The Mathematics of Money

182 Chapter 4 Annuities

present value or vice versa. While we might have a vague and conceptual idea of what is

going on as the payments are made—each payment chips away at the amount owed until the

debt is entirely paid off—we don’t yet have a way to look more specifically at the details.

Being able to do so would provide a better means of understanding of how all of this works.

Amortization tables are a tool to do this. The word amortization actually comes from the

French phrase à morte, meaning toward death, a fitting name for a look at how we kill off

the debt!^3 There are several different formats that can be used for amortization tables, but

any amortization table will provide a payment-by-payment detail of how each dollar paid

on a loan is allocated between paying the interest owed versus reducing the debt.

Setting Up an Amortization Table

The rows of an amortization table correspond to the scheduled payments for the loan.

Normally, there are columns for the number of the payment, the amount of the payment,

the amount of the payment that goes toward interest, the amount that goes toward princi-

pal, and the remaining balance after the payment is made. (Sometimes, the column for the

payment amount is left out. If all payments are assumed to be the same, repeating it is a

bit redundant. We will, however, include this column since later on we will be considering

tables where the payments can vary.)

To illustrate, let’s consider a loan of $1,000 at 8% interest for 4 years with payments due at

the end of each year. The annual payment (calculated by using the techniques of Section 4.4)

would then be $301.92. The amortization table for this loan would then look like this:

Payment

Number

Payment

Amount

Interest

Amount

Principal

Amount

Remaining

Balance

1 $301.92 $80.00 $221.92 $778.08

2 $301.92 $62.25 $239.67 $538.40

3 $301.92 $43.07 $258.85 $279.56

4 $301.92 $22.36 $279.56 $0.00

We can see from the table how each payment is split up between interest and principal, and

that the balance owed declines with each payment until it reaches zero after the last pay-

ment is made. How, though, is the split between interest and principal determined?

To answer this question, let’s walk through the calculation of the entries in the first two

rows. In the first year, the amount owed is the original $1,000, since no payments are made

until the end of that year. Interest is assumed to compound annually, and so no compounding

goes on during the year. The interest owed would be:

I 
 PRT

I 
 ($1,000)(0.08)(1)

I 
 $80.00

The first payment is $301.92, and since $80.00 of that must go to pay interest, the remain-

ing $301.92  $80.00 
 $221.92 is left to go toward principal. Taking that off of the prin-

cipal, the remaining balance would be $1,000  $221.92 
 $778.08, which is the amount

in the Remaining Balance column. Notice that these amounts agree with those shown in

the first row of the table.

Moving on to the second row, we repeat the same steps, though this time the principal is

no longer $1,000 but instead the $778.08 left after the first payment. Thus:

Interest: I 
 PRT 
 ($778.08)(0.08)(1) 
 $62.25

Principal: $301.92  $62.25 
 $239.67

Remaining Balance: $778.08  $239.67 
 $538.40

(^3) Or, alternatively and perhaps equally fi tting, for looking at how debt can kill you.