Understanding the Mathematics of Personal Finance

RULE OF 78

This rule is sometimes called the sum-of-digits method. I’ll explain where both of these names came from shortly. As you shall see, any loan bearing this type of prepayment penalty is to be avoided. Various states have a legislation limiting its use. Hopefully, by the time you’re reading this, the practice will have been banned universally.

The rule of 78 payoff calculation used to be very popular with automobile loans. I understand it is uncommon today. As long as it remains legal, however, you’ll see it being used—principally among somewhat unscrupulous dealers who offer “sub­prime” loans to people who don’t have good credit, probably at an exorbitant interest rate.

I’ll use the auto loan of Table 5.1 as my first example. The spreadsheet Ch5PrepaymentPenalties. xls is used to calculate these examples. The data input area to the left of the green line is identical to the data input area in the previous chapters’ spreadsheets, so I needn’t repeat it in detail. Also, the first six columns to the right of the green line are identical to the same columns in the previous chapters’ spread­sheets. Since there are so many numbers to calculate, this spreadsheet is quite wide and you’ll have to scroll a bit to see everything. If you like, you can shrink the width or hide the columns carrying intermediate calculates (such as shares) to see the results easily.

The first number I need is the sum of the digits of the payment numbers: 1 + 2 + 3 + ... + 35 + 36 = 666. If you were to do this for a 1-year loan, you’d get 1 + 2 + 3 + .. . + 11 +12 = 78, hence the name (although I’m pretty sure there aren’t many 1-year auto or other similar loans).

Next, assuming the loan was to be fully paid off, I need the sum of all the inter­est components of the monthly payments. In Table 5.1i this is the sum of all the numbers in the column labeled interest, which is equal to $1,921.64.

Divide the sum of the interest payments by the sum of the payment numbers and call the result the interest fraction:

1,921 64

Interest fraction = ’ ' = 2.885.

666

So far, all I’ve done is to calculate an odd parameter, the interest fraction. Now I’m going to multiply this interest fraction by the payment number, but in reverse order, and call this the number of shares.

For payment number 1, I’ll multiply the interest fraction by 36, from which I get (2.885)(36) = 103.87. For payment number 2, I’ll have (2.885)(35) = 100.99.

Table 5.2 is a repeat of Table 5.1 but with the fake balance column removed and several new columns, including a shares column, added.

This shares calculation is where the injustice comes about. If you add up all the entries in the shares column, you get $1,921.64. Similarly, if you add up all the entries in the interest column (new column #4), you get the same number. However, look at the total interest (Tot Int) column—this is a running total of the interest payments—and the “earned interest” column—a running total of the shares column. The name earned interest was coined by the people who came up with this procedure. It doesn’t mean what it says. The shares column loads the interest payments up front, early in the loan, so that “earned interest” isn’t really earned interest accrued; it’s a conveniently faked number.

The payoff number is calculated by adding together the true balance and the difference between the earned interest and the actual total interest. For example, after payment number 12,

Payoff = $10,392.96 + $1,056.03 - $1,033.51 = $10,415.49.

Compare this with the actual payoff at the same time, which is $10,392.96. The actual prepayment penalty is the difference between the payoff and the balance, which, after payment number 12, is $22.53.

Figure 5.1 shows the prepayment penalty for this loan versus the payment number. As the figure shows, 1 year into the loan, the prepayment penalty peaks at about $22.

At this point, you’re probably wondering if all of this was worth the trouble; $22 is not a huge amount of money, and I did say that the lender is often entitled to some prepayment penalty.

Now consider a $300,000, 15-year loan at the same interest rate (8%). Figure 5.2 shows the prepayment penalty as a function of payment number for this loan. About 5 years into the loan, this penalty is almost $12,000! When you consider that the administrative and bookkeeping costs of this loan aren’t much different from that of the previous example, this is one very, very large prepayment penalty.

While I’ve provided an online worksheet for calculating rule of 78 prepayment penalties, the rule of thumb is that you don’t want to be dealing with a lender who asks for this in the loan agreement. If it’ s a small several t year loan such as the example in Table 5.2’ then you can probably live with it because the amount isn’t very large. In general, however, it’s close to robbery.

Table 5.2 Auto Loan with Rule of 78 Prepayment Penalty Added

Pmt Nr Balance Interest Tot Int Shares Earned Payoff Penalty ($) ($) ($) interest ($) ($) ($)