Why Split a Loan Payment?
A fixed loan payment is not all principal repayment. Part pays the interest accrued since the previous payment; what remains reduces the amount owed. The split changes from one period to the next even when the scheduled payment does not: as the opening balance changes, so does the interest charge.
For any payment period, the essential order is calculate interest first, then subtract it from the payment. Reversing that order, or charging interest on the original loan amount every time, gives an incorrect amortization schedule. The method applies to level-payment loans with a specified rate and payment frequency; the rate used must be the effective rate for one payment period.
This definition assumes interest accrues at a fixed effective rate over each period, payments are made at the end of the period, and there are no fees, extra payments, or other balance adjustments. For a usual fully amortizing loan, each payment covers the interest due and includes a positive principal repayment. If a problem specifies a nominal annual rate, convert it to the effective rate for one payment period before calculating interest.
Why the Split Works
At the start of a period, the borrower owes \(B_{k-1}\). At an effective period rate \(i\), the interest accrued on that amount is \(iB_{k-1}\). Therefore, just before the scheduled payment, the amount owed is the opening balance plus the interest:
The payment \(R\) first covers the interest charge. The part left to reduce the amount owed is \(R-iB_{k-1}\), which is the principal portion. Equivalently, subtracting the entire payment from the amount owed after interest gives the same closing balance as subtracting only the principal portion from the opening balance:
This accounting identity explains why the payment split works. The interest portion compensates for the period’s interest accrual; only the remaining portion pays down the original balance. It also provides a useful check: interest portion plus principal portion must equal the scheduled payment.
A Reliable Calculation Process
Use the following sequence for each period. If a problem asks about a later payment, first determine that period’s opening balance from the information provided or from the earlier rows of the amortization schedule.
Identify the effective rate for one payment period. Convert a nominal rate if necessary.
Multiply the opening balance by the effective period rate.
Subtract the period’s interest from the scheduled payment.
Add the interest and principal portions and confirm that their sum equals the payment.
Keep full precision during the calculations and round monetary amounts only as the problem instructs. For a schedule stated in dollars and cents, a practical check is that the rounded interest and principal entries add to the payment to the cent.
Worked Example: Splitting Three Annual Payments
A hypothetical loan has an opening balance of $15,000. It charges an effective annual interest rate of 4%, and the borrower makes level annual payments of $4,000 at each year-end. Find the interest and principal portions of the first three payments, and check the balance after the third payment. No mortality or probability assumption is involved.
Method. Use 4% as the rate for each annual payment period. For each row, calculate interest from the opening balance, then subtract that interest from the $4,000 payment. The common error this avoids is applying the original $15,000 balance to every year’s interest calculation.
Payment 1. The opening balance is $15,000. Interest is \(0.04(15{,}000)=\$600\). The principal portion is \(\$4{,}000-\$600=\$3{,}400\). The balance after payment 1 is \(\$15{,}000-\$3{,}400=\$11{,}600\).
Payment 2. The opening balance is now $11,600. Interest is \(0.04(11{,}600)=\$464\). The principal portion is \(\$4{,}000-\$464=\$3{,}536\). The balance after payment 2 is \(\$11{,}600-\$3{,}536=\$8{,}064\).
Payment 3. The opening balance is $8,064. Interest is \(0.04(8{,}064)=\$322.56\). The principal portion is \(\$4{,}000-\$322.56=\$3{,}677.44\). The balance after payment 3 is \(\$8{,}064-\$3{,}677.44=\$4{,}386.56\).
Check. Each split adds to $4,000: \(\$600+\$3{,}400=\$4{,}000\), \(\$464+\$3{,}536=\$4{,}000\), and \(\$322.56+\$3{,}677.44=\$4{,}000\). Across the three payments, principal repaid is \(\$3{,}400+\$3{,}536+\$3{,}677.44=\$10{,}613.44\); subtracting this from $15,000 gives the same $4,386.56 closing balance.
Use the Opening Balance for Each Period
The balance used to calculate a period’s interest is the amount owed at the start of that period, not the balance after the payment. The payment includes interest accumulated during the period, so using a post-payment balance would charge interest on an amount that was not owed for the full period. Likewise, reusing the original loan amount would ignore the principal already repaid.
In the preceding example, the opening balance fell from $15,000 to $11,600 and then to $8,064. Accordingly, the interest charge fell from $600 to $464 and then to $322.56, even though the annual payment remained $4,000. The decrease in interest leaves more of each later payment available for principal.
Worked Example: Finding Monthly Components From the Initial Loan Amount
A hypothetical loan of $20,000 charges an effective monthly rate of 1%. The borrower pays $2,500 at the end of each month. Find the interest and principal portions of the first three payments and the balance after the third payment. No mortality or probability assumption is involved.
Method. Build the opening balance for each month by subtracting the previous month’s principal portion. The common mistake this avoids is using the original $20,000 balance to calculate interest in all three months.
Month 1. Interest is \(0.01(\$20{,}000)=\$200\). Principal is \(\$2{,}500-\$200=\$2{,}300\). The next opening balance is \(\$20{,}000-\$2{,}300=\$17{,}700\).
Month 2. Interest is \(0.01(\$17{,}700)=\$177\). Principal is \(\$2{,}500-\$177=\$2{,}323\). The next opening balance is \(\$17{,}700-\$2{,}323=\$15{,}377\).
