PORTFOLIO OPTIMIZATION, CAPM AND APT

Portfolio Optimization, CAPM and Arbitrage Pricing Theory (APT)

Abstract

Ever since the inception of Markowitz's modern portfolio theory, static portfolio optimization techniques were gradually phased out by dynamic portfolio management due to the growth of popularity in automated trading. In view of the intensive computational needs, it is common to use machine learning approaches on Sharpe ratio maximization for implementing dynamic portfolio optimization. In the literature, return-based approaches which directly used security prices or returns to control portfolio weights were often used. Inspired by the arbitrage pricing theory (APT), some other efforts concentrate on indirect modelling using hidden factors. On the other hand, with regard to the proper risk measure in the Sharpe ratio, downside risk was considered a better substitute for variance. In this paper, we investigate how the Gaussian temporal factor analysis (TFA) technique can be used for portfolio optimization. Since TFA is based on the classical APT model and has the benefit of removing rotation indeterminacy via temporal modelling, using TFA for portfolio management allows portfolio weights to be indirectly controlled by several hidden factors. Moreover, we extend the approach to some other variants tailored for investors according to their investment objectives and degree of risk tolerance.

Assignment

1. Introduction

Portfolio management has evolved as a core decision-making activity for investors and practitioners in the financial market nowadays. Prior to the inception of Markowitz's modern portfolio theory [11], theoretical research on investments has concentrated on modelling expected returns [2].

During the early stage of its development, portfolio optimization was often constrained by its static implementation. Unlike dynamic portfolio optimization by which the optimal portfolio weights were tracked over time based on updated market information, the weights determined using static optimization techniques could not adapt to market changes within the investment horizon.

Despite dynamic portfolio optimization being powerful, it turned out to be a problem that required intensive computation. Recall that the most natural technique for solving dynamic portfolio optimization problems was stochastic dynamic programming. However, this approach was often compromised by several factors such as the curse of dimensionality when too many state variables were involved [7]. In general, practical considerations such as taxes and transactions costs also increased the number of state variables in the objective function.

In fact, this problem could be better solved via some popular machine learning approaches [3, 12, 13 and 21] which required the optimal parameters to be adaptively learned over time, and consequently, we have the term adaptive portfolio management. Among the various methodologies suggested, the most popular one is based on maximizing the well-known Sharpe ratio [17]. In implementation, trading could be based on training a trading system on labelled data [12] or directly maximizing the expected profit via the so-called adaptive supervised learning decision networks [8 and 21]. In this paper, these approaches were generally referred to as return-based portfolio management because they either explicitly treated the weights as constants or depend directly on the security price or returns.

Inspired by the arbitrage pricing theory (APT) in finance, which assumes that the cross-sectional expected ...