Mathematical Models for Delegated Portfolio Management

该项目涉及两个主题：(i)研究管理其项目或投资组合风险的经理的最佳薪酬，并研究这些经理在场时金融市场的价格形成；（ii）对于一家提供共同基金的公司，研究什么样的基金最适合提供给投资者，以什么样的费用和合同特征，以满足不同信仰、风险偏好和消费需求的投资者的需求。关于主题(i)的发现将有助于理解哪种类型的薪酬方案最适合提供给管理公司或投资组合风险的经理，并理解管理层薪酬对资产价格形成的影响。关于主题（ii）的研究结果将解释基金家族如何构建其产品和相关的投资费用。调查结果将阐明在监管高管薪酬、监管对冲基金和共同基金、或向基金客户提供建议时应关注哪些问题。鉴于2007/2008年金融危机的原因之一是经理人的薪酬方式，这一研究方向很有意思。此外，由于危机，金融市场正面临新的变化和挑战，重要的是保持和改善对这一领域年轻研究人员的培训和支持，因为他们将在未来应对这些挑战方面发挥重要作用。支持和培训参与该项目的学生将有助于实现这一目标。用更专业的术语来说，本项目将：(a)利用随机分析的最新进展来发展最优合同的一般理论，包括管理项目风险的合同经理的情况，并将其应用于委托投资组合管理存在的均衡资产定价问题；(b)颠倒管理者和投资者的角色，从管理者的角度解决契约问题：如何优化向具有异质特征的投资者提供的基金菜单。从经济学的角度来看，主题(a)将部分地弥补契约理论中的一个空白：大多数动态模型只考虑管理者对项目回报的影响，而不考虑其风险。在方法上，它需要扩展最近的二阶倒向随机微分方程的复杂数学结果。主题(b)将涵盖另一个空白：大多数合同文献假设投资者向经理提供接受或放弃的合同。然而，在金融实践中，通常是另一种方式——对冲基金或共同基金家族提供具有精确合同特征的服务。在方法上，它需要解决困难的动态逆向选择问题，有时被表述为变分法问题，有时被简化为研究后向随机微分方程和相关的偏微分方程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project addresses two topics: (i) studying optimal compensation of managers who manage the risk of their projects or portfolios, and studying price formation in financial markets in the presence of such managers; (ii) for a firm offering mutual funds, studying what kind of funds are optimal to offer to investors, with what fees and contracting features, to satisfy the demand of investors of varying beliefs, risk preferences and consumption needs. The findings on topic (i) will help understand which type of compensation schemes are optimally offered to managers that manage risk of companies or portfolios, and to understand the effect of managerial compensation on the formation of asset prices. The findings on topic (ii) will explain how fund families structure their offerings and the associated investment fees. The findings will shed light on which issues to focus on when regulating compensation of executives, or regulating hedge funds and mutual funds, or giving advice to fund customers. This line of research is of interest given that one of the reasons for the 2007/2008 financial crisis was the way managers had been compensated. Moreover, because of the crisis, new changes and challenges are facing financial markets and it is important to maintain and improve the training and support available to young researchers in this field, as they will play an important role in facing those challenges in the future. This will be helped by supporting and training students working on this project.In more technical terms, this project will: (a) use recent advances in Stochastic Analysis to develop a general theory of optimal contracting that includes the case of contracting managers who manage the risk of their projects, and apply it to the problem of equilibrium asset pricing in the presence of delegated portfolio management; (b) reverse the roles of managers and investors, and solve contracting problems from the perspective of managers: how to optimally structure the menu of funds offered to investors with heterogeneous features. From the economic point of view, topic (a) will partially cover a gap in contract theory: most of the dynamic models only the effect of the managers on project return, and not on its risk. Methodologically, it requires extending recent sophisticated mathematical results for second order Backward Stochastic Differential Equations. Topic (b) will cover another gap: most of the contract literature assumes that investors offer take-it-or-leave-it contracts to managers. However, in financial practice, it is usually other way round - hedge funds or mutual fund families offer their services with precise contracting features. Methodologically, it requires solving difficult dynamic adverse selection problems, sometimes formulated as calculus of variations problems, sometimes reduced to studying Backward Stochastic Differential Equations and associated Partial Differential Equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.