Optimizing Portfolio Weights to Minimize Correlation with an Index
Summary
The document presents a market-neutral portfolio optimization objective that minimizes the absolute sample correlation between portfolio log returns and an index’s log returns. Long and short security weights determine portfolio value at each time, from which portfolio log returns are calculated; the objective compares their deviations from their respective means. Conventional holding limits are mentioned as constraints, and the source is identified as a thesis chapter on market-neutral portfolios.
The author notes that logarithmic returns make the objective non-convex and raise concerns that an optimizer may find different local solutions when inputs change. They ask whether simple returns could yield a convex formulation and why correlation constraints are uncommon, or whether another approach is preferable. The document offers no resolution, empirical results, or implementation guidance. It therefore serves as a formulation and a set of open questions rather than evidence that the optimization improves neutrality or portfolio performance.
Key ideas
- The objective minimizes the absolute correlation of portfolio and index log returns.
- Long and short security weights determine portfolio value and the resulting return series.
- The author identifies non-convexity as a challenge for finding stable solutions.
- The document raises, but does not answer, questions about simple returns and alternatives to correlation minimization.
Tags
Full text
# Minimizing Correlation to Index
# Minimizing Correlation to Index
In his PhD thesis in the chapter Market Neutral Portfolios, page 69, [1] Valle sets up an optimization problem which minimizes the absolute correlation of the portfolio log returns to the log returns of a given index.
The decision variables over which the optimization is performed are the portfolio weights $x_i^L$ on the long side and the portfolio weights $x_i^S$ on the short side for the securities $i=1, ..., n$. The price for a security $i$ at time $t$ is $V_{it}$. The overall value of the portfolio at time $t$ is $C_t$, based on which the log return $p_t$ is calculated. The return of the index is $R_t$. The mean of the log returns over the time span $t = 1, ..., T$ is $\overline{p}$ for the portfolio and $\overline{R}$ for the index.
The objective function is $$ \min \left| \frac{\sum_{t=1}^T(p_t-\overline{p})(R_t-\overline{R})}{\sqrt{\sum_{t=1}^T(p_t-\overline{p})^2\sum_{t=1}^T(R_t-\overline{R})^2}} \right| $$ where $$ \begin{align} p_t &= \ln(C_t/C_{t-1}) \\ C_t &= \sum_{i=1}^n x_i^L V_{it} - \sum_{i=1}^n x_i^S V_{it}. \end{align} $$
The absolute value can be removed by lifting the problem, as described in the thesis.
The constraints are conventional limit holding constraints.
- The objective function is clearly non-convex function due to the use of logarithmic returns. The fact that a global solution is unlikely to be found makes it problematic in real-world application, as a slight difference in inputs may converge to a different local optimum. Using simple returns, can the above be formulated as a convex optimization problem?
- When looking for other similar approaches, I was not able to find any. Is there a reason that constraints on return correlations against an index are not common? Is there a better alternative which renders the whole approach above moot?
[1] https://bura.brunel.ac.uk/bitstream/2438/10343/1/FulltextThesis.pdfShown 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.