Perturbation Analysis for Investment Portfolios Under Partial Information with Expert Opinions

Abstract

Abstract. We analyze the Merton portfolio optimization problem when the growth rate is an unobserved Gaussian process whose level is estimated by filtering from observations of the stock price. We use the Kalman filter to track the hidden state(s) of expected returns given the history of asset prices, and then use this filter as input to a portfolio problem with an objective to maximize expected terminal utility. Our results apply for general concave utility functions. We incorporate time-scale separation in the fluctuations of the returns process, and utilize singular and regular perturbation analysis on the associated partial information HJB equation, which leads to an intuitive interpretation of the additional risk caused by uncertainty in expected returns. The results are an extension of the partially-informed investment strategies obtained by the Black-Litterman model, wherein investors' views on upcoming performance are incorporated into the optimization along with any degree of uncertainty that the investor may have in these views.Key words. Filtering, control, Hamilton-Jacobi-Bellman equation, portfolio optimization, partial information, expert opinions. Subject classifications. 91G20, 60G35, 35Q93, 35C201. Introduction. It is well-known that optimal mean-variance portfolio weights are quite different from the weights used in real-life investment. As pointed out in [BL92], unconstrained optimization often results in large short positions among many asset classes, and constrained optimization results in zero weight in many assets and unreasonably large weights in assets with small capitalization. The model of Black and Litterman [BL91] is a practical solution to this problem, wherein investors' views on upcoming performance are incorporated into the optimization along with any degree of uncertainty that the investor may have in these views.In continuous time or multi-period settings, a natural extension of the Black and Litterman model is to include a separate stochastic process for the level of expected returns. Due to the large amount of data required for accurate estimation of this new process, much of the existing literature considers it to be unobserved, hence making this an investment problem with only partial information. Several papers consider expected returns to be an unobserved Markov process and then use filtering methods (i.e. Kalman filter or Wonham filter) to track the hidden state. Indeed, investment with partial information is studied in [Bre98,Bre06,ESB10,BR05,FPS14,Pap13,Pha09,SH04]. The effect of this added stochasticity is that optimal investment strategies will change from those suggested by the standard Merton problem [Mer69], with the effects of riskaversion characteristics and the investment horizon playing a role in the solution [Bre98,Wac02].The analysis in [FGW12] combines filtering with so-called expert opinions, a concept which is similar to the Black-Litterman model's consideration of investors' views. The idea is the following: in a dynamic investment problem, the nature of...