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论文编号：	15981
作者编号：	1120221294
上传时间：	2026/6/1 17:56:51
中文题目：	投资组合约束条件的设置对有效边界的影响研究：基于参数二次规划方法和有效边界的长度
英文题目：	Impacts of Portfolio Constraints on the Efficient Frontier: Based on Parametric Quadratic Programming and Efficient Frontier Length
指导老师：	齐岳
中文关键字：	投资组合选择模型；参数二次规划；约束条件；投资组合有效边界
英文关键字：	Portfolio selection model; Parametric quadratic programming; Constraint conditions; Portfolio efficient frontier
中文摘要：	随着金融市场的发展、投资数额和投资者数量的增加，如何科学引导投资，在权衡投资组合的风险和收益的同时满足投资者的需求，成为重要问题。Markowitz的投资组合选择理论作为现代金融学重要理论之一，是个人和机构投资者应用的重要投资工具。而有效边界是这一理论风险-收益权衡原则的重要体现，在股票配置和风险管理的过程中具有重要意义。 在理论上，传统的投资组合选择模型存在偏重风险收益权衡而忽略投资者现实需求等问题，学者也指出模型在约束条件方面存在不足；在实际投资中，公募基金会加入反映投资者资产特征偏好的限制，部分现实约束也有强制性，投资者有设置约束的需要。因此有必要在组合中加入更多贴合现实、符合投资者需求的约束。约束条件能够促进分散化、提升组合的应用性等，但也可能降低模型的有效性等。因此，对投资组合施加多少约束条件会带来有效边界怎样的变化，如何在满足投资者需求的同时寻求最优配置结果，是值得探究的问题。 本文利用参数二次规划算法精确、完整地求解有效边界的片状抛物线结构，从而创新性地提出反映有效组合股票配置范围的有效边界长度和长度比率，结合多样化比率、有效比率、有效边界抛物线段平均数量等，定量、具体地反映有效边界在选择范围、多样性、有效性和复杂性等方面的结构变化。通过梳理文献和基金的投资策略，筛选出可量化的线性等式或不等式约束条件，包括预算约束、禁止卖空、权重上下界约束，以及涵盖盈利能力、成长能力、市场表现、资本结构、营运能力、流动性等方面的约束。本文首先分析了每个约束的加入对不同规模组合有效边界结构的影响；然后探究同时变动股票数量和约束条件数量时，有效边界长度和其他结构特征的变化，以及不同类型、维度的约束起到的作用；进一步探究了变动股票数量和约束条件取值时有效边界的变化。本文的主要结论如下： 第一，本文利用参数二次规划完整求解有效边界的优势，精确计算有效边界的结构特征变化，衡量加入多种约束对有效边界的影响。已有关于投资组合有效边界求解的研究更多为近似或不完整的，而本文计算不同股票规模和不同约束条件下的有效边界长度和长度比率，以及其他基于精确完整的有效边界计算得出的结构特征，定量描述风险-收益权衡的结果，弥补了由于难以精确计算有效边界而难以准确反映其结构变化的研究不足。研究发现约束条件的加入能明显改变投资组合有效边界的结构特征：有效边界的长度明显缩短，即约束的引入对有效组合的证券配置范围产生了明显影响，投资决策的可选范围被缩小，但也降低了产生极端组合的可能性。约束的加入也明显提升了投资组合的多样化程度，增强了分散投资的优势，提升了投资的稳健性。但是，组合的有效性降低，有效边界的结构复杂性提高，对投资者的投资决策水平提出了更高要求。 第二，在加入不同约束时，随着股票数量增加，有效边界的各项结构特征的变化趋势总体一致：有效边界长度呈整体递增趋势，有效组合可选范围增加；多样化比率趋于下降，组合的多样化程度降低；有效比率趋于下降，组合有效性提升，相较于简单多样化组合的优越性得到提升；有效边界的抛物线段平均数量增加，有效边界结构复杂性上升。因此，对于常持有较大规模组合的公募基金而言，在股票配置范围更加广泛的同时，多样化难度提升，对投资者的专业素养要求也随之提高。在较大规模下，简单多样化的成本更高，投资组合选择模型有效性更明显，是良好的投资决策工具。 第三，随着约束数量的增加，有效边界结构特征指标的变动幅度总体上趋于降低，新增约束的影响相对减弱，有效边界长度趋于稳定，多样性的增加、有效性的减弱和结构复杂性的增加也并非无限的。从约束类型的角度看，权重上下界约束发挥着主导性作用，基于公司特征的约束也能够使有效边界的结构特征发生变化，但这种影响相对有限。在公司特征约束中，成长能力和盈利能力维度的约束对有效边界结构特征的影响相对较大。这说明投资者需要合理考虑对所加入约束条件的筛选。权重上界约束对有效边界的压缩、多样化的提升作用最为明显，若需降低风险应优先设置权重上限约束。公司特征约束可以引入基本面信息，其中可优先关注成长与盈利能力维度。 第四，在变动约束条件取值的情况下，仍能得到随着投资组合规模的扩大，总体上有效边界的长度提升，边界结构的复杂性上升，有效性有所提升，维持多样化的难度加大的结论。在权重上界、基于公司特征的上下限约束取值更加严格时，有效边界的结构变化总体更明显，其中权重上界的取值起主要作用，而基于公司特征的约束主要起到辅助调节作用。因此不同规模组合可采取约束强度差异化的策略，小规模组合应避免过严约束，从而避免对优化空间的过度限制，组合规模扩大时可通过收紧权重上界主动控制股票选择范围和分散程度。 本文的研究创新主要体现在以下方面： 第一，本文基于精确、完整求解有效边界的参数二次规划方法，创新地提出和计算有效边界的长度与长度比率，结合多样化比率、有效比率、有效边界抛物线段平均数量等，具体衡量加入约束对组合证券选择范围、多样性、有效性和复杂性等的影响。有效边界长度反映有效组合的选择范围，能直观反映有效边界的变化，从而体现风险与收益的权衡变化。现有对约束如何影响有效边界的研究多为定性的、位置变化的描述，少有研究从有效边界长度的角度，定量表明其具体变化了多少，无法准确体现约束对有效组合证券选择范围的影响，也未有充分探究有效边界其他具体结构特征变化。本文给出了精确求解有效边界并具体衡量其变化的方法，基于投资者的有效边界的片状抛物线结构，提出准确、量化地反映有效边界选择范围的完整的有效边界长度，结合代表组合多样化、有效程度、边界结构复杂性的指标，得出加入约束条件对有效边界的影响，具备一定的创新性。本文能够深化对投资组合有效边界的理论研究，弥补了对有效边界具体结构特征研究的不足，同时能够为投资者利用更清晰完整的有效边界及其结构特征进行投资实践提供思路。 第二，本文创新地对投资组合中的约束数量和股票数量进行同时的变动，探究有效边界的结构变化。现有将约束引入投资组合的研究多为讨论模型在计算方面的改进，或证明解的存在性。而在分析约束条件对有效边界影响的研究中，多采用单类型或者少数几个约束条件，多针对中小规模组合，且通常组合规模固定。鲜有研究从多个现实约束条件的角度，在变动的约束条件和变动的组合规模下探究有效边界的变化。而相关文献和基金实践都表明，投资者有在组合中设置多种多样的约束条件的需求。有效边界直接关系着投资者有效的股票配置方案和最优投资决策，本文选取研究文献和投资实践中较为常见的约束，加入投资组合选择模型并求解，发现约束数量、类型、取值等的变化能够带来有效边界的结构变化，能够弥补投资组合选择模型由于约束过于简单而影响现实适用性的不足，扩展关于约束条件对投资组合有效边界的理论影响的研究，并为不同规模的投资者提供设置约束条件的思路。
