In an installment loan, the borrower partially repays the loan with equal, regular payments, for a fixed amount of time. Typical installment loans are for college tuition, home mortgages, and car loans. Interest accrues on the principal, so the payment made is applied to the interest as well as the principal, as shown in the following example.
Example 9.9
A student wants to purchase a computer for $2,000. The store offers
an APR of 18% with monthly payments of $160. Here is a table showing the first
year of the loan.
The monthly interest rate is calculated by dividing the APR by the number of months, so 0.18/12 = 0.015.
End of Month, i |
Previous Principal
|
Payment
|
Interest paid
|
Payment toward Principal
|
New Principal
|
| 1 | 2000.00 | 160 | 2000 * 0.015 = 30 | 130.00 | 1870.00 |
| 2 | 1870.00 | 160 | 1870 * 0.015 =28.05 | 131.95 | 1738.05 |
| 3 | 1738.05 | 160 | 1738.05 * 0.015 = 26.07 | 133.93 | 1604.12 |
| 4 | 1604.12 | 160 | 1604.12 * 0.015 = 24.06 | 135.94 | 1468.18 |
| 5 | 1468.18 | 160 | 1468.18 * 0.015 = 22.02 | 137.98 | 1330.20 |
| 6 | 1330.20 | 160 | 1330.20 * 0.015 = 19.95 | 140.05 | 1190.15 |
| 7 | 1190.15 | 160 | 1190.15 * 0.015 = 17.85 | 142.15 | 1048 |
| 8 | 1048.00 | 160 | 1048.00 * 0.015 = 15.72 | 144.28 | 903.72 |
| 9 | 903.72 | 160 | 903.72 * 0.015 = 13.56 | 146.44 | 757.28 |
| 10 | 757.28 | 160 | 757.28 * 0.015 = 11.36 | 148.64 | 608.64 |
| 11 | 608.64 | 160 | 608.64 * 0.015 = 9.13 | 150.87 | 457.77 |
| 12 | 457.77 | 160 | 457.77 * 0.015 = 6.87 | 153.13 | 304.64 |
If we continue, we would see that the computer would be paid off after 14 months, with the last payment less than $160.
The loan payment formula given in the text is:
PMT = p * (APR/n) / ( 1 - (1+(APR/n) ) (-n*t))
An alternative formulation, which avoids the negative sign in the exponent is:
PMT = p * (APR/n) * ( 1 + (APR/n) ) nt/ ( ( 1 + (APR/n) ) nt - 1 )
Or, alternatively, these two equations can be written as:
Loans accumulated on credit cars differ from installment loans because the card-holder is not required to pay off the balance after a particular period of time. These are called open-end installment loans.
Mortgages are installment loans for purchasing homes. a mortgage with a constant interest rate is called a fixed rate mortgage, and a mortgage in which the rate changes with the prevailing interest rates is an adjustable rate mortgage.