A stock has an annual return of 10.2 percent and a standard deviation of 40 percent. What is the smallest expected loss over the next year with a probability of 5 percent?

To determine the smallest expected loss over the next year with a probability of 5%, we need to calculate the Value at Risk (VaR) at the 5% level. The VaR can be found using the z-score corresponding to the 5% probability in a normal distribution. 

Here are the steps to calculate it:

1. **Identify the z-score for the 5% probability:** The 5% tail corresponds to the lower 5% of the normal distribution. The z-score for this is approximately -1.645 (since it's negative on the left side of the distribution).

2. **Calculate the mean return and standard deviation:**
   - Mean return (\(\mu\)) = 10.2% = 0.102
   - Standard deviation (\(\sigma\)) = 40% = 0.40

3. **Apply the VaR formula:**
   \[
   \text{VaR} = \mu + (z \times \sigma)
   \]

4. **Substitute the values into the formula:**
   \[
   \text{VaR} = 0.102 + (-1.645 \times 0.40)
   \]

5. **Calculate the VaR:**
   \[
   \text{VaR} = 0.102 + (-0.658) = 0.102 - 0.658 = -0.556
   \]

So, the smallest expected loss over the next year with a probability of 5% is 55.6%. This means there is a 5% chance that the stock will lose 55.6% or more of its value over the next year.

Would you like more details on the calculation or have any questions?

Here are some questions you might consider next:
1. What is the interpretation of the Value at Risk (VaR)?
2. How do you calculate VaR for different confidence levels?
3. How does changing the standard deviation affect the VaR?
4. What are the limitations of using VaR in risk management?
5. Can VaR be applied to portfolios, and if so, how?

**Tip:** Always check the assumptions behind risk measures like VaR, as real-world returns may not follow a normal distribution exactly.

Learn how to calculate the smallest expected loss over the next year with a probability of 5% using Value at Risk (VaR). Understand the VaR formula and its application with a 10.2% annual return and 40% standard deviation.

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