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

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 $$\left(0.2,\ 1.5\right)$$

• Portfolio return of a two asset basket with perfect negative correlation would be 0. As an exercise, try and prove it.
– Kch
Mar 3 at 4:39
• Clearly not. the return is a function of a weights. Mar 3 at 15:33

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*}
• Thanks for pointing out the effect of the absolute value. However, I think the broken line's orientation should go left from (0.1, 1) to meet at $(0, \frac{\mu_1\sigma_2 + \mu_2\sigma_1}{\sigma_2+\sigma_1})$, $then go right and to (0.2, 1.5) Mar 3 at 15:32 • You are right: combining negatively correlated assets should reduce volatility. I have fixed the plot. – VDZ Mar 4 at 1:52 • Fun to see that, with$\rho=-1\$ the usual parabola has degenerated to a "wedge" shape, instead of rounded it has become pointy. Still vaguely the same shape. But it reaches all the way down to volatility = 0, which the parabola never does. Mar 4 at 2:33