First, the key terms.

Principal
The starting amount before interest or returns are added.
Compounding
Applying a rate to a balance that includes earlier growth.
Period
The interval to which a stated rate applies.

The idea, explained

For a fixed annual rate compounded annually, a starting amount P grows to P(1 + r)^n after n years, before fees and taxes. The formula assumes no deposits or withdrawals and a constant rate. Real investment returns can vary, so a projection is an illustration rather than a promised outcome.

STEP 1

Calculate the first period

Start with $1,000 at an assumed 5 percent annual rate and no contributions, withdrawals, fees or taxes. Five percent is 0.05 as a decimal. The first year adds $1,000 × 0.05 = $50, giving $1,050.

STEP 2

Apply the rate to the new balance

In year two the same assumed rate applies to $1,050, not just the original $1,000. Interest is $52.50, giving $1,102.50. The extra $2.50 is growth on the first year’s $50. Repeating this process produces the formula principal × (1 + rate) to the number of periods.

STEP 3

Match assumptions to the question

A quoted annual rate may use a particular compounding convention. Contributions arrive at different times, and investment returns usually vary. Inflation changes purchasing power; fees and taxes change what remains. The simple formula is a starting model, not a complete financial plan.

Follow a worked example.

With an illustrative $1,000 balance at 5% annually, the first year ends at $1,050 and the second at $1,102.50. The second year’s $52.50 includes growth on the first year’s interest. This example assumes the rate stays fixed.

Walk it through, one step at a time

  1. At 10 percent per year, a fictional $200 grows to $220 after one year.
  2. Year two adds $22, giving $242. Simple interest on the original principal would have added only $20 in each year, giving $240.
  3. The $2 difference comes from earning 10 percent on the first year’s $20. No money was added by the saver in this illustration.

Where people get stuck

Do not apply a hypothetical constant return as a guarantee for a volatile investment. Fees, inflation and taxes can change the result.

Now try it yourself.

Pause before opening each answer. Explain your reasoning in your own words.

1. Check the core idea

Why is the second year’s interest greater than the first in this example?

Reveal the answer & reasoning

The rate applies to a larger balance that includes the first year’s interest.

2. Apply what you learned

What is the balance after two years if $500 grows at an assumed 4 percent annually with no other changes?

Reveal the answer & reasoning

Year one: $500 × 1.04 = $520. Year two: $520 × 1.04 = $540.80. The second year adds $20.80 because the first year’s interest also participates in growth.