关键词: 多周期, 稀疏投资组合, CVaR, 交易频率, 多块交替方向乘子法

Abstract: With the development of the economy and the increase in the income level of the population, the demand for investment and financial management has been growing. Uncertainty is an essential attribute of financial activities. Return and risk often go hand in hand, and rational investors expect to maximize returns while minimizing risk. Therefore, portfolio selection has been a hot topic of research in the field of finance, where the relationship between risk and return is represented by a quantitative model, which is solved by adding various constraints to determine the specific proportion of each risky asset to be invested. In practice, investment activity is not static and requires consideration of multi-period decision making problems that are flexible and adaptable to market conditions. Technology has improved, financial products are being innovated, the number of risk assets available in the market is vast and the frequency of trading reflects the behaviours of investors’ buying and selling or adding or subtracting positions. In order to reduce management difficulties and transaction costs, it is of great theoretical and practical importance to study the sparse optimization problem of how to select very few assets for investment in the context of high-dimensional data. At the beginning of each new period, the choice is made again in response to new changes in the market.
However, frequent trading behaviour makes it more difficult for investors to manage, and the resulting transaction costs such as commissions and stamp duty are not conducive to achieving optimal returns at the end of the period. A portfolio with a sparse number of investments during the period and a sparse number of trades during the week is more consistent with the optimization objective and the actual situation. Therefore, we add a penalty term to the weight vector in the model to constrain the number of investments and the frequency of transactions during the investment process. A multi-period sparse portfolio optimization model with trading frequency constraints is proposed. The model uses the conditional value at risk (CVaR) as a measure of tail risk, and uses the Fused LASSO method to embody the trading frequency restriction. Applying the classical paradigm to the difference between the single-period asset position vector and the adjacent-period investment vector, we can obtain the optimal solution for different degrees of sparsity by adjusting the size of the canonical parameter. The objective function is set to a minimum CVaR value, while a constraint term on the frequency of trading is added and the investment wealth correlation between the inter-periods is specified. For the non-smooth optimization model with inequality constraints, a multi-block alternating directional multiplier method is applied to solve the problem, decomposing the original problem into multiple sub-problems, which are alternately updated until convergence. The algorithm combines the separability of the dyadic method with the relaxation of the Lagrange multiplier method to reduce computation time.
To verify the validity of the model computational method, the Fama and French database, which is available for download on a public website and consists of 48 industry assets, is first used for the relevant numerical tests. In addition, to better fit the domestic securities market and determine whether the model is practically feasible, 16 representative stocks from each sector are selected from the A-share market, their closing price data are downloaded from Wind Financial Terminal, cleaned and the returns are obtained by the logarithmic method. Through in-sample and out-of-sample empirical analysis, one year is chosen as the interval period in the experiment, and the initial wealth value is set to unit 1. Three indicators reflecting performance are calculated separately: dilution, number of changes in hands and Sharpe ratio. Based on the data, the following conclusions can be drawn: the CVaR-based multi-period sparse portfolio model with trading frequency restrictions proposed in this paper performs better than the equal-weighted model with sufficient risk diversification and the minimum CVaR model. The value-at-risk can be reduced and the sparse solution objective can be achieved under the set minimum end-of-period return condition, while also providing better results in terms of Sharpe ratio. By adjusting the regularisation parameters in the model, personalized investment strategy recommendations can be obtained based on specific investor preferences. The sparse optimization approach can be further improved in subsequent research, where non-convex optimizations such as SCAD and MCP can be substituted for parametric regularization. Alternatively, the idea of robust optimization can be combined with portfolio to find the worst-case optimal solution and eliminate estimation errors.

Key words: multi-period, sparse portfolio, CVaR, trading frequency, multi-block alternating direction method of multiplier