Introduction
The first row of an amortization schedule applies the loan’s periodic interest rate to the original principal. The scheduled payment must then be divided between the interest charge and the amount that reduces (or, in some cases, increases) the loan balance. Keeping these steps separate makes it easier to see exactly how the first payment affects the debt.
This tutorial considers a loan with payments at the end of each payment period. It focuses on the first payment only: the interest accrued during that period, the principal portion of the payment, and the resulting balance immediately after payment. The same accounting identities apply to many loan types, but the rate and payment period must first be matched correctly. This material is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
First-Payment Accounting
Assume interest is charged on the outstanding balance at a constant effective rate \(j\) over the first payment period, with no additional advances, fees, or payments during that period. The balance grows from \(L\) to \(L(1+j)\) before the payment is made. The payment \(R\) is then subtracted.
The principal portion is a signed amount. If \(R>jL\), some principal is repaid. If \(R=jL\), the payment covers exactly the interest and the balance does not change. If \(R<jL\), the payment does not cover the interest charge; the shortfall is added to the balance. That last case is called negative amortization.
Why the Balance Formula Holds
The formula follows directly from the order of events in an end-of-period payment. Interest is calculated on the opening balance \(L\), so the amount due immediately before payment is \(L+jL=L(1+j)\). Subtracting the payment gives the balance after payment. Equivalently, the payment first covers \(jL\) of interest, leaving \(R-jL\) to reduce principal.
This identity is useful as a check: calculating the balance by subtracting the principal portion from the opening balance must give the same result as adding interest to the opening balance and then subtracting the full payment. It does not depend on the loan’s later payments. It does depend on the payment being made at the end of the period; a payment at the beginning would occur before that period’s interest accrues.
For the first payment, this is the original principal \(L\).
Use the effective rate \(j\) for one interval between payments.
Multiply the opening balance by the periodic rate: \(jL\).
Subtract interest from the payment to find principal repaid, then subtract that principal portion from \(L\).
Worked Examples
Worked Example: A Quarterly Payment and Its First Balance
A borrower takes a loan of $25,000. Interest is 2% per quarter, and the scheduled payment of $2,300 is made at the end of each quarter. Find the first payment’s interest portion, principal portion, and the balance immediately after the payment.
Method. Apply the periodic rate to the opening principal, then split the payment. The stated rate is already per quarter, matching the payment interval.
Interest. The opening balance is $25,000, so the first quarter’s interest is \(0.02(25{,}000)=\$500\).
Principal portion. Of the $2,300 payment, $500 covers interest. The remainder repays principal: \(2{,}300-500=\$1{,}800\).
Balance after payment. Subtract the principal repaid from the opening balance: \(25{,}000-1{,}800=\$23{,}200\). As a check, the balance just before payment is \(25{,}000+500=\$25{,}500\); after subtracting the full payment, \(25{,}500-2{,}300=\$23{,}200\).
The first payment therefore contains $500 of interest and $1,800 of principal, leaving a balance of $23,200. The check avoids the common error of subtracting only the interest charge from the opening balance.
Worked Example: Find the Payment, Then Split It
A loan of $12,000 is repaid by 12 equal monthly payments at the end of each month. The nominal annual rate is 6% convertible monthly. Find the first payment to the nearest cent, then find its interest portion, principal portion, and the balance after payment. Use the level-payment loan model with no fees or extra payments.
Step 1: Find the monthly rate. A nominal annual rate of 6% convertible monthly gives \(j=0.06/12=0.005\), or 0.5% per month.
Step 2: Find the level payment. The present value of 12 end-of-month payments is the loan principal. Thus \(12{,}000=R a_{\overline{12}|0.005}\), where \(a_{\overline{12}|0.005}=[1-(1.005)^{-12}]/0.005\).
First, \((1.005)^{12}\approx1.061677812\), so \((1.005)^{-12}\approx0.941905340\). Therefore, \(a_{\overline{12}|0.005}\approx(1-0.941905340)/0.005=11.618932\). The unrounded payment is approximately \(12{,}000/11.618932=\$1{,}032.797\), so the payment rounded to cents is $1,032.80.
Step 3: Split the first payment. Interest for the first month is \(0.005(12{,}000)=\$60.00\). Using the contractual payment rounded to cents, the principal portion is \(1{,}032.80-60.00=\$972.80\).
