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The objective of this paper is to study the optimal consumption and portfolio choice problem of risk-controlled investors who strive to maximize total expected discounted utility of both consumption and terminal wealth. Risk is measured by the variance of terminal wealth, which introduces a nonlinear function of the expected value into the control problem. The control problem presented is no longer a standard stochastic control problem but rather, a mean field-type control problem. The optimal portfolio and consumption rules are obtained explicitly. Numerical results shed light on the importance of controlling variance risk. The optimal investment policy is nonmyopic, and consumption is not sacrificed.
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Inter‐temporal mutual‐fund management
Abstract Traditionally, mutual funds are mostly managed via an ad hoc approach, namely a terminal‐only optimization. Due to the intricate mathematical complexity of a continuum of constraints imposed, effects of the inter‐temporal reward for the managers are essentially neglected in the previous literature. For instance, the inter‐temporal optimal investment problem from the fund manager's viewpoint, who earns proportional management fees continuously (a golden rule in practice), has been outstanding for long. This article completely resolves this challenging question especially under generic running and terminal utilities, via the Dynamic Programming Principle which leads to a nonconventional, highly nonlinear HJB equation. We develop an original mathematical analysis to establish the unique existence of the classical solution of the primal problem. Further numerical calibrations and simulations for both the portfolio weight and the value functions illustrate the robustness of the optimal portfolio towards the manager's risk attitude, which allows different managers with various risk characteristics to sell essentially the same investment vehicle. Simulation studies also indicate that the policy of charging a substantial terminal‐only management fee can be replaced by another one with only a negligible amount over the interim period, which substantially reduces the total management fee paid by the clients without lowering the manager's satisfaction at all; this last observation echoes the magic of the alchemy of finance.
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- Award ID(s):
- 1905449
- PAR ID:
- 10444947
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Mathematical Finance
- Volume:
- 32
- Issue:
- 3
- ISSN:
- 0960-1627
- Page Range / eLocation ID:
- p. 825-877
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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