Calculation of $\sigma_{\text{diff}}$ gives a complex number, indicating a potential error in variance computation

Question

Given below is the problem I have solved but I am getting complex number as an answer for $\sigma_{\text{diff}}$ which is supposed to be real number. I need guidance. Have I made mistake in previous steps or I am making a mistake only in calculating $\sigma_{\text{diff}}$?

Problem 3: I am trying to decide between investing in one of two stocks. We will assume that daily returns are independent. I know that the annual return for stock A has a mean of $0.02$ and an annual standard deviation of $0.4$, while the respective numbers for stock B are a mean of $0.03$ and an annual standard deviation of $0.6$. Assume the covariance between the two yearly returns is $0.072$.

Part (a): Calculate the approximate distribution of the difference between the 50-day average return for stock A and the 30-day average return for stock B. Pay attention to the covariance between the two average returns.

Solution

Given:

• $\mu_{A,50} = \mu_{A,\text{daily}}$, $\sigma_{A,50} = \frac{\sigma_{A,\text{daily}}}{\sqrt{50}}$
• $\mu_{B,30} = \mu_{B,\text{daily}}$, $\sigma_{B,30} = \frac{\sigma_{B,\text{daily}}}{\sqrt{30}}$
• $\text{Cov}(R_A, R_B) = 0.072$

Task:

• Find distribution of $\mu_{\text{diff}} = \mu_{A,50} - \mu_{B,30}$
• Find $\sigma_{\text{diff}}$ using adjusted covariance

Solution:

1. Mean of Difference: $\mu_{\text{diff}} = \mu_{A,50} - \mu_{B,30} = 7.94 \times 10^{-5} - 0.000119$

2. Adjusted Covariance: $\text{Cov}(A_{50}, B_{30}) = \text{Cov}(R_A, R_B) \times \sqrt{\frac{30}{252}} \times \sqrt{\frac{50}{252}} = 0.072 \times \sqrt{\frac{30}{252}} \times \sqrt{\frac{50}{252}}$

3. Variance of Difference: $\sigma_{\text{diff}}^2 = \sigma_{A,50}^2 + \sigma_{B,30}^2 - 2 \times \text{Cov}(A_{50}, B_{30}) = \left(\frac{0.0252}{\sqrt{50}}\right)^2 + \left(\frac{0.0378}{\sqrt{30}}\right)^2 - 2 \times \text{Cov}(A_{50}, B_{30})$

4. Standard Deviation of Difference: $\sigma_{\text{diff}} = \sqrt{\sigma_{\text{diff}}^2}= \approx (9.10 \times 10^{-18} + 0.1486i) $

Calculation of $\sigma_{\text{diff}}$ gives a complex number, indicating a potential error in variance computation or data assumptions. The variance should be a positive real number.