英文摘要：	With the development of financial markets, the increase in investment amounts and the number of investors, how to scientifically guide investments, balance the risks and returns of portfolios while meeting the needs of investors has become an important issue. The portfolio selection model proposed by Markowitz, as one of the most important theories in modern finance, has become a common tool for individual and institutional investors. The efficient frontier, as an important manifestation of the risk-return trade-off principle in this theory, is of great significance in the process of investment and risk management. Theoretically, traditional portfolio selection models have problems such as emphasizing the trade-off between risk and return while ignoring investors' needs. Scholars have also pointed out deficiencies in the models' constraints. Practically, public funds add restrictions that reflect investors' preferences for asset characteristics, and some constraints are also mandatory, so investors need to set constraints. Therefore, it is necessary to add more constraints in line with reality and meet investors' needs. Constraints can promote diversification and enhance the applicability of the portfolio, but they may also reduce the effectiveness of the model. Therefore, how much the application of constraints will change the efficient frontier and how to seek the optimal result while meeting the needs of investors are issues worth exploring. This paper uses parameter quadratic programming that can accurately and completely calculate the efficient frontier to solve and analyze the piecewise parabolic segment structure of the efficient frontier. It innovatively proposes the length and length ratio of the efficient frontier that reflect the stock allocation range of the efficient portfolio. Combined with diversification ratio, efficiency ratio, average number of parabolic segments of the efficient frontier, it quantitatively and specifically reflects the structural changes of the efficient frontier in terms of selection range, diversity, effectiveness, and complexity. By reviewing the literature and investment strategies of funds, this paper selects quantifiable linear equality or inequality constraints, including budget constraints, short-selling prohibitions, upper and lower bound constraints, as well as constraints based on company characteristics covering profitability, growth, market performance, capital structure, operational capability, and liquidity. This paper first analyzes the impact of each constraint on the structure of the efficient frontier when the number of stocks in the portfolio changes. Then, it explores the changes in the length and other structural features of the efficient frontier when both the number of stocks and the number of constraints change, and analyzes the role of adding constraints of different dimensions. Further, this paper also explores the changes in the length and other structural features of the efficient frontier when both the number of stocks and the values of constraints change. The main conclusions of this paper are as follows: Firstly, this paper takes advantage of the parametric quadratic programming to fully solve the efficient frontier, accurately calculates its structural changes, and measures the impact of adding various constraints. Previous studies on solving the efficient frontier of portfolios are mostly approximate or incomplete. This paper calculates the length and length ratio of the efficient frontier under different stock scales and constraints, as well as other structural characteristics, and quantitatively describes the results of the risk-return trade-off. It makes up for the research deficiency that it is difficult to accurately reflect the changes of the efficient frontier due to the difficulty in precisely calculating the efficient frontier. Through the accurate measurement of the efficient frontier structure, this paper finds that the addition of constraints can significantly change the structural characteristics of the efficient frontier of the portfolio: the length of the efficient frontier is significantly