Zero-Interest Limits of Annuity Factors

Why Zero Interest Needs Separate Attention

Annuity formulas often appear as quotients whose numerator and denominator both become zero when the effective interest rate is zero. That does not mean the annuity has no value. It means the quotient form is not directly defined at zero and must be interpreted using the payments themselves or the limit of the factor as the rate approaches zero.

At a zero effective rate, money does not grow or shrink between payment dates. Consequently, a level stream of $n$ payments of amount $R$ has a value of $nR$, whether it is valued at the beginning or end of the stream. The immediate and due factors have the same zero-rate value, even though their payment dates differ.

Definition: An annuity-immediate with $n$ payments has payments at the ends of periods $1,2,\ldots,n$. An annuity-due with $n$ payments has payments at the beginnings of periods $0,1,\ldots,n-1$. The effective rate $i$ is the rate per payment period. Present values are measured at time 0; accumulated values are measured at time $n$.

The timing definitions matter even when $i=0$: an immediate annuity’s first payment is at time 1, while a due annuity’s first payment is at time 0. But because there is no interest, moving a payment from one date to another does not change its value at the chosen valuation date.

Factors at a Positive Rate and Their Zero-Rate Values

For $i>0$, let $v=1/(1+i)$. The standard factors for level annuities are:

a_{\overline{n}|}=\frac{1-v^n}{i}, \qquad \ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}, \qquad s_{\overline{n}|}=\frac{(1+i)^n-1}{i}, \qquad \ddot{s}_{\overline{n}|}=(1+i)s_{\overline{n}|}.

Here, $a_{\overline{n}|}$ is the present-value factor for an annuity-immediate, and $\ddot{a}_{\overline{n}|}$ is the present-value factor for an annuity-due. The factors $s_{\overline{n}|}$ and $\ddot{s}_{\overline{n}|}$ accumulate those respective payment patterns to time $n$. The due accumulation factor is written $\ddot{s}_{\overline{n}|}$; its relationship to $s_{\overline{n}|}$ follows because each due payment earns one additional period of interest relative to the corresponding immediate payment.

Theorem: At $i=0$, each of the four factors has value $n$: $$ a_{\overline{n}|}=\ddot{a}_{\overline{n}|}=s_{\overline{n}|}=\ddot{s}_{\overline{n}|}=n. $$ For a level payment $R$, the present value and accumulated value are therefore both $nR$, for either an annuity-immediate or an annuity-due.

Why the Zero-Rate Result Holds

The quotient formulas are derived from sums of discounted or accumulated payments. Those sums remain meaningful at $i=0$, even though the quotient forms do not. For an immediate annuity, the present-value factor is the sum of the discounts applied to payments at times $1$ through $n$. At zero interest, every discount factor is 1, so the sum contains $n$ ones.

a_{\overline{n}|}=\sum_{k=1}^{n}v^k \quad\longrightarrow\quad \sum_{k=1}^{n}1=n \qquad (i=0).

The same reasoning applies to a due annuity: its payments occur at times $0$ through $n-1$, and each has discount factor 1 at zero interest. For accumulated values, every payment also has accumulation factor 1. Thus, regardless of payment timing, each payment contributes its full amount to the value at the chosen date.

Equivalently, the quotient formulas have limits of $n$ as $i$ approaches zero. For example, the numerator $1-(1+i)^{-n}$ in $a_{\overline{n}|}$ is zero at $i=0$, as is the denominator $i$. Differentiating numerator and denominator with respect to $i$ gives a numerator derivative of $n(1+i)^{-n-1}$, which equals $n$ at zero, and a denominator derivative of 1. The limit is therefore $n$. The accumulated-value quotient has the same limit, and the due factors follow from multiplying by $1+i$, which approaches 1. The payment-sum argument also gives the value directly at zero without needing a limit.

Count the payments.
Identify the number $n$, not merely the number of periods named in the description. Check whether payments occur at times $1,\ldots,n$ or $0,\ldots,n-1$.

Confirm the rate is per payment period.
The zero-rate result applies when the effective rate for each payment period is zero.

Use the zero-rate factor.
Each applicable annuity factor is $n$, so multiply by the level payment $R$ to obtain the value $nR$.

Worked Examples

Worked Example: Present and Accumulated Values of an Immediate Annuity

A savings arrangement pays $275 at the end of each year for 18 years. Assume the effective annual interest rate is exactly 0%, with no fees or other cash flows. Find the present value at time 0 and the accumulated value at time 18.

Method. The payments occur at times $1,2,\ldots,18$, so this is an annuity-immediate with $n=18$ and $R=\$275$. At $i=0$, the present-value factor and accumulated-value factor are both 18.

The present value is:

PV=R a_{\overline{18}|} =\$275(18) =\$4{,}950.

For the accumulated value, each payment remains $275 through time 18 because no interest is earned. Thus:

AV=R s_{\overline{18}|} =\$275(18) =\$4{,}950.

