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4: Time Value of Money
4.5: Compounding - When Your Interest Starts Earning Interest

4.5: Compounding - When Your Interest Starts Earning Interest

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Page ID: 132023

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Learning Objectives

Explain how compound interest differs from simple interest.
Interpret the effect of time and frequency on compounding outcomes.
Apply compound interest formulas to solve for future or present value.

A Snowball

Compounding happens when interest isn’t just earned on the original amount you invested (the principal) - it’s also earned on the interest that has already accumulated. Think of it like this: It’s not just your money working for you. It is your interest working for you, too. This is the power of compounding interest - your money grows faster over time because it continues to build on itself.

What Is Compounding?

Compounding occurs when the interest you earned last year begins earning interest this year. It’s like a snowball rolling down a hill: as it rolls, it picks up more snow, which makes it grow bigger and faster. Let’s revisit an example to make this clearer: Imagine you deposit $1,000 into an account that earns 5 percent interest per year. Here’s what happens: Year 1: You earn 5 percent on $1,000 → Interest = $50 → Total = $1,050. Year 2: Now you earn 5 percent on $1,050 (not just the original $1,000). Interest = 5 percent of $1,050 = $52.50 → Total = $1,102.50. Year 3: You earn 5 percent on $1,102.50, and so on. Notice that each year, the interest grows because it’s calculated on a bigger balance. This is the magic of compounding interest: Your money grows on itself.

The Formula for Future Value with Compounding

When interest compounds, we calculate the future value (FV) of a lump sum like this:

$\displaystyle{FV = PV \times (1+r)^{n}}$

Where:

FV = Future value (amount in the future)

PV = Present value (amount today)

r = Annual interest rate (as a decimal)

n = Number of years

Plugging in the numbers:

$\displaystyle{FV = 1,000 \times (1+0.05)^{3}}$

$\displaystyle{FV = 1,000 \times 1.1576}$

$\displaystyle{FV = 1,157.63}$

What does this mean? After three years, your $1,000 has grown to $1,157.63. The extra $157.63 isn’t just from the original deposit - it’s also from interest earned on interest.

Why Compounding Matters: The Snowball Effect

Compounding works like a snowball rolling downhill: The longer it rolls, the bigger it grows. The key to harnessing this power is time. The more time your money has to compound, the greater its growth. Time is just as important as the interest rate when it comes to growing your money.

Summary

Compounding is the engine that powers wealth-building over time. Unlike simple interest, which pays only on the principal, compound interest rewards patience by paying on both the principal and the interest that has accumulated. The longer your money is invested, the more dramatic the growth—especially when reinvestment is frequent and uninterrupted.

Compound interest = interest on interest.
Time magnifies the effects of compounding dramatically.
The earlier you start, the more time does the work for you.

This section transitions from intuition to math, showing how compound interest accelerates the process of wealth accumulation. It’s not magic. It’s math and time.

Exercises

Why do people often underestimate how much compound interest can grow over time?
What does compound interest teach us about habits and timing in personal finance?
You invest $1,000 at 5% compounded annually for 10 years. What’s the total future value?

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