To find the standard deviation of the stock's returns, we first calculate the expected return and then use the formula for variance and standard deviation. Here's the step-by-step solution:

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1. Expected Return ($E[R]$):

The expected return is calculated as:

$E[R] = \sum (P_i \cdot R_i)$

Where:

• $P_i$: Probability of each scenario
• $R_i$: Return in each scenario

Substituting the given values: $E[R] = (0.25 \cdot -12.8\%) + (0.44 \cdot 11.5\%) + (0.31 \cdot 26.2\%)$ $E[R] = -3.2\% + 5.06\% + 8.122\% = 9.982\%$

The expected return is 9.982%.

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2. Variance ($\sigma^2$):

The variance formula is:

$\sigma^2 = \sum P_i \cdot (R_i - E[R])^2$

Substituting the given values: $\sigma^2 = (0.25 \cdot (-12.8\% - 9.982\%)^2) + (0.44 \cdot (11.5\% - 9.982\%)^2) + (0.31 \cdot (26.2\% - 9.982\%)^2)$

1. Calculate each deviation:

   • $-12.8\% - 9.982\% = -22.782\%$
   • $11.5\% - 9.982\% = 1.518\%$
   • $26.2\% - 9.982\% = 16.218\%$

2. Square the deviations:

   • $(-22.782)^2 = 518.77$
   • $(1.518)^2 = 2.31$
   • $(16.218)^2 = 262.98$

3. Multiply by the probabilities:

   • $0.25 \cdot 518.77 = 129.6925$
   • $0.44 \cdot 2.31 = 1.0164$
   • $0.31 \cdot 262.98 = 81.5298$

4. Sum them up: $\sigma^2 = 129.6925 + 1.0164 + 81.5298 = 212.2387$

5. Take the square root for standard deviation ($\sigma$): $\sigma = \sqrt{212.2387} \approx 14.57\%$

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Final Answer:

The standard deviation of the stock's returns is 14.57%.

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Do you want a deeper dive into this explanation or have additional questions? Here are five related questions you might find useful:

1. How does changing the probabilities affect the standard deviation?
2. What is the importance of standard deviation in portfolio management?
3. Can you explain how expected return influences decision-making?
4. How would these calculations change for a portfolio of multiple stocks?
5. How does skewness differ from standard deviation?

Tip: Always double-check your probabilities—they must sum to 100% for valid calculations!