Abstract: Portfolio selection is a practical financial problem. It is mainly concerned with determining a strategy for allocating the wealth among a set of financial assets under an uncertain environment. Markowitz (1952) proposed the Mean-Variance Theory and pioneered the quantitative research methodology in portfolio selection. This approach assumes that investors have the ability to achieve full statistic information of prices, which, however, violates the real world. The strong statistic assumption determines this approach's poor performance in practical application. Following the Kelly investment model (1956), Cover (1991) proposed the on-line portfolio selection model, which has none statistic assumption of price. In this paper, the on-line portfolio selection is followed and the improved on-line strategies are studied.
An on-line portfolio selection problem based on the expected utility maximization and L₁-median estimator is investigated in this paper. Considering the EG strategy estimate next price trend with only one-period price information which is not sufficient, the next price trend with multi-period price information is estimated and the performance of on-line strategy is improved.
Firstly, the Modified Weiszfed algorithm is applied to calculate the L₁-median estimator of multi-period price. Based on the L₁-median estimator, the estimation of future price trend is given by the ratio of L₁-median estimator to close price. Then, via maximizing utility, a novel on-line strategy named EGLM (Exponential Gradient via L₁-Median) is proposed. With portfolio vectors' distance defined by the relative entropy function, it is proved that EGLM is a universal portfolio. Therefore, as a universal portfolio, EGLM could achieve as much approximate return as BCRP. Meanwhile, by algorithm analyzing, it is found that EGLM's time complexity is O(mn)+O(Mn), which is linear with respect to m, number of securities, and is significant less that UP's O(n^m). Thus, the proposed EGLM is suitable for real-world large-scale applications.Finally, based on 6 datasets from domestic and foreign real markets, an experiment is given to illustrate the usefulness and effectiveness of EGLM. The results show that EGLM has not only better performance of return but also better trade-off between return and risk than UP and EG.
With the proposed EGLM strategy and its linear time complexity, investors can gather more precise information about securities' prices and save more calculating time. EGLM has not only the universal character but also good performance in practical performance.

Key words: on-line portfolio selection, universal portfolio, L₁-median estimator, relative entropy function, time complexity