Let's solve each part **step-by-step**: --- ## (a) **Apply**: Arithmetic and Geometric Mean of Investment Returns (Last 10 Years) ### **Step 1: Find the Annual Returns** Formula for annual return for year \( t \) (from year \( t-1 \) to \( t \)): \[ \text{Return}_t = \frac{\text{Price}_t - \text{Price}_{t-1}}{\text{Price}_{t-1}} \] Or, as a decimal: \[ r_t = \frac{\text{Price}_t}{\text{Price}_{t-1}} - 1 \] **We need the returns from 2015 to 2024** (10 yearly returns): | Year | Price (\$) | Return Calculation | Return (\( r_t \)) | |------|------------|----------------------------------------------------|--------------------| | 2015 | 125.92 | (125.92 / 104.02) - 1 | 0.2108 | | 2016 | 160.57 | (160.57 / 125.92) - 1 | 0.2754 | | 2017 | 131.98 | (131.98 / 160.57) - 1 | -0.1778 | | 2018 | 116.00 | (116.00 / 131.98) - 1 | -0.1211 | | 2019 | 98.48 | (98.48 / 116.00) - 1 | -0.1510 | | 2020 | 82.84 | (82.84 / 98.48) - 1 | -0.1587 | | 2021 | 79.71 | (79.71 / 82.84) - 1 | -0.0378 | | 2022 | 100.23 | (100.23 / 79.71) - 1 | 0.2577 | | 2023 | 118.78 | (118.78 / 100.23) - 1 | 0.1847 | | 2024 | 150.77 | (150.77 / 118.78) - 1 | 0.2693 | --- ### **Step 2: Arithmetic Mean** \[ \text{Arithmetic Mean} = \frac{1}{n} \sum_{i=1}^{n} r_i \] Sum of returns: \[ = 0.2108 + 0.2754 + (-0.1778) + (-0.1211) + (-0.1510) + (-0.1587) + (-0.0378) + 0.2577 + 0.1847 + 0.2693 \] \[ = 0.2108 + 0.2754 - 0.1778 - 0.1211 - 0.1510 - 0.1587 - 0.0378 + 0.2577 + 0.1847 + 0.2693 \] \[ = (0.2108 + 0.2754 + 0.2577 + 0.1847 + 0.2693) + (-0.1778 - 0.1211 - 0.1510 - 0.1587 - 0.0378) \] \[ = (1.198) + (-0.6464) = 0.5516 \] \[ \text{Arithmetic Mean} = \frac{0.5516}{10} = 0.0552 \] **Final Arithmetic Mean = 5.52%** --- ### **Step 3: Geometric Mean (Product Method)** Formula: \[ \text{Geometric Mean} = \left( \prod_{i=1}^{n} (1 + r_i) \right)^{1/n} - 1 \] Calculate each \((1 + r_i)\): | Year | \( 1 + r_i \) | |------|---------------| | 2015 | 1.2108 | | 2016 | 1.2754 | | 2017 | 0.8222 | | 2018 | 0.8789 | | 2019 | 0.8490 | | 2020 | 0.8413 | | 2021 | 0.9622 | | 2022 | 1.2577 | | 2023 | 1.1847 | | 2024 | 1.2693 | Now, multiply all: \[ \text{Product} = 1.2108 \times 1.2754 \times 0.8222 \times 0.8789 \times 0.8490 \times 0.8413 \times 0.9622 \times 1.2577 \times 1.1847 \times 1.2693 \] \[ = 1.2108 \times 1.2754 = 1.5456 \\ 1.5456 \times 0.8222 = 1.2715 \\ 1.2715 \times 0.8789 = 1.1177 \\ 1.1177 \times 0.8490 = 0.9495 \\ 0.9495 \times 0.8413 = 0.7998 \\ 0.7998 \times 0.9622 = 0.7696 \\ 0.7696 \times 1.2577 = 0.9683 \\ 0.9683 \times 1.1847 = 1.1475 \\ 1.1475 \times 1.2693 = 1.4571 \] So, the product is **approximately 1.4571**. Now, take the 10th root: \[ \text{GM} = (1.4571)^{1/10} - 1 \\ \] \[ = e^{\frac{1}{10} \ln(1.4571)} - 1 \\ \ln(1.4571) \approx 0.376 \\ 0.376 / 10 = 0.0376 \\ e^{0.0376} \approx 1.0383 \\ 1.0383 - 1 = 0.0383 \] **Final Geometric Mean = 3.83%** --- ### **Final Answers for (a):** - **Arithmetic Mean** = **5.52%** - **Geometric Mean** = **3.83%** --- ## (b) **Outline 3 Considerations When Investing** 1. **Risk Tolerance**: Assess your ability and willingness to accept losses or fluctuations in the value of your investment. Higher returns often come with higher risk. 2. **Investment Time Horizon**: Consider how long you plan to keep your money invested before needing it. Longer horizons can typically withstand more volatility. 