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Compound Interest Formula Explanation

Certainly! Let’s break down the compound interest formula in simple terms:

A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}A=P(1+nr​)nt

Here’s what each part of the formula represents:

  1. A: This is the amount of money you’ll have after a certain period, including the interest earned.
  2. P: This is your initial principal, the starting amount of money you invested or saved.
  3. r: This is the annual interest rate, expressed as a decimal. For example, if the interest rate is 5%, then rrr would be 0.05.
  4. n: This is the number of times the interest is compounded per year. For instance, if the interest is compounded monthly, then nnn would be 12.
  5. t: This is the number of years your money is invested or saved.

Step-by-Step Explanation

  1. Annual Interest Rate Conversion:
    • The term rn\frac{r}{n}nr​ converts the annual interest rate to the interest rate per compounding period. So if you have an annual rate of 5% and it’s compounded monthly, rn=0.0512\frac{r}{n} = \frac{0.05}{12}nr​=120.05​.
  2. Adding 1:
    • The 1+rn1 + \frac{r}{n}1+nr​ part means you’re taking the base value (1) and adding the interest rate per compounding period to it. This shows how your money grows in one compounding period.
  3. Exponentiation:
    • The ntntnt in the exponent means you’re raising the previous result to the power of the total number of compounding periods. If interest is compounded monthly for 10 years, ntntnt would be 12×10=12012 \times 10 = 12012×10=120. This shows how the money grows over multiple compounding periods.
  4. Multiplying by Principal:
    • Finally, you multiply the whole thing by PPP, your initial investment. This step scales up the growth factor to reflect how much your initial amount will grow after the specified number of years.

Simplified Example

Let’s say you invest $1,000 (P) at an annual interest rate of 6% (r = 0.06), compounded monthly (n = 12), for 5 years (t = 5).

  1. Calculate the monthly interest rate: 0.0612=0.005\frac{0.06}{12} = 0.005120.06​=0.005.
  2. Add 1 to the monthly interest rate: 1+0.005=1.0051 + 0.005 = 1.0051+0.005=1.005.
  3. Calculate the total number of compounding periods: 12×5=6012 \times 5 = 6012×5=60.
  4. Raise 1.005 to the power of 60: 1.00560≈1.348851.005^{60} \approx 1.348851.00560≈1.34885.
  5. Multiply by the initial principal: 1000×1.34885≈1348.851000 \times 1.34885 \approx 1348.851000×1.34885≈1348.85.

So, after 5 years, your $1,000 investment would grow to approximately $1,348.85.

This example shows how the formula accounts for the interest earned in each compounding period, leading to the exponential growth of your investment over time.