Assessing Nonlinear Dependence Through Joint Return Distributions
Summary
The note addresses how nonlinear dependence between asset returns can matter for portfolio analysis even though portfolio weights combine returns linearly. It recommends framing the question through the joint probability distribution and checking whether a proposed dependence pattern is compatible with the observed marginal distributions.
As an illustration, it proposes a curved relationship between two variables, adds Gaussian noise, and assumes one variable is normally distributed. The resulting second variable would have a distinct bimodal marginal distribution. If empirical data do not show that pattern, this specific relationship is contradicted. The example illustrates a way to evaluate a hypothesis; it does not establish that nonlinear dependence is absent in other forms, nor does it fully assess how such dependence affects portfolio optimization. The note also does not isolate normality as the sole source of dependence concerns.
Key ideas
- Linear portfolio weights do not imply that asset returns have only linear dependence.
- Dependence should be examined through the joint distribution and its marginal distributions.
- A proposed nonlinear relationship implies testable features in the marginal distributions.
- Rejecting one incompatible relationship does not rule out other forms of nonlinear dependence.
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Full text
# Is non-linear correlation an issue in portfolio optimization?
# 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?
## Answer by Attack68 (score 2)
https://quant.stackexchange.com/a/57239
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.svgShown 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.