RIsk-retun of 2-asset portfolio with perfect negative correlation

Question

Risk-retun of 2-asset portfolio with perfect negative correlation $(\rho=-1)$ is a straight line with slope of $\frac{|\mu_2 - \mu_1|}{\sigma_2+\sigma_1}$ since $\sigma_P=|\omega_1\sigma_1 -\omega_2\sigma_2|$ and $\omega_1+\omega_2=1$. so the equation is $\mu_P=\frac{|\mu_2 - \mu_1|}{\sigma_2+\sigma_1} \sigma_P +b$, I will get two values for $b$ if I plug in the two end points:

$b=\mu_1 - \frac{|\mu_2 - \mu_1|}{\sigma_2+\sigma_1} \sigma_1$

and

$b=\mu_2 - \frac{|\mu_2 - \mu_1|}{\sigma_2+\sigma_1} \sigma_2$

they are equal only if $\mu_1=\mu_2$ see the blue and purple lines in the plot.

Question: which line is the correct one? or neither is correct? is there any problem in the line function or is it okay plugging in both points?

There is no such issue for 2-asset portfolio with perfect positive correlation $(ρ=1)$ in which case the line equation is unique:

$\mu_P=\frac{\mu_2-\mu_1}{\sigma_2-\sigma_1} \sigma_P +\frac{\mu_1\sigma_2 - \mu_2\sigma_1}{\sigma_2-\sigma_1}$ see the red line in the plot.

plot with $\left(0.1,\ 1\right)$ and $\left(0.2,\ 1.5\right)$

Answer 1

The asset returns are \begin{align*} X_1 &= \mu_1 + \sigma_1 \varepsilon \\ X_2 &= \mu_2 - \sigma_2 \varepsilon \end{align*} where $\varepsilon \sim N(0,1)$. The returns of the portfolio are then \begin{align*} X &= w_1 X_1 + w_2 X_2 \\ &= w_1 \mu_1 + w_2 \mu_2 + (w_1 \sigma_1 - w_2 \sigma_2) \varepsilon \end{align*} and the expected returns and volatility are: \begin{align*} \mu &= w_1 \mu_1 + w_2 \mu_2 \\ \sigma &= | w_1 \sigma_1 - w_2 \sigma_2 |. \end{align*}

Because of the absolute value, it is not a line, but a broken line.