shortened, narrowing the optional range of investment decisions, but also reducing the possibility of generating extreme portfolios. Constraints significantly improves the diversification degree, but the effectiveness is reduced, and the structural complexity is increased. Secondly, when different constraints are added, as the number of stocks increases, the changing trends of various structural characteristics of the efficient frontier are generally consistent: the length of the efficient frontier shows an overall increasing trend, and the optional range of efficient portfolios increases; the diversification ratio tends to decline, and the diversification of the portfolio decreases; the efficiency ratio tends to decline, the effectiveness of the portfolio improves, and its superiority over the simple diversified portfolio is enhanced; the average number of parabolic segments of the efficient frontier increases, and the structural complexity of the efficient frontier rises. Therefore, for public funds that often hold large scale portfolios, while the stock selection range becomes more extensive, the difficulty of diversification increases, and the requirements for investors' professional qualities also increase. At a large scale, the cost of simple diversification is higher, and the effectiveness of the portfolio selection model is more obvious, making it a useful investment decision making tool. Thirdly, as the number of constraints increases, the fluctuation of the structural characteristic indicators of the efficient frontier generally tends to decrease, the marginal impact of the newly added constraints is relatively weakened, the length of the efficient frontier tends to be stable, and the increase in diversity, the decrease in effectiveness, and the increase in structural complexity are not infinite. From the perspective of constraint types, the bounds of weights play a leading role. Constraints based on company characteristics can also change the efficient frontier structure, but with relatively limited impacts. The constraints in dimensions of growth and profitability have a relatively greater impact on efficient frontier. This indicates that investors need to reasonably consider the screening of the constraints. The upper bound constraint of weights has the most obvious effect on compressing the efficient frontier and improving diversification. If risk reduction is needed, the upper bound constraint of weights should be set first. Company characteristic constraints can introduce fundamental, information among which the dimensions of growth and profitability can be given priority. Fourthly, when values the of the constraints are varied, it can still be concluded that as the scale of the portfolio expands, generally the length of the efficient frontier increases, the complexity of the frontier structure rises, the efficiency improves to some extent, and the difficulty of maintaining diversification increases. When the values of the upper bound of weights and the upper and lower bounds based on company characteristics are set more strictly, the structural changes of the efficient frontier are generally more obvious. The value of the upper bound of weights plays a major role, while the constraints based on company characteristics mainly have an auxiliary effect. Therefore, portfolios of different scales can adopt strategies with different constraint intensities. Small scale portfolios should avoid overly strict constraints to prevent excessive restriction of the optimization space. When the portfolio scale expands, the range of stock selection and the degree of diversification can be actively controlled by tightening the upper bound of weights. The