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

Question

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?

Answer 1

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.