Compound Interest

**Compound interest** is interest calculated on both the initial principal AND the accumulated interest from previous periods. Unlike simple interest, your money grows exponentially because you earn "interest on interest." ## The Compound Interest Formula $$A = P(1 + r)^t$$ Where: - $A$ = Final amount (principal + interest) - $P$ = Principal (starting amount) - $r$ = Annual interest rate (as a decimal) - $t$ = Time (in years) ## Finding Just the Interest To find only the interest earned: $$I = A - P = P(1 + r)^t - P$$ ## Why It Grows Faster With compound interest, each year's interest is added to the principal, so the next year you earn interest on a larger amount. This creates exponential growth!

Basic Compound Interest Calculation

You invest 1000 dollars at 5% compound interest for 3 years. How much will you have?

Identify the values: $P = 1000$, $r = 5\% = 0.05$, $t = 3$ = Values identified

Write the formula: $A = P(1 + r)^t$ = Formula ready

Substitute values: $A = 1000(1 + 0.05)^3 = 1000(1.05)^3$ = Values substituted

Calculate the power: $(1.05)^3 = 1.05 \times 1.05 \times 1.05 = 1.157625$ = $(1.05)^3 = 1.157625$

Multiply by principal: $A = 1000 \times 1.157625 = 1157.63$ = $A = 1157.63$

Answer: After 3 years, you will have 1157.63 dollars (earning 157.63 dollars in interest).

Comparing Simple vs Compound Interest

Compare 2000 dollars invested for 5 years at 6%: (a) simple interest (b) compound interest

Calculate simple interest: $I = 2000 \times 0.06 \times 5 = 600$ = Simple: 2600 dollars

Calculate compound interest: $A = 2000(1.06)^5$ = Need to compute $(1.06)^5$

Compute the power: $(1.06)^5 = 1.338226$ = Growth factor found

Find compound amount: $A = 2000 \times 1.338226 = 2676.45$ = Compound: 2676.45 dollars

Compare the difference: $2676.45 - 2600 = 76.45$ = Compound earns 76.45 dollars more

Answer: Simple interest: 2600 dollars. Compound interest: 2676.45 dollars. Compound interest earns 76.45 dollars more!

Long-Term Investment Growth

You invest 5000 dollars at 7% compound interest for 10 years. How much interest will you earn?

Identify values: $P = 5000$, $r = 0.07$, $t = 10$ = Values identified

Apply the formula: $A = 5000(1.07)^{10}$ = Formula set up

Calculate $(1.07)^{10}$: $(1.07)^{10} = 1.967151$ = Growth factor

Find final amount: $A = 5000 \times 1.967151 = 9835.76$ = $A = 9835.76$

Calculate interest earned: $I = 9835.76 - 5000 = 4835.76$ = Interest = 4835.76 dollars

Answer: You will earn 4835.76 dollars in interest, nearly doubling your money!

Finding the Required Principal

You want to have 10000 dollars in 8 years. If the interest rate is 4% compounded annually, how much should you invest now?

Identify what we know: $A = 10000$, $r = 0.04$, $t = 8$, $P = ?$ = Need to find P

Rearrange the formula: $A = P(1+r)^t \Rightarrow P = \frac{A}{(1+r)^t}$ = Formula rearranged

Calculate $(1.04)^8$: $(1.04)^8 = 1.368569$ = Growth factor

Divide to find P: $P = \frac{10000}{1.368569} = 7306.90$ = $P = 7306.90$

Answer: You need to invest 7306.90 dollars now to have 10000 dollars in 8 years.

Compound interest is one of the most powerful concepts in finance: - **Savings accounts**: Your money grows faster over time - **Retirement funds**: Small contributions become large sums over decades - **Investments**: The stock market uses compound returns - **Credit cards**: Debt can grow quickly if you only pay minimums - **Student loans**: Interest compounds, increasing what you owe Albert Einstein reportedly called compound interest "the eighth wonder of the world." Understanding it helps you build wealth and avoid debt traps!

Mistake: Forgetting to add 1 to the rate inside the parentheses

Why: Writing $(0.05)^3$ instead of $(1.05)^3$ gives a tiny number, not growth.

Correct: Always use $(1 + r)^t$. The 1 represents keeping your original money!

Mistake: Using simple interest formula for compound problems

Why: Simple interest is $I = Prt$. Compound interest uses exponents: $A = P(1+r)^t$.

Correct: Look for keywords: "compounded" means use the exponential formula.

Mistake: Not converting percentage to decimal

Why: Using $r = 5$ instead of $r = 0.05$ will give astronomical results.

Correct: Always divide percentage by 100: $5\% = 0.05$

Retirement Savings

Retirement accounts use compound interest to grow small contributions into large nest eggs over 30-40 years.

Investing 100 dollars monthly at 7% for 40 years can grow to over 260000 dollars!

Credit Card Debt

Credit cards compound interest monthly. High rates can quickly multiply what you owe.

A 1000 dollar balance at 20% APR becomes about 1220 dollars after one year if unpaid.

College Savings

529 plans and education savings accounts use compound interest to help families prepare for college costs.

Starting a 529 plan when your child is born gives 18 years of compound growth!

Compound interest formula: $A = P(1 + r)^t$

Interest is calculated on principal PLUS previously earned interest

Always add 1 to the rate: use $(1 + r)$, not just $r$

Convert percentage to decimal before calculating

Compound interest grows faster than simple interest over time

The longer the time, the bigger the difference becomes

Q: Why is compound interest more powerful than simple interest?

A: With simple interest, you only earn interest on the original principal. With compound interest, you earn interest on interest. Over time, this creates exponential growth that far exceeds simple interest.

Q: What does "compounded annually" mean?

A: It means interest is calculated and added to your account once per year. Interest can also be compounded monthly, daily, or even continuously - more frequent compounding leads to slightly more growth.

Q: Can I use this formula for monthly compounding?

A: For monthly compounding, adjust the formula: $A = P(1 + \frac{r}{12})^{12t}$ where you divide the rate by 12 and multiply time by 12.

Exit Presentation

Compound Interest

1 / 13

Financial Math / Interest

Compound Interest

Learn how compound interest grows your money faster by earning interest on interest.

Intermediate30 min

Use arrow keys or spacebar to navigate