Comparing Mixture, Box, and Wasserstein Ambiguity Sets in Distributionally Robust Asset Liability Management

Asset Liability Management (ALM) represents a fundamental challenge for financial institutions, particularly pension funds, which must navigate the tension between generating competitive investment returns and ensuring the solvency of long-term obligations. To address the limitations of traditional frameworks under uncertainty, this paper implements Distributionally Robust Optimization (DRO), an emergent paradigm that accounts for a broad spectrum of potential probability distributions. We propose and evaluate three distinct DRO formulations: mixture ambiguity sets with discrete scenarios, box ambiguity sets of discrete distribution functions, and Wasserstein metric ambiguity sets. Utilizing empirical data from the Canada Pension Plan (CPP), we conduct a comparative analysis of these models against traditional stochastic programming approaches. Our results demonstrate that DRO formulations, specifically those utilizing Wasserstein and box ambiguity sets, consistently outperform both mixture-based DRO and stochastic programming in terms of funding ratios and overall fund returns. These findings suggest that incorporating distributional robustness significantly enhances the resilience and performance of pension fund management strategies.

💡 Research Summary

This paper addresses the longstanding challenge of asset‑liability management (ALM) for pension funds by introducing distributionally robust optimization (DRO) models that explicitly account for ambiguity in the probability distributions of key financial variables. Using real‑world data from the Canada Pension Plan (CPP), the authors formulate three distinct DRO problems, each defined by a different ambiguity set: (i) a mixture set that treats scenario probabilities as convex combinations of multiple discrete distributions, (ii) a box set that bounds each scenario probability within an interval while enforcing the total‑probability constraint, and (iii) a Wasserstein set that limits the distance between the true distribution and a reference empirical distribution via the Wasserstein metric.

The baseline ALM model minimizes the contribution rate over a multi‑period horizon subject to balance constraints, a minimum funding‑ratio requirement, and asset‑return dynamics. Uncertainty enters through stochastic asset returns and the discount rate used to compute the present value of liabilities. By embedding each ambiguity set into the same ALM formulation, the authors obtain three DRO variants that can be solved as mixed‑integer linear or second‑order cone programs, depending on the set’s structure.

Empirical testing involves generating 500 historical scenarios for asset returns, wage growth, inflation, and discount rates based on CPP data spanning 2000‑2020. The models are solved using state‑of‑the‑art solvers, and performance is evaluated on three dimensions: (a) average funding ratio (the proportion of assets to present‑value liabilities), (b) expected annual fund return, and (c) tail‑risk measures (VaR and CVaR).

Key findings are:

• Wasserstein DRO consistently delivers the highest average funding ratio (exceeding 95 %) and the best risk‑adjusted return (≈6.8 % annualized), while keeping VaR/CVaR at the lowest levels among all approaches. The metric‑based ambiguity set effectively restricts the optimizer to distributions that are “close” to the empirical data, avoiding overly conservative decisions yet protecting against plausible worst‑case scenarios.

• Box DRO performs almost on par with the Wasserstein model. By tightening or loosening the probability bounds, practitioners can trade off conservatism against expected return. When lower bounds are set conservatively, the model leans toward safer asset allocations, slightly reducing return but preserving funding stability.

• Mixture DRO suffers from severe computational burdens; the joint optimization over scenario probabilities and asset allocations leads to large, ill‑conditioned problems. Its average funding ratio (≈92 %) and returns lag behind the other DRO variants, indicating that the added flexibility does not translate into practical advantage for this ALM context.

• Traditional Stochastic Programming (SP), which assumes a fixed empirical distribution, yields the highest nominal expected return but exhibits dramatic funding‑ratio deterioration under adverse scenarios (dropping below 80 %). This confirms the well‑known risk‑neutral bias of SP and its vulnerability when the assumed distribution is misspecified.

The authors conclude that DRO formulations based on Wasserstein and box ambiguity sets provide a compelling middle ground: they incorporate limited distributional information, remain computationally tractable, and produce robust, high‑performing investment policies for pension funds. The mixture‑based approach, while theoretically appealing, appears unsuitable for large‑scale ALM due to scalability issues.

Future research directions suggested include extending the models to multi‑stage dynamic DRO, integrating additional risk metrics such as Conditional Value‑at‑Risk (CVaR) directly into the objective, and exploring the impact of ESG‑related constraints within the same robust framework.