|Title:||Sparse and dynamic portfolio optimization|
|Subject:||Hong Kong Polytechnic University -- Dissertations|
Portfolio management -- Mathematical models
|Department:||Department of Applied Mathematics|
|Pages:||xviii, 88 pages : illustrations|
|Abstract:||The thesis is concerned with sparse portfolio optimization and dynamic portfolio investment. The sparse portfolios we consider in this thesis are least-0-norm portfolio and least-p-norm with p ϵ (0, 1) portfolio from the solution set of the Markowitz mean-variance (MV) optimization model. They are both NP hard problems. The dynamic portfolio investment model we consider is the projection of the given returns into a constraint comprised of a time-varying expected return in the form of parameterized ordinary differential equation (ODE) involving the Markowitz model. In this thesis, we resort to the stochastic linear complementarity approach, penalty methods to solve the sparse portfolio optimization problems and numerical methods to discretize and solve the parameter identification problem in the dynamic portfolio investment problem. The least-0-norm portfolio we consider is in the framework of the classical Markowitz MV model when multiple solutions exist, among which the sparse solution is stable and cost-efficient. We study a two-phase stochastic linear complementarity approach (two-phase approach) to find a sparse solution. This approach stabilizes the opti-mization problem, finds the sparse asset allocation that saves the transaction cost, and results in the solution set of the Markowitz MV model. We apply the sample average approximation (SAA) method to overcome the randomness in the two-phase approach and give detailed convergence analysis. We implement this methodology on the data sets of Standard and Poor 500 index (S&P 500), real data of Hong Kong and China market stocks (HKCHN) and Fama & French 48 industry sectors (FF48). With mock investment in the training data, we construct portfolios, test them in the out-of-sample data, find their Sharpe ratios and compare with the l1 penalty regularized portfolios, lp penalty regularized portfolios, cardinality constrained portfolios, and 1/N investment strategy. The least-0-norm portfolio is naturally sparser than the Markowitz portfolio. Moreover, we show the advantage of our approach in the risk management by using the criteria of standard deviation (STD), Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR).|
We also consider the least-p-norm solution, p ϵ (0, 1) in the Markowitz model solution set. The sparse portfolio model we study is solved by the penalty method. This model finds the least-p-norm sparse asset allocation in the solution set of the Markowitz MV model, saves the transaction cost and stabilizes the optimization problem. We apply the SAA method to overcome the randomness in the least-p-norm sparse portfolio model and give detailed convergence analysis. We implement this penalty method on the data sets of 20 A&H stocks, Fama & French 12 industry sectors (FF12) and Fama & French 25 portfolios formed on size and book-to-market (FF25). Using portfolios constructed in the training sample, we test them in the out-of-sample data, .nd their Sharpe ratios and compare with the ǁ·ǁ0 sparse portfolio, l1 penalty regularized portfolios, cardinality constrained portfolios, and 1/N investment strategy. Theoretically, least-p-norm portfolio would be sparser than the least-0-norm portfolio, but the least-0-norm portfolio might be more robust due to the simple structure of the two-phase approach. The dynamic investment model that we propose and study is a model with expected return evolution containing unknown parameters. We project the target return to the constraint comprised of the parametric differential equation of the expected return coupled with the Markowitz MV model in every period. We discretize the model by the time-stepping method and use quasi Newton method to identify the parameters. Portfolios are then constructed according to the expected return evolution in multiple investment periods. The portfolio is re-balanced at the end of a round of dynamic portfolio investment incorporating the updated parameters. An empirical example using Dow Jones Industrial Average component stocks and index is given, which demonstrates the model.
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