| 摘要: | 優化在當今世界無處不在。特別是，金融領域觀察到了控制和優化之間的明確聯繫，這些聯繫受投資動態框架的制約。更深入地說，二次規劃（quadratic programming, QP，優化的基礎）和黎卡迪方程（Riccati equation，最優控制的基礎）之間的關係可以通過封閉形式的方程來解釋，據作者所知，這種解釋僅出現在金融系統和控制的應用中。本論文使用了基於明確最佳化之求解方法應用於計算資本資產定價模型(Capital asset pricing model, CAPM)。參考之模型為零變異數CAPM，其透過零變異數投資組合解放對於無風險利率的假設。相較於主流文獻採用數值方法來完成，本文採用之等式限制二次規劃的明確解，此方法接近零變異數CAPM的設計精神，從而讓我們從效率前緣中選取最佳市場投資組合。除此之外，此方法對應於半正定的共變異數矩陣的解析解，這也符合馬可維茲對於投資多元化之設想。我們的研究成果有助於拓展金融經濟學的經典問題。最後，根據實證驗證，本研究結果初步的展示了對於金融市場實際情況的適用性，更進一步提高了人們對此方法的興趣。;Optimization is ubiquitous in the world today. In particular, the field of finance observes the explicit connection between control and optimization, which is subject to the framework of investment dynamics. To be more in-depth, the relation between quadratic programming (QP, a basis in optimization) and Riccati equation (a basis in optimal control) can be interpreted in terms of closed-form equations, which, to the best of author′s understanding, only appears in the application to financial systems and control. This paper employs an optimization-based method grounded in precision to apply to the Capital Asset Pricing Model (CAPM) in an application-oriented manner. The model referenced in this study is the Zero-Variance CAPM, which, through zero-variance portfolios, liberates itself from the assumption of a risk-free rate. In contrast to prevalent literature utilizing numerical methods, this paper adopts an equality-constrained QP approach. This method allows us to select market portfolios from the efficient frontier, bringing us closer to the theoretical thought process initially envisaged. Moreover, the methodology used corresponds to an analytical solution for the semi-positive definite covariance matrix, aligning with Markowitz′s conception of investment diversification. Consequently, the results may contribute to extending classical problems in financial economics. Finally, empirical validation suggests preliminary applicability to real-world financial market conditions, further augmenting interest in this approach. |
