In the business of Islamic finance, a quiet revolution is taking shape not in the form of flashy new instruments, but in a smarter way to distribute profits when two or more parties join forces. A team of researchers from ENSIIE and the Laboratoire de Mathématiques et Modélisation d’Evry (LaMME) at Université Évry Paris-Saclay has proposed a concept they call c-fair profit sharing ratios. Led by Abass Sagna, the work reframes profit and loss sharing contracts so that each partner’s payoff at the end of a project mirrors not just how much capital they put in, but also how much they contribute to the project’s day-to-day effort and risk. The result is a principled, adaptable way to balance labor and capital in structures like mudharabah and musharakah, potentially paired with an agency contract (wakalah).
What makes this idea compelling is not a single formula, but a practical way to align incentives with outcomes in real economies. The paper argues that you can design profit sharing so that the expected payoff to each partner, at maturity, is fair given their overall contribution and the unavoidable uncertainty of the venture. In plain terms: it treats the partnership as a shared voyage where crew skills, leadership, and capital are not just pooled, but weighed to reflect how risky the voyage is and how much each person actually contributes to steering it toward success. The study is a bridge between abstract math and concrete financial practice, and it explicitly notes that its framework can accommodate a broad spectrum of econometric models to estimate key quantities.
A fresh fairness rule for profit sharing
The central idea, sometimes described in the paper as c-fair profit sharing ratios, is to assign each partner a weight that combines two kinds of contribution. One is their share of the capital, which is a pure financial stake in the project. The other is their contribution to the project’s management and labor, which the authors quantify with a risk-related measure. The result is a simple but powerful decomposition: each partner’s share of the expected profit is the sum of a labor-based reward and a funding-based reward. In symbols, the paper shows that the profit share for partner ℓ can be thought of as a mix of two pieces: a portion that grows with the project’s overall success and a portion that scales with how much capital the partner has at risk. This view makes the allocation feel fair in two senses at once: it rewards the person who helps the project perform (labor) and it protects the person who puts up the money (capital), while still honoring the project’s risk profile.
To ground the idea, the authors split the analysis into concrete contract types. In a mudharabah arrangement, one partner supplies the capital while the other contributes labor or expertise. In a musharakah, all partners contribute capital and share profits according to pre-agreed ratios, but losses are absorbed proportionally to capital. The c-fair framework generalizes these setups so that profit sharing can flexibly reflect each partner’s relative emphasis on capital versus management. A neat takeaway is that the framework doesn’t prescribe a fixed split; instead it links the split to how risky the investment is and how much each partner contributes beyond money. If the project looks risky but likely profitable, the model shifts more of the upside to those who shoulder more labor. If the project is less risky, capital owners can receive a larger piece of the expected profit, while workers still get a fair share for their contribution.
From mudharabah to musharakah and back
At its heart, the mudharabah contract is nearly a two-person team: a capital owner and a laborer. The paper shows that under the fair sharing rule, the investor’s share grows with risk, while the worker’s share gets a counterbalancing adjustment tied to how much the project stands to improve the odds of success. When the two roles are symmetric in terms of management effort, the splits resemble the capital contributions. But when one partner shoulders more managerial load, the c-fair approach assigns a larger portion of the upside to that partner, reflecting the extra work and risk being taken on. In short, the model internalizes a simple intuition: the more you do to steer the project, the more you should be rewarded, especially when risk is high.
The musharakah contract—where multiple partners contribute capital—fits naturally into the c-fair framework. The authors derive a general method to compute each partner’s fair share in a setting with d partners. Each partner’s share combines two components: a labor-based weight that captures how much they helped manage the project, and a capital-based weight tied to their share of the initial funding. The math is crisp but the message is human: fair sharing should reflect both the headcount at the helm and the money on the line. When a manager from outside the core group is brought in and paid a fixed remuneration via a wakalah contract, the same principle applies, with the remuneration integrated into the overall fair allocation. The upshot is a family of formulas that can accommodate any plausible mix of leadership, capital, and external management, all while ensuring the total shares sum to one and each partner’s payoff aligns with the project’s real-world dynamics.
