## Balance Inte

INTEREST PRINCIPAL REMAINING INTEREST PRINCIPAL REMAINING TO-t : FOR 1-1 TO 7 : LPRINT " --------"» : NEXT I s LPRINT

End three units (1 + 2 = 3) of the total 78. or %s of the original finance charge. It's fairly easy to derive the numerator and denominator of the fraction for a one-year loan:

Because one of the terms (n or n + 1) is always even, you can divide it in your head. For a 12-month loan, n is 12 and n + 1 is 13: divide the 12 by 2 to get 6 and multiply 6 by 13. The result? Good old 78.

For a one-year loan with six months remaining, S equals 21 (6*7/2 = 3*7 = 21). so your rebate is a148 of your finance charge. You can find the payoff amount by subtracting the rebate and the payments-to-date from the total scheduled payments.

That's all you need to calculate Rule of 78s ratios for any term and for early payoff. For example, a 30-month loan has 30*31/2=15*31 =465 units. If you paid it off 17 months early, the rebate would be 17*18/2= 17*9= 153 units.

### Enter the Computer

All of the above Is fine, but what you really need to know is the difference between paying a loan by simple interest and by the Rule of 78s. The Program Listing does that for you.

When you run Loan, it prompts you to choose printer or screen output or both. Next, you're prompted for the loan amount, term, and interest rate. Loan cal culates and prints a table showing monthly interest, principal, and balance remaining by simple interest and by the Rule of 78s.

The Figure shows a sample output for a loan of $10,000 financed over 36 months at 18 percent annual interest. As you can see, the payoff calculations for the two methods differ. The closer you are to the end of the loan term, the closer the Rule of 78s approximates the simple-interest formula. Notice that the bank is always ahead by this approximation. In this example. the overpayment by the Rule of 78s Is the most at the 13th month ($77.10).

The totals at the bottom of the Figure show that the total interest and principal are the same by either method if you pay the loan over the full term. The balance remaining is an error caused by rounding off to the nearest cent.

The program calculates the monthly payment by the simple-interest formula in line 240: the sum of monthly principal and interest by either method equals the monthly payment. The loop (lines 300-410) calculates and prints the interest and principal for each month as shown in the Figure. The formula defined In line 120 rounds off all payments to the nearest cent.

The interest according to the simple-in-terest formula (I) is the principal balance remaining (B) times the monthly interest rate (R). The interest according to the Rule of 78s (17) is the finance charge (F) times the units assigned to the month in question (T-N + 1) divided by the total number of units for the loan (U = T*(T + l)/2).

The finance charge is the total of payments minus the loan amount, as calculated in line 250.

### A Closing Note

The Rule of 78s penalty is vastly outweighed by the financial advantage of paying off a loan early. However, the penalty isn't negligible. Calculating your loan payments by the Rule of 78s won't help you avoid that penalty, but at least you'll know where you really stand with the bank.l

David McAnaney. a senior project engineer with a Boston firm, designs wastewater collection and treatment systems for municipalities. He's been using computers since 1963for mathematical modeling, design, and word processing. You can write to him at 70 Court St.. Exeter. NH 03833.

Related Articles

Conhaim. R.L., "The Rule of 78s," June 1981, p. 296. A Model I program that calculates annual percentage rate.

Conhaim, R.L., "The Real Rule of 78s," July 1981, p. 289. A Model I program that calculates Interest savings for early payoff.

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