Matching the Interest Period to the Payment Period

Why the Interest Period Must Match the Payment Period

An annuity factor counts interest from one payment date to the next. Its rate must therefore describe interest over that same interval. If payments are monthly and the stated rate is effective monthly, the rate goes directly into the monthly annuity factor. If the rate is effective annually, it must first be converted to an equivalent monthly rate. Using an annual rate as though it applied to every month changes the assumed accumulation and produces an inconsistent value.

The key is to distinguish the payment interval from the way a rate is quoted. A rate described as effective for each payment interval is ready to use in the annuity factor. A rate effective over a different interval needs conversion before that factor is applied.

Definition: The payment interval is the time between consecutive payments. In an ordinary annuity-immediate, payments occur at the ends of successive intervals; in an annuity-due, they occur at the beginnings. An effective rate per payment interval is the rate $j$ such that one unit invested at the start of an interval accumulates to $1+j$ at its end.

For $n$ equal payment intervals and a rate $j$ effective per interval, the present-value and accumulated-value factors for an annuity-immediate are $a_{\overline{n}|}$ and $s_{\overline{n}|}$, evaluated using $j$. In the formulas below, $v=1/(1+j)$. The number $n$ counts payment intervals, not necessarily years.

a_{\overline{n}|}=\frac{1-v^n}{j} =\frac{1-(1+j)^{-n}}{j}, \qquad s_{\overline{n}|}=\frac{(1+j)^n-1}{j}.

These formulas assume that each interval has the same effective rate $j$, that the payments are level, and that payment dates are separated by exactly one such interval. The first formula values payments at the time immediately before the first payment; the second accumulates payments to the time of the final payment. If rates vary by interval, a single constant-rate annuity factor does not apply; the individual cash flows must instead be valued using the relevant accumulation factors.

Why the Annuity Factor Uses the Per-Interval Rate

The annuity-immediate present-value factor is the sum of the discounted values of one unit paid at the end of each interval. The first payment is discounted for one interval, the second for two, and the final payment for $n$ intervals. With an effective rate $j$ for each interval, the discount factor for $k$ intervals is $(1+j)^{-k}$.

a_{\overline{n}|} =\sum_{k=1}^{n}(1+j)^{-k} =\frac{1-(1+j)^{-n}}{j}.

To justify the final equality, let $v=(1+j)^{-1}$. Multiplying the finite sum $v+v^2+\cdots+v^n$ by $1-v$ cancels the interior terms and leaves $v-v^{n+1}$. Thus the sum is $v(1-v^n)/(1-v)$. Since $1-v=jv$, this becomes $(1-v^n)/j$. The accumulated-value factor follows by accumulating each payment to the final payment date: the payments earn interest for $n-1,n-2,\ldots,0$ intervals, respectively, so the resulting sum is $1+(1+j)+\cdots+(1+j)^{n-1}$. This geometric sum is $((1+j)^n-1)/j$.

Converting a Rate to the Payment Interval

Suppose $i$ is an effective annual rate and payments are made $m$ times per year. Under compound interest with a consistent annual accumulation basis, the equivalent effective rate $j$ for one payment interval satisfies

(1+j)^m=1+i, \qquad j=(1+i)^{1/m}-1.

The conversion makes the accumulation over a full year agree: applying the interval rate $m$ times gives the same annual accumulation as applying the annual effective rate once. Do not divide an annual effective rate by $m$; that generally gives a different annual accumulation.

A nominal rate convertible $m$ times per year is different. By definition, the rate per conversion period is $i^{(m)}/m$; it is not the effective annual rate. When payments occur once per conversion period, that periodic rate is the rate to use in the annuity factor. If the payment period does not match the conversion period, first find the accumulation over a payment interval from the applicable interest model.

Worked Example: Present Value of Monthly Payments

A hypothetical equipment fund will receive 24 monthly payments of $250, each at the end of a month. The effective interest rate is 0.5% per month. Find the present value immediately before the first payment. Assume deterministic payments, no mortality contingency, and the same effective rate for every month.

Method. The payment interval is one month, and the quoted rate is effective per month, so use $j=0.005$ directly and $n=24$. The payments form an annuity-immediate.

PV=250a_{\overline{24}|} =250\left(\frac{1-(1.005)^{-24}}{0.005}\right).

Compute the power and discount factor:

(1.005)^{24}\approx1.127159776, \qquad (1.005)^{-24}\approx0.887185669.

Then

a_{\overline{24}|} \approx\frac{1-0.887185669}{0.005} =\frac{0.112814331}{0.005} \approx22.5628662,

and

PV\approx250(22.5628662)=\$5{,}640.7166 \approx\boxed{\$5{,}640.72}.

Check. There are 24 payments of $250, so the undiscounted total is $6,000. The present value is lower, as expected at a positive interest rate. The method avoids treating the monthly rate as annual or changing the 24 monthly intervals into two years inside the annuity factor.

Worked Example: Converting an Annual Effective Rate for Quarterly Payments

A hypothetical reserve account receives four quarterly deposits of $1,000 at the end of each quarter. Its effective annual interest rate is 4.060401%. Find the accumulated value immediately after the fourth deposit. Assume deterministic deposits, no mortality contingency, and a constant compound-interest basis.

Method. Four payment intervals make up one year. Convert the annual effective rate to an effective quarterly rate. The stated annual rate is chosen so that the equivalent quarterly rate is exactly 1%.

j=(1+0.04060401)^{1/4}-1 =(1.01^4)^{1/4}-1 =0.01.

