Understanding the Mathematics of Personal Finance

# INITIAL CHARGES AND EFFECTIVE INTEREST RATE

Very often, a bank or loan company will charge some sort of loan initiation fee at the outset of the loan (I call these up-front costs). Suppose, still using Table 2.2 as our example, that the bank wants \$100 up front for setting up the loan. One way to handle this would be for the bank to just give you \$9,900 while recording the loan as \$10,000; then Table 2.2 would still be correct.

Usually, however, this is not what happens. When you take a \$10,000 loan, you probably want to walk out of the bank with \$10,000 for whatever your purpose is. In this case, the bank adds the extra \$100 into the loan. That is, it pretends that it really loaned you \$10,100. Columns 2 and 3 of Table 2.4 shows the new balance worksheet for this loan. Naturally, at the end of 2 years when you go to repay the loan, you owe more than Table 2.2 predicts.

In the last two columns, I reset the principal to \$10,000 and then adjusted the interest rate until the balance after 2 years was approximately the same as the balance at the same time in columns 2 and 3. I had to set the interest rate to 10.50% to get this result. As you can see, the loan with the \$100 initial charge may be thought of as the same loan without this charge, but with an effective interest rate that’s a bit higher than the originally stated interest rate.

Two points here: First, you can see the value of using a spreadsheet for these calculations. I was able to adjust the input parameters quite easily until I got the results I was looking for, using the Basic tab of my spreadsheet. If I wanted to be a little fancier, I could have set up both examples side by side—that’s actually what I did to create Table 2.4.

Second, this effective interest rate calculation is a very good way to compare loans from two different lenders. Suppose you are offered two loans with the same number of payments but at different interest rates and with different initial charges. Which one is the better loan? By breaking the two loans down to the money you’re actually borrowing and an effective interest rate, you can see which deal is better.

This is actually charging simple interest for prorations inside one compounding interval.

Table 2.4 The Same Loan as Shown in Table 2.2 but with a \$100 Initial Charge

 Compounding interval # Initial charge as charge Initial charge as effective interest Interest (\$) Balance (\$) Interest (\$) Balance (\$) 0 0.00 10,100.00 0.00 10,000.00 1 84.17 10,184.17 87.50 10,087.50 2 84.87 10,269.03 88.27 10,175.77 3 85.58 10,354.61 89.04 10,264.80 4 86.29 10,440.90 89.82 10,354.62 5 87.01 10,527.91 90.60 10,445.22 6 87.73 10,615.64 91.40 10,536.62 7 88.46 10,704.10 92.20 10,628.81 8 89.20 10,793.30 93.00 10,721.82 9 89.94 10,883.25 93.82 10,815.63 10 90.69 10,973.94 94.64 10,910.27 11 91.45 11,065.39 95.46 11,005.73 12 92.21 11,157.60 96.30 11,102.03 13 92.98 11,250.58 97.14 11,199.18 14 93.75 11,344.34 97.99 11,297.17 15 94.54 11,438.87 98.85 11,396.02 16 95.32 11,534.20 99.72 11,495.74 17 96.12 11,630.32 100.59 11,596.32 18 96.92 11,727.23 101.47 11,697.79 19 97.73 11,824.96 102.36 11,800.15 20 98.54 11,923.50 103.25 11,903.40 21 99.36 12,022.87 104.15 12,007.55 22 100.19 12,123.06 105.07 12,112.62 23 101.03 12,224.08 105.99 12,218.60 24 101.87 12,325.95 106.91 12,325.52

Since both loans have the same number of payments, it is the loan with the lower effective interest rate.

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