Type:

Master's Thesis

Corporate Partner:

OLZ & Partners Asset and Liability Management AG

Date Published:

March 30, 2017

The aim of this thesis is to identify a systematic advantage of investing in diversified portfolios instead of in concentrated ones. This idea is inspired by the identification, in Baker and Haugen (2012), of the low volatility anomaly—that is to say, the fact that low volatility stocks outperform high volatility stocks.
We consider several definitions of diversification and present the corresponding diversified portfolios. Firstly, as it takes full advantage of the aforementioned low volatility anomaly, we briefly present the minimum variance portfolio. Secondly, we introduce some examples of diversified portfolios in the Markowitz framework, which is the setting in which the modern concept of diversification was first developed. In particular, we focus on the optimal portfolio, the market portfolio, and the EW portfolio. Then, we consider portfolios based on the concept of risk contribution with respect to both assets and uncorrelated factors. To begin with, we choose volatility as a measure of risk. We define risk contributions with respect to both assets and uncorrelated factors. In the case of uncorrelated factors, we choose to focus on principal components and minimum torsion bets. The first set of uncorrelated factors, defined in Meucci (2010), are derived operating a principal component decomposition of the covariance matrix of the asset return. The second, defined in Meucci, Santangelo, and Deguest (2015), is the set of uncorrelated factors optimized to closely track the original assets. We then attempt to extend the idea of risk contribution to more general risk measures, and in particular to expected shortfall. In doing so, we follow the reasoning suggested in Shi (2015) and find our first important result. In fact, despite the strong theoretical background of this generalization, when trying to implement the resulting portfolio in practice we discover that it is not possible to obtain a reliable result. Indeed, the construction of such a portfolio relies on the estimation of the Hessian of the expected shortfall, which is systematically extremely poor and strongly dependent on the assumptions we make with regard to the distribution of the asset return.
We then choose the most interesting diversified portfolios and test them on several markets in order to empirically evaluate them. In particular, we focus on the equally weighted portfolio and the three equal risk contribution portfolios, based on assets, principal components bets, and minimum torsion bets, respectively. As we have already mentioned, we compare these portfolios to the minimum variance portfolio. We test them both in the long run—that is to say, over a time period of 15 years, and in the short run focusing on the financial crisis. In our analysis, we take into particular consideration annualized return, Sharpe ratio, value at risk, expected shortfall, and maximum drawdown as measures of portfolio performance. In fact, we are willing to state that a portfolio outperforms another only if it does so with respect to all these measures. What we find is that two portfolios significantly outperform all the others: the minimum variance portfolio and the equal risk contribution portfolio based on principal components bets. In addition, the second of the two is usually the best portfolio. This finding becomes even more interesting when we focus on diversification measures. In fact, the two best portfolios in terms of performance also appear to be the most concentrated. Consequently, we can conclude that not only is there no clear evidence of a “diversification anomaly” but, in the cases we considered, the most diversified portfolios are also the worst performing. It seems to be more reasonable to select a few stocks really well than to just take everything that the market offers.
Finally, we try to improve the equal risk contribution portfolio based on principal components
bets, using dimensionality reduction. This should both speed up the solution of the related optimization problem and help to get rid of some noise in the estimated covariance matrix. However, when tested, even if the computation time is actually shorter, the performance of the portfolio decreases significantly. Consequently, even if dimensionality reduction usually gives good results with principal components, this is not the case when using them to build an equal risk contribution portfolio.