Correlation Limits in Multi-Asset Portfolios

Summary

The document addresses whether portfolio assets should be negatively correlated or mostly uncorrelated. Its central lesson is a mathematical constraint: correlations across many assets cannot all be moderately negative while still forming a valid correlation matrix. The equicorrelation example, with a common pairwise correlation and identity diagonal, illustrates how matrix validity restricts the permissible correlation value as the number of assets changes.

The answer concludes that a portfolio can contain only a limited number of mutually negatively correlated assets, whereas many assets can have zero pairwise correlation. It does not provide a numerical threshold, a portfolio construction procedure, or empirical evidence that one correlation pattern produces better investment outcomes. The discussion therefore clarifies feasibility rather than establishing that zero correlation is optimal; expected returns, individual risks, and portfolio objectives still matter.

Key ideas

• A valid correlation matrix limits how many assets can all be negatively correlated.
• In an equal-correlation model, the allowable common correlation depends on the number of assets.
• Many assets can be pairwise uncorrelated, but mutually negative correlations cannot scale without constraint.
• The document explains a mathematical limitation rather than proving which portfolio structure is best.

Tags

# Should portfolios have zero or negative correlation between assets?


# Should portfolios have zero or negative correlation between assets?












- Is it more optimal to have a portfolio whose assets are negatively correlated? (I am not requiring all assets to be negatively correlated in this case, nor (-1) perfectly negative correlation either. I just mean moderately negative $\rho$ values, with little to no zero or positive correlations)

- and is it more realistic, or smarter than the previous, to construct a portfolio whose majority of assets have a correlation of 0?

Why, and how to reconcile the answers to the above two?

## Answer by Forgottenscience (score 1)

https://quant.stackexchange.com/a/58318

While the close vote might be reasonable, there is mathematical arguments that show there is a limit to how negatively correlated a set of assets can be. It is even a classic quant interview question: Let the correlation matrix be $$\Omega = \rho \mathbf{1} + (1-\rho)I_d,$$

where $\mathbf 1$ is the $d \times d$ matrix with 1's everywhere. What is the range of $\rho \in (-1,1)$ that are valid?

Essentially, you can have only a few number of truly negatively correlated assets, but infinitely many with zero correlation.

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.