Step 4: Find and check the balance. The balance after payment is \(12{,}000-972.80=\$11{,}027.20\). Alternatively, the amount due just before payment is $12,060.00, and \(12{,}060.00-1{,}032.80=\$11{,}027.20\).
The first payment is $60.00 interest and $972.80 principal, leaving $11,027.20. The payment is rounded to two decimal places; calculations use that stated cents amount for the schedule entry.
Worked Example: Convert the Quoted Rate Before Calculating
A borrower owes $50,000 on a loan with a nominal annual rate of 7.2% convertible semiannually. The first scheduled payment is $4,000 at the end of six months. Find its interest and principal portions and the balance immediately afterward. Round money to the nearest cent.
Method. Convert the nominal rate to the rate for one six-month payment period. Because the nominal rate is convertible twice a year, the periodic rate is \(j=0.072/2=0.036\), or 3.6% per half-year.
Interest. The first period’s interest is \(0.036(50{,}000)=\$1{,}800.00\).
Principal portion. The $4,000 payment covers $1,800.00 of interest, leaving \(4{,}000-1{,}800=\$2{,}200.00\) to repay principal.
Balance and check. Subtracting principal repaid gives \(50{,}000-2{,}200=\$47{,}800.00\). The pre-payment amount is \(50{,}000+1{,}800=\$51{,}800\), and \(51{,}800-4{,}000=\$47{,}800.00\), which agrees.
The first payment contains $1,800.00 of interest and $2,200.00 of principal. The common mistake here is to apply 7.2% as the six-month rate. That would give $3,600 of interest, but 7.2% is the nominal annual rate, not the rate for one half-year.
Rate Periods, Timing, and Common Traps
The periodic rate is the rate that applies over the time from the opening balance to the next scheduled payment. If a loan quotes a nominal annual rate \(i^{(m)}\) convertible \(m\) times per year and payments occur at that same frequency, then the rate per payment period is \(i^{(m)}/m\). If an effective annual rate \(i\) is given and there are \(m\) equal payment periods per year, the equivalent effective rate per period is \(j=(1+i)^{1/m}-1\). Do not use either conversion without checking that the quoted compounding convention and payment timing support it.
A rate stated directly per payment period needs no conversion. For example, “2% per quarter” already gives the rate to use for quarterly payments. In contrast, a nominal annual rate is not automatically the rate per payment period. The rate must always be interpreted before multiplying it by the opening balance.
| Situation | First-period interest | Effect of payment |
|---|---|---|
| \(R>jL\) | \(jL\) | Principal portion is positive; balance falls |
| \(R=jL\) | \(jL\) | Principal portion is zero; balance is unchanged |
| \(R<jL\) | \(jL\) | Principal portion is negative; balance rises |
A frequent timing error is to assume that any payment called “the first payment” is made at the end of the first period. The formulas here assume that interest accrues on \(L\) for one full period before the payment. If the contract instead requires a payment immediately at the start, that payment is not preceded by a full period of interest; the end-of-period calculation above does not apply unchanged.
Check Your Understanding
For each problem, find the requested first-payment quantities using the stated payment timing. Money answers are rounded to the nearest cent where needed.
- A $8,000 loan charges 1.25% per month. The first end-of-month payment is $700. Find its interest portion, principal portion, and the balance after payment. (Answer: interest $100.00; principal $600.00; balance $7,400.00)
- A $30,000 loan has a nominal annual rate of 9% convertible quarterly, and the first quarterly payment is $2,500. Find the first-period interest, principal portion, and balance after payment. (Answer: interest $675.00; principal $1,825.00; balance $28,175.00)
- A $9,000 loan has an effective annual rate of 12.6825%, with monthly payments. Use the equivalent monthly rate of 1% and a first payment of $950. Find the interest portion, principal portion, and post-payment balance. (Answer: interest $90.00; principal $860.00; balance $8,140.00)
- A $20,000 balance accrues interest at 0.75% per month. The borrower makes a $120 payment at month-end. Find the signed principal portion and the balance afterward; state whether the balance falls or rises. (Answer: interest $150.00; principal portion −$30.00; balance $20,030.00, so it rises)
- A $16,000 loan has a nominal annual rate of 8% convertible quarterly. The first payment is $1,500 at the end of the quarter. A borrower mistakenly uses 8% as the quarterly rate. Find the correct interest and balance after payment, and identify the mistaken interest charge. (Answer: correct quarterly rate 2%; interest $320.00; principal $1,180.00; balance $14,820.00; mistaken interest charge $1,280.00)