4.	You have been saving money for the past few years and have saved $5000 in cash. The goal has been to save up enough money to buy a house. You decided that it would be better to invest the money in a bank until you are ready to make your big purchase. After speaking with a bank representative, you are given two different investment options.

Option 1- pays 6.5% per year, compounded quarterly for 10 years.
Option 2- pays 6.5% per year, compounded continuously for 10 years.


To analyze which investment option is better, we need to calculate the future value of your $5000 investment under both scenarios.

### **Option 1: Compounded Quarterly**
The formula for compound interest when compounded $ n $ times per year is:

\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]

Where:
- $ A $ = the amount of money accumulated after $ t $ years, including interest
- $ P $ = the principal amount (initial investment)
- $ r $ = annual interest rate (in decimal form)
- $ n $ = number of times the interest is compounded per year
- $ t $ = number of years

For Option 1:
- $ P = 5000 $
- $ r = 6.5\% = 0.065 $
- $ n = 4 $ (quarterly compounding)
- $ t = 10 $ years

Plugging these values into the formula:

\[
A = 5000 \left( 1 + \frac{0.065}{4} \right)^{4 \times 10}
\]

### **Option 2: Compounded Continuously**
The formula for continuously compounded interest is:

\[
A = P e^{rt}
\]

Where:
- $ A $ = the amount of money accumulated after $ t $ years, including interest
- $ P $ = the principal amount (initial investment)
- $ e $ = Euler's number (approximately 2.71828)
- $ r $ = annual interest rate (in decimal form)
- $ t $ = number of years

For Option 2:
- $ P = 5000 $
- $ r = 0.065 $
- $ t = 10 $ years

Plugging these values into the formula:

\[
A = 5000 e^{0.065 \times 10}
\]

Let me calculate the results for both options.After 10 years, the future value of your $5000 investment will be:

- **Option 1 (Compounded Quarterly)**: \$9527.79
- **Option 2 (Compounded Continuously)**: \$9577.70

### Conclusion:
Option 2 (compounded continuously) yields slightly more, with a future value of \$9577.70, compared to Option 1's \$9527.79. Although the difference is not huge, continuously compounded interest gives you a better return.

Do you want further details or have any questions?

Here are 5 related questions to think about:
1. How much would you have after 15 years with both options?
2. What if the interest rate was higher, say 7%?
3. How would monthly compounding compare with continuous compounding?
4. What if you added monthly contributions to your investment?
5. How does the choice of investment option change for shorter time periods, like 5 years?

**Tip**: Compounding frequency matters more over longer time horizons!

You have been saving money for the past few years and have saved $5000 in cash. The goal has been to save up enough money to buy a house. You decided that it would be better to invest the money in a bank until you are ready to make your big purchase. After speaking with a bank representative, you are given two different investment options.

Option 1- pays 6.5% per year, compounded quarterly for 10 years.
Option 2- pays 6.5% per year, compounded continuously for 10 years.

Compare two investment options for a $5000 deposit: one compounded quarterly at 6.5% and the other compounded continuously at 6.5%. Learn how these interest models affect future values over a 10-year period, with detailed calculations and formulas.

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