Calculating a Scheduled Loan Payment

What Determines a Scheduled Loan Payment?

A level-payment loan exchanges an amount advanced now for a fixed sequence of equal payments. To calculate each payment, the key is to value the payments at the same date as the loan principal. If the payment stream’s present value equals the principal, the scheduled payment balances the loan under the stated interest assumptions.

The interest rate must correspond to the time between payments. For annual payments, use an effective annual rate and count years. For monthly payments, use the effective rate per month and count months. A rate expressed on a different basis must be converted before applying the payment formula.

Definition: A level-payment loan with payments in arrears has a principal $L$ advanced at time 0 and $n$ equal payments $R$ made at the ends of periods 1 through $n$. If the effective interest rate per payment period is $i$, and there are no fees, skipped payments, or other cash flows, the scheduled payment is $$ R=\frac{L}{a_{\overline{n}|}}=\frac{Li}{1-(1+i)^{-n}}, $$ where $a_{\overline{n}|}=(1-v^n)/i$ and $v=1/(1+i)$. This formula assumes $i>0$ and that the loan is fully repaid by the $n$th payment.

The payment period is the interval between scheduled payments. Thus, $n$ is the number of payments, not automatically the number of years. If a loan has monthly payments over five years, then $n=60$. If the stated rate is nominal annual $i^{(m)}$, convertible $m$ times per year, and payments are made at each conversion date, the effective rate per payment period is $i^{(m)}/m$.

Why the Payment Formula Works

At origination, each payment is discounted back to time 0. The payment at time $k$ has present value $Rv^k$, so the total present value is the sum of those discounted payments. Equating that value to the amount borrowed gives the result.

L=Rv+Rv^2+\cdots+Rv^n =R\frac{v(1-v^n)}{1-v} =R\frac{1-v^n}{i} =Ra_{\overline{n}|}.

For the geometric sum, $1-v=i/(1+i)=iv$. Therefore, $v/(1-v)=1/i$, which gives the displayed annuity-immediate factor. Solving $L=Ra_{\overline{n}|}$ for $R$ gives the scheduled-payment formula. This valuation applies when payments are at the ends of periods and the same effective rate applies in every period; a different payment timing or changing interest rates requires a different valuation.

Identify the payment dates.
Decide whether payments are annual, monthly, quarterly, or another frequency, and count the total number of payments.

Find the rate per payment period.
Convert the stated rate if its compounding basis does not match the payment interval.

Calculate the payment.
Use $R=L/a_{\overline{n}|}$ for end-of-period payments, retaining precision until the final rounding.

Check the valuation.
Discount the $n$ payments to origination. Their present value should equal $L$, apart from a small difference caused by rounding the payment to cents.

Worked Payment Calculations

Worked Example: Four Annual Payments at an Effective Annual Rate

A hypothetical loan of $\$5{,}000$ is repaid by four equal annual payments at year-end. The effective annual interest rate is $8\%$. Find the scheduled annual payment to the nearest cent and verify it at origination.

Method. The payments are annual and occur at the end of each year, so use $i=0.08$, $n=4$, and the annuity-immediate formula. The present-value factor is $$ a_{\overline{4}|}=\frac{1-(1.08)^{-4}}{0.08} =\frac{1-0.735029853}{0.08} =3.31212684\text{ (approximately)}. $$

The payment before rounding is $$ R=\frac{5{,}000}{3.31212684} \approx \$1{,}509.6040. $$ Rounded to the nearest cent, the scheduled payment is $\$1{,}509.60$.

To check, multiply the unrounded payment by the annuity factor: $$ \$1{,}509.6040(3.31212684)\approx \$5{,}000.00. $$ Using the cent-rounded payment instead gives $\$1{,}509.60(3.31212684)\approx \$4{,}999.99$, a difference of about one cent from rounding. The full-precision payment satisfies the origination value exactly, up to the displayed precision. The common mistake to avoid is reporting $\$1{,}509.00$: the division does not produce that amount, and rounding to cents gives $\$1{,}509.60$.

Worked Example: Converting a Nominal Rate for Semiannual Payments

A hypothetical one-year loan of $\$2{,}100$ has a nominal annual interest rate of $20\%$, convertible semiannually. It is repaid by two equal payments, each made at the end of a half-year. Calculate each payment and verify the result.

