To calculate the standard deviation of the stock's returns, we first need to calculate the variance, and then take the square root of the variance to obtain the standard deviation.

Steps:

1. Expected Return (E[R]) is given as 3.32%, but let's calculate it from the probabilities and return rates to confirm the provided value:

$E[R] = (P_1 \times R_1) + (P_2 \times R_2) + (P_3 \times R_3)$

Where:

• $P_1 = 33\%$, $P_2 = 31\%$, $P_3 = 36\%$ are the probabilities,
• $R_1 = -6\%$, $R_2 = 2\%$, $R_3 = 13\%$ are the respective returns.

Substitute the values:

$E[R] = (0.33 \times -6) + (0.31 \times 2) + (0.36 \times 13)$

$E[R] = -1.98 + 0.62 + 4.68$

$E[R] = 3.32\%$

This matches the given expected return.

2. Variance ($\sigma^2$) is calculated as:

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

Substitute the values for each situation:

• For $R_1 = -6\%$: $(R_1 - E[R])^2 = (-6 - 3.32)^2 = (-9.32)^2 = 87.06$ $P_1 \times (R_1 - E[R])^2 = 0.33 \times 87.06 = 28.74$

• For $R_2 = 2\%$: $(R_2 - E[R])^2 = (2 - 3.32)^2 = (-1.32)^2 = 1.7424$ $P_2 \times (R_2 - E[R])^2 = 0.31 \times 1.7424 = 0.539$

• For $R_3 = 13\%$: $(R_3 - E[R])^2 = (13 - 3.32)^2 = (9.68)^2 = 93.5424$ $P_3 \times (R_3 - E[R])^2 = 0.36 \times 93.5424 = 33.694$

Now, sum up these weighted squared differences to find the variance:

$\sigma^2 = 28.74 + 0.539 + 33.694 = 62.973$

3. Standard Deviation ($\sigma$) is the square root of the variance:

$\sigma = \sqrt{62.973} \approx 7.95\%$

Final Answer:

The standard deviation of the stock's return is approximately 7.95%.

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Would you like more details on the steps? Here are some related questions to explore further:

1. How is expected return calculated in portfolio theory?
2. What is the difference between variance and standard deviation?
3. How can we use the standard deviation in risk management for stocks?
4. What is the significance of probabilities in stock returns?
5. How can we apply this standard deviation to compute value-at-risk (VaR)?

Tip: Always ensure that your returns are in the same units (percentages or decimals) throughout your calculations to avoid errors.