How to Calculate Compound Interest
Compound interest is often called "interest on interest." Unlike simple interest, where you only earn money on your original principal, compound interest allows you to earn returns on the principal plus the interest you've already accumulated.
1. The Basic Formula
The standard formula for calculating the future value of a lump sum investment with compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
What the Variables Mean:
- A (Future Value): The total amount of money you will have at the end of the time period.
- P (Principal): The starting amount of money (your initial investment).
- r (Annual Interest Rate): The interest rate expressed as a decimal (e.g., 5% becomes 0.05).
- n (Compounding Frequency): The number of times interest is compounded per year.
- Annually: n = 1
- Quarterly: n = 4
- Monthly: n = 12
- Daily: n = 365
- t (Time): The number of years the money is invested.
2. Step-by-Step Example
Let's calculate the future value for the following scenario:
- Principal (P): $10,000
- Interest Rate (r): 5% (0.05)
- Compounding (n): Monthly (12 times per year)
- Time (t): 10 years
Step 1: Plug the numbers into the formula
\[ A = 10,000 \left(1 + \frac{0.05}{12}\right)^{12 \times 10} \]
Step 2: Simplify the variables inside the parentheses
Divide the rate by the frequency:
\[ 0.05 / 12 = 0.0041667 \]
Add 1 to this result:
\[ 1 + 0.0041667 = 1.0041667 \]
Step 3: Calculate the total number of periods (exponent)
Multiply n by t:
\[ 12 \times 10 = 120 \]
Step 4: Raise the base to the power of the exponent
\[ (1.0041667)^{120} \approx 1.647009 \]
Step 5: Multiply by the Principal
\[ 10,000 \times 1.647009 = 16,470.09 \]
Result: After 10 years, your investment would grow to $16,470.09.
3. Calculating with Regular Contributions
If you add money regularly (like $100 a month), the formula becomes much more complex because each individual deposit earns interest for a different amount of time.
This is often calculated using the Future Value of an Annuity formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} + PMT \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\left(\frac{r}{n}\right)} \]
- PMT: The regular monthly contribution amount.
- Note: This assumes deposits are made at the end of each period.
4. The "Rule of 72" (Quick Estimate)
For a quick mental estimate of how long it takes to double your money, divide 72 by your interest rate.
- Formula: \( 72 / \text{Interest Rate} \)
- Example: With a 6% return: \( 72 / 6 = 12 \) years to double.