Understanding the Context

What Is Compound Interest?

Compound interest is the interest calculated on the initial principal and the accumulated interest from previous periods. Unlike simple interest — which is calculated only on the original principal — compound interest allows your money to grow faster because interest earns interest. This effect, often called “interest on interest,” dramatically enhances long-term growth.

The Compound Interest Formula

Key Insights

The standard formula for compound interest is:

$$
A = P \left(1 + \frac{r}{n}\right)^{nt}
$$

Where:

• $ A $ = the future value of the investment or savings (total amount)
  - $ P $ = the principal amount (initial investment)
  - $ r $ = annual interest rate (expressed as a decimal)
  - $ n $ = number of times interest is compounded per year
  - $ t $ = the time the money is invested or borrowed, in years

Final Thoughts

Breaking Down the Formula

1. Principal (P)
This is the starting amount of money. For example, if you invest $1,000, $ P = 1000 $.

2. Interest Rate (r)
This is the annual rate expressed as a decimal. To use the formula, divide your percentage by 100. For example, a 5% annual rate is $ r = 0.05 $.

3. Compounding Frequency (n)
Interest can be compounded daily, monthly, quarterly, or annually. The frequency, $ n $, tells how often interest is calculated and added to the principal:

| Compounding Frequency | Example Value (n) |
|----------------------|-------------------|
| Annually | 1 |
| Semi-annually | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |

4. Time (t)
Expressed in years. For a 10-year investment, $ t = 10 $.