Your sister just deposited $14,000 into an investment account. She believes that she will earn an annual return of 10.5 percent for the next 9 years. You believe that you will only be able to earn an annual return of 9.8 percent over the same period. How much more must you deposit today in order to have the same amount as your sister in 9 years?

To find out how much more you need to deposit today to match your sister's future amount, we can use the future value formula to determine the amount she will have after 9 years, and then use the present value formula to find out how much you need to invest today to reach that same future value at your lower return rate.

### 1. **Future Value of Your Sister's Investment:**
Given:
- $ PV_{\text{sister}} = 14,000 $ dollars,
- $ r_{\text{sister}} = 10.5\% = 0.105 $,
- $ n = 9 $ years.

The future value $ FV $ is calculated using:

\[
FV = PV \times (1 + r)^n
\]

\[
FV_{\text{sister}} = 14,000 \times (1 + 0.105)^{9}
\]

### 2. **Present Value for Your Investment:**
Given:
- $ FV = FV_{\text{sister}} $ (from above),
- $ r_{\text{you}} = 9.8\% = 0.098 $,
- $ n = 9 $ years.

We solve for the present value $ PV $:

\[
PV_{\text{you}} = \frac{FV_{\text{sister}}}{(1 + 0.098)^{9}}
\]

### 3. **Difference in Required Investment:**
The difference is given by:

\[
\text{Difference} = PV_{\text{you}} - PV_{\text{sister}}
\]

Let's calculate these values.### Results:
- **Future Value of Your Sister's Investment**: \$34,386.54
- **Present Value You Need to Invest**: \$14,824.07
- **Difference**: \$824.07

To match the amount your sister will have in 9 years, you need to deposit **\$824.07 more** today than she did.

Would you like more details or have any questions?

Here are some related questions you might consider:
1. What if the investment period was longer, say 15 years? How would that affect the difference?
2. How sensitive is the required deposit to small changes in the interest rate?
3. What is the effect of quarterly or monthly compounding on the future value?
4. How much less would you need to deposit if your expected return was higher?
5. How would the required deposit change if you both planned to withdraw the funds periodically?
6. What is the impact of fees or taxes on the required investment?
7. How would different risk levels affect the choice between these two interest rates?
8. What would happen if the market conditions caused the return rates to change after a few years?

**Tip**: Even small differences in interest rates can significantly affect the required initial investment over time.

Explore how to calculate the difference in required investment when comparing different interest rates over a period using the future value and present value formulas. Get a detailed solution for an investment scenario with specific interest rates and time periods.

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