Abstract: Enhanced index tracking problem is a bi-objective optimization problem of selecting a portfolio outperforming the benchmark index while subjecting to a limited additional risk. It receives extensive attention from both theoretical researches and financial practices. Considering that asymmetric risks coincide more with the investors' perception of risks, an enhanced index tracking model is constructed based on the asymmetric active risks measure in this paper. The asymmetric active risk is gauged by the downside tracking error, and at the same time the short-sale constraint is adopted that is more consistent with the Chinese stock markets. By transform of the constraints on excess returns, our model is equivalent to the non-negative weighted lasso problem; hence, sparse portfolios can be built. Further, the generalized least angle regression (GLARS) algorithm is proposed to solve the model. GLARS algorithm is able to provide the solution of portfolio's coefficients which minimize the downside tracking error when excess returns change within a reasonable range and then the effective frontier can be derived, depicting the trade-off of the portfolio's excess return and downside tracking error.
The empirical analysis is conducted by daily closing prices of Shanghai Stock Exchange (SSE) 50 index and its constituent stocks from Jan.4^th to Dec.30^th in 2016. Comparing our model with the existing enhanced index tracking strategy based on the excess return-tracking error model, the following results are reached. Controlling the number of stocks, the portfolio based on our model can obtain higher unit risk-return. Moreover, the downside median deviation and maximum drawdown of our portfolio' excess return are both lower and the excess return is more right-skewed. Under the requirement of sparseness, the portfolio based on our model is able to obtain a higher accumulated return than the benchmark index in terms of out-of-sample performance, which is of great value for the both institutional and individual investors and enriches the existing research of enhanced index tracking model.

Key words: enhanced index tracking, downside tracking error, efficient frontier