Tyler Trucks stock has an annual return mean and standard deviation of 13 percent and 42 percent, respectively. Michael Moped Manufacturing stock has an annual return mean and standard deviation of 13.0 percent and 68 percent, respectively. Your portfolio allocates equal funds to Tyler Trucks stock and Michael Moped Manufacturing stock. The return correlation between Tyler Trucks and Michael Moped Manufacturing is 0.5. What is the smallest expected loss for your portfolio in the coming month with a probability of 16 percent?

To find the smallest expected loss for your portfolio with a probability of 16% over the coming month, we will follow these steps:

1. **Convert annual statistics to monthly statistics:**
   - Mean monthly return for Tyler Trucks (\(\mu_T\)) and Michael Moped Manufacturing (\(\mu_M\)):
     \[
     \mu_T = \frac{0.13}{12} \approx 0.01083
     \]
     \[
     \mu_M = \frac{0.13}{12} \approx 0.01083
     \]

   - Standard deviation of monthly returns for Tyler Trucks (\(\sigma_T\)) and Michael Moped Manufacturing (\(\sigma_M\)):
     \[
     \sigma_T = \frac{0.42}{\sqrt{12}} \approx 0.1213
     \]
     \[
     \sigma_M = \frac{0.68}{\sqrt{12}} \approx 0.1962
     \]

2. **Determine the combined portfolio mean return (\(\mu_P\)) and standard deviation (\(\sigma_P\)):**
   - The portfolio mean return (since it is equally weighted):
     \[
     \mu_P = \frac{\mu_T + \mu_M}{2} = \frac{0.01083 + 0.01083}{2} = 0.01083
     \]

   - The portfolio standard deviation (since it is equally weighted):
     \[
     \sigma_P = \sqrt{\left(\frac{\sigma_T^2}{2} + \frac{\sigma_M^2}{2} + 2 \times 0.5 \times \frac{\sigma_T}{\sqrt{2}} \times \frac{\sigma_M}{\sqrt{2}}\right)}
     \]
     Simplifying this:
     \[
     \sigma_P = \sqrt{\left(\frac{(0.1213)^2 + (0.1962)^2}{2} + 2 \times 0.5 \times \frac{0.1213}{\sqrt{2}} \times \frac{0.1962}{\sqrt{2}}\right)}
     \]
     \[
     \sigma_P = \sqrt{\left(\frac{0.01471 + 0.03846}{2} + 2 \times 0.5 \times 0.0858 \times 0.1388 \right)}
     \]
     \[
     \sigma_P = \sqrt{\left(0.02659 + 2 \times 0.5 \times 0.01189 \right)}
     \]
     \[
     \sigma_P = \sqrt{0.02659 + 0.01189} \approx \sqrt{0.03848} \approx 0.1962
     \]

3. **Identify the z-score for the 16% probability:** The z-score corresponding to the lower 16% of the normal distribution is approximately -0.994.

4. **Calculate the Value at Risk (VaR) for the portfolio:**
   \[
   \text{VaR} = \mu_P + (z \times \sigma_P)
   \]
   \[
   \text{VaR} = 0.01083 + (-0.994 \times 0.1962)
   \]
   \[
   \text{VaR} = 0.01083 - 0.1950 \approx -0.1842
   \]

So, the smallest expected loss for your portfolio in the coming month with a probability of 16% is approximately 18.42%. This means there is a 16% chance that the portfolio will lose 18.42% or more of its value in the next month.

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

Here are some questions you might consider next:
1. How does correlation between stocks affect portfolio risk?
2. What are the advantages of diversifying a portfolio?
3. How can Value at Risk (VaR) be used in risk management?
4. What are other risk measures besides VaR?
5. How do changes in the time period affect VaR calculations?

**Tip:** Always consider the correlation between assets in a portfolio, as it can significantly impact the overall risk and potential returns.

Learn how to calculate the smallest expected loss for a portfolio with given stock returns and correlations. Understand the application of statistics, probability, and finance principles in determining the Value at Risk (VaR) with a probability of 16%.

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