Portfolio Optimization and Parameter Uncertainty
Introducing portfolio optimization with fully general parameter uncertainty using the Resampled Portfolio Stacking approach.
Portfolio optimization remains one of the most researched areas in quantitative investment management. Yet it is also one of the most criticized aspects due to its inherent sensitivity to parameter estimates or more generally market model uncertainty.
Some practitioners are so critical that they do not use portfolio optimization explicitly, but most still try to build portfolios with a good trade-off between risk and return, albeit in implicit ways. This article argues that there are elegant and practically feasible ways to handle the problem.
The article gives a gentle introduction to resampled portfolio optimization, which is a heuristic used to alleviate the effects of parameter sensitivity. It also introduces the Exposure Stacking method and gives references to the more general Resampled Portfolio Stacking approach introduced in the Portfolio Construction and Risk Management book¹, which you can get access to through the crowdfunding campaign².
Resampled portfolio optimization and Exposure Stacking
Resampled portfolio optimization was first introduced by Michaud and Michaud (1998)³ for mean-variance efficient frontier optimization. The idea is to sample B parameter samples and compute B different efficient frontiers with the sampled parameter values. The resampled portfolio is then simply an average over the B samples, with the portfolio risk aligned by its index on the efficient frontier.
Since the original suggestion, resampled portfolio optimization has been used in many different ways by investment practitioners. An elegant feature of the resampled approach is that it gives us great flexibility in relation to how we introduce and analyze parameter uncertainty.
A critique of the original resampled approach is that it had very little mathematical justification. It just seemed to work well in practice. Kristensen and Vorobets (2024)⁴ provide the fundamental perspectives for understanding the resampled approach and introduces the Exposure Stacking method, which allows us to go beyond uniform sample weights and can improve the risk-adjusted out-of-sample performance.
The video below gives a walkthrough of the Exposure Stacking article and the accompanying Python code.
Derivatives Portfolio Optimization and Parameter Uncertainty
While derivatives are easy to handle in general portfolio management through a proper separation between relative market values v and relative exposures e, see Vorobets (2022)⁵, they introduce significant complexity when it comes to resampled portfolio optimization.
Derivatives can be elegantly handled using Entropy Pooling and CVaR optimization⁶, see Vorobets (2024)⁷ and the video below going through the article and accompanying Python code.
Resampled Portfolio Stacking generalization
While Exposure Stacking correctly focuses on portfolio exposures and not weights, it gives equal importance to exposures that might have very different marginal risks and returns.
Chapter 6 of the Portfolio Construction and Risk Management book² introduce the Resampled Portfolio Stacking generalization, which allows us to stack marginal risk and return contributions.
The new approach enables us to account for the differences in individual risk and return contributions including complex diversification interactions.
You can access the book and the accompanying Python code here². See also the video giving more information about the book below.
[1]: Portfolio Construction and Risk Management book Substack article: https://substack.com/home/post/p-145811595
[2]: Portfolio Construction and Risk Management crowdfunding campaign: https://igg.me/at/pcrm-book
[3]: Efficient Asset Management, Oxford University Press, 1998.
[4]: Kristensen, Laura and Vorobets, Anton, Portfolio Optimization and Parameter Uncertainty (January 30, 2024). Available at SSRN: https://ssrn.com/abstract=4709317
[5]: Portfolio Management Framework for Derivative Instruments (September 14, 2022). Available at SSRN: https://ssrn.com/abstract=4217884
[6]: Entropy Pooling and CVaR Optimization in Python Substack article: https://substack.com/home/post/p-145811596
[7]: Vorobets, Anton, Derivatives Portfolio Optimization and Parameter Uncertainty (May 13, 2024). Available at SSRN: https://ssrn.com/abstract=4825945