Paying off a loan
A loan to buy a home is usually repaid by equal payments, every month for many years. Borrow $200,000 at 0.5% a month, which is 6% a year compounded monthly, and repay $1200 at the end of every month.
The amount still owed is called the balance. At the end of each month the lender charges interest on the balance, and the payment covers that interest first. Whatever is left of the payment is called the principal part, and it reduces the balance. Paying off a loan this way, in equal payments that each cover the interest and repay part of the loan, is called amortizing it.
Month one
The interest for month 1 is 0.5% of $200,000: 200000 × 0.005 = 1000 dollars. Of the $1200 payment, $1000 pays that interest, and only 1200 − 1000 = 200 dollars comes off the loan. The balance falls to 200000 − 200 = 199800 dollars.
Month two
Month 2 charges interest on the new balance, $199,800, not on the $200,000 first borrowed: 199800 × 0.005 = 999 dollars. So 1200 − 999 = 201 dollars comes off the loan, and the balance falls to 199800 − 201 = 199599 dollars.
The interest fell by $1, so the principal part rose by $1. The payment of $1200 did not change; only the split inside it moved.
The first four payments of $1200, to the nearest dollar. The interest part falls by about $1 a month and the principal part rises by about $1 a month.
Why the balance falls slowly at first
In month 1, only $200 of the $1200 reduces the debt, one sixth of the payment; the other five sixths is interest. The balance is large, so its interest is large, and little of the payment is left over for the loan itself.
Each month the balance falls by that month's principal part, so the next month's interest is 0.5% of that principal part less, and the principal part grows by the same amount. That makes each principal part 1.005 times the one before: $200, then 200 × 1.005 = 201 dollars, then 201 × 1.005 = 202.005 dollars. The principal part grows like a compound interest balance, slowly at first and faster and faster.
By month 240 the payment is $541.25 of interest and $658.75 of principal, and by month 336 it is $136.69 of interest and $1063.31 of principal, to the nearest cent. After 180 months, half of the 30 years the loan runs, $141,836.26 is still owed: more than two thirds of the loan.
Three payments of $1200. The gold part is interest and the white part repays the loan: $1000 and $200 in month 1, about $541 and $659 in month 240, and about $137 and $1063 in month 336.
The gold curve is the balance, in thousands of dollars, month by month over the 360 payments. The white line is a balance that fell by the same amount every month. The gold curve stays above it the whole way. The dot marks month 180, halfway through the loan, when $141,836.26 is still owed: most of the loan is repaid in the second half.
Three steps, repeated
Every month repeats the same three steps. The interest is the balance times the monthly rate. The principal part is the payment minus the interest. The new balance is the old balance minus the principal part. A table of these steps, one row for each payment, is called an amortization schedule.
Carried on to the end without rounding, this schedule pays off the loan with 359 payments of $1200 and a smaller last payment of $296.99, 30 years in all. The borrower pays 359 × 1200 + 296.99 = 431096.99 dollars, so the interest over the whole loan is the total paid minus the amount borrowed, 431096.99 − 200000 = 231096.99 dollars.
The usual mistakes
Charging interest on the first amount borrowed. The interest is charged on what is still owed. When this loan is down to $70,000, the month's interest is 0.5% of $70,000, which is $350, not 0.5% of $200,000.
Charging interest on the payment. 0.5% of the $1200 payment is $6, but the rate applies to the balance, not to the money handed over.
Taking the whole payment off the balance. Only the part left after the month's interest reduces the loan: $200 of the first $1200.
Thinking the split stays the same. The payment is fixed, but the interest inside it is a percentage of a falling balance, so the interest part shrinks and the principal part grows.
Worked example: A Loan Repaid in Three Equal Payments, Read Line by Line
Question Nadia borrows $3310 at 10% a year and repays it in three equal yearly payments of $1331. Interest is charged at the end of each year on the amount still owed at the start of that year, and whatever is left of the payment comes off the loan. (a) Find the interest, the amount repaid and the balance left for the first payment and for the second. (b) Do the same for the third payment, and find the total interest paid over the whole loan.
1.Year 1 opens with $3310 owed. The interest is 10% of $3310, which is $331, so of the $1331 payment only 1331 − 331 = $1000 comes off the loan.
