6.2 Calculating the Principal and Interest Components of a Loan Payment

NOTE

As described above, you can find the interest paid and principal paid components of any payment by using the amortization worksheet on your calculator.  To calculate these values by hand, use the following formulas.

• [latex]\mbox{Interest Paid for a Payment}=\mbox{Balance from the Previous Row} \times i[/latex]
• [latex]\mbox{Principal Paid for a Payment}=\mbox{Payment}-\mbox{Interest Paid}[/latex]

where [latex]i[/latex] is the periodic interest rate for the loan.

But, these formulas require two things:  that you know the balance from the previous row and that the payment frequency and the compounding frequency for the loan are equal. Although it is possible to solve both of these problems, such as doing an interest conversion to find the equivalent rate with the compounding frequency equal to the payment frequency, the easiest way to find the interest and principal paid components is to use the amortization worksheet on a financial calculator.

NOTE

The calculator thinks in terms of payment numbers, not years.  That is, P1 and P2 must be payment numbers.  In the last part of the previous example, a common mistake is to set P1=3 and P2=3 to find the balance at the end of year 3.  But P1=P2=3 tells the calculator to find the balance after payment number 3, not after year 3.  To find the balance at the end of a particular year, you need the number of the last payment to happen in that year.  You can find this payment number by multiplying the payment frequency by the year.

For example, suppose a loan has quarterly payments and you want to find the balance at the end of year 5. The last payment to happen in year five is 20 (4×5).  The balance at the end of year 5 is the same as the balance for payment 20.  In the amortization worksheet you would set P1=20 and P2=20.

NOTES

1. The PRN entry on the amortization worksheet is the sum of the principal paid entries in the amortization schedule starting at payment number P1 and ending at payment number P2.  For example, if P1=4 and P2=7. the PRN entry tells you the sum of the principal paid column in the amortization schedule starting with payment number 4 and ending with payment number 7.
2. The INT entry on the amortization worksheet is the sum of the interest paid entries in the amortization schedule starting at payment number P1 and ending at payment number P2.  For example, if P1=4 and P2=7. the INT entry tells you the sum of the interest paid column in the amortization schedule starting with payment number 4 and ending with payment number 7.

NOTE

A common mistake occurs in translating years into payment numbers. You often need to find the total interest paid or total principal paid for a particular year.  To do this, you need to set P1 equal to the number of the first payment that occurs in that year and P2 equal to the number of the last payment that occurs in that year.

For example, suppose you have monthly payments and you want to know the total interest paid in the fourth year. In error, you might calculate that the fourth year begins with payment 36 and ends with payment 48, and so enter P1=36 and P2=48. But the 36th payment is actually the last payment of the third year. The first payment to occur in year four is the 37th.  So, if you wanted to find the total interest paid in year 4, P1=37 and P2=48.

When you need to find the first payment number and last payment number for a particular year, there are two methods you can use to calculate the correct payment numbers.

1. Calculate the payment number at the end of the year in question by multiplying P/Y by the year number:  P/Y×year number. To find the payment number at the start of the year, subtract the payment frequency less one (P/Y – 1) from the last payment number of the year. For example, suppose the you have monthly payments and want to find the interest paid for year 4.  The last payment in year 4 is 12×4=48.  The first payment in year 4 is 48-(12-1)=37.  
2. Calculate the payment number at the end of the year in question by multiplying P/Y by the year number:  P/Y×year number.  To find the payment number for the first payment in the year, multiply P/Y by the previous year number and then add one to it:  P/Y×(year-1)+1. For example, suppose the you have monthly payments and want to find the interest paid for year 4.  The first payment in year 4 is 12×3+1=37. The last payment of the fourth year remains at payment 48.