Perfectly Negative Correlation Produces a Broken Portfolio Risk-Return Line
Summary
The document examines the risk-return locus for a two-asset portfolio when the assets have perfect negative correlation. It represents the assets as returns driven by the same standard normal shock with opposite signs. For portfolio weights summing to one, expected return is the weighted average of asset means, while portfolio volatility is the absolute value of the difference between their weighted volatilities.
That absolute value is the key: as weights vary, volatility first falls toward zero and then rises, so the risk-return plot is a broken line rather than one straight line. This resolves the apparent inconsistency in the question, where calculating a single line through both endpoint portfolios can produce different intercepts. The explanation is algebraic and assumes perfect correlation of negative one; it does not discuss estimation error, short-sale or weight constraints, or portfolios with imperfect correlation.
Key ideas
- With perfect negative correlation, portfolio volatility is the absolute difference between the weighted asset volatilities.
- Portfolio expected return remains the weighted average of the two asset means.
- The absolute value creates a kink where the weighted volatilities offset, yielding a broken-line risk-return locus.
- A single straight-line equation through both endpoint portfolios need not describe the full locus.
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# RIsk-retun of 2-asset portfolio with perfect negative correlation
# 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 $\left(0.2,\ 1.5\right)$
## Answer by VDZ (score 4, accepted)
https://quant.stackexchange.com/a/61491
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.