Question: A portfolio consists of two stocks, $A$ and $B$. $A$ has expected return $.15$, volatility $.2$, proportion of portfolio $.5$. $B$ has expected return $.18$, volatility $.25$, proportion of portfolio $.5$. $\beta_A^P = .8.$ Annual effective risk-free interest rate is $.05.$ Determine the correlation of stock $A$ with portfolio $P$.
In my attempt, I use the following identity: $\beta_A^P = \frac{\mathrm{Cov}(R_P,R_A)}{\mathrm{Var}(P)}$. Since $\mathrm{Var}(P) = \mathrm{Var}(0.5A+0.5B) = .025625 + .025\rho_{A,B}$, we can say $.8 = \frac{ \mathrm{Cov}(R_P,R_A) }{\sqrt{.025625+.025\rho_{A,B}}} $
If I know that $\rho_{R_P,R_A} = \rho_{P,A}$, I can solve for $\mathrm{Cov}(R_P,R_A)$ in terms of $\rho_{A,B}$ and solve from there. Is this step justifiable?