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## 6.4 Compound Interest

PLEASE NOTE: This book is currently in draft form; material is not final.

### Learning Objectives

1. Define compound interest.
2. Calculate compound interest for an investment.

Now what happens if Martha takes the following line of thought: “Mary owes \$14 at the end of week 1, so the second week is as if she is borrowing \$14 to pay her first week’s debt. I should be charging her 55.56% of \$14, or \$7.78, interest for the second week!”

Where does the extra \$2.78 of interest (\$7.78−\$5) come from? Martha is now assuming that in the second week, additional interest needs to be paid on the first week’s interest of \$5. Since 55.56% of \$5 is \$2.78, Mary would owe even more interest for each week she doesn’t repay the loan, including interest-on-interest, interest-on-interest-on-interest, etc. This state of affairs, where interest is determined in each period based upon principal and accrued interest is called compound interestDetermining interest in each period based upon principal and accrued interest..

Compound interest is much more common than simple interest; unless specifically stated otherwise, it is best to assume interest will be compounded. Compounding also introduces another complexity, the idea of a compounding periodHow frequently interest-on-interest is determined.: how frequently interest-on-interest is determined. In this case, our compounding period is weekly, so we need to use the weekly rate of interest, and charge interest-on-interest appropriately.

Equation 6.4 Compound Interest for 2 Periods

principal + first week’s interest + second week’s interest + interest-on-interest = FV
PV + (PV × r) + (PV × r) + [(PV × r) × r)] = FV PV × (1 + r + r + r2) = FV PV × (1 + r)2 = FV

Equation 6.5 Compound Interest for 2 Periods (alternative)

amount owed at week 1 + interest on amount owed at week 1 = FV
[PV + (PV × r)] + [PV + (PV × r)] × r = FV [PV × (1 + r)] + [PV × (1 + r)] × r = FV [PV × (1 + r)] × (1 + r) = FV PV × (1 + r)2 = FV

Considering that the total owed at period n must be (1 + r) times what is owed the prior period gives us the generalized form for compound interest:

Equation 6.6 Compound Interest for n Periods

PV × (1 + r)n = FV

### Geometric Growth

Compound Interest for n Periods is one of the most important equations in all of finance and, actually, many of the sciences as well. Anything that grows at a constant rate based on the amount the period before (for example, bacteria in a petri dish) can be modeled by this equation.

Now that we have a generalized equation, we can rearrange it to solve for more than just FV. This equation is so important that most financial calculators and spreadsheet programs have dedicated functions that can be used to solve for any of the variables given the other three.

### Key Takeaways

• PV × (1 + r)n = FV
• Compound interest includes interest-on-interest.

### Exercise

1. If \$300 is invested at a 4% interest rate, how much compound interest will be earned in 4 years? How much more compound interest is earned vs. simple interest (in other words, what was the interest on interest)? What is the total FV of the investment?