Applications of forward performance processes in dynamic optimal portfolio management
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The classical optimal investment models are cast in a finite or infinite horizon setting, assuming an a priori choice of a market model (or a family of models) as well as a priori choice of a utility function of terminal wealth and/or intermediate consumption. Once these choices are made, namely, the horizon, the model and the risk preferences, stochastic optimization technique yield the maximal expected utility (value function) and the optimal policies wither through the Hamilton-Jacobi-Bellman equation in Makovian models or, more generally, via duality in semi-martingale models. A fundamental property of the solution is time-consistency, which follows from the Dynamic Programming Principle (DPP). This principle provides the intuitively pleasing interpretation of the value function as the intermediate (indirect) utility. It also states that the value function is a martingale along the optimal wealth trajectory and a super-martingale along every admissible one. These properties provide a time-consistent framework of the solutions, which ``pastes" naturally one investment period to the next. Despite its mathematical sophistication, the classical expected utility framework cannot accommodate model revision, nor horizon flexibility nor adaptation of risk preferences, if one desires to retain time-consistency. Indeed, the classical formulation is by nature ``backwards" in time and, thus, it does not allow any ``forward in time" changes. For example, on-line learning, which typically occurs in a non-anticipated way, cannot be implemented in the classical setting, simply because the latter evolves backwards while the former progresses forward in time. To alleviate some of these limitations while, at the same time, preserving the time-consistency property, Musiela and Zariphopoulou proposed an alternative criterion, the so-called forward performance process. This process satisfies the DPP forward in time, and generalizes the classical expected utility. For a large family of cases, forward performance processes have been explicitly constructed for general Ito-diffusion markets. While there has already been substantial mathematical work on this criterion, concrete applications to applied portfolio management are lacking. In this thesis, the aim is to focus on applied aspects of the forward performance approach and build meaningful connections with practical portfolio management. The following topics are being studied. Chapter 2 starts with providing an intuitive characterization of the underlying performance measure and the associated risk tolerance process, which are the most fundamental ingredients of the forward approach. It also provides a novel decomposition of the initial condition and, in turn, its inter-temporal preservation as the market evolves. The main steps involve a system of stochastic differential equations modeling various stochastic sensitivities and risk metrics. Chapter 3 focuses on the applications of the above results to lifecycle portfolio management. Investors are firstly classified by their individual risk preference generating measures and, in turn, mapped to different groups that are consistent with the popular practice of age-based de-leveraging. The inverse problem is also studied, namely, how to infer the individual investor-type measure from observed investment behavior. Chapter 4 provides applications of the forward performance to the classical problem of mean-variance analysis. It examines how sequential investment periods can be ``pasted together" in a time-consistent manner from one evaluation period to the next. This is done by mapping the mean-variance to a family of forward quadratic performances with appropriate stochastic and path-dependent coefficients. Quantitative comparisons with the classical approach are provided for a class of market settings, which demonstrate the superiority and flexibility of the forward approach.