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论文编号：	14645
作者编号：	1120201097
上传时间：	2024/6/5 16:41:16
中文题目：	投资组合有效边界非光滑点的存在性与对CAPM的影响研究：基于参数二次优化方法
英文题目：	Research on the Existence of Kinks on Portfolio Efficient Frontier and Implications on CAPM: Based on Parametric Quadratic Programming Method
指导老师：	齐岳
中文关键字：	投资组合有效边界；片状抛物线结构；参数二次优化；非光滑点；切线不存在比率；资本资产定价模型
英文关键字：	Portfolio efficient frontier; Piecewise-segment parabolic structure; Parametric quadratic programming method; Kinks; Tangency-nonexistence ratio; Capital asset pricing model
中文摘要：	现代投资组合选择理论提出了有效边界的概念，即在给定风险水平下，有效边界是投资者可以获得最大收益的投资组合集合。学者们通常通过计算并连接孤立的点形成有效边界，并且假设过无风险资产rf的一条直线与有效边界相切，相切在资本资产定价模型（Capital Asset Pricing Model，CAPM）中扮演着重要的角色。然而，当有效边界存在非光滑点（kinks）时，相切假设可能不再成立。因此，有必要通过对有效边界非光滑点的系统性求解考察其性质和对CAPM的影响，现有研究对该问题的关注仍不充分。在此背景下，如何基于符合现实约束的投资组合选择模型，利用参数二次优化法对模型优化求解并深入分析以非光滑点为代表的投资组合有效边界片状抛物线结构特征是本文关注的核心问题。 本文首先从约束条件的视角对传统的均值-方差模型进行扩展，构建了符合现实需求的投资组合选择模型，并利用参数二次优化法求解了完整的有效边界而非孤立的点。然后，以不同股票规模和不同约束条件下的完整有效边界为研究对象，探索其片状抛物线结构特征对投资组合选择的影响。进一步，基于历史数据求解的有效边界片状抛物线结构特征证明了非光滑点的存在性，通过定理和推论的形式论证了有效边界非光滑点的相关性质，并提出切线不存在比率的概念和计算方式。最后，以市场投资组合的确定为切入点阐述了非光滑点的存在性对CAPM产生影响的具体表现，并通过定理的形式提出了最大夏普比率投资组合的性质，利用有效边界结构特征给出了求解最大夏普比率投资组合的具体算例。本文的研究结论主要如下： 第一，已有关于投资组合模型求解的研究更多是对有效边界的近似描述而非精确求解，导致有效边界难以反映完整的求解结果以及有效边界上关键点的内在联系，特别是无法实现非光滑点的系统求解。本文基于参数二次优化法实现了不同股票规模和不同约束条件下投资组合有效边界的完整求解，投资者可以根据自身的风险偏好在完整的有效边界上选择投资组合并求解非光滑点。 第二，抛物线段的平均数量、拐点投资组合的分布频率、拐点投资组合的多样性以及非光滑点的存在比率等指标较为全面地刻画了有效边界的片状抛物线结构特征。约束条件的变动会改变有效边界的片状抛物线结构特征，相较于股息率约束条件，权重上界约束条件的变动对有效边界的片状抛物线结构产生的影响更明显和突出。投资者可以根据不同约束条件下的有效边界片状抛物线结构特征选择符合自身投资偏好的投资组合。 第三，非光滑点是有效边界上左右导数不相等的点，并且只能通过参数二次优化法实现求解。随着投资组合选择的股票规模从5增加到2000只，投资者可以发现非光滑点存在于不同股票规模和不同约束条件的有效边界上。由于非光滑点的存在，当无风险收益率落入某些区间时资本配置线与投资组合有效边界无法相切。投资者在利用有效边界进行资产配置时要考虑非光滑点的存在可能对投资决策产生的影响。 第四，非光滑点的存在性对CAPM产生影响的核心是使理论上的市场投资组合（切点投资组合）无法确定。CAPM假设市场投资组合是资本市场线与有效边界相切时的切点，非光滑点导致这种关系存在不成立的可能，可能性的大小通过切线不存在比率进行度量。由此，投资者在使用CAPM进行投资实践时要充分考虑与有效边界有关的信息，避免由于信息的不足导致决策偏差。 本文的研究创新主要体现在以下三方面： 首先，本文首次利用参数二次优化法对以A股为样本的不同股票规模和不同约束条件的有效边界非光滑点进行系统性求解，证明了非光滑点存在于不同股票规模和不同约束条件的有效边界上，并非已有研究指出的“非光滑点是稀有的”，有助于投资者进一步理解投资组合选择理论在资产配置中的应用。 其次，关注到非光滑点的存在性可能导致资本配置线与有效边界无法相切，进而使切点投资组合无法确定，本文基于对定理和推论的证明创新性地提出切线不存在比率对资本配置线与有效边界无法相切的可能性进行量化。基于对非光滑点和切线不存在比率的求解，通过数理推导的形式证明了期望收益率坐标轴由分段连接的切线存在区间和切线不存在区间构成，当无风险收益率落入某些区间时资本配置线与有效边界无法相切。 最后，本文以市场投资组合和最大夏普比率投资组合的确定为切入点，阐述了有效边界非光滑点的存在性及结构特征在资产定价领域的应用场景，强调了有效边界结构特征所包含信息的重要性，丰富了投资组合选择理论的相关研究。
英文摘要：	Modern portfolio selection theory introduces the concept of the efficient frontier, which represents the set of investment portfolios that offer the maximum returns for a given level of risk. Researchers traditionaly construct the efficient frontier by computing and connecting isolated points, assuming that a straight line passes through the risk-free asset rf and is tangent to the efficient frontier. The tangency plays a crucial role in the Capital Asset Pricing Model (CAPM). However, when the efficient frontier contains kinks, the tangency assumption may not hold. Therefore, it is necessary to systematically analyze the properties of kinks on the efficient frontier and their implications for CAPM through systematic solving. Existing research has not adequately addressed this issue. The core focus of this dissertation lies in investigating how to develop portfolio selection models based on realistic constraints, optimize the models using parametric quadratic programming methods, and thoroughly analyze piecewise-segment parabolic structure features of the efficient frontier of portfolios represented by kinks. The dissertation extends the traditional mean-variance model from the perspective of constraints, constructing a portfolio selection model that meets real-world demands. The efficient frontier is completely solved instead of isolated points by parametric quadratic programming. Then, this dissertation explores the impact of the parabolic structure of the complete efficient frontier under different stock scales and constraints on portfolio selection. Based on historical data, the parabolic structure of the efficient frontier validates the existence of kinks. The relevant properties of kinks on the efficient frontier are demonstrated, and the concept and calculation method of the tangency-nonexistence ratio of are proposed. Lastly, the dissertation elucidates the specific manifestations of the influence of kinks on CAPM, utilizing the features of the efficient frontier structure to provide specific examples of solving the maximum Sharpe ratio portfolio. The main research conclusions of this dissertation are summarized as follows: Firstly, existing research on portfolio optimization has predominantly focused on approximating