research innovations of this paper mainly lie in the following two aspects: Firstly, based on the precise and complete parameter quadratic programming, this paper innovatively proposes and calculates the length and length ratio of the efficient frontier. Combined with the diversification ratio, the efficiency ratio, and the average number of parabolic segments, it specifically measures the impact of adding constraints on the range, diversity, efficiency, and complexity of portfolio selection. The length of the efficient frontier reflects the range of choice for efficient portfolios and can reflect the changes in the efficient frontier, thereby demonstrating the trade-off between risk and return. Existing research on how constraints affect the efficient frontier is mostly qualitative and positional change descriptions. Few studies have quantitatively indicated the specific changes from the perspective of the length of efficient frontier, unable to accurately reflect the impact of constraints on the range of choice for efficient portfolio securities, nor have they fully explored the changes in other specific structural features of the efficient frontier. This paper provides a precise method for solving the efficient frontier and specifically measuring its changes, proposes a complete and accurate length of the efficient frontier that reflects the range of choice based on the piecewise-segment structure of the efficient frontier for investors, and combines indicators representing portfolio diversification, efficiency, and the complexity of the boundary structure to determine the impact of adding different constraints on the efficient frontier, which is innovative. This paper can deepen the theoretical research on the efficient frontier of portfolios, fill the research gap on specific structural features of the efficient frontier, and provide ideas for investors to utilize the clearer and more complete effective frontier and the structural features for investment practices. Secondly, this paper innovatively designs portfolio selection models with different numbers of constraints and different numbers of stocks to measure the impact of constraint settings on the efficient frontier. Currently, most studies on constraints incorporate them as a type of condition into the portfolio and discuss improvements in model calculation or prove the existence of solutions. In existing research on the impact of constraints on the efficient frontier, scholars mostly use single types or a few types of constraints, mostly targeting small and medium-sized portfolios, and the portfolio size is usually fixed. Few studies have explored the changes in the efficient frontier from the perspective of multiple realistic constraints. However, both relevant literature and fund practices indicate that investors have the need to set various constraint conditions in their portfolios. The efficient frontier directly relates to investors' optimal investment decisions. This paper selects the constraints that are relatively common in research and practice, constructs and solves the portfolio selection model. It is found that changes in the number, type, and values of the constraints can lead to structural changes in the efficient frontier. This paper can fill the gap of the deficiency of portfolio selection models due to overly simplistic constraints and affect their practical applicability. It can also expand the theoretical research on the impact of setting multiple constraints on the efficient frontier of portfolios of different sizes and provide theoretical references for constructing portfolio selection models of different scales.
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