Understanding the Mathematics of Personal Finance

# PRORATION—WORKING INSIDE A COMPOUNDING INTERVAL

Suppose I want to pay off a loan 4 months and 20 days after taking the loan.[7] The bank isn’t going to just use the results of the fourth compounding interval calculation and give me the interest they earned in the last 20 days. On the other hand, I don’t think it’ s fair for the bank to use the fifth compounding interval calculation and charge me 10 days’ worth of interest that the bank hasn’t earned.

Let’s say that there are 30 days in the current month. Take the interest rate per

month ^ R j and divide it by 30, giving us an interest rate per day, then we multiply

this by the number of days, giving us an effective interest rate for the 20 days of the month:

The interest is just the above number times the balance after 4 months. The balance as of 20 days after the fourth payment is therefore \$10,337.52 + 20(0.000278) (\$10,337.52) = \$10,394.95.

The procedure is referred to as prorating; you convert the monthly interest rate to an effective daily rate and then calculate the interest earned in the appropriate number of days.5

On my spreadsheet Ch2CompountInterest. xls, click on the Prorate tab. To the left of the green line you’ll see three new entries: number of days in month, day of proration, and payoff month number. For the example above, using the same \$10,000 loan as in previous examples, enter 30, 20, and 4 for these variables, respectively.

To the right of the green line you’ll see the same columns as in the Basic tab. On the top row, you’ll see the daily interest rate calculated. Then, look at the new column, payoff. This column shows the payoff number for the twentieth day of a 30-day month after the fourth payment.

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