Let's solve both questions step by step: ### **Given Data** - Initial investment: $1,000,000 - Monthly returns (in decimal): \( R_1 = -0.4, \; R_2 = 0.67, \; R_3 = 1.0, \; R_4 = -0.5, \; R_5 = 0.2, \; R_6 = -0.165 \) --- ## **Question 1: Total worth after 6 months** The formula for compounded growth each month is: \[ \text{Final Amount} = \text{Initial Amount} \times \prod_{i=1}^{6} (1 + R_i) \] Plug in the values: \[ \text{Final Amount} = \$1,000,000 \times (1 + R_1) \times (1 + R_2) \times (1 + R_3) \times (1 + R_4) \times (1 + R_5) \times (1 + R_6) \] \[ = \$1,000,000 \times (1 - 0.4) \times (1 + 0.67) \times (1 + 1.0) \times (1 - 0.5) \times (1 + 0.2) \times (1 - 0.165) \] \[ = \$1,000,000 \times 0.6 \times 1.67 \times 2.0 \times 0.5 \times 1.2 \times 0.835 \] Now, calculate step by step: 1. \( 0.6 \times 1.67 = 1.002 \) 2. \( 1.002 \times 2.0 = 2.004 \) 3. \( 2.004 \times 0.5 = 1.002 \) 4. \( 1.002 \times 1.2 = 1.2024 \) 5. \( 1.2024 \times 0.835 = 1.004004 \) So, \[ \text{Final Amount} = \$1,000,000 \times 1.004004 = \$1,004,004 \] **Final Answer for Q1:** \[ \boxed{\$1,004,004} \] --- ## **Question 2: Geometric mean of monthly returns** The geometric mean \( G \) for the 6 returns is: \[ G = \left[ \prod_{i=1}^6 (1+R_i) \right]^{1/6} - 1 \] From above, we already calculated \( \prod_{i=1}^6 (1+R_i) = 1.004004 \). So, \[ G = (1.004004)^{1/6} - 1 \] Calculate \( (1.004004)^{1/6} \): \[ 1.004004^{1/6} \approx e^{\frac{\ln(1.004004)}{6}} \] \[ \ln(1.004004) \approx 0.003999 \] \[ 0.003999 / 6 \approx 0.0006665 \] \[ e^{0.0006665} \approx 1.000667 \] \[ G = 1.000667 - 1 = 0.000667 \] Rounded to 3 decimal places: \[ \boxed{0.001} \] --- ## **Summary of Answers** 1. **Total worth after 6 months:** \(\boxed{\$1,004,004}\) 2. **Geometric mean of returns:** \(\boxed{0.001}\) (rounded to 3 decimal places)