the efficient frontier rather than precise solutions. Consequently, the efficient frontier often fails to reflect complete solution results and the intrinsic connections of key points on the efficient frontier, particularly the inability to achieve systematic solving of kinks.This dissertation, utilizing parametric quadratic programming, achieves a complete solution for the efficient frontier under different stock scales and various constraints. Investors can thus select investment portfolios based on their risk preferences and scrutinize kinks from the comlete efficient frontier. Secondly, metrics such as the average number of parabolic segments, the frequency distribution of corner portfolios, the diversity of corner portfolios, and the ratio of kinks existence comprehensively characterize the parabolic structure features of the efficient frontier. Changes in constraints alter the piecewise-segment parabolic structure features of the efficient frontier. Compared to constraints of dividend yield, variations in upper-bound constraints have a more noticeable and pronounced impact on the piecewise-segment parabolic structure features of the efficient frontier. Investors can select portfolios that align with their investment preferences based on the piecewise-segment parabolic structure features of the efficient frontier. Thirdly, kinks are points on the efficient frontier where the left and right derivatives are not equal, and can only be scrutinized by parametric quadratic programming. As the number of stocks for portfolio construction increases from 5 to 2000, investors may observe the existence of kinks on the efficient frontier under different stock scales and various constraints. The presence of kinks may lead to inconsistencies between the efficient frontier and the assumptions of the CAPM. Due to the existence of kinks, the capital allocation line may not be tangent to the efficient frontier when the risk-free rate falls within certain intervals. When utilizing the efficient frontier for asset allocation, investors should consider the potential impact of the existence of kinks on investment decisions. Fourthly, the core of the implication of the existence of kinks for CAPM is that it makes the theoretical market portfolio (tangent portfolio) uncertain. The CAPM assumes that the market portfolio is the point of tangency between the capital allocation line and the efficient frontier. However, kinks may render this relationship invalid, and the magnitude of this possibility is characterized by the non-tangency ratio. Therefore, when investors use the CAPM in investment practice, they should fully consider the information related to the efficient frontier to avoid decision biases caused by insufficient information. The research innovation of this dissertation is mainly reflected in the following three aspects: Firstly, this dissertation systematically solves the kinks on the efficient frontier using parametric quadratic programming method for different stock scales and various constraint conditions, utilizing A-shares as samples. It demonstrates that kinks exist on the efficient frontier under different stock scales and constraint conditions, contrary to the notion suggested by previous research that "kinks are rare." This finding contributes to a deeper understanding of the application of portfolio theory in asset allocation for investors. Secondly, recognizing that the existence of kinks may prevent the capital allocation line from being tangent to the efficient frontier, this dissertation innovatively introduces the concept of the tangency-nonexistence ratio to quantify the possibility of the capital allocation line failing to be tangent to the efficient frontier. Based on the solution for kinks and the tangency-nonexistence ratio, this dissertation formally proves, through theorems and corollaries, that the expected return axis consist of tangent intervals and non-tangent intervals. When the risk-free rate falls within certain intervals, the capital allocation line cannot be tangent to the efficient frontier. Finally, this dissertation innovatively takes the determination of the market portfolio and the maximum Sharpe ratio investment portfolio as the starting point, and elaborates on the application scenarios of kinks and piecewise-segment parabolic structure features of the efficient frontier in the field of asset pricing. It emphasizes the importance of information regarding the structural features of the efficient frontier, enriching the relevant research on portfolio selection theory.
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