The main reason for selecting portfolio models is to reduce investment risk through diversification. Mean Variance model was introduced for this purpose. However, it was soon realized that errors in estimates of means and co variances can result in wrong selection of portfolio and increase the investment risk. To cater this problem of estimation error which may result in selection of wrong portfolio, several models have been developed the appropriateness of which have been discussed in the research paper An Does greater diversification really improve performance in portfolio selection?

Following six data sets have been used for this analysis: 1. FF25, containing 25 Fama and French portfolios of firms sorted by size and book-to market from July 1963 to December 2004 2. 48Ind, containing 48 industry portfolios representing the U.S. stock market from July 1963 to December 2004 3. 100Ind, containing 100 industry portfolios representing the U.S. stock market from July 1963 to December 2004 4. Ftse100, containing 63 assets of the Ftse100 Market Index from January 2007 to May 2013 5. FtseMib, containing 34 assets of the FtseMib Market Index from January 2007 to May 2013 6. Stoxx50, containing 32 assets of the Eurostoxx50 Market Index from January 2007 to May 2013

Six models that are part of the research to determine the most appropriate portfolio diversification model: 1. Cardinality Constrained Minimum Variance Equally-Weighted Portfolios 2. Cardinality Constrained Minimum Variance portfolios 3. Cardinality Constrained Minimum CVaR portfolios 4. Cardinality Constrained Minimum Semi-MAD portfolios 5. Equally weighted portfolio

Equally weighted portfolio which has been normally recognized as the safest and most appropriate way to develop a portfolio is compared against the performances of portfolio developed with the help of other models.

To summarizing in-sample results, if we take median of standard deviation of all the models we find that EW model has the highest median of standard deviation among all the models and on the contrary all the cardinality range models have lowest median of their standard deviation, meaning thereby that EW model is comparatively more risk than other models. Following graph reflects the results of this finding:

Now coming to how the portfolios have performed we evaluate the out of sample results of the portfolios selected under each model. Since reporting the results for all data sets and for all cardinality is impractical, therefore only two cardinalities have been reported which are 5 and 10. The choice is based on the observation that K = 5, 10 generally belong to the optimal ranges in which the various models achieve the in-sample lowest risk for each data set.

It is remarkable that the EW portfolio has almost always the worst performance, with the single exception of the 100Ind market, where the CCMCVaR portfolios generate the highest standard deviation. The results are shown below in the table:

The last performance measure considered in our analysis is Max drawdown, which is the worst out-of-sample loss achieved by a model. Again, carrying on results of standard deviation as discussed above the EW portfolio always has the worst performance for both of the considered cardinalities (5 and 10), except for the 100Ind market where the CCMCVaR portfolios provide the worst loss.

Although there is not a clear superiority of a single model, it can be observed that the CCMV portfolios present the best values for 3 data sets out of 6 for K = 5 and for 4 data sets out of 6 for K = 10, as reflected in the table above

Sample results negate the idea EW model is the best portfolio selection method, rather CCMV models give the best results as far as sample results are to be evaluated. The research paper goes on to conclude that in most cases limiting the number of allowed assets in a portfolio improves the performance of the portfolios selected by the models. The author names this result as “Small portfolio effect.” However, research paper ends with the disclaimer that further research is needed to determine the validity of this small portfolio effect.