Year 1 opens with $3310 owed, so the interest is $331 and only $1000 of the payment comes off the loan. 2.(a) The balance is 3310 − 1000 = $2310. Year 2 opens with that, so its interest is 10% of $2310, which is $231, the amount repaid is 1331 − 231 = $1100, and the balance is 2310 − 1100 = $1210.
(a) The balance is $2310, and year 2 charges 10% of that, so $231 is interest and $1100 is repaid. 3.Year 3 opens with $1210 owed. The interest is $121, so the amount repaid is 1331 − 121 = $1210, and the balance is 1210 − 1210 = 0: the loan is cleared exactly.
Year 3 opens with $1210 owed, charges $121 and repays $1210, which clears the loan exactly. 4.(b) The interest paid is 331 + 231 + 121 = $683. Check: the three payments come to 3 × 1331 = $3993, and 3993 − 3310 = $683, which is the same total.
(b) The interest paid is 331 + 231 + 121 = $683, and 3 × 1331 − 3310 = $683 as well. 5.Read down the table: the interest falls $331, $231, $121 while the amount repaid rises $1000, $1100, $1210. The first payment is mostly interest and the last is almost all loan.
Read down the columns: the interest shrinks while the amount coming off the loan grows, because the interest is charged on what is still owed.
Answer: (a) Year 1 pays $331 interest and repays $1000, leaving $2310; year 2 pays $231 interest and repays $1100, leaving $1210; (b) year 3 pays $121 interest and repays $1210, leaving nothing owed, and the interest over the loan is $683
Common mistakes
- Splitting the loan into three equal parts of $1103.33 and charging 10% on each. The interest is charged on the balance still owed, and that balance falls as the loan is repaid, so the interest falls from $331 to $231 to $121 while the payment stays at $1331.
- Charging 10% of the original $3310 for each of the three years, giving 3 × 331 = $993 of interest. That would be the charge if nothing were repaid until the end. Because the balance falls each year, the true total is $683.
Worked example: The Same Mortgage Over Twenty Years and Over Twenty-Five
Question A bank lends $240000 at 5% a year, with the interest charged each month on the amount still owed. The bank quotes a monthly payment of $1583.89 over 20 years, which is 240 payments, and $1403.02 over 25 years, which is 300 payments. (a) Find the total paid and the total interest for the 20-year loan. (b) Do the same for the 25-year loan, and say how much more interest the longer loan costs and how much smaller its monthly payment is.
1.Over 20 years there are 12 × 20 = 240 payments of $1583.89, so the cash handed over is 240 × 1583.89 = $380133.60.
Every payment is the same, so the cash handed over is 240 × 1583.89 = $380133.60. 2.(a) Of that, $240000 is the amount borrowed, so the interest is 380133.60 − 240000 = $140133.60.
(a) Taking out the $240000 borrowed leaves $140133.60 of interest. 3.Over 25 years there are 12 × 25 = 300 payments of $1403.02, so the cash handed over is 300 × 1403.02 = $420906, and the interest is 420906 − 240000 = $180906.
The 25-year loan hands over 300 × 1403.02 = $420906, of which $180906 is interest. 4.(b) The longer loan costs 180906 − 140133.60 = $40772.40 more in interest, and its monthly payment is 1583.89 − 1403.02 = $180.87 smaller.
(b) The longer loan costs $40772.40 more in interest, for a payment $180.87 a month smaller. 5.Check by following the borrower: for the first 240 months the longer loan saves $180.87 a month, which is 240 × 180.87 = $43408.80, and then it runs on for 60 months more at 60 × 1403.02 = $84181.20. The difference is 84181.20 − 43408.80 = $40772.40, the same extra interest.
Following the borrower gives the same figure: 240 × 180.87 = $43408.80 saved, then 60 × 1403.02 = $84181.20 still to pay.
Answer: (a) $380133.60 is paid in all, of which $140133.60 is interest; (b) $420906 is paid in all, of which $180906 is interest, so the longer loan costs $40772.40 more in interest for a payment $180.87 a month smaller
Common mistakes
- Choosing the longer loan because it costs less. It costs $180.87 less each month and $40772.40 more in all, so the two sentences describe the same loan and the choice has to be made with both in view.
- Working the interest out as 5% of the whole $240000 for every year, giving 25 × 12000 = $300000. The interest is charged on the amount still owed, which falls with every payment, so the true total over 25 years is $180906.