Selecting Futures for Diversification: Minimum Variance and Log-Determinant Search

# How can I select the least correlated portfolio of assets?


# How can I select the least correlated portfolio of assets?












Can anyone explain the process and the calculations needed to select a portfolio of liquid futures assets with the least correlation? Given a set of returns for a series of assets, how do I select the best subset such that I minimize their correlation with each other?

## Answer by Patrick Burns (score 5, accepted)

https://quant.stackexchange.com/a/528

Since you are asking for low correlation of the assets, I'm guessing that you are really trying to get a low (or minimum) volatility portfolio. If that is the case, then the steps for one approach are:

- estimate the variance matrix of the universe of assets

- use a portfolio optimizer to select the minimum variance portfolio given your constraints

This assumes that you don't have preferences in terms of expected returns of some assets over others. That seems to be implied from your question.

You don't indicate the size of your universe. If it is large, then you'll want to use a factor model or shrinkage model rather than the sample estimate to estimate the variance matrix.

## Answer by ZAxisMapping (score 4)

https://quant.stackexchange.com/a/1720

The question is somewhat vague (lacking a well-defined objective), so this advice may not apply.

Be mindful that you may be simultaneously considering multiple futures contracts that contain overlapping underlying constituents (e.g. futures that track the EuroStoxx 600 and DAX). If you are using a risk model, the idiosyncratic risk may not, in fact, be uncorrelated across constituents. This phenomenon is true for other 'composite' assets, such as ETFs, as well.

Dan diBartolomeo, of Northfield, motivates this concept clearly in his 1998 paper:

"Optimization with Composite Assets Using Implied Covariance Matrices"

## Answer by user410 (score 3)

https://quant.stackexchange.com/a/523

I would start by looking at calculating the efficient frontier. which maximizes return given a specified risk.

Here is the Wikipedia article which specifies how you would do the calculation:

http://en.wikipedia.org/wiki/Modern_portfolio_theory

-Ralph Winters

## Answer by Ram Ahluwalia (score 3)

https://quant.stackexchange.com/a/1739

Mean-Variance optimization is the standard finance answer to this question.

However, solutions can be costly since the weights will likely be dispersed across many instruments raising fixed transactions costs. I would consider Sparse PCA as another solution where you can specify cardinality constraints on the number of securities in your basket to better manage the transasctions costs vs. diversification trade-off

## Answer by FJ Hsu (score 1)

https://quant.stackexchange.com/a/85519

I know this is an old thread, but I came across the same problem recently, and would like to add my take on it.

If your goal is to choose a subset of futures contracts whose return streams are as independent as possible, I would not formulate it first as a minimum-variance weighting problem.

That is a related problem, but not the same one. Minimum variance asks:

> given a fixed universe, what weights minimize portfolio variance?

Your question sounds more like:

> from a large universe, which k contracts should I choose so that the selected return streams are as uncorrelated / non-redundant as possible?

For that, a clean formulation is:

- compute a return matrix $R \in \mathbb{R}^{T \times N}$, where columns are assets

- estimate the correlation matrix $C$

- choose a subset $S$ of size $k$

- maximize some measure of “joint independence” of the selected assets

A useful objective is to maximize the determinant of the selected correlation submatrix:

$\max_{S: |S|=k} \det(C_S)$

where $C_S$ is the correlation matrix of the chosen assets.

Why determinant?

- if two selected assets are highly redundant, the determinant falls

- if the selected assets span distinct directions in return space, the determinant rises

- it captures joint diversification, not just pairwise low correlations

For numerical stability, in practice I maximize:

$\log \det(C_S)$

instead.

The hard part is that this is a combinatorial search problem. If you have N contracts and want k of them, the number of possible subsets is:

$\binom{N}{k}$

which becomes very large very quickly. So for realistic universes, brute force is usually only feasible as a benchmark on small cases.

In practice, this means we need a search method that can explore the space of subsets efficiently.

A simple approach is greedy selection: start with the best pair of assets, then repeatedly add the asset that increases the log-determinant the most. This is very fast but can get stuck in locally good combinations that are not globally optimal.

I've found that a method that works well in practice is beam search. Instead of keeping only one candidate portfolio at each step, beam search keeps the best B partial portfolios. Each step expands these candidates by adding one new asset, evaluates the resulting log-determinants, and keeps the top B again. This allows the search to explore multiple promising directions while keeping the computational cost manageable.

In my experiments, beam search provides a good tradeoff between runtime and solution quality: it is dramatically faster than brute force but typically finds portfolios that are very close to optimal.

I ended up packaging this idea into a small Python library called Diversifind, which searches for maximally diversified subsets from a correlation matrix. The basic usage looks like this:

```
from diversifind import beam

results = beam(corr, symbols, k=8, beam_width=500)
print(results.pretty())
```

The library implements greedy, beam search, and brute-force methods for benchmarking, along with some diagnostics (eigenvalue spectrum, effective rank, pairwise correlations) to help evaluate the diversification of the resulting portfolio.

So if the goal is to select a subset of futures with the least redundancy in their return streams, the problem can be framed as a subset selection problem on the correlation matrix, where maximizing the log-determinant of the selected submatrix encourages jointly independent assets. In practice the main challenge is computational, since the number of possible subsets grows combinatorially, which is why heuristic searches such as beam search can be useful for exploring the space efficiently.

Perhaps some of you may find this alternative way of approaching the problem useful.