3. **Diversification**: Spreading your investments across various asset classes or sectors reduces risk, as poor performance in one area may be offset by gains in another. --- ### **Summary Table for (a)** | Year | Price | Return (\( r_t \)) | |------|---------|--------------------| | 2015 | 125.92 | 0.2108 | | 2016 | 160.57 | 0.2754 | | 2017 | 131.98 | -0.1778 | | 2018 | 116.00 | -0.1211 | | 2019 | 98.48 | -0.1510 | | 2020 | 82.84 | -0.1587 | | 2021 | 79.71 | -0.0378 | | 2022 | 100.23 | 0.2577 | | 2023 | 118.78 | 0.1847 | | 2024 | 150.77 | 0.2693 | --- ### **Final Answers** **a)** - Arithmetic Mean: **5.52%** - Geometric Mean: **3.83%** **b)** 1. Risk tolerance 2. Investment time horizon 3. Diversification

Let's approach each part step-by-step, explaining the process clearly. --- ### (a) Calculating the Historical Returns and Means **Step 1: Determine the annual returns for each year** Using the given stock prices for the last 10 years, we calculate the year-over-year returns by taking the ratio of the current year's price to the previous year's price, subtracting 1 to get the return as a decimal. **Step 2: Find the Arithmetic Mean** Add all these annual returns together and divide by the number of years (10). This gives the average return per year, known as the arithmetic mean. **Step 3: Find the Geometric Mean** For the geometric mean, multiply all the factors (each year's price ratio plus 1). Then, take the 10th root of this product and subtract 1. This method accounts for compounding over the years, providing a more accurate average growth rate. **Final Results:** - The **Arithmetic Mean** of the annual returns is approximately **5.52%**. - The **Geometric Mean** of the returns, which reflects the compounded annual growth rate, is approximately **3.83%**. --- ### (b) Considerations When Investing 1. **Risk Tolerance:** Understand how much risk you're willing to accept. Higher potential returns usually come with higher risk, so assess your comfort level with market fluctuations. 2. **Investment Time Horizon:** Decide how long you plan to keep your money invested. Longer investment periods can help recover from market downturns and capitalize on growth. 3. **Diversification:** Spread your investments across different assets or sectors. This reduces risk because poor performance in one area may be offset by gains in another. --- **Summary:** By calculating the average returns, you gain insight into the stock's historical performance. When investing, it's crucial to consider your risk appetite, how long you'll invest, and spreading investments to manage risk effectively.

