Understanding the Mathematics of Personal Finance

IN THE LIMIT-CONTINUOUS COMPOUNDING

This section just shows a point that is probably interesting for those comfortable with the math. It’s not a necessary section. However, I recommend looking at the graph and reading through the description of the axes.

The balance of a loan that’s compounded once a year at, say, 10% annual interest grows by 10% a year. As I have shown above, if the same loan is compounded monthly (12 times a year), the balance grows by 10.47% a year. What if the loan is compounded more often—weekly, daily, or even hourly? Does the effective interest rate just keep growing?

1 10 100 1,000 10,000

Number of compounding intervals per year Figure 2.1 Effective interest rate versus the number of compounding intervals per year.

The answer to this question is shown in Figure 2.1. When 10% annual interest is compounded just once a year, the balance grows by exactly 10%. When it’s com­pounded 12 times a year (monthly), the balance grows by about 10.45%. When it’s compounded 100 or 1,000 or 10,000 or more times a year, the balance grows by about 10.52%. Once you’re compounding more often than weekly, the EAPR stops growing by any meaningful amount.

The format of Figure 2.1 is a bit different from anything I’ve shown before. The horizontal axis doesn’t start at 0 and the tic marks don’t represent equal size steps (a “linear” axis). This graph is known as a “semilogarithmic” graph. (Don’t let the name scare you.) In this graph, every tic mark on the horizontal axis represents a factor of 10 larger than the previous tic mark. If I didn’t do this, I would have to choose between just showing what’s happening for the Nr of Compounding Intervals Per Year varying between 1 and about 20—and therefore not showing how the curve “flattens out,” or showing it between, say, 1 and 1,000 so I could show the flattening, and thereby squeezing the early growth details so tightly to the vertical axis that they would be unintelligible.

As to where the data for this graph came from, if you have a bit of a calculus background, you’ll recognize that what I’m doing is just letting y approach infinity in the expression

where e is the base of natural logarithms, 2.718....

PROBLEMS

1. For a simple interest loan, calculate the following:

(a) The interest on a 1-year loan of $6,700 at 10% interest per year.

(b) The interest on a 3-year loan of $500 at 6% interest per year.

(c) The interest on a 7-month loan of $1,000 at 8% interest per year.

(d) The annual interest rate on a loan of $12,000.00 that paid back $15,600.00 after 3 years.

2. The APR of a compound interest loan = 18.0%. What is the compounding rate if the inter­est is compounded once, twice, three times, or four times a year?

3. A loan with an APR of 7.5% is compounded annually. If $1,250.00 is borrowed, find the balance each year for 3 years.

4. I took a loan for $10,000 that is accruing interest at an APR of 7.5%, compounded monthly. What is the balance every month for 18 months?

5. For the same loan as above, I decide to pay the loan back 13 days after my eleventh payment, in a 28-day month. What is my balance on the payoff day?

6. For the loan of problem 4, assume that my lender wanted an up-front fee of $250 to set up the loan. I want the lender to fold this fee into the loan. First, repeat problem 4 showing this up-front fee as additional principal. Second, return the loan to the original principal but show the effective interest rate due to the higher payments.

7. For the calculations in this problem, assume that the interest is compounded annually. Suppose you are able to take a 10-year loan at the loan rates shown in the table, and you are able to save the money for 10 years at the savings rates shown in the table. You borrow $10,000 and put it all into the savings account. Fill out the table. (This is known as working with “other people’s money.”)

Loan rate 10-year Savings 10-year Profit

(%) balance ($) rate (%) balance

Chapter