Fraunhofer Institute for Industrial Mathematics ITWM
Fraunhofer Institute for Industrial Mathematics ITWM

Multi-Criteria Portfolio Optimization Under Uncertainty

Robust Strategies for Dynamic Markets

Modern asset allocation is still largely based on Markowitz’s mean-variance approach. However established this model may be, it has a critical weakness: it assumes precise knowledge of parameters such as expected returns. In practice, these inputs must be estimated, and even minimal deviations often lead to unstable, non-implementable decisions.

As part of a doctoral research project, we investigate these sensitivities and develop strategies that explicitly incorporate uncertainty into the optimization process – without falling into excessive conservatism.

Regret Robustness: Preserving Opportunities Instead of Merely Avoiding Risks

Common hedging strategies, such as strict min-max robustness, primarily focus on the worst-case scenario. However, in normal market phases this often leads to very conservative results that fall significantly short of their potential. An effective alternative is the concept of regret robustness:

  • Focus on relative loss: Typically, »regret« is defined as the distance to the best possible decision one could have made in hindsight (with knowledge of the actual market development).

  • Benchmarks as a reference point: Decision-makers do not evaluate their solutions against hypothetical optima, but against real competitors or market indices. We adopt this idea by minimizing performance deviations relative to real benchmark portfolios.

A Universal Approach Beyond the Financial World

The mathematical concept of a »portfolio« can be applied far beyond traditional investments. For this reason, we deliberately design our theoretical approaches to be application-agnostic: the concepts support a wide range of decision-making problems under uncertainty – for example in the energy sector, logistics, or the management of strategic project portfolios.

By investigating general mathematical properties, we establish a theoretical foundation that remains valid independent of the specific field of application and provides a stable basis for various disciplines.

Our institute director, Prof. Dr. Anita Schöbel, as an expert in robust optimization, and Dr. Pascal Halffmann, as an expert in multi-criteria optimization, supervise the doctoral research. Both contribute their extensive experience in translating mathematical foundations into reliable tools for practice.

Crisis Scenario
© Fraunhofer ITWM
Scatter plot for regime C based on the criteria of expected portfolio return, variance, and solvency ratio.
Base-Case Scenario
© Fraunhofer ITWM
Scatter plot for regime N based on the criteria of expected portfolio return, variance, and solvency ratio.
Growth Scenario
© Fraunhofer ITWM
Scatter plot for regime G based on the criteria of expected portfolio return, variance, and solvency ratio.

Results highlighted in blue or purple satisfy the specified minimum solvency ratio according to the solvency ratio criterion from Dächert et al. The figure shows that the entire feasible set varies considerably across the scenarios, particularly when accounting for the minimum solvency ratio criterion requirement. In addition, the Pareto front, which is the primary focus of our analysis, differs substantially between regimes C and N. By contrast, regimes N and G exhibit largely comparable results, as they differ only in their covariance matrices.

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