Understanding the Mathematics of Personal Finance

# Loan Amortization and Savings

3.1 LOANS

Many, if not most, loans are repaid with periodic payments. You make monthly mortgage payments, monthly car loan payments, and so on. Although it’s not math­ematically necessary, it’s typical for the payment intervals and dates (how often and when you make payments) to coincide with the interest compounding intervals and dates. Keep in mind that the interest is calculated on the money you have owed for the past month (assuming monthly compounding) and does not reflect the payment you’re about to make.

In some cases, your payments are credited on the day that they are received, and accrued interest is calculated by the day. Credit card companies (as discussed in Chapter 6) do this. I’m not going to explore this situation here because it doesn’t really change the numbers very much while, at the same time, it does complicate things enough to make the understanding of the basic factors involved annoyingly more difficult. If you’re having trouble getting statements from a lender to exactly match your own calculations based on this chapter, take a close look at your state­ments and see whether this situation is causing the problem.

Table 3.1 shows a \$30,000 loan taken at an annual rate of 8%, compounded monthly for 4 years. The interest earned after the first month of this loan is \$200.00, as shown. I didn ’ t show all of the rows of the table, but I think the message is clear.

Table 3.2 is a repeat of the first few rows of Table 3.1 but with a payment column added. Assume that each month, on the same day that the interest is calculated, you make a payment of \$200.00. Since this is exactly the amount of the first month’s interest, the balance never changes, as shown in the table.

Paying exactly the month’s accrued interest and thereby freezing the balance is sometimes called making interest-only payments on a loan. It’s useful for loans that you intend to pay off in full soon. For example, if you buy a new house before you

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

Copyright © 2009 John Wiley & Sons, Inc.

Table 3.1 Amortization Table for a \$30,000 Loan with 8% Annual Interest, Compounded Monthly for 4 Years

 Compounding interval Interest (\$) Balance (\$) 0 0.00 30,000.00 1 200.00 30,200.00 2 201.33 30,401.33 3 202.68 30,604.01 4 204.03 30,808.04 5 205.39 31,013.42 6 206.76 31,220.18 7 208.13 31,428.31 42 262.63 39,657.03 43 264.38 39,921.41 44 266.14 40,187.55 45 267.92 40,455.47 46 269.70 40,725.17 47 271.50 40,996.67 48 273.31 41,269.98

Table 3.2 Amortization Table for a \$30,000 Loan with 8% Annual Interest, Compounded Monthly for 4 Years and \$200 a Month Payments

 Compounding interval Interest (\$) Payment (\$) Balance (\$) 0 0.00 0.00 30,000.00 1 200.00 200.00 30,000.00 2 200.00 200.00 30,000.00 3 200.00 200.00 30,000.00 4 200.00 200.00 30,000.00 5 200.00 200.00 30,000.00

sell your old house, you might just pay the interest on the mortgage taken on the new house to keep your monthly expenses as low as possible while not letting your balance grow. As soon as your old house sells, you use the proceeds of the sale to pay off the loan.

If you make a monthly payment of more than the (first month’s) interest, the balance will decrease. Table 3.3 shows this happening when a \$300 payment is made each month. As you can see, both the monthly interest and the balance are decreasing with time. You are now paying off the loan. It is not necessary for all the payments to be equal to pay off a loan. As long as each payment is larger than the interest accrued since the last payment, the balance will decrease.

Table 3.3 Amortization Table for a \$30,000 Loan with 8% Annual Interest, Compounded Monthly for 4 Years and \$300 a Month Payments

 Compounding interval Interest (\$) Payment (\$) Balance (\$) 0 0.00 0.00 30,000.00 1 200.00 300.00 29,900.00 2 199.33 300.00 29,799.33 3 198.66 300.00 29,698.00 4 197.99 300.00 29,595.98 5 197.31 300.00 29,493.29

Table 3.4 Amortization Table for a \$30,000 Loan with 8% Annual Interest, Compounded Monthly for 4 Years with \$732.09 a Month Payments

 Compounding interval Interest (\$) Payment (\$) Balance (\$) 0 0.00 0.00 30,000.00 1 200.00 732.39 29,467.61 2 196.45 732.39 28,931.67 3 192.88 732.39 28,392.16 4 189.28 732.39 27,849.05 5 185.66 732.39 27,302.32 42 33.28 732.39 4,293.48 43 28.62 732.39 3,589.71 44 23.93 732.39 2,881.25 45 19.21 732.39 2,168.07 46 14.45 732.39 1,450.13 47 9.67 732.39 727.41 48 4.85 732.26 0.00

Table 3.4 shows what happens when the regular monthly payment is \$732.39. I’m just showing the first few and the last few rows of the table. The last row of the table shows the point of this example. The regular monthly payment of \$732.39 brings the balance to zero at the end of 48 months. The loan has been paid off in 48 months by a regular monthly payment of \$732.39. This procedure is called amortiz­ing the loan, and Table 3.4 is called an amortization table.

Before discussing about how I calculated the \$732.39 payment, let me show some other results that occur when you pay off a loan with regular payments. First, look at the interest amounts corresponding to the first and the last payments. The first interest amount is \$200.00; the last interest amount is only \$4.85. This means that at the first payment, the balance is reduced by \$732.39 - \$200.00 = \$532.39, while at the last payment, the balance is reduced by \$732.39 - \$4.85 = \$727.54.

The calculated interest starts high and decreases with each payment, while the amount that the balance is reduced starts low and increases with each payment. This is a basic characteristic of how compound interest is calculated and the loan amor­tization process. It has nothing to do with a bank “front-loading the interest to get its profit out quickly” or anything of the sort. Early in the amortization of a loan, you still owe most of the principal, so the interest for each payment period is high. As you approach paying off the loan, you don’t owe very much money, so the inter­est for each payment period is low.

