modern portfolio theory multidimensional extension ESG asset class solvency

The thesis extends classical 2D MPT to a 4D framework that also includes ESG and Market SCR. It builds 4D profiles for 14 asset classes using returns, volatility, ESG scores and Solvency II SCR. First, an A Posteriori model generates millions of portfolios and extracts a 4D Pareto-efficient frontier. Second, an A Priori model applies investor profile to pick preferred portfolios, revealing distinct allocation patterns. Third, the 4D set is compared to 2D results.

  From 2D to 4D: Rethinking Modern Portfolio Theory

Modern Portfolio Theory (MPT) has shaped the way we think about investing for more than sixty years. Since Markowitz, portfolio construction has essentially been described in a two-dimensional world: investors combine assets to maximise expected return for a given level of risk, or to minimise risk for a given level of return. The result is the well-known efficient frontier, a curve where each point represents the “best possible” trade-off between return and volatility (cf. appendix 1). In practice, this representation is extremely powerful, but it no longer fully matches the reality of institutional investors in 2025.

Two developments are particularly important. First, environmental, social and governance (ESG) criteria have moved to the core of investment mandates. Asset owners – insurers, pension funds, banks – are increasingly expected to demonstrate that their portfolios are not only profitable and well diversified, but also aligned with broader sustainability objectives. Second, for regulated investors such as insurance companies under Solvency II, market risk carries a very concrete cost in the form of the Market Solvency Capital Requirement (SCR). Each euro invested in a given asset class translates into a certain amount of regulatory capital that must be held on the balance sheet. In many institutions, these two dimensions – ESG and SCR – are still treated “around” the optimisation model rather than inside it: ESG is often handled through exclusion lists or filters, and SCR is checked ex post to verify that the final allocation remains compatible with capital constraints.

The central question is simple to state: to what extent can ESG and Market SCR be integrated into portfolio optimisation as formal dimensions, and how do these additions reshape the efficient frontier and the resulting asset allocation? The intention is not to abandon classical MPT, but to extend it so that the optimisation framework reflects more faithfully the actual objectives and constraints of institutional investors today (cf. appendix 2 for a detailed overview of the research question and sub-questions).

Building the 4D Investment Universe

To explore this question, the thesis builds a realistic but tractable multi-asset universe composed of 14 asset classes. The universe covers European large-cap and small-cap equities, emerging market and developed world equities, euro government bonds, euro investment grade and high-yield corporate bonds, emerging market bonds, global convertible bonds, European core real estate, global core infrastructure, commodities, private equity and euro cash. For each asset class, four key characteristics are constructed over the period from 31 January 2019 to 31 January 2025. The first is expected return, estimated from realised past performance and annualised. The second is risk, measured by the annualised standard deviation of returns, in line with standard MPT practice. The third is an ESG score derived from issuer-level ratings provided by Refinitiv Eikon and Morningstar/Sustainalytics; these scores are aggregated to the level of each asset class. When one provider does not cover a given asset class, simple statistical relationships with the other provider are used to avoid excluding it. The fourth characteristic is Market SCR, computed using the Solvency II standard formula: each asset class is assigned regulatory stress factors for equity, spread or currency risk, and these are combined via the official correlation matrix to obtain a diversified capital charge per euro invested (cf. appendix 3).

How the 4D Framework Rewrites the Efficient Frontier

The result is a four-dimensional “identity card” for each asset class: expected return, volatility, ESG score and Market SCR per euro invested. These vectors form the raw material for the empirical part of the thesis, which compares the traditional 2D efficient frontier (return–volatility) with an extended 4D frontier that also takes ESG and SCR into account. The empirical strategy proceeds in two steps that correspond to two ways of thinking about optimisation. In a first, preference-free step (A Posteriori model), the thesis maps the entire opportunity set implied by the data and a simple set of allocation rules. Portfolio weights are discretised in steps of 3.75%, each asset class must receive at least a small positive weight, and the weights must sum to 100%. This tranche-based enumeration may sound restrictive, but it has the advantage of remaining completely transparent: for every feasible combination of weights, it is straightforward to compute the four characteristics of the corresponding portfolio.

Mapping the 4D Opportunity Set: The A Posteriori Model

On this basis, more than ten million distinct portfolios are generated and evaluated using a Python program. For each, the expected return, volatility, ESG score and Market SCR are known. The next step is to isolate those portfolios that are truly “efficient” when all four dimensions are considered together. A portfolio is kept if no other portfolio exists that is strictly better on all four criteria at the same time. Economists would keep the Pareto-efficient set; in more intuitive terms, the portfolios that are clearly dominated by others in every respect are removed. After this filtering, about 1.2 million portfolios remain. Together, they form an empirical 4D efficient frontier: a multidimensional surface that generalises the traditional risk–return curve.

Several results emerge from this empirical exercise. A first important finding is that ESG and Market SCR are not redundant with return and volatility. One might worry that ESG scores simply mirror sector or regional exposures already captured by risk and return, and that SCR is just an imperfect reflection of volatility. In reality, the four dimensions generate genuine trade-offs. Portfolios with excellent ESG scores are rarely those that minimise Market SCR. Very high-return portfolios tend to perform poorly both in terms of volatility and SCR. Conversely, portfolios that look attractive from a purely risk–return perspective can be weak on sustainability or capital efficiency. This confirms that including ESG and SCR explicitly as objectives, rather than as soft constraints, is not merely a cosmetic change but modifies the structure of the opportunity set (cf. appendix 4).

