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I have been given the expected returns and standard deviations of 2 stocks A and B, as well as the standard deviation of the market portfolio and correlation between security A and the market portfolio. I am asked to determine the idiosyncratic risks of both A & B as well as the value of Beta for B given it is positive.

I am finding it difficult to calculate the idiosyncratic risk of B without its covariance, correlation, or the risk-free rate or expected return of the market.

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Check out page 24 of Active Portfolio Management (mine is Second Edition) if you can get your hands on it. Problem 3 is effectively this exact problem. Or see here:

Calculating the correlation of stock A with stock B

This should be all the tools you need to solve it!

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