Method. There are two payment periods. Because the nominal rate is convertible twice per year and payments coincide with the conversion dates, the effective rate per half-year is $$ i=\frac{0.20}{2}=0.10,\qquad n=2. $$ The annuity-immediate factor is $$ a_{\overline{2}|}=\frac{1-(1.10)^{-2}}{0.10} =\frac{1-1/1.21}{0.10} =\frac{0.173553719}{0.10} =1.73553719. $$

The payment is $$ R=\frac{2{,}100}{1.73553719}=\$1{,}210. $$ The present-value check, keeping the two half-year dates explicit, is $$ \frac{1{,}210}{1.10}+\frac{1{,}210}{1.21} =1{,}100+1{,}000 =\$2{,}100. $$

The scheduled payment is $\$1{,}210.00$ every half-year. This calculation avoids treating $20\%$ as the rate for each half-year: that would apply the nominal annual rate on the wrong time basis and produce an incorrect payment.

Worked Example: Monthly Payments With a Monthly Effective Rate

A hypothetical short-term loan of $\$2{,}050$ is repaid with two equal payments at the ends of the next two months. The effective interest rate is $5\%$ per month. Find the payment to the nearest cent and check its value.

Method. The rate is already stated per payment period, so $i=0.05$ and $n=2$. The annuity factor is $$ a_{\overline{2}|} =\frac{1-(1.05)^{-2}}{0.05} =\frac{1-1/1.1025}{0.05} =\frac{0.092970522}{0.05} =1.85941043. $$

Then $$ R=\frac{2{,}050}{1.85941043}\approx \$1{,}102.50. $$ For a direct check, discount each payment separately: $$ \frac{1{,}102.50}{1.05}+\frac{1{,}102.50}{1.1025} =\$1{,}050+\$1{,}000 =\$2{,}050. $$

The scheduled payment is $\$1{,}102.50$ per month. The common mistake here is to count two payments but discount using an annual rate or annual periods. Both the exponent and the rate must use months in this problem.

Payment Timing and Practical Checks

A payment at the beginning of a period is not equivalent to a payment at its end: it is discounted for one fewer period. For $n$ equal payments made at times $0,1,\ldots,n-1$, the present-value factor is the annuity-due factor $\ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}$. Consequently, the payment for an annuity due is $L/\ddot{a}_{\overline{n}|}$, not $L/a_{\overline{n}|}$. Confirm the first payment date before selecting the factor.

A scheduled payment is a level amount in the model, but a contract may require rounding each installment to cents. In that case, the present value of the rounded installments can differ slightly from the principal. Actual loan contracts may address this with a final adjusted payment. Do not alter the formula to force an exact equality after rounding; instead, state the rounding and distinguish the unrounded mathematical payment from the amount scheduled in cents.

Several quick checks help catch errors:

For $i>0$, each installment must exceed the amount obtained by dividing principal by the number of payments, since some of each payment covers interest.
Increasing the rate while holding principal and term fixed increases the payment; increasing the number of payments while holding principal and rate fixed generally reduces it.
Use the number of payment periods as $n$, and use the effective rate for one such period as $i$.
When possible, verify using $R(v+v^2+\cdots+v^n)=L$, not just by repeating the same formula with the same inputs.

These checks are useful for calculations aligned with the SOA Exam FM (Financial Mathematics) syllabus and for reviewing a quoted level payment. They do not replace checking contract terms such as fees, a different first payment date, or a final payment that is not level.

Key takeaway: For a loan repaid by $n$ equal end-of-period payments, first match the effective rate to the payment period, then calculate $R=L/a_{\overline{n}|}$. The defining check is that the present value at origination of all scheduled payments equals the principal.

Check Your Understanding

For each problem, assume there are no fees or other cash flows and round the requested payment to the nearest cent.

A hypothetical $\$7{,}200$ loan is repaid by two annual end-of-year payments at an effective annual rate of $25\%$. Find the payment. (Answer: $\$5{,}000.00$)
A $\$4{,}200$ loan is repaid by two equal year-end payments at an effective annual rate of $10\%$. Calculate the payment and verify its present value. (Answer: $\$2{,}420.00$; present value $=\$2{,}420/1.10+\$2{,}420/1.21=\$2{,}200+\$2{,}000=\$4{,}200$)
A hypothetical $\$6{,}000$ loan has two month-end payments, with an effective rate of $5\%$ per month. Find each payment. (Answer: $\$3{,}226.83$)
A loan of $\$1{,}941.56$ is repaid with two equal quarter-end payments. The effective rate is $2\%$ per quarter. Calculate the payment and show the unrounded present-value factor used. (Answer: $\$1{,}000.00$; $a_{\overline{2}|}=(1-1/1.02^2)/0.02\approx1.94156094$)
A $\$4{,}200$ loan is repaid by two equal payments, the first immediately and the second one year later. The effective annual rate is $10\%$. Find each payment, identifying the correct timing factor. (Answer: $\$2{,}200.00$; $\ddot{a}_{\overline{2}|}=1+1/1.10=21/11$, so $4{,}200/(21/11)=2{,}200$)

What You'll Learn