Compound Interest
Compound interest is interest earned on both the money you put in and the interest it has already earned. The formula is A = P\left(1 + \dfrac{r}{n}\right)^{nt}, where P is the starting amount, r is the yearly rate as a decimal, n is how many times a year interest is added, and t is the number of years.
Before this lesson: You should know simple interest, exponential growth and decay, and the exponent rules.
Interest on Interest
Put $1,000 in a savings account that pays 10% a year. After one year, you earn $100, so you have $1,100.
With simple interest, you earn $100 every year, because the interest is always 10% of the first $1,000. With compound interest, the second year pays 10% of $1,100, which is $110. Your interest starts earning interest too.
After 5 years, simple interest gives $1,500. Compound interest gives $1,610.51. The longer you wait, the bigger the gap.
Compound Interest Formula
A = P\left(1 + \dfrac{r}{n}\right)^{nt}| Symbol | Meaning |
|---|---|
| A | Amount at the end (starting money plus interest) |
| P | Principal: the starting amount |
| r | Yearly interest rate, as a decimal (6% is 0.06) |
| n | Number of times per year interest is added |
| t | Number of years |
Interest is “added” or compounded at the end of each period. The interest earned is A - P.
| Compounded | n | Rate each period |
|---|---|---|
| Annually (yearly) | 1 | r |
| Semiannually (twice a year) | 2 | r \div 2 |
| Quarterly | 4 | r \div 4 |
| Monthly | 12 | r \div 12 |
| Daily | 365 | r \div 365 |
How to Use the Formula
- Write down P, r, n and t. Change the percent to a decimal.
- Find the rate per period, \tfrac{r}{n}, and add 1.
- Find the number of periods, nt.
- Raise the factor to that power. Then multiply by P.
- Round money to the nearest cent at the end, not in the middle.
When interest is added once a year (n = 1), the formula becomes A = P(1 + r)^t. That is the growth formula from the last lesson.
Worked Examples
Example 1: Compounded Annually
You put $2,000 in an account that pays 5% interest, compounded annually. How much is in the account after 3 years? How much interest did you earn?
Step 1: P = 2000, r = 0.05, n = 1, t = 3.
Step 2: A = 2000(1.05)^3. Work out the power: 1.05^2 = 1.1025 and 1.1025 \cdot 1.05 = 1.157625.
Step 3: A = 2000 \cdot 1.157625 = 2315.25. The interest is 2315.25 - 2000 = 315.25.
Answer: $2,315.25 in the account, with $315.25 in interest.
Example 2: Compounded Semiannually
You invest $1,000 at 10% interest, compounded semiannually. How much do you have after 2 years?
Step 1: P = 1000, r = 0.10, n = 2, t = 2.
Step 2: The rate per period is 0.10 \div 2 = 0.05. The number of periods is 2 \cdot 2 = 4. So A = 1000(1.05)^4.
Step 3: 1.05^4 = 1.1025 \cdot 1.1025 = 1.21550625. Then A = 1215.50625, which rounds to $1,215.51.
Answer: $1,215.51. Compounded once a year, the same deposit would give $1,210, so adding interest more often earns $5.51 more.
Example 3: Compounded Quarterly
Find the balance after 1 year if $5,000 earns 8% interest, compounded quarterly.
Step 1: n = 4, so the rate per period is 0.08 \div 4 = 0.02. There are 4 \cdot 1 = 4 periods.
Step 2: A = 5000(1.02)^4. 1.02^2 = 1.0404 and 1.0404^2 = 1.08243216.
Step 3: A = 5000 \cdot 1.08243216 = 5412.1608, which rounds to $5,412.16.
Answer: $5,412.16
Example 4: Simple vs. Compound Interest
Compare $1,000 at 10% for 3 years with simple interest and with interest compounded annually.
Simple: I = Prt = 1000 \cdot 0.10 \cdot 3 = 300. The total is $1,300.
Compound: A = 1000(1.1)^3 = 1000 \cdot 1.331 = 1331.
Answer: Compound interest gives $1,331, which is $31 more than the $1,300 from simple interest.
Example 5: Word Problem
Maya has $10,000 to save for one year. Bank A pays 6% compounded annually. Bank B pays 6% compounded semiannually. Which bank gives more, and by how much?
Bank A: A = 10{,}000(1.06)^1 = 10{,}600.
Bank B: The rate per period is 0.06 \div 2 = 0.03, for 2 periods. A = 10{,}000(1.03)^2 = 10{,}000 \cdot 1.0609 = 10{,}609.
Step 3: 10{,}609 - 10{,}600 = 9.