A second result concerns the shape of the efficient frontier itself. When the traditional 2D frontier (return against volatility) is plotted and the portfolios on the 4D frontier are projected into this same plane, the central part of the curve is essentially preserved. Median return and volatility change only marginally: in the simulations, the typical efficient portfolio goes from around 8.4% to roughly 8.1% in expected return, while volatility stays close to 6.4% (cf. appendix 5). The main differences appear at the extremes. In the high-return, high-volatility region, a number of portfolios that seem attractive in two dimensions – because they push expected return above 10–11% – are systematically eliminated once ESG and SCR are taken into account. The additional criteria do not drag down the entire frontier; instead, they prune the most aggressive, risk-seeking allocations that would be difficult to justify from a

Bringing the Investor Back In: The A Priori Model

In a second step, the thesis reintroduces investor preferences. From the point of view of practice, it is not enough to know that a portfolio lies on the 4D frontier; one also needs to decide which point on this frontier is most suitable for a given type of investor. To mimic the way mandates are formulated in the real world, the thesis uses simple and interpretable investor profiles. Portfolios are first normalised on each of the four dimensions and then assigned a composite score based on the relative importance attached to return, risk, ESG and SCR. A “return-maximising” profile will place more weight on expected return, a “low-risk” profile on volatility, an “ESG-focused” profile on sustainability scores, and an “SCR-conservative” profile on capital efficiency. For that, a lexicographic logic is applied: a profile might, for instance, first require that Market SCR remains low and then, among all low-SCR portfolios, seek the best possible ESG score and return. This A Priori approach translates qualitative preferences into a transparent selection rule applied to the 4D frontier.

The A Priori model provides the third set of insights of the thesis. Portfolios that score particularly well on ESG tend to allocate more to European core real estate and global core infrastructure, which combine acceptable returns with strong sustainability ratings, and are willing to accept somewhat higher volatility and SCR to achieve this. SCR-efficient portfolios, by contrast, tilt heavily towards euro government bonds and cash, which carry low Solvency II capital charges, and thus exhibit both low risk and low expected return. More balanced profiles, which seek to perform “well enough” on all four dimensions (above-median return, below-median volatility, above-median ESG and below-median SCR), converge toward diversified allocations combining developed-market equities, investment-grade credit, infrastructure and a limited exposure to high-SCR segments such as high-yield bonds, emerging markets or private equity. These patterns illustrate how a multidimensional framework can make explicit the trade-offs that investment committees currently handle in a more informal way.

Final Words and Contribution for Practitioners

For practitioners, the message is that it is possible to move beyond the traditional risk–return view without losing the clarity, discipline and intuition that made MPT successful. Regulated investors can explicitly optimise for capital efficiency by treating Market SCR as an optimisation axis in its own right, rather than discovering its impact afterwards in regulatory reports. ESG-oriented investors can place sustainability at the heart of portfolio construction, rather than using it solely as a filter that shrinks the investible universe. Product designers and asset managers can derive different, consistent model portfolios – “balanced”, “ESG-plus”, “capital-light” – from the same underlying 4D frontier. In this sense, the efficient frontier ceases to be a simple curve and becomes a multidimensional surface describing the intersection of profit, purpose and prudence.

Appendices

Appendix 1: Classical Markowitz efficient frontier

07 BFWD 2025 10 El Azri Appendix 1

Appendix 2: Research Question and Sub-questions

“To what extent can the Modern Portfolio Theory be extended into a multidimensional framework that simultaneously integrates return, risk, ESG performance, and market SCR constraints, while remaining computationally tractable and practically applicable to institutional portfolio management and how does a multidimensional portfolio optimization model incorporating ESG scores and market SCR risk compare to the classical Markowitz model in terms of efficiency and asset allocation outcomes?”

A series of sub-questions also helped to better grasps the real meaning of the work. These sub-questions directly stem from the general research question.

Theoretical Foundations – sub-questions

Q1. What are the theoretical underpinnings and limitations of the classical Modern Portfolio Theory?

Q2. How have researchers and practitioners attempted to extend MPT to incorporate sustainability and liquidity dimensions?

Data and Metrics – sub-questions

Q3. How can ESG scores and SCR be quantitatively integrated into a portfolio optimization model?

Q4. What data limitations exist when integrating ESG and SCR scores into portfolio construction?

Model Construction and Feasibility – sub-questions

Q5. What kind of optimization techniques are suitable for a 4D portfolio model?

Q6. Is the proposed multidimensional model computationally tractable for real-world application?

Empirical Comparison and Evaluation – sub-questions

Q7. How does the asset allocation produced by the 4D model differ from that of the traditional 2D model (return-risk)?

Q8. Does the 4D model generate portfolios with superior or more balanced outcomes in terms of ESG and market SCR, without significantly compromising return and risk levels?

Appendix 3: Final vectors list

07 BFWD 2025 10 El Azri Appendix 3

Appendix 4: 2D projections with ESG and SCR as colors

07 BFWD 2025 10 El Azri Appendix 4

Appendix 5: Output comparison for the 2D and 4D set

07 BFWD 2025 10 El Azri Appendix 5

References

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Authors

07 BFWD 2025 10 Foto Samy El Azri

Samy El Azri

PhD Candidate in Finance, HEC Liège
Samy El Azri
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