Analysing assets' performance inside a portfolio: From crossed beta to the net risk premium ratio

This paper is focused on enlarging the performance inside a portfolio that provides the Treynor ratio by relating portfolio weights with performance indicators. Intuition suggests that the higher the weight of an asset, the higher should be its expected performance. These weights, and the information that we can obtain from their analysis, are not only relevant for investors but also for corporate managers. Nevertheless, the available performance indicators are not linked to portfolio weights. In order to fulfil this gap we answer three questions: which is the minimum risk premium that justifies holding an asset in long position? How can we analyse if the performance of an asset justifies the budget's weight invested in it? And, how can we apply ex-post optimisation to performance analysis? Methodologically, we centre the analysis on the definition of crossed beta and the net risk premium ratio that stems from it. The latter fulfils the axioms of risk/reward performance measures. The three answers to the questions are related to the net risk premium. The analysis in developed for the Mean-Variance and Mean-Gini models. The empirical illustration, based on DJIA assets, that completes the paper shows how the analysis of portfolio weights provides relevant information about the performance of assets.