Let’ s look at another example where these factors are much more apparent. Consider a \$300,000 loan, at 8% annual interest compounded monthly, paid back with regular monthly payments over a 20-year period. This could be a mortgage on a home. I still haven’t shown you how I came up with the payment amount but just by looking at the amortization table, you can see that it’s correct. Table 3.5 shows this amortization table. I’ve added a few columns to the table that I’ll discuss soon. Note also that I’ve replaced the title compounding interval with the title payment #. In this example, they’re interchangeable.

Figure 3.1 shows the interest and the amount of each payment going to reduce the balance over the 20-year life of the loan. As you can see, not until a little past halfway through the 20-year life of the loan does the monthly balance reduction get greater than the monthly interest payment.

The second and third columns of Table 3.5 show some dates, starting with July 2008 (7/08). I assumed that the loan was taken on July 1, and the first payment was due on August 1, 2008. In the last column on the right, I add up all the interest payments

 Payment number Figure 3.1 Example of payment allocation to interest and balance reduction in a regular payment loan.

Table 3.5 Amortization Table for a \$300,000 Loan with 8% Annual Interest, Compounded Monthly for 20 Years with \$2,509.32 a Month Payments

 Payment # Month Year Interest (\$) Balance reduction (\$) Balance (\$) Total interest per year (\$) 0 7 2008 0.00 0.00 300,000.00 0.00 1 8 2008 2,000.00 509.32 299,490.68 2,000.00 2 9 2008 1,996.60 512.72 298,977.96 3,996.60 3 10 2008 1,993.19 516.13 298,461.83 5,989.79 4 11 2008 1,989.75 519.57 297,942.26 7,979.54 5 12 2008 1,986.28 523.04 297,419.22 9,965.82 6 1 2009 1,982.79 526.53 296,892.69 1,982.79 7 2 2009 1,979.28 530.04 296,362.66 3,962.08 8 3 2009 1,975.75 533.57 295,829.09 5,937.83 9 4 2009 1,972.19 537.13 295,291.96 7,910.02 10 5 2009 1,968.61 540.71 294,751.25 9,878.64 11 6 2009 1,965.01 544.31 294,206.94 11,843.65 12 7 2009 1,961.38 547.94 293,659.00 13,805.03 13 8 2009 1,957.73 551.59 293,107.41 15,762.75 14 9 2009 1,954.05 555.27 292,552.14 17,716.80 15 10 2009 1,950.35 558.97 291,993.16 19,667.15 16 11 2009 1,946.62 562.70 291,430.47 21,613.77 17 12 2009 1,942.87 566.45 290,864.01 23,556.64 18 1 2010 1,939.09 570.23 290,293.79 1,939.09 19 2 2010 1,935.29 574.03 289,719.76 3,874.39 231 10 2027 161.31 2,348.01 21,849.13 2,300.66 232 11 2027 145.66 2,363.66 19,485.47 2,446.32 233 12 2027 129.90 2,379.42 17,106.05 2,576.23 234 1 2028 114.04 2,395.28 14,710.77 114.04 235 2 2028 98.07 2,411.25 12,299.52 212.11 236 3 2028 82.00 2,427.32 9,872.20 294.11 237 4 2028 65.81 2,443.51 7,428.69 359.92 238 5 2028 49.52 2,459.80 4,968.90 409.45 239 6 2028 33.13 2,476.19 2,492.70 442.57 240 7 2028 16.62 2,492.70 0.00 459.19

made in each calendar year. I did this because on some loans, the interest paid each year is deductible from your income taxes.1 Alternatively, if you are calculating interest paid to you for a loan that you made, you might have to pay taxes on your interest.[8] [9]

Table 3.6 Amortization Table for a \$30,000 Loan with 8% Annual Interest, Compounded Monthly for 4 Years with \$732.09 a Month Payments; “Simplified” Amortization Table

 Payment # Payment (\$) Balance (\$) 0 0.00 35,154.72 1 732.39 34,422.33 2 732.39 33,689.94 3 732.39 32,957.55 4 732.39 32,225.16 5 732.39 31,492.77 6 732.39 30,760.38 7 732.39 30,027.99 8 732.39 29,295.60 9 732.39 28,563.21 10 732.39 27,830.82 11 732.39 27,098.43 12 732.39 26,366.04 13 732.39 25,633.65 14 732.39 24,901.26 15 732.39 24,168.87 16 732.39 23,436.48 17 732.39 22,704.09 18 732.39 21,971.70 19 732.39 21,239.31 20 732.39 20,506.92 21 732.39 19,774.53

In order to avoid having to explain why early payments on a loan go mostly to interest (and for some more insidious reasons that I’ll get into in Chapter 5), some lenders multiply the monthly payment by the number of payments and call this the amount you owe (your balance) the day you take the loan. In the example of Table 3.4, this amounts to (\$732.39)(48) = \$35,154.72. They then present a table that looks like a true amortization table in which they simply take this very large number and subtract a payment each month, as is shown in Table 3.6. The arithmetic here is very simple. Since I got \$35,154.72 by multiplying \$732.39 by 48, when I subtract \$732.39 from this “balance” 48 times, I get to 0, that is, the loan is paid off. As I have said, I’ll discuss this further in Chapter 5. The biggest problem with this type of table is that it seems to be telling you that you owe \$35,154.72 immediately after taking the loan. This is a lie (actually, it’s just the first of many lies in this table). It is impossible for you to owe so much more than you borrowed almost immediately after you took the loan.

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