Find the accumulated value of an investment of $20,000 for five years at an interest rate of 1.15% if the money is Compounded semi annually, compounded, quarterly, compounded, monthly and compounded continuously

To find the accumulated value of an investment of $20,000 for five years at an interest rate of 1.15%, we can calculate the amount based on the different compounding frequencies: semi-annually, quarterly, monthly, and continuously.

The general formula for compound interest (except for continuous compounding) is:
\[
A = P \left(1 + \frac{r}{n}\right)^{n \cdot t}
\]
where:
- $ A $ is the accumulated amount,
- $ P $ is the principal (initial amount, here $ P = 20,000 $),
- $ r $ is the annual interest rate (here $ r = 0.0115 $),
- $ n $ is the number of compounding periods per year, and
- $ t $ is the time in years (here $ t = 5 $).

For continuous compounding, the formula is:
\[
A = P \cdot e^{r \cdot t}
\]

### 1. Compounded Semi-Annually
Here, $ n = 2 $.

\[
A = 20000 \left(1 + \frac{0.0115}{2}\right)^{2 \times 5}
\]

### 2. Compounded Quarterly
Here, $ n = 4 $.

\[
A = 20000 \left(1 + \frac{0.0115}{4}\right)^{4 \times 5}
\]

### 3. Compounded Monthly
Here, $ n = 12 $.

\[
A = 20000 \left(1 + \frac{0.0115}{12}\right)^{12 \times 5}
\]

### 4. Compounded Continuously
For continuous compounding, we use the formula:
\[
A = 20000 \cdot e^{0.0115 \times 5}
\]

Let’s calculate each of these amounts.Here are the accumulated values for each compounding method:

1. **Compounded Semi-Annually**: $21,180.22
2. **Compounded Quarterly**: $21,181.96
3. **Compounded Monthly**: $21,183.12
4. **Compounded Continuously**: $21,183.71

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions you may find useful:

1. How does compounding frequency affect the total accumulated amount?
2. What would be the effect of a higher interest rate on these calculations?
3. How does continuous compounding compare to other compounding methods in different scenarios?
4. What would happen if the investment period was extended to 10 years?
5. How do other investment factors, like inflation, impact the effective return?

**Tip:** The more frequently interest is compounded, the greater the accumulated amount will be due to the effect of interest on interest.

Find the accumulated value of an investment of $20,000 for five years at an interest rate of 1.15% if the money is compounded semi-annually, quarterly, monthly, and continuously.

Discover how to calculate the accumulated value of a $20,000 investment over five years at a 1.15% interest rate, compounded semi-annually, quarterly, monthly, and continuously. This example explores how compounding frequency impacts total growth, ideal for high school students studying compound interest and exponential growth.

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