Risk, reward, and the language of two numbers
The paper builds on two key quantities that try to capture the essence of uncertainty in a future payoff: the investment risk ρ and the investment profit Δ. The investment profit Δ captures how much the project’s end result might exceed its initial capital, while the risk ρ measures how much of the upside is matched by downside exposure. In the simplest terms, ρ is the ratio of expected downside to expected upside. If ρ is small, the project looks robust; if ρ is large, gains and losses move more together, and the upside is quickly offset by the downside. A crucial insight is that the c-fair sharing ratio of each partner is not a static percentage. It depends on ρ and Δ, the heartbeats of the investment. When risk is moderate and the expected profit is large, labor contributions tend to be rewarded more richly; when risk is heavy, capital receives a bigger slice of the expected payoff. The intuition is elegant: the framework couples risk-bearing with returns in a way that feels fair across a spectrum of scenarios.
To make these ideas tangible, the authors give an accessible interpretation: γℓ, the share for partner ℓ, can be thought of as a weighted mix of two terms. One term represents the partner’s capital-based reward, proportional to their stake κℓ. The other term represents the labor-based reward, proportional to the pair through the investment risk and opportunity. The result is a formula that reads as a balance between risk and reward, a kind of modern algebra for teamwork and trust. The paper also emphasizes that these quantities—ρ and Δ—aren’t simply abstract numbers. They can be computed from a range of models, from the classic Black–Scholes framework to more flexible econometric setups that better capture real-world asset dynamics.
Why this matters beyond the equation box
What makes this work exciting is not just the math, but the political economy of it. Islamic finance prizes profit-and-loss sharing as a way to tether finance to real economic activity and to limit speculative or debt-driven risk. The c-fair framework supports that philosophy by offering a principled way to adjust contracts as conditions change, rather than locking partners into a fixed split that may become unfair as markets move. In practical terms, this could help evolution in microfinance, small-business lending, and project finance by providing a transparent, model-based method to decide who gets what when the probability of success tilts in one direction or another.
The authors are explicit about applying their framework with a broad set of econometric tools. They point out that while the Black–Scholes world gives clean, tractable numbers, real markets are messier: distributions can be skewed, volatility can shift, and the timing of cash flows matters. The proposed approach is not a rigid template but a design principle that can be tuned with data. That flexibility is a feature, not a bug, because it means practitioners can calibrate c-fair shares to actual project characteristics, regulatory expectations, and the risk appetite of the participants.
What this could mean for the real economy
If this framework gains traction, the design of partnership contracts could become more dynamic and more aligned with actual contributions. For lenders and investors, it offers a language to describe how much of the upside they are taking on relative to their capital and management effort. For entrepreneurs and labor contributors, it provides a structured way to argue for fair compensation when the project’s risk profile shifts or when the team’s leadership evolves. And because the model honors the principle that losses are shared proportionally to capital, it preserves a core Shariah-compatible feature: risk sharing without guaranteed returns.
In a world where teams increasingly blend finance with hands-on labor, the idea of c-fair profit sharing is appealing because it rewards the labor that often drives innovation and execution. It also helps explain why some partnerships feel fair even when the math behind their profits is complex: the framework translates the intangible value of leadership, teamwork, and risk-taking into a concrete, auditable share of returns. The authors even hint at ongoing work to test these ideas against real-world data and to extend the system to more elaborate financial arrangements, signaling that this is not a one-off theoretical exercise but a stepping stone toward more resilient, transparent financing practices.
Where the conversation goes from here
The paper is careful to frame its contribution as a foundation for future work. It acknowledges that the calculation of ρ and Δ will hinge on how we model the dynamics of the underlying asset or project. The authors sketch how a Black–Scholes-type model can yield explicit expressions for these quantities, but they also point out that many asset returns do not follow a neat log-normal distribution in the real world. That acknowledgement is important: the strength of the proposed framework is its compatibility with a broad suite of econometric tools, from ARIMA to GARCH and beyond. The practical takeaway is not a single recipe but a set of design principles that finance teams can adapt to their data and their regulatory context.
The study, a collaboration rooted in the institutions behind it—ENSIIE and the LaMME group at Université Évry Paris-Saclay—offers a thoughtful, human approach to sharing risk and reward. It also serves as a reminder that mathematical models can be harnessed not just to chase profits, but to engineer fairness and trust in collective ventures. As the authors note, the real test will be how these ideas perform when confronted with real-world contracts, negotiation frictions, and evolving regulatory landscapes. If they pass that test, a future of more transparent, balanced, and resilient Islamic finance contracts could be within reach, one c-fair step at a time.
Lead researchers and the institutions behind the study: The work comes from a collaboration between ENSIIE and the Laboratoire de Mathématiques et Modélisation d Evry (LaMME), Université Évry Paris-Saclay (UMR CNRS 8071). The author listed as the lead is Abass Sagna.
