投資組合之均異模型及估計係數模型改良

Abstract

投資組合最佳化希望透過數學計量模型以決定資金在資產投組的配置權重，以期達成最大化報酬及最小化風險之投資目標，此研究在投資領域相當重要。現代投資組合的研究始於Markowitz(1952)所提出之均異準則(mean-variance criteria)及其形成之均異模式(mean-variance model)；此外，相信市場具備效率性之學者，認為投組績效欲優於市場表現並不容易，因此設定追隨市場指數為目標，發展出各種指數投組模式(index portfolio model)，其中最廣為使用之模型為Sharpe(1989)之估計係數模型(estimated coefficient model)。 然而上述兩種模式，基本上屬於非線性數學規劃模式，只有在特定條件下才可求得全域最佳解，更因此無法考慮形成投組過程中不同的主觀與客觀之操作條件限制。 為此，本論文提出以上兩種投組最佳化模式之線性模式。由於所提模式為線性模型，因此可以保證得到全域最佳解，並藉此考慮上述主客觀限制條件於模式中。在客觀條件限制方面，包括了如整股投資限制、交易成本限制及限制投資股數問題等；在主觀條件限制方面，則是投資人追求除了最小化風險及最大化報酬兩大目標外，可能進一步希望考慮更多樣化之投組目標，這些目標之導入將使原數學規劃模式形成一多目標規劃的題型，本研究中亦利用多目標規劃的現有解題模式發展包含四種投組目標之投組最佳化模式。 在投組最佳化模式之非線性限制被解除後，投資組合模式將可更廣泛的加入其它更為多樣化之限制條件，而不僅限於本文模式中所考慮到的主客觀限制。此外，投資組合模式之線性化方法亦可用於解決資源規劃等其它數學規劃問題。 為驗證所提模式之可行性，本論文以臺灣股票市場上市公司為例，進行所提各模式與原有模式之實證比較。

The research of portfolio optimization problems intends establishing mathematical models to decide the optimal capital allocation weights among assets. In 1952, Markowitz proposed the mean variance model to determine the optimization allocation weights of portfolios. However, there are scholars who believe the financial market to be efficient and then trying to outperform the market index is not a easy task. They proposed index portfolio models trying to determine the optimal investment weights to follow the index trend. The most popular index portfolio model is the estimated coefficient model proposed by Sharpe (1989). No matter the mean variance model or the estimated coefficient model, they are both nonlinear mathematical programming models and cannot promise to obtain the global optimal solution. Furthermore, the nonlinear model is difficult to solve when considering more objective and subjective constraints in the original model. In this thesis, two linear portfolio optimization models are proposed to improve the traditional models. And then, objective and subjective constraints are taken in consideration in the developed proposed models. The objective constraints considered in this research contain: (1) minimal investment unit limitation, (2) transaction cost consideration, and (3) limited investment category constraint. The subjective constraints come from more objectives other than original minimizing portfolio risk objective and maximizing portfolio return objective in the optimization process. After these objectives have been included in the proposed model, the programming models become multiple objective programming models, which are provided in this thesis as well. Empirical tests in the Taiwan stock market are provided to verify that the proposed models are superior to the traditional models.