Fair profit sharing ratios of Islamic investment contracts[Clé de répartition équitable des contrats islamiques d'investissement]

We consider Islamic investment contracts, also known as Profit and Loss (PL) sharing contract, which may be mixed with an agency contract. Our aim is to determining both the profit sharing ratios and the induced expected payoff (at maturity) of each partner $\ell$ of the considered contracts according to its contribution to the project's overall success. To this end, we introduce the notion of {\em $c$-fair} profit sharing ratios ($c = (c_1, \ldots,c_d) \in (\mathbb R^{\star})^d$, where $d$ is the number of partners) which is an equilibrium approach that considers the contributions of the counterparts of these contracts in terms of both capital and labour. We show several new results that elucidate the relationship between these profit-sharing ratios and various important economic factors, such as {\em investment risk}, {\em investment opportunity}, contribution factors to both {\em capital} and {\em labour}, providing accordingly a way of choosing the profit sharing ratios in connection with the real economy. The design of our approach allows all econometrics or stochastic diffusion models to be used to compute or approximate the quantities of interest. In practice, this work could help Islamic Finance institutions to better understand these contracts, while opening up new possibilities in the engineering and the industry of Islamic Finance.