The geometric average return for a stock over multiple periods is calculated using the formula:

$R_g = \left( \prod_{i=1}^n (1 + R_i) \right)^{\frac{1}{n}} - 1$

Where:

• $R_i$ is the return for each period,
• $n$ is the number of periods.

For the given returns: $9.31\%$, $-6.72\%$, $22.38\%$, and $15.07\%$, the steps are:

1. Convert percentages to decimals:

   • $R_1 = 0.0931$
   • $R_2 = -0.0672$
   • $R_3 = 0.2238$
   • $R_4 = 0.1507$

2. Compute $1 + R_i$ for each period:

   • $1 + R_1 = 1.0931$
   • $1 + R_2 = 0.9328$
   • $1 + R_3 = 1.2238$
   • $1 + R_4 = 1.1507$

3. Multiply these together: $\text{Product} = 1.0931 \times 0.9328 \times 1.2238 \times 1.1507$

4. Take the fourth root ($n = 4$): $R_g = \left( \text{Product} \right)^{\frac{1}{4}} - 1$

Let’s calculate.

$\text{Product} = 1.0931 \times 0.9328 \times 1.2238 \times 1.1507 = 1.439068$

$R_g = \left(1.439068\right)^{\frac{1}{4}} - 1$

$R_g = 1.095 \, \text{(approximately)} - 1 = 0.095 \text{ or } 9.5\%.$

Among the options, the closest value is 9.47%.

Answer: 9.47%

Would you like a step-by-step breakdown or have further questions?

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Related Questions:

1. What is the difference between arithmetic and geometric average returns?
2. How do compounding effects influence geometric returns compared to arithmetic returns?
3. Can geometric average returns ever be higher than arithmetic average returns?
4. How do negative returns impact geometric averages?
5. Why is the geometric average return preferred for multi-year investments?

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Tip:

When analyzing returns, always compare the arithmetic and geometric averages to understand both the overall trend and the compounding effects of volatility.