the perpetuity’s first payment is at $n+1$. If instead the perpetuity’s first payment is due at time $n$, the value at time $n$ includes that payment and is $R+R/i$, not merely $R/i$. Always list the payment dates before selecting a factor.

\begin{aligned} V_0 &=\sum_{t=1}^{n} C v^t+\sum_{t=n+1}^{\infty} Rv^t\\ &=C a_{\overline{n}|}+v^n\left(\frac{R}{i}\right). \end{aligned}

This direct-sum expression also gives a useful check. The first sum contains exactly the $n$ finite payments. In the second sum, factoring out $v^n$ leaves $R\sum_{j=1}^{\infty}v^j=R/i$. The two sums do not overlap, so no payment is counted twice.

Mark the payment dates.
Record the last finite payment and the first perpetuity payment. Check whether a payment occurs at the focal date.

Choose a common focal date.
For payments at times $1,\ldots,n$ followed by payments at $n+1$ onward, time $n$ is convenient: the perpetuity is worth $R/i$ there.

Value each segment and combine.
Use the finite-annuity value for the first stage and the perpetuity value for the continuation. Discount the combined value to the requested date if needed.

Worked Example: Different Payment Amounts Before and After the Switch

A fund pays \$300 at each year-end for years 1 through 3, then pays \$500 at each year-end from year 4 forever. Find its value at time 0 at an effective annual interest rate of 10%. Method: value the finite annuity at time 0 and the perpetuity at time 3, then discount the latter. This avoids treating the year-4 payment as if it were due at time 3.

Finite segment. The three payments of \$300 occur at times 1, 2, and 3. At 10%, $v=1/1.10=0.9090909$, and $v^3=1/(1.10)^3=0.7513148$. Thus $a_{\overline{3}|}=(1-v^3)/0.10=2.4868520$, giving $300(2.4868520)=\$746.06$.

Perpetuity segment. The first \$500 payment is at time 4, so at time 3 the perpetuity is worth $500/0.10=\$5{,}000$. Discounting three years gives $5{,}000(0.7513148)=\$3{,}756.57$.

Total and check. The value is $\$746.0556+\$3{,}756.5740=\$4{,}502.6296$, or \$4,502.63 to the nearest cent. The time-0 expression is also $300(v+v^2+v^3)+500(v^4+v^5+\cdots)$, which contains the same payment dates and confirms the split.

Changing the Timing of the Finite Segment

The formula above assumes that finite payments occur at times $1$ through $n$, with the perpetuity beginning at $n+1$. If the finite payments are instead due at times $0,1,\ldots,n-1$, the finite segment is an annuity-due. When the perpetuity begins at time $n$, its first payment is due at the transition date.

At time $n-1$, that first perpetuity payment is one year away, so the continuation is worth $R/i$. At time $n$, the perpetuity is worth $R+R/i$, because a payment is due immediately and the later payments continue at times $n+1,n+2,\ldots$. Either focal date works if both segments are valued at that same date.

Worked Example: An Annuity-Due Followed by a Perpetuity

A grant pays \$200 at times 0, 1, and 2, then pays \$360 at times 3, 4, and so on forever. Find its value at time 0 at an effective annual interest rate of 6%. Method: use time 2 as the common focal date. This makes the perpetuity-immediate value apply one year before its first payment.

Finite segment at time 2. The payments made at times 0, 1, and 2 accumulate to time 2 as $200(1.06)^2=\$224.72$, $200(1.06)=\$212.00$, and $\$200.00$. Their total at time 2 is $\$636.72$.

Perpetuity at time 2. The first \$360 payment is at time 3, one year after the focal date. Its value at time 2 is therefore $360/0.06=\$6{,}000$.

Discount to time 0. The combined value at time 2 is $\$636.72+\$6{,}000=\$6{,}636.72$. Since $v^2=1/(1.06)^2=0.8899964$, its present value is $6{,}636.72(0.8899964)=\$5{,}906.6572$, or \$5,906.66 to the nearest cent.

Check. The finite payments discounted separately are $200+200/1.06+200/(1.06)^2=\$566.68$. The perpetuity’s time-0 value is $6{,}000/(1.06)^2=\$5{,}339.98$. Together they give \$5,906.66, confirming the focal-date calculation.

Why the Perpetuity Factor Works

The perpetuity value used here follows from the geometric series of discounted payments. For a perpetuity-immediate whose first payment is one year after its valuation date, the payments have present values $R v, R v^2,\ldots$. Since $0<v<1$, the infinite geometric series converges:

R v+R v^2+\cdots =R\frac{v}{1-v} =R\frac{1/(1+i)}{1-1/(1+i)} =\frac{R}{i}.

The condition $i>0$ matters: it ensures discounted payments shrink and the infinite series has a finite value. The formula also depends on level payments continuing indefinitely and on a constant effective annual rate. If payment amounts or interest rates vary, this particular perpetuity factor does not directly apply.

For the finite segment, $C a_{\overline{n}|}$ is simply $Cv+Cv^2+\cdots+Cv^n$. Thus the combined formula is not a special extra annuity factor: it is the sum of the present values of two non-overlapping sets of payments.

Common Timing Mistakes

A frequent error is to put the perpetuity at the wrong date. Suppose finite payments occur at years 1 through 3 and the perpetuity begins at year 4. The perpetuity is worth $R/i$ at time 3, then must be discounted by $v^3$ to time 0. Treating $R/i$ as if it were already a time-0 value omits three years of discounting.

Another trap is changing the first perpetuity date without changing the valuation. If the perpetuity begins at time 3 rather than time 4, then it pays at the same time as the last finite payment. Using $R/i$ at time 3 would omit the time-3 payment, because $R/i$ at that date represents payments starting at time 4. The time-3 value of a perpetuity with an immediate payment is $R+R/i$.

Key takeaway: Write the payment times first. For finite payments at times $1,\ldots,n$ followed by perpetuity payments at $n+1,n+2,\ldots$, the value at time $n$ of the perpetuity is $R/i$, and the time-0 value of the full cash flow is $C a_{\overline{n}|}+v^n(R/i)$.

This segmentation method is useful whenever a contract, fund, or other hypothetical cash flow changes from a specified payment schedule to a continuing level payment. It keeps each segment’s timing visible and makes the result easy to verify by direct discounting. The method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the payment dates before calculating. Unless stated otherwise, use a constant effective annual interest rate and round money to the nearest cent.

A fund pays \$120 at the ends of years 1 and 2, then \$240 at each year-end from year 3 forever. Find its value at time 0 at 5%. (Answer: \$4,576.87)
A trust pays \$80 at times 0, 1, and 2, then \$150 at times 3, 4, and so on forever. Find its value at time 0 at 10%. (Answer: \$1,458.51)
A contract pays \$500 at the ends of years 1 through 5, then \$700 at each year-end from year 6 forever. Find its value at time 0 at 4%. (Answer: \$16,609.64)
At 8%, a fund pays \$200 at each year-end for years 1 through 3, then pays $Q$ at each year-end from year 4 forever. If its value at time 0 is \$6,000, find $Q$. (Answer: \$552.72)
At 5%, a fund pays \$100 at times 1, 2, 3, and 4, then \$250 at times 5, 6, and so on forever. Find its value at time 4. (Answer: \$5,431.01)

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