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I was trying to solve the following exercise:

"Stocks A have $\mu_A=8\%$, $\sigma_A=2,5\%$ and stocks B have $\mu_B=6\%$, $\sigma_B=1,2\%$. Let us suppose that expexted returns are independent. What is the standard deviation of a portfolio made up of one stock A and one stock B?"

Since expected returns are independent, I immediately thought of the formula $$\sigma_P^2=w\sigma_A^2+(1-w)\sigma_B^2,$$ where $w$ represents the weight of stock A in the portfolio.

So I was wondering whether in the text there is a missing value, such as the weight of one stock in the portfolio. I don't think I can assume stocks are equally weighted. Anyway the final result is $2.77\%$. Thanks in advance.

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If you buy a position in stock A, the variance is $0.025^2$. If now you buy an additional position of the same size in stock B, the variance is $0.025^2+0.012^2=0.000769$. Independent variances can be added. However I have to say this is not well explained in the problem statement, which is very unclear. We are adding a new position with additional money, not splitting a given amount of money between the 2 stocks. $\sqrt{0.00769}=0.02773$

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