The deposits form an annuity-immediate with $n=4$ and a 1% rate per quarter. Accumulating to the date of the fourth deposit gives

AV=1{,}000s_{\overline{4}|} =1{,}000\left(\frac{(1.01)^4-1}{0.01}\right).

Since $(1.01)^4=1.04060401$,

s_{\overline{4}|} =\frac{1.04060401-1}{0.01} =4.060401, \qquad AV=1{,}000(4.060401) =\boxed{\$4{,}060.40}.

Check and trap avoided. The accumulated value must exceed the total deposits of $4,000 because the earlier deposits earn interest. A common wrong calculation inserts the annual effective rate directly as a quarterly rate:

1{,}000\left(\frac{(1.04060401)^4-1}{0.04060401}\right) \approx1{,}000(4.250286) =\$4{,}250.29.

That calculation assumes the account earns 4.060401% in each quarter, which would imply a much higher annual accumulation than the stated 4.060401% effective annual rate. The rate must be converted to 1% per quarter before applying the quarterly annuity factor.

Annuity-Due Payments Use the Same Rate-Matching Rule

Changing payment timing from the end to the beginning of each interval does not change which rate belongs in the factor. The rate still must be effective per payment interval. Timing changes which periods the payments earn interest, not the meaning of the rate.

For level payments at the beginnings of $n$ successive intervals, the accumulated value at the end of the $n$th interval is $R\ddot{s}_{\overline{n}|}$, where $R$ is the payment and $\ddot{s}_{\overline{n}|}=(1+j)s_{\overline{n}|}$. This follows because each due payment is made one interval earlier than its corresponding immediate payment and therefore earns one additional interval of interest.

Worked Example: Accumulating Monthly Payments Made in Advance

A hypothetical account receives 12 monthly deposits of $800, with the first deposit made immediately and each later deposit made one month after the previous one. The effective interest rate is 1% per month. Find the account value at the end of month 12, just after the last month of interest. Assume deterministic deposits, no mortality contingency, and the same effective rate each month.

Method. The payments are at the beginnings of their monthly intervals, so this is an annuity-due. The payment interval is one month, making $j=0.01$, and there are $n=12$ deposits.

AV=800\ddot{s}_{\overline{12}|} =800(1.01)s_{\overline{12}|} =800(1.01)\left(\frac{(1.01)^{12}-1}{0.01}\right).

Evaluate the power and immediate-annuity factor:

(1.01)^{12}\approx1.126825030, \qquad s_{\overline{12}|} \approx\frac{1.126825030-1}{0.01} =12.6825030.

Therefore,

\ddot{s}_{\overline{12}|} \approx1.01(12.6825030) =12.8093280, \qquad AV\approx800(12.8093280) =\boxed{\$10{,}247.46}.

Check. The deposits total $9,600, and the accumulated value is greater than that amount. The first deposit earns 12 months of interest and the last earns one month; the annuity-due factor reflects those extra months compared with an annuity-immediate. Using an annual rate as a monthly rate would not.

Common Rate-Period Errors

Most errors arise from a rate label being read without checking its time unit. Before substituting into an annuity factor, write down the payment interval and the interval represented by the rate. If they differ, convert the rate first.

Using an annual effective rate in a monthly factor. Convert it using $j=(1+i)^{1/12}-1$; do not use $i$ itself as the monthly rate.
Dividing an effective annual rate by the number of payments. This simple division does not generally preserve annual accumulation. Use the equivalent-rate equation.
Confusing effective and nominal rates. A nominal rate convertible monthly gives the monthly rate by dividing the nominal rate by 12. An annual effective rate needs an equivalent-rate conversion instead.
Using the right rate but the wrong count. If there are 18 monthly payments, the factor uses $n=18$, not $n=1.5$ because the stream lasts 1.5 years.
Assuming one factor works for changing rates. A constant-rate annuity factor presumes the same rate each payment interval. If that assumption fails, value each payment using the applicable period-by-period accumulation or discount factors.

Key takeaway: An annuity factor’s rate is the effective rate for one payment interval, and its $n$ is the number of those intervals. If the quoted rate covers a different period, convert it to an equivalent rate before using the factor.

Check Your Understanding

For each problem, identify the effective rate per payment interval before choosing the annuity factor.

A fund pays $1,000 at the end of each half-year for six half-years. The effective rate is 2% per half-year. Find the present value immediately before the first payment. (Answer: $1{,}000a_{\overline{6}|}$ at 2%, approximately $5,601.43.)
A hypothetical account receives five quarterly deposits of $700 at each quarter-end. The annual effective rate is 8.243216%. Find the accumulated value just after the fifth deposit. (Answer: the equivalent quarterly rate is 2%; $700s_{\overline{5}|}$ at 2%, approximately $3,642.83.)
Eight monthly deposits of $250 are made at the beginning of each month. The effective rate is 0.75% per month. Find the value at the end of month eight. (Answer: $250(1.0075)s_{\overline{8}|}$ at 0.75%, approximately $2,068.69.)
An annual effective rate is 6.16778%, and payments are monthly. Find the equivalent monthly effective rate and state the rate to use in a monthly annuity factor. (Answer: $(1.0616778)^{1/12}-1\approx0.005$, or 0.5% per month.)
A nominal rate of 6% convertible monthly applies to an account receiving 12 end-of-month payments of $400. State the monthly effective rate and write, but do not evaluate, the accumulated-value expression. (Answer: the monthly rate is $0.06/12=0.005$; the expression is $400s_{\overline{12}|}$ at 0.5%.)

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