Lesson 7.3

Compound Interest: Discrete and Continuous

Albert Einstein reputedly called compound interest the "eighth wonder of the world." Whether that's true or not, the math behind money growing on itself is the most practical application of exponential functions you will ever learn.

Introduction

At its core, interest is the cost of borrowing money or the reward for saving it. "Simple interest" pays you only on your initial deposit. "Compound interest" pays you on your initial deposit plus the interest you've already earned.

Prerequisite Connection

You need to be comfortable entering complex exponential expressions into a calculator and converting percentages to decimals (e.g., ).

Today's Increment

We introduce two formulas: one for interest compounded periodically (monthly, quarterly) and one for interest compounded continuously, using the natural base .

Why This Matters

The transition from "monthly" to "continuous" compounding is your first real glimpse of a limit—the foundational tool of calculus. We take discrete steps and make them infinitely small to discover a smooth, continuous reality.

Key Concepts

Discrete Compounding Formula

Use this when interest is calculated a specific number of times per year.

: The final Amount (balance).

: The Principal (initial deposit).

: The annual rate (as a decimal).

: Number of times compounded per year (e.g., 12 for monthly).

: Time in years.

Compounding Frequency Key

Annually

Quarterly

Monthly

Daily

Continuous Compounding Formula

Use this when interest is compounded "continuously" (essentially every instant). It represents the mathematical limit as .

Memory Aid: "Pert"

Worked Examples

Example 1: Monthly Compounding

Basic

You invest $1,000 at an annual interest rate of 5% compounded monthly. How much money will be in the account after 10 years?

Identify Variables

(not 5!)
(monthly)

Set Up Equation

Calculate

Answer

$1,647.01

Example 2: The Impact of Frequency

Intermediate

Compare the balance of a $10,000 investment at 6% for 20 years if compounded:
a) Annually
b) Daily.

Scenario A: Annual ()

Scenario B: Daily ()

Conclusion

Compounding daily earned an extra $1,126.51 ($33,197.86 - $32,071.35) over the 20-year period compared to annual compounding.

Example 3: Continuous Compounding

Advanced

Using the same numbers as Example 2 ($10,000 at 6% for 20 years), calculate the balance if compounded continuously. How does this compare to the Daily figure?

Select Formula

The word "continuously" signals us to use .

Substitute and Solve

Answer

$33,201.17. This is only $3.31 more than the daily compounding figure! As gets very large, the difference between discrete and continuous becomes negligible.

Common Pitfalls

Decimal Conversion Errors

Remember that must be a decimal. 4.5% is , not 0.45 (which would be 45%!).

Calculator Order of Operations

When calculating , make sure to put the exponent in parentheses if you are typing it all at once. Otherwise, the calculator might raise it to the power of and then multiply the result by .

Real-World Application

Credit Card Debt

Compound interest works against you when you borrow money. Credit cards typically compound interest daily.

If you have a $5,000 balance at 20% APR and ignore it for 5 years, the debt will swell to over $13,590. You end up paying more in interest than the original items cost!