# Minimizing Correlation to Index

In his PhD thesis in the chapter Market Neutral Portfolios, page 69,  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.

1. 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?

2. 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?

• I can't comment on his method. I use a much simpler approach. The long portfolio L has a certain dollar Beta with respect to the index, and the short portfolio S also has a (usually negative) dollar Beta. With a back of the envelope calculation (no optimization) I adjust the size of L and S so the Betas perfectly offset each other. And voila! a portfolio with no correlation to the index. – noob2 Aug 8 '18 at 15:48
• @noob2 Your approach and the comparison with the approach above is laid out on pages 71-72 of the thesis. – hps Aug 8 '18 at 16:20