Ebook Dynamic asset allocation and latent variables
The solution to a multi-period portfolio problem can differ substantially from the solution to a static or single-period portfolio problem, as demonstrated originally by Samuelson (1969) and Merton (1969,1971,1973). This paper offers an explicit solution to a basic multi-period dynamic portfolio problem when the return dynamics are described by a multivariate time-series model and the investor is concerned with maximizing the expected utility of wealth at a given horizon. The modeling framework encompasses return generating models where some of the basic state variables are unobserved, and where the investor is faced with a filtering problem as part of the overall dynamic asset allocation problem. Our solution makes it possible to address, e.g., the portfolio implications of an estimated VAR-model that involves return-predictability for investors with different risk aversion and time horizons in a simple and consistent manner. Furthermore, when some of the state-variables are unobserved, we establish a close link between how unobserved state-variables can be handled consistently in the econometric estimation of the model as well as in a subsequent analysis of optimal asset allocation choice by using a Kalman filtering approach. This is explored in a realistic model calibration.
The multivariate discrete-time modeling of return dynamics is basically similar to the multivariate VAR-setting used by Campbell, Chan and Viceira (2003), but extended to the situation where some of the state-variables may not be directly observed by the investor. The general version of our return generating model is based on a state-space representation which consists of a transition equation and a measurement equation. The transition equation describes the return dynamics, and this is exactly the Campbell et al. (2003) multivariate VAR-model. The measurement equation describes what is being observed (and what is not).
The Campbell et al. (2003) return generating model is the special case where all state variables are directly observed. Campbell et al. (2003) investigates the optimal asset allocation and consumption policy of an infinitely-lived investor with Epstein-Zin recursive preferences, and relies on an approximate solution methodology in order to solve for the optimal policies. Also, Campbell et al. (2003) must assume that the VAR-model is time-homogenous in order to solve the infinite-horizon investment and consumption problem. Due to the more simple assumption of power-utility of terminal wealth, we can relax these assumptions and still address the effects of time horizon and risk aversion on optimal asset allocation. When state-variables are directly observed, we thus obtain an explicit solution to the relevant dynamic portfolio problem and a simple recursive solution algorithm for implementing the solution. The involved recursive solution algorithm solves a particular system of difference equations; this system of difference equations is analogues to the multi-dimensional Ricatti equation that arises in related continuous-time contexts when solving the relevant Hamilton-Jacobian-Bellman equation; see e.g. Liu (1999).
When some of the state-variables are not directly observed, the same solution procedure applies. However, in this case the relevant state-variables are now the perceived values of the possibly latent state-variables. The perceived values of the state-variables can recursively be found by Kalman filtering, which is applicable to any model that has a state-space representation. As we demonstrate, the dynamics of the Kalman filtered state-variables is also described by a VAR-model, and the solution methodology and recursive algorithm used for the case without observation noise can also be applied to this general case.
In a continuous-time framework, Williams (1977), Detemple (1986), Dothan and Feldman (1986), and Gennotte (1986) have previously demonstrated that the dynamic portfolio problem of an investor, who cannot directly observe the state-variables, separates into a filtering problem, in which the investor estimates the state-variable position, and an investment problem where the filtered estimates are treated as the relevant state-variables. Furthermore, in a setting with a single risky asset, Detemple (1986) and Gennotte (1986) show that the uncertainty about the instantaneous excess return on the risky asset is not affecting the optimal portfolio choice, which simply takes the form of Merton (1969) by substituting the instantaneous expected excess return with its perceived value. While our discrete-time solution also allows the filtering problem and the investment problem to be handled separately and consecutively, the uncertainty about the exact positions of the unobserved state-variables affects the optimal portfolio choice in our discrete-time setting. For example, even in a one-period model with a single risky asset, the relevant variance that must be used along with the perceived value of the expected excess return in order to determine the optimal portfolio (in the Markowitz (1952), Merton (1969), and/or Samuelson (1969) formulas) is affected by the uncertainty about the true expected excess return. In such a one-period setting, our solution resembles and coincides with the Bayesian approach of incorporating parameter uncertainty into the portfolio choice problem, as originally carried out by e.g. Klein and Bawa (1976) and Bawa, Brown and Klein (1979).
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