Tyler Trucks stock has an annual return mean and standard deviation of 12.0 percent and 35 percent, respectively. Michael Moped Manufacturing stock has an annual return mean and standard deviation of 10.8 percent and 53 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 −.5. What is the smallest expected loss for your portfolio in the coming month with a probability of 2.5 percent? (A negative value should be indicated by a minus sign. Do not round intermediate calculations. Round the z-score value to 3 decimal places when calculating your answer. Enter your answer as a percent rounded to 2 decimal places.)

Answer 1

To find the smallest expected loss for the portfolio, we need to calculate the portfolio return and then find the corresponding z-score.

First, calculate the portfolio return mean:

Portfolio Return Mean = (1/2)(12.0% + 10.8%) = 11.4%

Next, calculate the portfolio return standard deviation using the formula for a portfolio of two assets:

Portfolio Return Standard Deviation = √(0.5^2 * 35%^2 + 0.5^2 * 53%^2 + 2 * 0.5 * 0.5 * -0.5 * 35% * 53%) ≈ 36.88%

Now, we can use the z-score formula to find the z-score for the 2.5% probability:

z = (X - μ) / σ

rearranging the formula to solve for X:

X = z * σ + μ

where X is the portfolio return, μ is the portfolio return mean, σ is the portfolio return standard deviation, and z is the z-score.

For a probability of 2.5%, the z-score is -1.96 (from a standard normal distribution table). Plugging in the values:

X = -1.96 * 36.88% + 11.4% ≈ -4.6208%

Since a negative value indicates a loss, the smallest expected loss for the portfolio in the coming month with a probability of 2.5% is approximately -4.62%.