Let's carefully go through each part with detailed calculations and then present the final answers. --- ### Part (a): Calculating the Arithmetic and Geometric Mean of the Investment Returns **Step 1: Determine the annual returns** Given the stock prices for the last 10 years, we calculate the yearly returns by dividing the current year's price by the previous year's price, subtracting 1. The prices are as follows: - 2015: \$125.92 - 2016: \$160.57 - 2017: \$131.98 - 2018: \$116.00 - 2019: \$98.48 - 2020: \$82.84 - 2021: \$79.71 - 2022: \$100.23 - 2023: \$118.78 - 2024: \$150.77 Now, compute each year's return: - 2016 over 2015: (160.57 / 125.92) - 1 ≈ 1.2754 - 1 = 0.2754 or 27.54% - 2017 over 2016: (131.98 / 160.57) - 1 ≈ 0.8222 - 1 = -0.1778 or -17.78% - 2018 over 2017: (116.00 / 131.98) - 1 ≈ 0.8789 - 1 = -0.1211 or -12.11% - 2019 over 2018: (98.48 / 116.00) - 1 ≈ 0.8490 - 1 = -0.1510 or -15.10% - 2020 over 2019: (82.84 / 98.48) - 1 ≈ 0.8413 - 1 = -0.1587 or -15.87% - 2021 over 2020: (79.71 / 82.84) - 1 ≈ 0.9622 - 1 = -0.0378 or -3.78% - 2022 over 2021: (100.23 / 79.71) - 1 ≈ 1.2577 - 1 = 0.2577 or 25.77% - 2023 over 2022: (118.78 / 100.23) - 1 ≈ 1.1847 - 1 = 0.1847 or 18.47% - 2024 over 2023: (150.77 / 118.78) - 1 ≈ 1.2693 - 1 = 0.2693 or 26.93% The list of returns in decimal form: - 2016: 0.2754 - 2017: -0.1778 - 2018: -0.1211 - 2019: -0.1510 - 2020: -0.1587 - 2021: -0.0378 - 2022: 0.2577 - 2023: 0.1847 - 2024: 0.2693 **Step 2: Calculate the Arithmetic Mean** Sum all the returns: 0.2754 + (-0.1778) + (-0.1211) + (-0.1510) + (-0.1587) + (-0.0378) + 0.2577 + 0.1847 + 0.2693 = approximately 0.5516 Divide by 9 (since there are 9 returns, corresponding to 10 years of prices): Arithmetic Mean ≈ 0.5516 / 9 ≈ 0.0613 or **6.13%** *(Note: Since the returns are from year-over-year, the average is over 9 intervals, but the question asked for last 10 years, so the mean is based on these 9 returns. Alternatively, if considering 10 years and returns aligned accordingly, the calculation might vary slightly. For simplicity, we consider the average of these 9 returns.)* **Final approximate Arithmetic Mean = 6.13%** --- **Step 3: Calculate the Geometric Mean** The geometric mean considers the compounded growth rate over the period: \[ \text{GM} = \left( \prod_{i=1}^n (1 + r_i) \right)^{1/n} - 1 \] Calculate each factor: - 1 + 0.2754 = 1.2754 - 1 - 0.1778 = 0.8222 - 1 - 0.1211 = 0.8789 - 1 - 0.1510 = 0.8490 - 1 - 0.1587 = 0.8413 - 1 - 0.0378 = 0.9622 - 1 + 0.2577 = 1.2577 - 1 + 0.1847 = 1.1847 - 1 + 0.2693 = 1.2693 Multiply all these factors: 1.2754 × 0.8222 ≈ 1.049 1.049 × 0.8789 ≈ 0.921 0.921 × 0.8490 ≈ 0.782 0.782 × 0.8413 ≈ 0.658 0.658 × 0.9622 ≈ 0.633 0.633 × 1.2577 ≈ 0.796 0.796 × 1.1847 ≈ 0.944 0.944 × 1.2693 ≈ 1.198 Now, take the 9th root: \[ \text{GM} = (1.198)^{1/9} - 1 \] Calculate: \[ \ln(1.198) ≈ 0.180 \] Divide by 9: \[ 0.180 / 9 ≈ 0.020 \] Exponentiate: \[ e^{0.020} ≈ 1.0202 \] Subtract 1: \[ 1.0202 - 1 = 0.0202 \text{ or } 2.02\% \] **Final Geometric Mean ≈ 2.02%** --- ### **Final Answers for Part (a):** - **Arithmetic Mean:** approximately **6.13%** - **Geometric Mean:** approximately **2.02%** --- ### Part (b): Considerations When Investing Your Money 1. **Risk Tolerance:** Understand how much fluctuation in your investment returns you are willing to accept. Higher returns often come with higher risk, so assess your ability to handle potential losses. 2. **Investment Time Horizon:** Decide how long you plan to keep your money invested. Longer periods can help ride out market volatility and benefit from compounding growth. 3. **Diversification:** Spread your investments across different assets, sectors, or regions. This reduces risk because poor performance in one area can be offset by gains in others. --- ### **Summary:** - The **average annual return** (arithmetic mean) over the last 10 years is about **6.13%**. - The **average compounded annual growth rate** (geometric mean) is about **2.02%**. - When investing, consider your risk appetite, investment duration, and diversification to make informed decisions. --- If you need further clarification, feel free to ask!