# Is non-linear correlation an issue in portfolio optimization?

Portfolio weights are linear combinations of assets. How can it be true then for there to be, and how can someone prove that there is any, non-linear correlation issues in portfolio optimization? Is the normality assumption of Markowitz all there is to blame for non-linear portfolio correlation/co-dependencies if they do exist?

Try and think about linear and non-linear correlations in terms of joint probability density functions. What does it mean for two assets to be linearly or non-linearly correlated?

Suppose we hypothesize a non-linear relationship between two (asset returns) variables as the following:

$$Y = \pm \sqrt{4 - X^2} + \epsilon , \quad \epsilon \sim \mathcal{N}(0, \sigma^2)$$

Now suppose that $$X \sim \mathcal{N}(0, 1)$$. Then naturally $$Y$$ is derived from this and we can plot the joint distribution and the marginal distributions:

Notice that $$X$$ has the traditional Guassian distribution that we expect, but $$Y$$ is forced by the non-linear relationship to have a distinct bi-modal distribution.

If $$X$$ and $$Y$$ were two asset returns it would be easy to disprove the hypothesized relationship by considering the empirical marginalised distribution and discovering that $$Y$$ does not at all have a bi-modal distribution whilst $$X$$ is Guassian. Therefore, the specific non-linear relationship cannot exist.

Asserting a non-linear relationship means considering how the joint distribution and the marginalized distributions can co-exists whilst satisfying the empirical data.

Wikipedia gives a nice view as well on linear correlation and how joint pdfs can look for different values. https://en.wikipedia.org/wiki/Correlation_and_dependence#/media/File:Correlation_examples2.svg

• lets try empirical examples instead (they always tend to be unimodal). How much of the daily correlation between AAPL and NFLX is non-linear? – develarist Aug 10 '20 at 12:53
• What non-linear relationship are you hypothesizing exists between AAPL and NFLX? – Attack68 Aug 10 '20 at 12:56
• Im trying to detect non-linearity to disprove that there is a purely linear cross-correlation – develarist Aug 10 '20 at 13:05
• @develarist what do you mean by how much of correlation is non-linear? Correlation is a linear dependence measure between assets, it doesn't capture non-linearity. But your correlation might be spurious due to the presence of non-linear dependence. – Evgenii Aug 10 '20 at 14:23
• Your right, i meant to say how much of the dependency or interaction between AAPL and NFLX returns are due to linear correlation and how much due to non-linear dependence/comovement? Given that it likely isnt all purely due to linear correlation, and that there must be some nonlinear dependence occuring as well – develarist Aug 10 '20 at 18:23