Fair profit sharing ratios of Islamic investment contracts

The aim of this work is to calculate the fair profit-sharing ratios and the expected payoffs at maturity for each partner in islamic investment contrats (or instruments), based on profit and loss-sharing (PL-sharing). These investment contracts, known as {\em mudarabah} and {\em musharakah}, can be compared to {\em limited partnerships} and {\em joint ventures} (including all types of venture, such as joint-stock companies, partnerships, etc) in conventional finance. To compute these quantities, we introduce the notion of c-fair profit-sharing ratios, where $c = (c_1, \ldots,c_d) \in (\mathbb R^{\star})^d$ and $d$ is the number of partners. This constitutes an equilibrium approach that accounts for the contributions of the contracting parties in terms of both capital and labour. We show that the $c$-fair profit-sharing ratio of each partner is the sum of their contributions to capital and labour, weighted by some economic factors that we identify as {\em investment risk and opportunity} respectively. We deduce that, in the $c$-fair model, the expected investment profit is distributed among the contracts partners according to the shares $\varpi_{\ell} = c_{\ell} \, / \, (c_1 + \ldots + c_d)$, that correspond to their respective contribution weights to the venture's overall success. We extend these results to mixed contract that combine one or both previous contracts with an agency (known as {\em wakalah}) contract.