Negative Correlation and Portfolio Risk

Article Quant Q&A · Author: Marty B.

Summary

The document explains why a two-asset portfolio’s standard deviation is not automatically zero when the assets have correlation of negative one. Substituting that correlation into the variance formula yields a perfect square: the absolute difference between the two assets’ weighted standard deviations. Risk reaches zero only when the weights make those weighted contributions equal.

The example uses sample asset returns and reports their standard deviations, then gives the weight that produces zero portfolio risk under perfect negative correlation. The result illustrates a special case, not a general portfolio hedge: zero risk depends on the exact correlation and matching weights, and the example does not address estimation error or correlations changing over time.

Key ideas

With correlation of negative one, portfolio variance reduces to a squared difference of weighted standard deviations.
Portfolio standard deviation is the absolute value of that difference.
Zero risk requires weights that equalize the two weighted standard deviations.
The example’s zero-risk result assumes perfect negative correlation and the stated asset standard deviations.

Tags

Portfolio construction Risk management Statistics

Full text

# Correlation -1 and standard deviation


# Correlation -1 and standard deviation












My book says that for a portfolio of two stocks:

$\sigma_p = \sqrt{w_A^2 \sigma_A^2 + (1-w_A)^2 \sigma_B^2 + 2 w_A (1 - w_A) \rho_{AB} \sigma_A \sigma_B}$

Elsewhere it says that if the correlation is -1, then the standard deviation is 0.

However, when I substitute $\rho_{AB}$ with $-1$ clearly $\sigma_p \neq 0$.

What am I missing here?

## Answer by Chris Degnen (score 4)

https://quant.stackexchange.com/a/24734

$\sigma_p=\sqrt{\omega_a^2 \sigma_a^2+(1-\omega_a)^2 \sigma_b^2+2 \omega_a (1-\omega_a) \rho_{ab} \sigma_a \sigma_b}$

with

$\rho_{ab}=-1$

the term under the square root simplifies to

$(\omega_a \sigma_a-(1-\omega_a) \sigma_b)^2$

which is equivalent to $(-\omega_a \sigma_a+(1-\omega_a) \sigma_b)^2$

therefore

$\sigma_p=\omega_a \sigma_a-(1-\omega_a) \sigma_b$

or $\sigma_p=-\omega_a \sigma_a+(1-\omega_a) \sigma_b$

"Each equation is only valid when the right-hand side is positive. Since one is always positive when the other is negative (except when both equations equal zero), there is a unique solution for the risk and return of any combination of securities A and B."

Ref. Modern Portfolio Theory & Investment Analysis, page 72 (Case 2)

Running some test data, with perfect negative correlation the minimum portfolio s.d. is zero.

Test data

```
a = {0.9624, 1.6462, -0.0378, -4.0397, 0.2045}
b = {-3.6569, -4.5494, -2.2938, 3.1099, -2.6359}
```

$\sigma_a=2.21804$

$\sigma_b=2.99359$

$\omega_a1=\frac{\sigma_b+\sigma_p}{\sigma_a+\sigma_b}$

$\omega_a2=\frac{\sigma_b-\sigma_p}{\sigma_a+\sigma_b}$

with $\sigma_p=0$

$\omega_a1=\omega_a2=0.574406$

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.

When Perfect Negative Correlation Can Eliminate Portfolio Risk

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