Essays on Optimal Management of Portfolios

Individuals, endowments, pension funds, and sovereign wealth funds constantly face portfolio management decisions ultimately affecting the lives of billions. This dissertation addresses two crucial challenges in the context of portfolio management which are to identify the optimal portfolio at any given point in time and to transition the portfolio accordingly. The first essay considers the dynamic consumption/portfolio allocation problem and develops solution techniques allowing us to identify the optimal portfolio allocation (between a risky and risk-free asset) and consumption level over time. Deciding how much to consume, versus how much to save is a question affecting nearly everyone. This essay develops two perturbation methods that yield approximate closed-form solutions to dynamic portfolio allocation/consumption problems under general preferences and a time-varying investment opportunity set. These solution methods are illustrated with examples involving time-varying expected returns, volatility, and interest rates. The second essay examines this monolithic "risky asset" more closely, developing a robust diversification method to define the optimal allocation of assets within the risky portfolio component. This diversification method relies solely on the covariance structure of the investment opportunity set which can be much more precisely estimated than expected returns. Using this new approach leads to an alternative to the mean-variance efficient set of portfolios as traditionally implemented. The resulting optimally diversified portfolio equalizes the correlation between each asset and the overall portfolio. This approach provides individuals and institutions with a robust and stable asset allocation rule which avoids the typical over-concentrations associated with substantial losses during market downturns. The third essay analyzes the question of how to optimally transition a portfolio from one set of allocations to another in the face of transaction costs (which are quantity-sensitive) and a penalty (utility) function for not holding the target portfolio. This improves on existing models by allowing for portfolio mandates that may not be mean-variance efficient, e.g., if they are sub-portfolios and do not represent the total wealth of an investor, or if political or reputational concerns lead to restricting investments. This essay identifies a formula for the optimal rebalancing path and shows that there are several general factors influencing this path.