Scaling and Interpreting Variance and Correlation Objectives
Summary
The document evaluates a portfolio objective that adds portfolio variance and a correlation-based term while requiring weights to sum to one. It explains that the formulation is mathematically permissible, but the two terms have different scales. Without an explicit preference or scaling between them, the larger-magnitude term can dominate the optimization, so the result may not reflect a balanced tradeoff.
It also distinguishes financial exposure from statistical similarity: portfolio profit and loss is driven by covariance, whereas correlation alone does not measure the magnitude of co-movement. If the concern is correlation under downside conditions, the response recommends modeling that scenario with stochastic optimization. Multi-term objectives are common, but minimizing variance first may leave a unique minimum-variance portfolio, making a secondary correlation objective irrelevant. No data or empirical comparison is provided.
Key ideas
- Adding variance and correlation terms is mathematically valid, but their relative scales affect the optimizer.
- Without explicit scaling or preference weights, the larger term may dominate the result.
- Portfolio profit and loss exposure is represented by covariance rather than correlation alone.
- Downside correlation concerns call for modeling adverse scenarios, potentially with stochastic optimization.
- A secondary objective may be irrelevant when the first objective already determines a unique solution.
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Full text
# Double objective in portfolio optimization # 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? ## Answer by kurtosis (score 5, accepted) https://quant.stackexchange.com/a/57681 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.
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