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?
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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
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$\begingroup$ lets try empirical examples instead (they always tend to be unimodal). How much of the daily correlation between AAPL and NFLX is non-linear? $\endgroup$ Aug 10, 2020 at 12:53
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$\begingroup$ What non-linear relationship are you hypothesizing exists between AAPL and NFLX? $\endgroup$– Attack68 ♦Aug 10, 2020 at 12:56
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$\begingroup$ Im trying to detect non-linearity to disprove that there is a purely linear cross-correlation $\endgroup$ Aug 10, 2020 at 13:05
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1$\begingroup$ @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. $\endgroup$– EvgeniiAug 10, 2020 at 14:23
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$\begingroup$ 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 $\endgroup$ Aug 10, 2020 at 18:23