# 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.

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