(C) If your investment of \( \$ 1,000 \) declined to \( \$ 400 \) in 8 years, what would have been your compounded annual rate of return? Be sure to use cell references and to convert your answer to a percent rounded to two places after the decimal point. I

To determine the **compounded annual rate of return** when an investment declines from \$1,000 to \$400 over 8 years, you can use the following formula: \[ \text{Final Value} = \text{Initial Investment} \times (1 + r)^n \] Where: - **Final Value (FV)** = \$400 - **Initial Investment (PV)** = \$1,000 - **Number of Years (n)** = 8 - **Annual Rate of Return (r)** = ? ### Step-by-Step Calculation: 1. **Rearrange the Formula to Solve for \( r \):** \[ \frac{\text{FV}}{\text{PV}} = (1 + r)^n \] \[ \frac{400}{1000} = (1 + r)^8 \] \[ 0.4 = (1 + r)^8 \] 2. **Take the 8th Root of Both Sides:** \[ 1 + r = 0.4^{\frac{1}{8}} \] 3. **Calculate \( 0.4^{\frac{1}{8}} \):** \[ 0.4^{\frac{1}{8}} \approx 0.8928 \] 4. **Solve for \( r \):** \[ r = 0.8928 - 1 = -0.1072 \] \[ r \approx -10.72\% \] ### Using a Spreadsheet (e.g., Excel): If you're using a spreadsheet like Excel, you can calculate the rate using the following formula. Assume: - **Initial Investment** is in cell `A1` (\$1,000) - **Final Value** is in cell `A2` (\$400) - **Number of Years** is in cell `A3` (8) **Formula:** ```excel =((A2/A1)^(1/A3) - 1) * 100 ``` This will give you **-10.72%** as the compounded annual rate of return. ### **Final Answer:** The compounded annual rate of return is **–10.72 %**.