Compound interest calculated more than once a year involves compounding at regular intervals within a year, such as semi-annually, quarterly, or monthly. Here's an example:

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Example: Quarterly Compounding

You invest $5,000 at an annual interest rate of 8% compounded quarterly for 3 years.

Formula for Compound Interest:

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

Where:

• $A$ = Final amount (future value)
• $P$ = Principal amount (initial investment)
• $r$ = Annual interest rate (decimal form)
• $n$ = Number of compounding periods per year
• $t$ = Time in years

Given:

• $P = 5000$
• $r = 0.08$ (8% annual rate)
• $n = 4$ (quarterly compounding)
• $t = 3$ years

Step 1: Plug Values Into the Formula

$A = 5000 \left( 1 + \frac{0.08}{4} \right)^{4 \cdot 3}$

$A = 5000 \left( 1 + 0.02 \right)^{12}$

Step 2: Simplify the Expression

$A = 5000 \left( 1.02 \right)^{12}$

Step 3: Calculate $1.02^{12}$

$1.02^{12} = 1.26824$

Step 4: Multiply by the Principal

$A = 5000 \cdot 1.26824 = 6341.20$

Final Answer:

After 3 years, the investment will grow to $6,341.20 with quarterly compounding.

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Key Point:

The more frequently the interest compounds, the greater the final amount due to the effect of earning "interest on interest."

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Would you like further explanations or examples? Here are five related questions to explore further:

1. What happens to the final amount if compounding frequency is changed to monthly?
2. How does compound interest differ from simple interest?
3. What is the formula for continuously compounded interest?
4. Can you calculate the total interest earned in this example?
5. How does increasing the annual interest rate affect the future value?

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Tip: To maximize returns, aim for investments with higher compounding frequencies at competitive interest rates.