# Cluster Random Subspace- A Process Diagram

In the last post, I introduced a method for improving portfolio optimization called the Cluster Random Subspace Method (CRSM). The paper was written by Michael Guan for his thesis in computer science, and in case you missed it a link to the paper can be found here: CRSM Thesis. CRSM has demonstrated considerable advantages versus using conventional Markowitz tangency/max-sharpe (MSR) portfolios especially on either large-scale or homogeneous universes. This can be expected because CRSM is designed to reduce variance by aggregation or use of a statistical ensemble approach. Conventional MSR suffers from the “curse of dimensionality” in these situations and tends to serve more as an “error-maximizer” rather than produce effective portfolio allocations. Since MSR is used within CRSM to maximize the portfolio sharpe ratio in this paper, it can be directly compared to using a standard MSR approach along with using RSM/RSO– the original random subspace method which also uses MSR, and just an equal weight portfolio. CRSM-R is just a variant of CRSM that uses replacement in the sampling process. I took this chart from the paper which aggregates the results across six different universes using each algorithm type. For a more in depth breadown, I would suggest reading the original paper:

In terms of the objective function- maximizing the sharpe ratio- CRSM is by far the best algorithm and vastly superior to MSR across the universes tested in the paper. While I don’t want to spoil the surprise, this does not come at the cost of a reduced CAGR- as is typically the case with say Michaud resampling etc. In fact CRSM also has the highest CAGR- beating out equal weight, which is impressive especially for homoegeneous universes. To get a better sense of how CRSM works, it is useful to look at the process diagram below:

read the paper and equal weight was the best when dealing with homogenous population SP-100. how would you optimize for a homogenous population.. weighting for instance SP, NDX and RUT.. that would consistently beat an equal weighting benchmark? I imagine the approach would be somewhat different in this regard.

hi marcus, one thing that is important to note is that the CRSM and RSM results versus equal weight will change with each run since the process is stochastic. typically i have seen superior results or at the very least results that are equal to equal weight. the key point is that the algorithm is adaptive to the universe– MSR does very poorly versus equal weight in some universes -especially homogeneous ones, while it tends to exceed in heterogeneous universes. in contrast CRSM matches or exceeds (or comes close to equal weight) in the one case and outperforms in the other case and also in mixed universes. the point is, we can’t know in advance when equal weight will outperform, so given an abitrary universe choice we are probably better off with CRSM.

As for your good question, i think that there are other approaches that are superior to CRSM for either type of universe. the correlation matrix is the biggest issue in my opinion, and the key reason why simple ranking often outperforms. addressing this issue is the key to solving the problem in my opinion—at least in an optimization framework.

best

david

I’d call this stratified random sampling (http://en.wikipedia.org/wiki/Stratified_sampling) where the clusters are the subpopulations. Using that argument the strategy looks pretty good to me.