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Compound Interest Formula

The Anatomy of the Formula

To calculate the future value of an investment or loan with compound interest, we use a specific formula. It accounts for the initial principal, the interest rate, how often it's compounded, and the total time.

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

Let’s look at each piece. The principal PP is your starting point—the money you deposit or borrow. The annual interest rate rr must be converted from a percentage to a decimal. For example, 5% becomes 0.05.

The variable nn represents the compounding frequency. If interest is compounded annually, n=1n=1. For semi-annually, n=2n=2. Quarterly is n=4n=4, and monthly is n=12n=12. Finally, tt is the duration of the investment in years.

The core of the formula is two-fold. First, rn\frac{r}{n} calculates the interest rate for a single compounding period. Then, ntnt calculates the total number of compounding periods over the investment's lifetime.

Putting the Formula to Work

Let's walk through an example. Suppose you invest $10,000 in an account with an annual interest rate of 6%, compounded quarterly. You plan to leave the money in for 5 years. What will the future value be?

First, identify your variables:

  • P=10000P = 10000
  • r=0.06r = 0.06 (6% as a decimal)
  • n=4n = 4 (compounded quarterly)
  • t=5t = 5 (years)

Now, we substitute these values into the formula.

A=10000(1+0.064)4×5A = 10000(1 + \frac{0.06}{4})^{4 \times 5}

To solve this, follow the order of operations. Start with the calculations inside the parentheses.

A=10000(1+0.015)20A=10000(1.015)20A=10000(1.346855)A≈13468.55\begin{align*} A &= 10000(1 + 0.015)^{20} \\ A &= 10000(1.015)^{20} \\ A &= 10000(1.346855) \\ A &\approx 13468.55 \end{align*}

After 5 years, the investment will have grown to approximately $13,468.55. This calculation shows how the principal grows by earning interest on previously earned interest, period after period.

Continuous Compounding

What happens if interest is compounded not just monthly or daily, but infinitely many times? This concept is called continuous compounding. It represents the theoretical limit of compounding frequency, and it has its own, simpler formula.

A=PertA = Pe^{rt}

Let's reuse our previous example, but with continuous compounding:

  • P=10000P = 10000
  • r=0.06r = 0.06
  • t=5t = 5
A=10000×e0.06×5A=10000×e0.3A≈10000×1.349858A≈13498.58\begin{align*} A &= 10000 \times e^{0.06 \times 5} \\ A &= 10000 \times e^{0.3} \\ A &\approx 10000 \times 1.349858 \\ A &\approx 13498.58 \end{align*}

With continuous compounding, the final amount is $13,498.58. As you can see, this is slightly more than the $13,468.55 from quarterly compounding. While not a huge difference in this case, it illustrates that a higher compounding frequency results in a larger future value.

Quiz Questions 1/5

Which formula correctly calculates the future value, A, of a principal amount, P, with an annual interest rate, r, compounded n times per year over t years?

Quiz Questions 2/5

If you invest $5,000 at an annual interest rate of 4%, compounded semi-annually, what will be the approximate value of the investment after 10 years?

These formulas are the engines behind calculating growth over time, whether in savings, investments, or loans.