Abstract
Portfolio selection is undoubtedly one of the most challenging topics in the area of finance. Since Markowitz's initial contribution in 1952, portfolio allocation strategies have been intensely discussed in the literature. With the development of online optimization techniques, dynamic learning algorithms have proven to be an effective approach to building portfolios, although they do not assess the risk related to each investment decision.In this work, we compared the performance of the Online Gradient Descent (OGD) algorithm and a modification of the method, that takes into account risk metrics controlling for the Beta of the portfolio. In order to control for the Beta, each asset was modeled using the CAPM model and a time-varying Beta that follows a random walk. We compared both the traditional OGD algorithm and the OGD with Beta constraints with the Uniform Constant Rebalanced Portfolio and two different indexes for the Brazilian market, composed of small caps and the assets that belong to the Ibovespa index. Controlling the Beta proved to be an efficient strategy when the investor chooses an appropriate interval for the beta during bull markets or bear markets. Moreover, the time-varying beta was an efficient metric to force the desired correlation with the market and also to reduce the volatility of the portfolio during bear markets.
Highlights
The portfolio selection problem (PSP) is a decisive process in which the investor must allocate a quantity of wealth to a set of assets within a time horizon
As an initial analysis we tested the performance of the Online Gradient Descent (OGD) algorithm without controlling for the Beta of the portfolio and compared with the index of small caps (SMALL) and the uniform
The UCRP portfolio is constructed fixing a uniform weight for each asset that belongs to the index small caps including the risk free asset
Summary
The portfolio selection problem (PSP) is a decisive process in which the investor must allocate a quantity of wealth to a set of assets within a time horizon. Each asset represents a distinct investment opportunity and a decision made for an allocation is a portfolio. In this problem, the investor seeks to allocate his money in a stock market to get a good relationship between expected return and risk. Choosing the optimal portfolio is as old a problem as the stock market itself. It was from the work of Markowitz [23] in 1952 that this question became a mathematical problem. As an alternative to this problem, Cover [8] proposed a portfolio optimization model that did not rely on statistical assumptions. The Universal Portfolios (UP) algorithm, introduced by Cover, marks the beginning of a new dynamic investment strategy called Online Portfolio Selection (OPS)
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