Abstract: The index-linked trades have become especially prevalent in the asset management industry in the last decades, as investors tend to require benchmarking as a mechanism to evaluate portfolio performance.Index tracking models and Enhanced indexation models are two particularly popular index trading models. Index tracking models aim to reproduce the performance of a stock market index, but without purchasing all of the stocks that make up the index.Enhanced indexation models (EIM) are related to index tracking, in the sense that they also consider a specific indexas a reference. They however aim to outperform the index by generating excess return.The prior EIM are mostly concentrated on overcoming the computational difficulty raised by restricting the cardinality of the portfolios-not on answering the question if they do attain their stated purpose, i.e. obtain returnin excess of the index. In this paper, the enhanced indexation investment strategies that adopt subset of the constituent stocks to construct portfolios to track the benchmark index trend are studied and excess returns relative to index is obtained. As a risk measure, an attractive propertiesof Lower Partial Moment (LPM) is that most other downside risk measures can be specified as special cases of LPM. However, the LPMs has not been used to study the index-linked trade models.Therefore the concept of LPM is extended to suit to the enhanced indexation model by using the index return as target return. In addition, the definition of the Upper Partial Moment (UPM) is given, and then enhanced indexation models are constructed based on the UPM-LPM ratio.In order to solve the complex nonlinear problem and tackle ‘dimensional curse’ induced by large scale portfolio selection, kernel estimation method is used to obtain tracking portfolios' density function directly without considering the high dimensional joint distribution of assets' returns, which can overcome the ‘dimensional curse’ problem. Furthermore, the LPM and UPM kernel estimators, which are smooth functions of the positions and have arbitrary order derivatives, can significantly reduce the complexity of solving these models. Finally, we adopt five common indicesand their constituent stocks from Shanghai and Shenzhen securities markets to test our models' performance. The period from 4 January 2005 to 31 December 2016 is concentrated on, spanning the global and post-global financial crisis. The empirical results show that our models outperform the indices and the tracking portfolios selected by the model with considering target index return is very robust. Our approach has potential applications to better manage pension funds, public funds, private equity funds and fund of funds.

Key words: enhanced indexation, lower partial moment, upper partial moment, kernel estimation