# Double objective in portfolio optimization

Is there anything infeasible or ethically wrong about optimizing portfolios like this?

$$\min_w \enspace w' \Sigma w + w' C w$$

where $$\Sigma$$ is the asset return covariance matrix, and $$C$$ is the asset return correlation matrix. subject to the individual portfolio weights in vector $$w$$ summing to 1.

It says, minimize portfolio variance as well as portfolio correlation. If there is something numerically wrong about this, why aren't dual objectives like this ever used?

• Why would you want that? Also, these terms are not of the same order. – Bob Jansen Aug 28 '20 at 17:40
• i've corrected it to minimize portfolio correlation – develarist Aug 28 '20 at 17:45

There is nothing wrong mathematically (nor ethically) with this objective function. However, this objective is weird in a couple of ways.

First, there is no weighting on these which implies you prefer to minimize these terms in accordance with their orders of magnitude. As has been pointed out, the correlation term is likely much larger so your optimization would be tilted toward minimizing correlation.

Second, from a financial perspective, what you are exposed to (in terms of P&L) is covariance, not correlation. If you are trying to minimize correlation in some downside scenario, you should model that (and use stochastic optimization instead of this deterministic setup).

Are dual-term objectives like you have used? Sure; mean-variance or mean-ES optimizations have similar multi-term objectives. Are dual objectives like "minimize $$f(X,A)$$ subject to maximizing $$g(X|A)~\forall A\in\Omega$$" possible? Sure; those are multi-criteria optimizations where you have staging or conditioning.

What you have is not exactly a multi-criteria optimization. If you rewrote this as "minimize portfolio correlation for a portfolio minimizing portfolio variance," that would be a degenerate solution -- since there is only one minimum-variance portfolio so the correlation objective would irrelevant.

• Hi: $C_{ij}$ is just $\Sigma_{ij}$ scaled by the product of the respective standard deviations ($\sigma_{i} \sigma_{j}$ ) so what that optimization will probably do is put larger weights on the stocks with larger standard deviations which is probably not what you want. I would think of $C$ and $\Sigma$ as basically the same thing and therefore not do what you're doing. – mark leeds Aug 29 '20 at 1:31
• Except the $\sigma_i$s and $\sigma_j$s are $<1$, so the $w^TCw$ term will likely dominate. Either way, I agree that it is probably not what we want. – kurtosis Aug 29 '20 at 5:03
• to be frank, I thought about it some more and I'm not sure what will happen during the optimization but it doesn't seem like a good idea. – mark leeds Aug 30 '20 at 2:07