Answer: Bank B gives $9 more, because its interest is added twice and starts earning sooner.
Common Mistakes
| Mistake | Example of the error | Fix |
|---|---|---|
| Using the percent, not the decimal | 1000(1 + 5)^3 | 5% is 0.05: 1000(1.05)^3 |
| Forgetting to divide r by n | Quarterly at 8%: (1.08)^4 | Divide first: (1.02)^4 |
| Using t instead of nt | Semiannual for 2 years: (1.05)^2 | There are 2 \cdot 2 = 4 periods: (1.05)^4 |
| Multiplying before the exponent | 2000(1.05)^3 = 2100^3 | Do the power first, then multiply by P |
| Rounding too early | Rounding 1.1025 to 1.10, then squaring | Keep all the digits. Round to cents only at the end |
| Giving A when the question asks for interest | Interest = $2,315.25 | Interest is A - P = 315.25 |
Compounding More Often
Here is $1,000 at 12% for 1 year, compounded different ways. More periods give a little more money, because interest starts earning sooner.
| Compounded | n | Formula | Balance |
|---|---|---|---|
| Annually | 1 | 1000(1.12)^1 | $1,120.00 |
| Semiannually | 2 | 1000(1 + 0.12/2)^{2} | $1,123.60 |
| Quarterly | 4 | 1000(1 + 0.12/4)^{4} | $1,125.51 |
Loans and credit cards use the same formula. When you owe money, compound interest makes the debt grow faster, so paying early saves money.
Practice Problems
- $3,000 is invested at 4% interest, compounded quarterly, for 5 years. Give P, r, n and t, and write the formula with the numbers in it.
- $500 earns 6% interest, compounded annually. Find the balance after 2 years.
- How much interest was earned in problem 2?
- $1,000 earns 4% interest, compounded semiannually. Find the balance after 1 year.
- $800 earns 12% interest, compounded quarterly. Find the balance after 1 year. Round to the nearest cent.
- $4,000 earns 5% interest, compounded annually. Find the balance after 2 years.
- $2,000 is saved for 2 years at 10%. How much more does compound interest (annually) give than simple interest?
- What is n for interest compounded monthly? For daily?
- $7,500 is invested at 3.6% interest, compounded monthly, for 10 years. Write the formula with the numbers in it. Do not solve.
- A credit card balance of $1,000 has 24% yearly interest, compounded monthly. No payments are made. What is the balance after 2 months?
Show answers
- P = 3000, r = 0.04, n = 4, t = 5. A = 3000\left(1 + \tfrac{0.04}{4}\right)^{4 \cdot 5} = 3000(1.01)^{20}.
- 500(1.06)^2 = 500 \cdot 1.1236 = 561.80. The balance is $561.80.
- 561.80 - 500 = 61.80. The interest is $61.80.
- 1000(1.02)^2 = 1000 \cdot 1.0404 = 1040.40. The balance is $1,040.40.
- 800(1.03)^4 = 800 \cdot 1.12550881 = 900.407048. The balance is $900.41.
- 4000(1.05)^2 = 4000 \cdot 1.1025 = 4410. The balance is $4,410.
- Simple: 2000 + 2000 \cdot 0.10 \cdot 2 = 2400. Compound: 2000(1.1)^2 = 2420. Compound gives $20 more.
- Monthly: n = 12. Daily: n = 365.
- A = 7500\left(1 + \tfrac{0.036}{12}\right)^{12 \cdot 10} = 7500(1.003)^{120}
- The monthly rate is 0.24 \div 12 = 0.02. 1000(1.02)^2 = 1040.40. The balance is $1,040.40.
FAQ
What does compounded mean?
It means the interest is added to your balance. After that, the interest earns interest too. Compounded quarterly means this happens 4 times a year.
What is the difference between simple and compound interest?
Simple interest is always figured on the starting amount, so it grows by the same dollar amount each year. Compound interest is figured on the current balance, so it grows faster each year.
Does compounding more often always give more money?
Yes, for the same rate and time. But the gain gets small. Going from yearly to monthly adds a little, and going from monthly to daily adds even less.
Is the compound interest formula an exponential function?
Yes. The factor 1 + \tfrac{r}{n} is the base, and the number of periods is the exponent. The balance grows exponentially over time.
Summary
- Compound interest: A = P\left(1 + \tfrac{r}{n}\right)^{nt}.
- P is the principal, r the yearly rate as a decimal, n the periods per year and t the years.
- Divide the rate by n. Multiply the years by n.
- The interest earned is A - P. Round to the cent only at the end.
- Compound interest grows faster than simple interest, because interest earns interest.
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Next lesson: Geometric Sequences
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