Multiscale Markowitz

TL;DR

This paper proposes a multiscale Markowitz portfolio optimization framework that incorporates the relationship between variance at different time scales, enabling more flexible and robust risk management aligned with investor preferences across multiple frequencies.

Contribution

It introduces a novel multifrequency optimization method that considers the scaling behavior of variance, extending traditional Markowitz models to better handle market complexities like crashes and regime changes.

Findings

Enables specifying target variance across multiple frequencies

Addresses limitations of traditional Markowitz in volatile markets

Provides a practical implementation framework

Abstract

Traditional Markowitz portfolio optimization constrains daily portfolio variance to a target value, optimising returns, Sharpe or variance within this constraint. However, this approach overlooks the relationship between variance at different time scales, typically described by $\sigma(\Delta t) \propto (\Delta t)^{H}$ where $H$ is the Hurst exponent, most of the time assumed to be \(\frac{1}{2}\). This paper introduces a multifrequency optimization framework that allows investors to specify target portfolio variance across a range of frequencies, characterized by a target Hurst exponent $H_{target}$, or optimize the portfolio at multiple time scales. By incorporating this scaling behavior, we enable a more nuanced and comprehensive risk management strategy that aligns with investor preferences at various time scales. This approach effectively manages portfolio risk across multiple frequencies and adapts to different market conditions, providing a robust tool for dynamic asset allocation. This overcomes some of the traditional limitations of Markowitz, when it comes to dealing with crashes, regime changes, volatility clustering or multifractality in markets. We illustrate this concept with a toy example and discuss the practical implementation for assets with varying scaling behaviors.