Matching the Rate Period to the Payment Period

Why the Payment Interval Determines the Rate

A level-payment loan schedule applies interest between successive payments. The rate in the balance recurrence must therefore describe the accumulation over one payment interval. A stated annual rate is not automatically the rate to use when payments are monthly, quarterly, or otherwise more frequent or less frequent than annually.

The conversion depends on how the annual rate is stated. An effective annual rate describes the accumulation over a full year; a nominal rate specifies a conversion frequency; and a force of interest describes continuous accumulation. Each can be translated to an equivalent effective rate for the payment period. The essential check is that the selected periodic rate reproduces the stated annual accumulation when compounded across a year.

Definition: If payments are made $p$ times per year, the payment-period rate is the effective rate $j$ for the interval $1/p$ year between successive payments. A loan schedule with end-of-period level payments uses this rate in $B_k=B_{k-1}(1+j)-R$, where $B_k$ is the balance just after payment $k$ and $R$ is the payment. This assumes a fixed rate, payments at regular intervals, and no fees or other balance adjustments.

Converting the Stated Rate

For an effective annual rate $i$, the annual accumulation is $1+i$. The equivalent rate per payment period is the rate which, when applied for $p$ successive periods, produces that same annual accumulation. For a nominal annual rate $i^{(m)}$ convertible $m$ times per year, first identify the accumulation implied by its conversion convention. A force of interest $\delta$, when constant throughout the year, gives annual accumulation $e^\delta$.

Rate-conversion formulas: For $p$ payments per year:

If the effective annual rate is $i$, then $j=(1+i)^{1/p}-1$.
If the nominal annual rate is $i^{(m)}$, then $j=(1+i^{(m)}/m)^{m/p}-1$. This uses the equivalent annual accumulation $(1+i^{(m)}/m)^m$ and assumes an equivalent constant yield for the payment intervals.
If the force of interest is constant at $\delta$, then $j=e^{\delta/p}-1$.
If the effective annual discount rate is $d$, annual accumulation is $1/(1-d)$, so $j=(1-d)^{-1/p}-1$.

If the rate is already stated as an effective rate per payment period, use it directly. For a nominal rate, do not use the formula without first confirming its conversion frequency and the period to which the payments relate.

The effective annual-rate formula follows directly from equivalent accumulation. If $j$ is the rate for one of $p$ equal payment intervals, the accumulation over the year is $(1+j)^p$. Equivalence to the stated effective annual rate requires

(1+j)^p=1+i \quad\Longrightarrow\quad j=(1+i)^{1/p}-1.

Taking a root, rather than dividing the annual effective rate by $p$, is necessary because the intervals compound. Once $j$ is found, the standard level-payment formula applies for $n$ end-of-period payments on a loan of amount $L$: $R=L/a_{\overline{n}|}$, where $a_{\overline{n}|}=[1-(1+j)^{-n}]/j$. This formula assumes $j>0$, a fixed rate, and payments at the end of each period.

Count payment periods per year.
For example, monthly payments give $p=12$, while payments every three months give $p=4$.

Identify the rate convention.
Determine whether the statement gives an effective annual rate, a nominal rate with a stated conversion frequency, a constant force, an effective discount rate, or an already-periodic rate.

Convert to the payment interval.
Use the corresponding formula and retain sufficient precision until the payment and schedule values are calculated.

Check a full year of accumulation.
Confirm that $p$ payment-period accumulations equal the annual accumulation implied by the stated convention.

Worked Loan Examples

Worked Example: Quarterly Payments at an Effective Annual Rate

A borrower receives $10,000 and repays it with four equal payments at the ends of successive quarters. The effective annual interest rate is 21%. Find the quarterly rate, the payment, and the interest and principal portions of the first payment. Assume no fees or other adjustments.

Method. Convert the effective annual rate to an equivalent quarterly effective rate, then use the level-payment annuity formula. The four payments span one year, so the conversion check is $(1+j)^4=1.21$.

There are four payment periods per year, so $p=4$. Therefore,

j=(1.21)^{1/4}-1\approx0.0488088482.

The quarterly rate is approximately 4.88088482%. The annuity factor for four quarterly payments is

a_{\overline{4}|} =\frac{1-(1+j)^{-4}}{j} =\frac{1-(1.21)^{-1}}{0.0488088482} \approx3.55578395.

Thus the level payment is $R=10{,}000/3.55578395\approx \$2{,}812.32$. First-quarter interest is calculated on the original balance:

10{,}000(0.0488088482)\approx \$488.09.

The first payment's principal portion is $\$2{,}812.32-\$488.09=\$2{,}324.23$. The balance immediately after that payment is $\$10{,}000-\$2{,}324.23=\$7{,}675.77$. Values are rounded to cents only at the final step. As a check, $(1+0.0488088482)^4\approx1.21$, as required.

Common mistake avoided. Using 5% per quarter would be incorrect: $1.05^4=1.21550625$, which implies an effective annual rate of 21.550625%, not 21%.

