Institutional investors and reserve managers are increasingly recognising the importance of integrating climate risk considerations into their portfolios. This shift reflects a broader movement to embed sustainability into portfolio construction. Traditional methods often employ constrained optimisation frameworks that treat carbon footprints as deterministic values based on historical data or single-scenario projections. This research identifies a significant limitation in this approach: it fails to account for the inherent uncertainty in projections, which is comparable to conducting asset allocation based on a single path of future returns.

Carbon returns

The paper introduces the novel concept of “carbon returns”, defined as the negative percentage change in a carbon footprint measure. This framing mirrors financial returns, ensuring that a reduction in emissions is treated as a positive outcome while an increase reflects an undesirable state. Unlike standard methods that focus on static emission levels, carbon returns are treated as random variables. This stochastic approach offers two distinct advantages. Firstly, it incentivises actual decarbonisation rather than simply favouring current low emitters. Secondly, it enhances robustness, as return distributions are found to vary less across different emission measurement methodologies than absolute levels.

Empirical analysis was conducted on production-based CO2 emissions for 23 advanced economies from 1982 to 2021. The results indicate that while absolute emission levels vary depending on the accounting method (such as production-based versus consumption-based), the distributions of historical carbon returns remain remarkably consistent across these methods. European economies, including Germany, France, Italy, and the United Kingdom, tended to exhibit higher average carbon returns and less extreme values in the tails, reflecting effective domestic policies and structural shifts towards low-carbon economies.

Portfolio construction with carbon returns

While classical mean-variance optimisation can be applied to carbon returns, it often leads to highly concentrated “corner solutions” that may be unrealistic for institutional practitioners. To address this, the researchers propose a framework inspired by the Hierarchical Risk Parity (HRP) algorithm. This machine-learning-based approach focuses on risk diversification through hierarchical clustering and risk allocation, avoiding the numerical instability often caused by covariance matrix inversion in traditional models.

A key modification in this study is the use of Expected Shortfall (ES) instead of volatility as the primary measure of risk. Given that carbon risk is characterised by infrequent but extreme events, managing tail risk is considered more appropriate for climate risk management. Because historical carbon time series are relatively short, the authors utilise t-copula methods to simulate 10,000 scenarios. This process preserves empirically observed cross-country dependence patterns while providing greater visibility into the tails of the distribution, allowing for more stable estimates of “footprint shortfall”.

Optimisation results

The framework supports eight distinct implementation configurations by alternating between financial (F) and carbon (C) inputs at three workflow stages: clustering countries, intra-cluster risk allocation, and inter-cluster risk allocation. The (F,F,F) configuration represents standard HRP focusing on financial metrics, while (C,C,C) fully prioritises the carbon return distribution. In-sample results confirm that portfolios prioritising carbon returns are more effective at mitigating downside risk in the environmental domain.

Out-of-sample performance was evaluated beginning in January 2008. The results indicate that the (C,C,C) optimised portfolio demonstrated superior performance in carbon spaces, showing higher cumulative carbon returns and lower left-tail risk. Furthermore, portfolios with more “F” inputs exhibited lower financial tail risk, while those with more “C” inputs displayed lower carbon tail risk. This demonstrates that the algorithm achieves its intended objectives and allows investors to select implementations based on their specific policy goals or risk preferences.

Concluding remarks

This paper provides a practical framework for decarbonising sovereign bond portfolios by treating transition outcomes as random variables. By moving beyond deterministic carbon budgets, the framework offers a more transparent and adaptable path for reserve managers and other institutional investors. The HRP structure supports diversified allocations and reduces sensitivity to noisy estimates, while the Expected Shortfall objective focuses attention on highly adverse transition outcomes. Future research may extend this analysis to broader asset classes, such as corporate bonds, or incorporate more systematic forward-looking scenario narratives into the return distributions.