Calculating interest on consumer loans

The federal Truth in Lending Act (TIL) requires lenders to disclose to credit applicants both the interest rate expressed as an annual percentage rate (APR) and the finance charge. As explained in Chapter 6, the APR is the relative cost of credit on a yearly basis expressed as a percentage rate. Any interest rate quoted by a lender must be the APR. The finance charge is the cost of credit expressed in dollars.

Calculating an Installment Loan Payment
Installment credit typically comes with a fixed interest rate, meaning that the rate will not change over the life of the loan. Lenders often offer variable-rate loans (also called adjustable-rate loans) to borrowers. When the loan rate varies, the monthly payment will go up or down, allowing the loan to be paid off by the same date as originally established in the contract.

To help you figure out the required monthly payment for different loan amounts, Table 7.1 shows various monthly installment payments used to repay a $1000 loan at commonly seen APR interest rates and time periods. For loans of other dollar amounts, divide the borrowed amount by 1000 and multiply the result by the appropriate figure from the table. For example, an automobile loan for $12,000, financed at 10 percent interest, might be repaid in 36 equal monthly payments of $387.24 ($32.27 12). A loan for $3550 at 16 percent for 24 months will require monthly payments of $173.81 ($48.96 3.550).

The finance charge must include all mandatory charges to be paid by the borrower. In addition to interest, lenders may charge fees for a credit investigation; a loan application; or credit life, credit disability, or credit unemployment insurance. When fees are required, the lender must include them in the finance charge in dollars as part of the APR calculations. When the borrower elects these options voluntarily, the fees are not included in the finance charge and APR calculations, even though they raise the actual cost of borrowing. It is easy to calculate the finance charge on a consumer loan. First, multiply the monthly payment by the number of months and subtract the original amount borrowed. In the 36-month automobile loan example given earlier, the finance charge would be $1,940.64 [($387.24 x 36) $12,000]. Second, add any other mandatory charges.


Finance Charge and APR Calculations for Installment Loans
Interest accounts for the greatest portion of the finance charge. Three methods are used to calculate interest on installment and noninstallment credit: the declining- balance (sometimes called the simple-interest) method, the add-on interest method, and the discount method. The add-on method predominates at banks, savings and loan associations, and consumer finance companies for installment loans for automobiles, furniture, and other credit requiring collateral. The declining-balance method is widely used by credit unions to calculate interest on loans. The declining-balance method is always used for credit cards and for most home mortgages. The following discussion illustrates the calculation of the annual percentage rate for installment loans using each of the three methods.

The Declining-Balance Method With the declining-balance method, the interest assessed during each payment period (usually each month) is based on the outstanding balance of the installment loan. The lender initially calculates a schedule (such as that given in Table 7.2) to have the balance repaid in full after a certain number of months. The borrower may vary the rate of repayment by making payments larger than those scheduled or may repay the loan in full at any time.

Here is an illustration of the declining-balance method for an installment loan. As shown in Table 7.2, at the end of the first month, a periodic interest rate (the monthly rate applied to the outstanding balance of a loan) of 1.5 percent (18 percent annually divided by 12 months) is applied to the beginning balance of $1000, giving an interest charge of $15. Of the first monthly installment of $91.68, $15 goes toward the payment of interest and $76.68 ($91.68 $15.00) goes toward payment of the principal.

For the second month, the outstanding balance is reduced to $923.32 ($1000 - $76.68). Since the balance is $78.68 lower, the interest portion of the payment drops to $13.85 (0.015 x $923.32). Because the declining-balance method of calculating interest on installment loans applies the periodic interest rate to the outstanding loan balance, the APR and the simple interest rate will differ only if fees (such as an application fee) boost the finance charge. (This method of paying off a loan, called amortization, is discussed in Chapter 9 when we examine home mortgage loans.) Note that decliningbalance loans carry no prepayment penalties.


