# Non-linear correlation (co-dependence) and the efficient frontier

The graph below shows how the efficient frontier for 2 assets bends into a sharp bisection as correlation decreases from $$1$$ to $$-1$$, with $$\rho=-1$$ being the most diversified, and highly unattainable since negatively correlated assets are hard to find.

Since correlation $$\rho$$ only measures linear co-dependency between two assets, the graph is ignoring any non-linear dependence between the same assets, meaning that the true efficient frontier could be very different if based on a non-linear measure, rather than linear $$\rho$$. How then does the efficient frontier change if we take the non-linear relationship betwen assets $$A$$ and $$B$$ into account? What would be a good measure for non-linear dependence, and how would the efficient frontier based on that measure look as its value changes?

## 1 Answer

What you show here as an efficient frontier for a two-asset portfolio is presumably the usual return versus risk profile, where the vertical axis represents expected portfolio return $$\mathbb{E}(r_P) := \mu_P$$ and the horizontal axis represents the standard deviation of portfolio return $$\sqrt{var(r_P)} :=\sigma_P$$. These quantities are given analytically in terms of $$w$$, the weight of asset A, and the expected returns $$\mu_A, \mu_B$$, and the standard deviation of returns, $$\sigma_A, \sigma_B$$ for the individual assets, according to

$$\tag{*}\sigma_P = w \mu_A + (1-w) \mu_B,\\ \sigma_P^2 = w^2 \sigma_A^2 + (1-w)^2 \sigma_B^2 +2\mathbb{E}[(r_A-\mu_A)(r_B- r_b)]$$

The third term on the right-hand side contributing to $$\sigma_P^2$$ is the covariance of returns of the individual assets, which by definition of the Pearson correlation coefficient $$\rho$$ is

$$\mathbb{E}[(r_A-\mu_A)(r_B- r_b)] = \rho \sigma_A \sigma_B$$

It is a mathematical fact that $$-1 \leqslant \rho \leqslant 1$$, and (*) determines $$\mu_P$$ and $$\sigma_P$$ as functions of $$w$$ and also of $$\rho$$ (for $$\sigma_P$$ only).

Hence, with $$\mu_A, \mu_B, \sigma_A, \sigma_B$$ fixed, the efficient frontier is the locus of points $$(\sigma_P(w,\rho), \mu_P(w))$$ with the parameter $$\rho$$ fixed as $$w$$ varies between $$0$$ and $$1$$.

Nothing about the joint return distribution for $$r_A$$ and $$r_B$$ other than the means and variances of the marginal distributions and the Pearson correlation was used here. As long as risk is represented by the standard deviation of portfolio return, then the graph is unchanged regardless of the presence or absence of nonlinear dependency.

If you want to see something different, you have to specify (1) a different measure for risk such as $$VaR_\alpha$$, the worst loss that can be expected with a prescribed level of confidence $$\alpha$$, and (2) introduce a specific joint distribution of returns.

For (2) we could use, for example, a joint return distribution with normal marginal distributions and a dependence structure specified by a (Student's) t copula. In addition to a correlation parameter $$\rho$$ there will be a degree-of-freedom parameter $$\nu$$ which introduces tail dependency as it is varied. Here tail dependency means that the correlation of returns deviates from $$\rho$$ when conditioned on extreme returns.

Now your "efficient frontier" would be a surface representing $$VaR_\alpha(w,\rho, \nu)$$ versus $$\mu_P(w)$$, parameterized by both $$\rho$$ and $$\nu$$.

There is no limit to the number of parameters and model complexity you could introduce here, and the determination of the frontier would no doubt require a numerical procedure. It is not clear what would be gained.

• no mention of non-linear codependence in this answer. If the t-copula is parameterized by $\rho$, linear correlation, then it too doesn't address non-linearity does it – develarist Sep 29 '20 at 1:59
• @develarist: Is that a statement or a question? I just said above that the t-copula has an additional parameter, degrees of freedom, that controls tail dependence. Perhaps you are not familiar with that aspect. See here for some details. This answer is not just about the details of a t-copula or any other model of non-linear dependence per se. – RRL Sep 29 '20 at 5:19
• you said t-copula is a function of two parameters: $\rho$ and $\nu$. I definitely was not talking about $\nu$. you still haven't addressed non-linearity. if you say this answer is not about any model of non-linear dependence, then this answer doesn't belong to the original question does it – develarist Sep 29 '20 at 5:21
• What non-linearity are you taking about? Can you describe that in detail? My answer also points out that your question with unlabeled axes is not even clear about what you mean by efficient frontier. As I said if that means a plot of standard deviation versus expected return, then nothing changes regardless of nonlinearity. If you are confused why I labeled the second t-copula parameter $\rho$, then call it something else. – RRL Sep 29 '20 at 5:27
• If you read about how it is constructed in a multivariate setting then there is both a dispersion matrix $\Sigma$ and a DOF parameter. For a bivariate distribution this involves two parameters $\nu$ and whatever you want to call the second parameter. – RRL Sep 29 '20 at 5:27