To solve this problem, we'll need to compute the returns for both the stock market investment and the money market account, then compare the results to find out how much more Hailey will earn from the stock market investment.
### Step-by-Step Solution
#### 1. Initial Setup
- Initial investment: \[tex]$500
- Stock market annual yield: 12% (0.12)
- Money market monthly yield: 4.7% (0.047)
#### 2. Stock Market Investment Calculation
The stock market yields an annual return of 12%. Using the formula for compound interest, we can calculate the value of the investment after 1 year and 10 years.
For 1 year:
\[ A_{\text{stock}, 1} = P \times (1 + r)^t \]
\[ A_{\text{stock}, 1} = 500 \times (1 + 0.12)^1 \]
\[ A_{\text{stock}, 1} = 500 \times 1.12 \]
\[ A_{\text{stock}, 1} = 560 \]
For 10 years:
\[ A_{\text{stock}, 10} = P \times (1 + r)^t \]
\[ A_{\text{stock}, 10} = 500 \times (1 + 0.12)^{10} \]
\[ A_{\text{stock}, 10} = 500 \times (1.12)^{10} \]
\[ A_{\text{stock}, 10} \approx 500 \times 3.1058 \]
\[ A_{\text{stock}, 10} \approx 1552.9 \]
#### 3. Money Market Account Calculation
The money market account yields a monthly return of 4.7%. First, we need to convert this to an annual yield by compounding monthly interest.
\[ r_{\text{annual}} = (1 + r_{\text{monthly}})^{12} - 1 \]
\[ r_{\text{annual}} = (1 + 0.047)^{12} - 1 \]
\[ r_{\text{annual}} \approx (1.047)^{12} - 1 \]
\[ r_{\text{annual}} \approx 1.7378 - 1 \]
\[ r_{\text{annual}} \approx 0.7378 \]
So, the effective annual interest rate is approximately 73.78%.
For 1 year:
\[ A_{\text{money}, 1} = P \times (1 + r_{\text{annual}})^t \]
\[ A_{\text{money}, 1} = 500 \times (1 + 0.7378)^1 \]
\[ A_{\text{money}, 1} = 500 \times 1.7378 \]
\[ A_{\text{money}, 1} \approx 868.9 \]
For 10 years:
\[ A_{\text{money}, 10} = P \times (1 + r_{\text{annual}})^t \]
\[ A_{\text{money}, 10} = 500 \times (1 + 0.7378)^{10} \]
\[ A_{\text{money}, 10} \approx 500 \times (1.7378 )^{10} \]
\[ A_{\text{money}, 10} \approx 500 \times 90.144 \]
\[ A_{\text{money}, 10} \approx 45072 \]
#### 4. Comparison
Now, we compare the earnings from both investments after 1 year and 10 years.
For 1 year:
\[ \text{Difference}_{1 \text{ year}} = A_{\text{money}, 1} - A_{\text{stock}, 1} \]
\[ \text{Difference}_{1 \text{ year}} = 868.9 - 560 \]
\[ \text{Difference}_{1 \text{ year}} \approx 308.9 \]
For 10 years:
\[ \text{Difference}_{10 \text{ years}} = A_{\text{money}, 10} - A_{\text{stock}, 10} \]
\[ \text{Difference}_{10 \text{ years}} = 45072 - 1552.9 \]
\[ \text{Difference}_{10 \text{ years}} \approx 43519.1 \]
### Conclusion
After 1 year, Hailey will earn approximately \$[/tex]308.9 more from the money market account than the stock market. After 10 years, the money market account yields approximately \$43519 more than the stock market investment.