Check. There are 18 payments, and $18\times275=4{,}950$. Both values are $4,950. This method avoids the mistake of treating the zero in the quotient formula for $a_{\overline{n}|}$ or $s_{\overline{n}|}$ as a zero annuity value.

Worked Example: A Due Annuity With Deposits at the Beginning of Each Period

A hypothetical account receives 24 deposits of $125, one at the beginning of each month for 24 months. Assume the effective monthly interest rate is 0%, with no fees. Find the value at time 0 and the accumulated value at the end of month 24.

Method. The first deposit is made immediately, at time 0, and the final deposit is made at time 23. This is an annuity-due with $n=24$ and $R=\$125$. Since the effective rate per month is zero, each due factor equals 24.

At time 0, the deposit made immediately is already present, and every later deposit has no discount applied:

PV=R\ddot{a}_{\overline{24}|} =\$125(24) =\$3{,}000.

At time 24, none of the deposits has accumulated interest:

AV=R\ddot{s}_{\overline{24}|} =\$125(24) =\$3{,}000.

Check. The deposits total $24\times125=\$3{,}000$, and a zero monthly rate leaves that total unchanged. This method avoids counting 25 deposits: the first deposit is at time 0, but the complete schedule still contains only 24 deposits.

Worked Example: Comparing Immediate and Due Payment Timing

Two hypothetical plans each pay $400 once per year for 10 years. Plan A pays at each year-end; Plan B pays at each year-beginning. Assume the effective annual interest rate is 0%, with no fees. For each plan, find the present value at time 0 and the accumulated value at time 10.

Method. Plan A is an annuity-immediate with payments at times $1$ through $10$. Plan B is an annuity-due with payments at times $0$ through $9$. Each plan has 10 payments. At zero interest, each of the four relevant factors is 10.

For Plan A:

PV_A=\$400a_{\overline{10}|} =\$400(10)=\$4{,}000, \qquad AV_A=\$400s_{\overline{10}|} =\$400(10)=\$4{,}000.

For Plan B:

PV_B=\$400\ddot{a}_{\overline{10}|} =\$400(10)=\$4{,}000, \qquad AV_B=\$400\ddot{s}_{\overline{10}|} =\$400(10)=\$4{,}000.

Check. Each plan makes $10\times400=\$4{,}000$ of payments. The dates differ, but with no interest all payments retain their full value. This method avoids assuming that the earlier payment dates in Plan B create a larger value when the rate is zero; earlier timing matters for interest accumulation only when interest is present.

Common Mistakes and Why the Result Matters

The zero-rate case is a useful boundary check for annuity calculations. If a formula or spreadsheet returns anything other than the total of the level payments when the effective rate is zero, the payment count, rate period, or timing setup should be reviewed.

Substituting zero into a quotient. The formulas $(1-v^n)/i$ and $((1+i)^n-1)/i$ have zero denominators at $i=0$. Do not report the resulting undefined expression as a value; use the payment sum or the limiting factor $n$.
Confusing factor value with dollar value. At zero interest, the factor is $n$, not $nR$. Multiply the factor by the payment amount to obtain the annuity’s value.
Miscounting due payments. A due annuity with $n$ payments at times $0$ through $n-1$ has $n$ payments, not $n+1$. Including both time 0 and time $n$ would describe a different stream.
Using the wrong period for the rate. A rate must match the payment period. If monthly payments are being valued, the applicable rate is the effective monthly rate; a stated annual rate cannot be treated as a monthly rate without conversion.

Key takeaway: For a level annuity with $n$ payments and zero effective interest per payment period, the immediate and due present-value and accumulated-value factors all equal $n$. Therefore each value is simply $nR$. The quotient formulas are undefined at zero, but the payment stream and its limiting factor are not.

Check Your Understanding

For each problem, assume there are no fees, taxes, or additional cash flows.

An annuity-immediate pays $180 at the end of each quarter for 16 quarters. The effective quarterly rate is 0%. Find its present value at time 0 and accumulated value at the end of quarter 16. (Answer: $2,880 for each value.)
An annuity-due makes 9 monthly payments of $250 at an effective monthly rate of 0%. State the payment dates and find the present value at time 0 and accumulated value at time 9. (Answer: Payments at times 0 through 8; $2,250 for each value.)
A fund receives 20 beginning-of-year deposits of $325. The effective annual rate is 0%. Find the accumulated value at time 20 and identify the applicable accumulated-value factor. (Answer: $\ddot{s}_{\overline{20}|}=20$; accumulated value $6,500.)
A level annuity-immediate has 7 payments of $500 and an effective rate of 0%. What is $a_{\overline{7}|}$, and what is the present value? (Answer: $a_{\overline{7}|}=7$; present value $3,500.)
Two streams each have 6 annual payments of $700 at an effective annual rate of 0%. One stream is immediate and the other is due. Compare their present values at time 0 and accumulated values at time 6. (Answer: Both streams have present value $4,200 and accumulated value $4,200.)

This zero-rate boundary case is aligned with the SOA Exam FM (Financial Mathematics) syllabus: identify the payment pattern, count the payments, and value the stream consistently with its stated interest assumption.

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