Month 3. Interest is \(0.01(\$15{,}377)=\$153.77\). Principal is \(\$2{,}500-\$153.77=\$2{,}346.23\). The balance after the third payment is \(\$15{,}377-\$2{,}346.23=\$13{,}030.77\).
Check. The first three interest charges total \(\$200+\$177+\$153.77=\$530.77\), and principal repaid totals \(\$2{,}300+\$2{,}323+\$2{,}346.23=\$6{,}969.23\). The total of the payments is \(\$7{,}500\), equal to \(\$530.77+\$6{,}969.23\). Also, \(\$20{,}000-\$6{,}969.23=\$13{,}030.77\), matching the balance obtained period by period.
Match the Rate to the Payment Frequency
A nominal annual rate convertible \(m\) times per year, written \(i^{(m)}\), gives a rate per conversion period of \(i^{(m)}/m\). When loan payments occur at those same intervals, this is the rate to use in the payment split. A nominal annual rate is not automatically the effective rate for each payment period; applying the annual percentage directly to a monthly or quarterly opening balance can substantially overstate interest.
Worked Example: Using a Semiannual Rate
A hypothetical loan has a nominal annual rate of 8% convertible semiannually. Immediately before payment 5, its opening balance is $43,200. Level semiannual payments are $5,000. Find the interest and principal portions of payments 5 and 6, and the balance after payment 6. No mortality or probability assumption is involved.
Method. There are two conversion periods per year, and payments are semiannual. The effective rate for one payment period is therefore \(0.08/2=0.04\), or 4%. The common mistake this avoids is treating the nominal annual rate of 8% as the rate for each half-year.
Payment 5. Interest is \(0.04(\$43{,}200)=\$1{,}728\). Principal is \(\$5{,}000-\$1{,}728=\$3{,}272\). The opening balance for payment 6 is \(\$43{,}200-\$3{,}272=\$39{,}928\).
Payment 6. Interest is \(0.04(\$39{,}928)=\$1{,}597.12\). Principal is \(\$5{,}000-\$1{,}597.12=\$3{,}402.88\). The balance after payment 6 is \(\$39{,}928-\$3{,}402.88=\$36{,}525.12\).
Check. For payment 5, \(\$1{,}728+\$3{,}272=\$5{,}000\). For payment 6, \(\$1{,}597.12+\$3{,}402.88=\$5{,}000\). Using 8% for payment 5 would incorrectly give \(\$43{,}200(0.08)=\$3{,}456\) of interest. That is not the half-year interest charge under the stated nominal-rate convention.
When the Payment Does Not Cover Interest
The principal portion is payment minus interest, so it need not be positive for every possible payment amount. If \(R>iB_{k-1}\), the payment covers the interest and repays some principal. If \(R=iB_{k-1}\), the payment covers exactly the interest and makes no principal repayment. If \(R<iB_{k-1}\), the payment does not cover all the interest; the calculated principal portion is negative. Under a loan arrangement that adds unpaid interest to the balance, this is negative amortization rather than principal repayment.
For example, if the opening balance is $10,000, the effective monthly rate is 1.2%, and the payment is $100, interest is \(0.012(\$10{,}000)=\$120\). The principal portion is \(\$100-\$120=-\$20\). The payment is $20 short of the interest charge. Do not describe that result as a $20 principal repayment.
Common Mistakes to Avoid
- Charging interest on the original principal every period. Use the opening balance for the period. Principal repaid by earlier payments has already reduced what is owed.
- Using the wrong side of the payment date. Calculate interest from the opening balance for the period, not from the balance after the current payment.
- Applying an annual rate to a shorter payment period. Convert a nominal rate to the rate per payment period when required by its conversion convention.
- Calling the whole payment principal. Subtract the interest charge first; only the remainder reduces principal.
- Assuming every principal portion is positive. Check whether the payment exceeds the interest due. If it does not, the split indicates no principal reduction or negative amortization.
Check Your Understanding
For each problem, identify the effective rate per payment period before calculating the split. Assume payments occur at the end of each period and there are no fees or extra payments.
- A hypothetical loan opens the year with a balance of $12,000. The effective annual rate is 5%, and the annual payment is $2,800. Find the first payment’s interest and principal portions and the balance after payment. (Answer: interest $600; principal $2,200; closing balance $9,800.)
- A hypothetical loan has an opening balance of $31,250 for a month, an effective monthly rate of 0.8%, and a monthly payment of $2,100. Find the interest and principal portions and the closing balance. (Answer: interest $250; principal $1,850; closing balance $29,400.)
- Immediately before payment 5 on a hypothetical annual-payment loan, the opening balance is $17,500. The effective annual rate is 6% and the payment is $2,000. Find the interest and principal portions and the balance after payment 5. (Answer: interest $1,050; principal $950; closing balance $16,550.)
- A hypothetical quarterly-payment loan has a nominal annual rate of 9% convertible quarterly. Its opening balance before payment 3 is $24,000, and each payment is $2,700. Find the period rate, interest and principal portions, and the balance after payment 3. (Answer: period rate 2.25%; interest $540; principal $2,160; closing balance $21,840.)
- A hypothetical loan has an opening balance of $10,000, an effective monthly rate of 1.2%, and a monthly payment of $100. Find the interest and principal portions. If unpaid interest is added to the balance, state the balance after payment. (Answer: interest $120; principal portion \(-\$20\); closing balance $10,020.)