linear equation of Perfect positive correlation portfolio (two-asset case)

Question

When risky assets X and Y are perfectly positive correlated (i.e. $\rho_{XY}=1$), the standard deviation of this portfolio is: $$\sigma_p=w_X\sigma_X+(1-w_X)\sigma_Y$$And the expected return of this portfolio is:$$\mathbb{E}[R_p]=w_X\mathbb{E}[R_X]+(1-w_X)\mathbb{E}[R_Y]$$From these two equations above, we can derive a linear function about $E[R_p]$ and $\sigma_p$,which is:\begin{equation}\mathbb{E}[R_p]=\left(\mathbb{E}[R_Y]-\frac{\mathbb{E}[R_X]-\mathbb{E}[R_Y]}{\sigma_X-\sigma_Y}\sigma_Y \right)+ \left(\frac{\mathbb{E}[R_X]-\mathbb{E}[R_Y]}{\sigma_X-\sigma_Y} \right)\sigma_P\end{equation} My question is, how can we derive this equation?

Answer 1

Since $\sigma_p=w_X\sigma_X+(1-w_X)\sigma_Y$, we can solve for $w_X$ when $\sigma_X \neq \sigma_Y$, obtaining

$$\tag{1}w_X = \frac{\sigma_P - \sigma_Y}{\sigma_X - \sigma_Y}$$

We have,

$$\tag{2}\mathbb{E}[R_p]=w_X\mathbb{E}[R_X]+(1-w_X)\mathbb{E}[R_Y] = \mathbb{E}[R_Y] + w_X (\mathbb{E}[R_X]- \mathbb{E}[R_Y])$$

Substituting for $w_X$ in (2) using (1), we get

$$\mathbb{E}[R_p]= \mathbb{E}[R_Y] + \frac{\sigma_P - \sigma_Y}{\sigma_X - \sigma_Y}(\mathbb{E}[R_X]- \mathbb{E}[R_Y]) \\ =\mathbb{E}[R_Y] - \frac{\mathbb{E}[R_X]- \mathbb{E}[R_Y]}{\sigma_X - \sigma_Y}\sigma_Y+ \frac{\mathbb{E}[R_X]- \mathbb{E}[R_Y]}{\sigma_X - \sigma_Y}\sigma_P$$