Worked Example: Quarterly Payments With a Nominal Rate Convertible Quarterly

A $10,000 loan is repaid by four equal end-of-quarter payments. The stated nominal annual rate is $i^{(4)}=10\%$, convertible quarterly. Find the quarterly rate, payment, and first-payment split. Assume the rate stays fixed and there are no fees.

Method. Because the rate is nominal convertible quarterly and payments are also quarterly, the rate per conversion period is already the rate per payment period:

j=\frac{i^{(4)}}{4}=\frac{0.10}{4}=0.025.

The quarterly rate is 2.5%. Its implied annual effective accumulation is $1.025^4\approx1.10381289$, or an effective annual rate of approximately 10.3813%. The annuity factor is

a_{\overline{4}|} =\frac{1-(1.025)^{-4}}{0.025} \approx3.761974.

The payment is $R=10{,}000/3.761974\approx \$2{,}658.18$. First-quarter interest is $\$10{,}000(0.025)=\$250.00$, so the principal repaid is $\$2{,}658.18-\$250.00=\$2{,}408.18$. The balance after the first payment is $\$10{,}000-\$2{,}408.18=\$7{,}591.82$. Rounded to cents, the first payment components sum to $\$2{,}408.18+\$250.00=\$2{,}658.18$.

Common mistake avoided. Dividing 10% by four is appropriate here because the nominal rate is specifically convertible quarterly. It would not be appropriate to apply that same division automatically to an effective annual rate.

Worked Example: Monthly Payments at a Constant Force of Interest

A $12,000 loan is repaid with 12 equal end-of-month payments. The force of interest is constant at $\delta=0.12$ per year. Find the equivalent monthly rate, payment, and first-payment interest and principal portions. Assume no fees and retain full precision until rounding the money to cents.

Method. With a constant force, accumulation over $t$ years is $e^{\delta t}$. A month is $1/12$ year, so the effective monthly rate is

j=e^{0.12/12}-1=e^{0.01}-1\approx0.0100501671.

The monthly rate is approximately 1.00501671%. Its annual check is $(1+j)^{12}=e^{0.12}$. The 12-payment annuity factor can also be written using $1+j=e^{0.01}$:

a_{\overline{12}|} =\frac{1-(1+j)^{-12}}{j} =\frac{1-e^{-0.12}}{e^{0.01}-1} \approx11.251510.

The payment is $R=12{,}000/11.251510\approx \$1{,}066.52$. First-month interest is $12{,}000(0.0100501671)\approx \$120.60$. The first payment repays $\$1{,}066.52-\$120.60=\$945.92$ of principal, leaving a balance of $\$12{,}000-\$945.92=\$11{,}054.08$. The rounded payment splits into $\$120.60+\$945.92=\$1{,}066.52$.

Common mistake avoided. Using $\delta/12=0.01$ as the effective monthly rate confuses a force with an effective rate. The force accumulates continuously, so the monthly effective rate is $e^{0.01}-1$, not $0.01$.

Checks and Common Rate-Period Traps

Before using a rate in a loan schedule, state what interval it applies to. For an effective annual rate $i$, the check is $(1+j)^p=1+i$. For a nominal annual rate $i^{(m)}$, compare with $(1+i^{(m)}/m)^m$. For a constant force $\delta$, compare with $e^\delta$. The rate conversion and the balance recurrence must use the same payment interval.

A frequent error is to divide an annual rate by the number of payments per year without checking its convention. That division is correct for a nominal rate when the conversion frequency matches the payment frequency. It is not generally correct for an effective annual rate or a force. Another trap is confusing the nominal conversion frequency with the payment frequency: if they differ, first find the equivalent accumulation over the payment interval rather than selecting the nominal rate's conversion-period rate unchanged.

Key takeaway: A loan schedule needs an effective rate for the interval between payments. Convert the stated rate convention to that interval, verify the annual accumulation, and only then calculate the payment or interest and principal portions.

Check Your Understanding

For each problem, identify the payment-period rate before using it in any loan calculation.

A loan has an effective annual rate of 6% and semiannual end-of-period payments. Find the effective rate per payment period. (Answer: $(1.06)^{1/2}-1\approx0.029563$, or 2.9563% per half-year.)
A loan has a nominal annual rate of 9% convertible quarterly, with monthly payments. Find the equivalent effective monthly rate. (Answer: $(1+0.09/4)^{1/3}-1\approx0.007444$, or 0.7444% per month.)
A loan has a constant force of interest of 0.09 per year and payments every two months. Find the effective rate per payment period. (Answer: $e^{0.09/6}-1=e^{0.015}-1\approx0.0151136$, or 1.5114% per two months.)
A loan has an effective annual rate of 12.550809% and quarterly payments. Find the quarterly rate and verify the annual accumulation. (Answer: 3% per quarter, since $1.03^4=1.12550809$.)
A loan has an effective annual rate of 8% and payments every four months. A candidate uses $0.08/3$ as the payment-period rate. Give the correct rate and explain why that division is not valid. (Answer: $(1.08)^{1/3}-1\approx0.0259856$, or 2.5986% per four months; the effective annual rate compounds, so it must be converted by taking a cube root rather than divided by three.)

What You'll Learn