A Krasnoselskii-Mann Proximity Algorithm for Markowitz Portfolios with Adaptive Expected Return Level

TL;DR

This work proposes an exact, convergent, and efficient Krasnoselskii-Mann Proximity Algorithm based on the proximity operator and Krasnoselskii-Mann momentum technique that achieves significant improvements over state-of-the-art methods in portfolio optimization.

Abstract

Markowitz's criterion aims to balance expected return and risk when optimizing the portfolio. The expected return level is usually fixed according to the risk appetite of an investor, then the risk is minimized at this fixed return level. However, the investor may not know which return level is suitable for her/him and the current financial circumstance. It motivates us to find a novel approach that adaptively optimizes this return level and the portfolio at the same time. It not only relieves the trouble of deciding the return level during an investment but also gets more adaptive to the ever-changing financial market than a subjective return level. In order to solve the new model, we propose an exact, convergent, and efficient Krasnoselskii-Mann Proximity Algorithm based on the proximity operator and Krasnoselskii-Mann momentum technique. Extensive experiments show that the proposed method achieves significant improvements over state-of-the-art methods in portfolio optimization. This finding may contribute a new perspective on the relationship between return and risk in portfolio optimization.

Figures (2)

• Figure 1: Cumulative wealths of different strategies with respect to trade time on 6 benchmark data sets.
• Figure 2: Final cumulative wealths of different strategies with respect to different transaction cost rates on 6 benchmark data sets.

Theorems & Definitions (14)

• Lemma 1
• Theorem 2
• Proposition 3
• Lemma 4: Moreau decomposition
• Theorem 5: KM theorem
• Proposition 6
• Lemma 7
• Corollary 8
• Proposition 9
• Theorem 10