Understanding the Mathematics of Personal Finance

# REGULAR SAVINGS

The other side of making regular payments to amortize a loan is making regular payments into a savings account so as to build up resources for your retirement, for a child’s college education, and so on.

Suppose you could start saving \$350 a month regularly. Savings bank interest rates will probably vary, but just to get an idea of what will happen, assume that they

Table 3.10 A Savings Plan Spreadsheet Example

 Start Month: 7 Start Year: 2005 Nr Mnthly Pmts 360 Rate 4.50% Monthly Pmt \$350.00
 Pmt Nr Mnth Year Balance (\$) Payment (\$) Interest (\$) Tot Int/Year (\$) 1 7 2005 350.00 350.00 0.00 0.00 2 8 2005 701.31 350.00 1.31 1.31 3 9 2005 1,053.94 350.00 2.63 3.94 4 10 2005 1,407.89 350.00 3.95 7.89 5 11 2005 1,763.17 350.00 5.28 13.17 6 12 2005 2,119.79 350.00 6.61 19.79 7 1 2006 2,477.74 350.00 7.95 7.95 8 2 2006 2,837.03 350.00 9.29 17.24 9 3 2006 3,197.67 350.00 10.64 27.88 10 4 2006 3,559.66 350.00 11.99 39.87 11 5 2006 3,923.01 350.00 13.35 53.22 12 6 2006 4,287.72 350.00 14.71 67.93 184 10 2020 92,506.80 350.00 344.30 3,327.40 185 11 2020 93,203.70 350.00 346.90 3,674.30 186 12 2020 93,903.21 350.00 349.51 4,023.81 187 1 2021 94,605.35 350.00 352.14 352.14 356 2 2035 260,448.50 350.00 971.73 1,938.51 357 3 2035 261,775.18 350.00 976.68 2,915.19 358 4 2035 263,106.84 350.00 981.66 3,896.85 359 5 2035 264,443.49 350.00 986.65 4,883.50 360 6 2035 265,785.15 350.00 991.66 5,875.17

stay constant at 4.50% APR. (Another way to approach this is to save regularly for a year and then to use the year’s savings to buy a higher interest certificate of deposit.)

Table 3.10 shows the results of using the Save tab on the same Ch3.Amortization. xls spreadsheet used above for loans. Again, I’ve put the items to the left of the green line on the spreadsheet on top, merely to fit on the printed page.

The input variables are similar to the loan input variables. The only difference is that the Principal variable on the Loan sheet has been replaced with a Monthly Pmt variable on the Save sheet.

The variables to the right of the green line are the same on both spreadsheets. In the Savings sheet, however, the balance and the interest grow each month. I’ve

 Figure 3.2 Example of growth of a regular savings plan.

included the total interest per year numbers in the Savings sheet because some savings’ interest is taxable.

In the example shown, the monthly interest starts out very small. In the first full year of savings, the total interest for the year is not even \$100. But look at what happens when you keep at it. At payment number 187 (about 15 years after starting), the monthly interest is greater than \$350. In other words, the savings account itself is contributing more each month than you are. It’s taken you 10 years to reach this point, but the rate of growth of your savings is getting large.

If you keep this up for 30 years as a savings plan toward your retirement, the monthly interest is over \$900 and your balance is about \$266,000. This is truly “the power of compound interest.”

Figure 3.2 shows the balance versus the number of months that deposits have been made. As you can see, the instantaneous rate of growth is growing with time; that is, this savings account is “taking on a life of its own.”

PROBLEMS

1. Find the regular monthly payments for the following loans (that accrue interest monthly):

(a) Principal = \$100,000; rate = 6.00%; 15 years

(b) Principal = \$230,000; rate = 9.10%; 20 years

(c) Principal = \$10,000; rate = 12.00%; 3 years

(d) Without using a spreadsheet, principal = \$250,000; rate = 6.00%; 15 years

2. Using the loan from problem 1a above, assume that the loan originated in July of 2005 and that the interest is deductible from your taxes. How much interest did you pay in the calendar years 2005, 2010, 2015, and 2020?

3. Using the loan from problem 2 above, after which payment (payment number, month, and year) did you pay off more than half of the principal?

4. Again using the loan from problem 2, in May of 2013, instead of your regular payment, you send in \$10,000.00. Assuming that your lender is agreeable to recalculating, what will your regular payments be for the duration of the loan?

5. Again using the loan from problem 2, but start the loan in January for convenience, you have a job that pays you a small subsistence salary monthly and then sends you your com­mission checks quarterly. You negotiated a loan that allows you to pay interest only for 2 months and then to make a payment large enough to “catch up” on the third month. Show the first year’s amortization table for this loan.

6. You have decided that you can make \$1,000 a month payments on a 20-year loan. Calculate and plot a graph of the amount you can borrow versus the interest rate you can get, for interest rates varying from 0% to 10%. Remember that on a graph, you can’t resolve too many significant figures, so it’s all right to approximate the calculations.

Many of the calculators on the Web will let you solve this problem directly—entering what you know and getting what you want. My spreadsheet is set up to calculate the payment from the other variables. To solve this problem on my spreadsheet, you’ll have to enter the number of monthly payments and the rate, and then “adjust” the principal until the payment gets very close to \$1,000. This problem illustrates the importance of getting as low an interest rate as you can to maximize your borrowing power for a fixed payment amount.

7. You are buying a new car for \$31,800.00. You have \$5,000 for a down payment and you wish to finance the remainder over 4 years. The car dealer offers you a 7% loan with up­front costs of \$250. You take possession of the car on May 1, 2009 and your first loan payment is due the same day. How much are your monthly payments?

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