Question

Describe how compound interest is figured.

Answer 1

Compound interest is interest calculated on the initial principal and also on the interest that has been added to that principal from prior periods — i.e., “interest on interest.” Mathematically:

- Standard compound formula:
A = P (1 + r/n)^(n t)
where
- A = future value (amount after t years)
- P = principal (initial amount)
- r = annual nominal interest rate (as a decimal, e.g. 5% = 0.05)
- n = number of compounding periods per year (e.g. 12 for monthly)
- t = number of years

- Interest earned = A − P.

- Continuous compounding:
A = P e^(r t)

- Effective annual rate (EAR), which shows the true yearly rate given compounding frequency:
EAR = (1 + r/n)^n − 1

How to compute step-by-step:
1. Convert the annual rate to a decimal (r).
2. Divide r by n to get the periodic rate.
3. Add 1 to the periodic rate.
4. Raise that to the power n·t.
5. Multiply by the principal P to get A.

Example:
P = $1,000, r = 5% = 0.05, n = 12 (monthly), t = 3 years.
A = 1000 (1 + 0.05/12)^(12·3) = 1000 (1.0041667)^36 ≈ $1,161.62.
Interest earned ≈ $161.62 (vs. simple interest $150 for 3 years).

You can also rearrange the formula to solve for r or t when A and P are known. For example for t:
t = ln(A/P) / (n ln(1 + r/n)).

That’s the essence: compounding increases the amount because each period’s interest is added to the base that earns interest in the next period.