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.)
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%.
To calculate the smallest expected loss for your portfolio, we will use the formula for the portfolio return given by:
ρ_r = w₁ρ₁ + w₂ρ₂
Where:
ρ_r is the portfolio return correlation
w₁ and w₂ are the weights allocated to Tyler Trucks and Michael Moped Manufacturing stocks, respectively
ρ₁ and ρ₂ are the returns mean of Tyler Trucks and Michael Moped Manufacturing stocks, respectively.
Given:
ρ_r = -0.5 (correlation between Tyler Trucks and Michael Moped Manufacturing)
ρ₁ = 0.12 (annual return mean of Tyler Trucks)
ρ₂ = 0.108 (annual return mean of Michael Moped Manufacturing)
σ₁ = 0.35 (standard deviation of Tyler Trucks)
σ₂ = 0.53 (standard deviation of Michael Moped Manufacturing)
Since the portfolio allocates equal funds to both stocks, the weights are equal.
Using the formula, we can calculate the portfolio standard deviation (σ_p):
σ_p = sqrt(w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ_rσ₁σ₂)
Since w₁ = w₂ = 0.5:
σ_p = sqrt(0.5² * 0.35² + 0.5² * 0.53² + 2 * 0.5 * 0.5 * (-0.5) * 0.35 * 0.53)
Now, we can calculate the z-score for a probability of 2.5 percent. The z-score represents the number of standard deviations away from the mean:
z = invNorm(1 - p) = invNorm(1 - 0.025)
Using a z-table or a calculator, the z-score for a probability of 2.5 percent is approximately 1.96.
Finally, we can calculate the smallest expected loss (L):
L = ρ² - zσ_p
L = 0.5² - 1.96 * σ_p
Let's calculate:
σ_p = sqrt(0.5² * 0.35² + 0.5² * 0.53² + 2 * 0.5 * 0.5 * (-0.5) * 0.35 * 0.53)
σ_p ≈ 0.413
L = 0.5² - 1.96 * 0.413
L ≈ -0.333
Therefore, the smallest expected loss for your portfolio in the coming month with a probability of 2.5 percent is approximately -0.333 percent.