The Add-On Method The add-on method is a widely used technique for computing interest on installment loans. With this method, the interest is calculated and added to the amount borrowed to determine the total amount to be repaid. Equation (7.1) is used to calculate the dollar amount of interest. Note that the interest rate used in this equation for the add-on method is an add-on rate and should not be confused with the APR. With the add-on interest method, interest is calculated by applying an interest rate to the amount borrowed times the number of years. The add-on interest formula given in Equation (7.1) is used as follows:
azzzz878

 mky
PRT (7.1)*
where
I Interest or finance charges
P Principal amount borrowed
R Rate of interest (simple, add-on, or discount rate)
T Time of loan in years

 For example, assume that Megan Broman of New York City borrows $2000 fortwo years at 9 percent add-on interest to be repaid in monthly installments. Using Equation (7.1), her finance charge in dollars is $360 ($2000 x0.09 x2). Adding the finance charge ($360) to the amount borrowed ($2000) gives a total amount of $2360 to be repaid. When this amount is divided by the total number of scheduled payments (24), we find that Megan must make 24 monthly payments of $98.33.


Calculating the APR When the Add-On Method Is Used Add-on
rates and APRs are not equivalent. This is because the add-on calculation assumes the original debt is owed for the entire period of the loan. But of course the debt does go down as the debt is repaid. In the example just given, Megan does not have use of the total amount borrowed for the full two years. Equation (7.2) shows the n-ratio method of estimating the APR on her add-on loan

where
APR Annual percentage rate
Y Number of payments in one year
F Finance charge in dollars (dollar cost of credit)
D Debt (amount borrowed or proceeds)
P Total number of scheduled payments
Using Equation (7.2), the APR is 16.4 percent. Note that the APR is approximately double the add-on rate because, on average, Megan has use of only half of the borrowed money during the entire loan period.


Applying the Rule of 78s to Determine Prepayment Penalties Most
installment loan contracts that use the add-on method include a prepayment penalty a special charge assessed to the borrower for paying off a loan early. Prepayment penalties take into consideration the reality that borrowers should pay more in interest early in the loan period when they have the use of more money and increasingly less interest as the debt shrinks over time. With an add-on method loan, however, the interest is spread evenly across all payments rather than declining as the loan balance falls. If an add-on method loan is paid off early, the lender will use some penalty method to compensate for the lower interest component applied in the early months.

The rule of 78s method (also called the sum of the digits method) is the most widely used method of calculating a prepayment penalty. Its name derives from the fact that, for a one-year loan, the numbers between 1 and 12 for each month add up to 78 (12 + 11 + 9 +8 +7  6+ 5 + 4 + 3 + 2 + 1). For a two-year loan, the numbers between 1 and 24 would be added, and so on for loans with longer time periods. To illustrate the use of the rule of 78s method, consider the case of Devin Grigsby from Berea, Kentucky. He borrowed $500 for 12 months plus an additional $80 finance charge, and is scheduled to pay equal monthly installments of $48.33 ($580 12). Assume Devin wants to pay the loan off after only six months.

 He might assume  incorrectly that he would owe only $250 more because after six months he had paid  $250 (one-half) of the $500 borrowed and $40 (one-half) of the finance charge, for atotal of $290 in payments ($48.33 6). Actually, Devin still owes $268.46, including a prepayment penalty of $18.46. To calculate this amount using the rule of 78s method, the lender adds together all of the numbers between 12 and 7 (12 for the first month, 11 for the second, and so on for six months): 12 + 11 + 10 + 9 + 8 + 7 57. The lender assumes that during the first six months $58.46 [(57 - 78) $80]—not $40—of the finance charges was received from the $290 in payments Devin had made on the  loan.* Consequently, only $231.54 ($290.00 $58.46) was paid on the $500 borrowed, leaving $268.46 ($500.00 - $231.54) still owed, for a prepayment penalty of $18.46 ($268.46 - $250.00).

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