Understanding the Mathematics of Personal Finance


The list of how many different forms of prepayment penalties can be contrived is endless. Here are a few examples I have encountered:

1. Six months’ interest on 80% of the balance. Looking at the rule of 78 example shown in Figure 5.2, after 5 years, the remaining balance is approxi­mately $236,000. Eighty percent of this is approximately $189,000, and 6 months’ interest on this is about $7,560. This is considerably better than the rule of 78 prepayment penalty but still a lot of money.


0 20 40 60 80 100 120 140 160

Payment number

Figure 5.3 Comparison of different penalties for various prepayment penalty formulas (large loan example).

2. A flat 2% of the outstanding balance. In the above example, this would be about $3,600. This is still a lot, but we’re getting better.

3. Three percent of the balance for the first year of the loan, 2% for the second year, 1% for the third year, then 0 for the remainder of the loan term. This is very expensive for the first year of the loan ($9,000) but obviously not of concern after the third year.

4. Innumerable variations on the above. That is, a declining penalty that goes to 0 after some reasonable (i. e., short) period of time. The lender’s goal here is to keep you from jumping ship if rates drop before the lender has recovered at least a reasonable piece of the up-front expenses.

Figure 5.3 shows the rule of 78 prepayment penalty and also the first three of the above prepayment penalties (labeled PP1, PP2, and PP3). Interestingly enough, very early in the loan, the rule of 78 penalty is the best of the group (best = lowest penalty). For most of the loan, this is approximately from 36 to 145 months; the rule of 78 penalty is the worst of the group.

Many loans, particularly mortgages, allow for a fairly substantial partial prepay­ment, with the accompanying drop in interest amount, without triggering a prepay­ment penalty (if there is one). Also, many state laws have something to say about prepayment penalties on mortgages. Another point: Some prepayment penalties are triggered either by the sale of the home or the prepayment of the mortgage without the sale, and some are triggered only by the prepayment of the mortgage. You’ll have to do some research to find out what applies to you. Please do all of these the day before you sign the loan contract, not the day after. Also, remember that you’re the customer; when you take a loan—negotiate.


1. Consider a $25,000, 5-year auto loan with a fixed interest rate of 7.6%.

(a) After which monthly payment does the worst case rule of 78 prepayment penalty occur and how much is this penalty?

(b) If you haven’t made the monthly payment yet, using your answers from problem 1a, how much would it cost you to pay off this loan?

(c) Again using your answer from problem 1a, what is the payoff penalty as a function of the unpaid balance, both before and after you’ ve made the regular twenty - first payment?

(d) Again using your answer from problem 1a, what is the penalty based on 3 months’ interest on the remaining balance (after the monthly payment is made)? How does this compare to the rule of 78 penalty?

2. Assume that you have a savings account that pays 7.00% annual percentage rate (APR), compounded monthly, with $100,000 in it on the day you bought your car. The best money­managing philosophy would have been to take the money from your savings account and to buy your car with cash because you are paying more on your loan than your money was earning.1 For whatever reason, you took the loan and are making your payments by withdrawing money from your savings. You now decide that you made a mistake and want to pay off the loan. In both the prepayment penalty examples shown above and also the case of no prepayment penalty, when is the best time to pay off your loan?

3. Repeat the same problem but with a savings interest rate of 8.0%.

I’m ignoring subtle issues such as inflation and taxation in this problem.

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