# Fair profit sharing ratios of Islamic investment contracts

Abass Sagna

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Abass Sagna. Fair profit sharing ratios of Islamic investment contracts. 2025. hal-05132054v2

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# Fair profit sharing ratios of Islamic investment contracts

ABASS SAGNA *

## Abstract

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 $l$ of the considered contracts according to its contribution to the project's overall success. To this end, we introduce the notion of *c-fair* profit sharing ratios ($c = (c_1, ..., c_d) \in (\mathbb{R}^*)^d$, 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 *investment risk*, *investment opportunity*, contribution factors to both *capital* and *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.

## 1 Introduction

This work aims to determine the *c*-fair profit-sharing ratios and expected profit shares of partners in Islamic investment contracts (*mudharabah* and *musharakah* contracts), possibly combined with *wakalah* (or agency) contract. The results presented in this paper are novel and are based on an equilibrium approach (employed, for example, in [8] for the specific case of a *mudharabah* contract) between the counterparties of the combination of the considered contracts. This approach takes into account the contribution of each partner to the project's overall success (which distinguishes it from the approach in [8]). The partners' contribution to the global success of the project is reflected in a weighting allocation system

<footnote>*ENSIIE & labouratoire de Mathématiques et Modélisation d'Evry (LaMME), Université Évry Paris-Saclay, UMR CNRS 8071, 23 Boulevard de France, 91037 Evry. E-mail: abass.sagna@ensiie.fr / abass.sagna@univ-evry.fr.</footnote>
that applies a weight coefficient of $1/c_\ell$ to each partner $\ell$, according to their contribution to capital and labour. We then aim to determine the profit-sharing ratios that equalise the associated rated contributing payoffs at maturity, i.e. $Pay_\ell / c_\ell$, for $\ell = 1, \dots, d$, where $Pay_\ell$ is the maturity payoff of partner $\ell$. We then present several new results that elucidate the relationship between profit-sharing ratios and several important economic factors, such as investment risk, investment opportunity, the contribution factor to labour and the contribution factor to capital, thereby connecting the choice of profit-sharing ratios to the real economy.

Before revisiting the definition of these contracts, it is important to note that a *mud-harabah* contract is a form of active partnership in which the capital contributor or limited partner (known as the *rabul maal*) provides the capital, while the entrepreneur or general partner (known as the *mudharib*) provides their expertise in the form of management or labour to help the funds grow. If the investment is successful, the profit is shared according to a profit-sharing ratio set by mutual agreement between the partners at the conclusion of the contract. In the event of a loss not attributable to the manager, the capital contributor loses their capital up to the amount of the loss incurred. However, the manager, who is not paid under the contract, loses his contribution to the work and the reward for his labour.

A *mucharakah* contract, or partnership agreement, is an active partnership between two or more partners who contribute the necessary capital to carry out a specific project, which may take the form of a company. Profits are shared according to the profit-sharing ratios agreed upon by the partners at the conclusion of the contract. Losses are shared by all partners in proportion to their capital contribution.

To be more explicit about our contribution, we will now consider an example to help the lecturer understand the context of this work.

Consider a corn farmer who sows a quantity $L$ of corn seeds with a harvesting time horizon $T$. The picked quantity $R_T$ (we suppose that $R_T$ is not identically equal to $L$: $R_T \neq L$) may be considered as a random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. The farmer's payoff at time $T$ will be $(R_T - L)^+$ (for any real number $x$, we denote $x^+ = \max(x, 0)$) in the event of successful harvest, and, $(L - R_T)^+$ in the event of a failed harvest. A (commonly acceptable) rational behaviour from the farmer is to invest in the labour when he expects a success of its operation rather than its failure, i.e., when

$$ \mathbb{E}(R_T - L)^+ \geq \mathbb{E}(L - R_T)^+ \quad \text{or, equivalently} \quad \Delta(R_T, L) := \mathbb{E}(R_T) - L \geq 0. \quad (1) $$

Any $T$ horizon and $R$-asset based investment with initial capital (which could be physical, as in the previous example) $L$ satisfying the Equation (1) at time $T$ will be called a *viable investment*.

Now, introduce an investor who can provide the quantity $L$ of seed that the farmer (who cannot get hold of it) needs in order to receive corn in return.
Here's what we might expect from the investor's behaviour:

1. If the investor strongly suspected that the farming operation would fail (e.g., $R_T < L$), he would likely not participate in it. This naturally imposes the viability constrain (1) that is the framework considered in this paper.

2. Now, if he expects the operation to be successful (as does the farmer, if we suppose that they have the same information), e.g., $\mathbb{E}(R_T) \ge L$, and takes part in corn farmer's (physical) capital, two situations may then occur at time $T$:

* the operation succeeds $R_T \ge L$: the farmer receives a reward for his labour, which he must share with the investor in proportion to their respective contributions to the capital,

* the operation fails $R_T < L$: the farmer loses the reward for his labour and cannot reasonably claim the harvest $R_T$ that belongs to the investor. Moreover, except in cases of mismanagement by the farmer, in which the investor may claim a refund of the difference between $L$ and $R_T$, it is unjustifiable for the investor to claim more than the entire crop, $R_T$. In fact, any additional advantage, in cash or any other kind in favour of the investor in this context is a ribâ (a precept which includes the principle of interest-bearing loans, but not only). This is prohibited by the Qur'an.

Islamic financing is based on the application of *Shariah* law and operates within the framework of ribâ-free business transactions. Its major basic principles include (see e.g. [1, 2, 3, 4, 5, 6, 7, 8]): the gharar-free (and maysir-free) financing principle (seeking to avoid excessive contractual uncertainty and speculative or gambling transactions); the asset-backed financial principle (that any financial transaction must be linked to tangible assets to promote real economic activities and discourage speculative practices); the risk-sharing principle in sales transactions (e.g. avoiding the sale of an item that is not actually held); and the profit and loss sharing principle in partnership contracts. The latter two principles can clearly be seen as tools that allow alternative trading activities to be built around the first two principles (the ribâ-free and gharar-free or maysir-free principles). This paper deals with c-fair profit-sharing ratios in profit-and-loss-sharing contracts, namely PL-sharing mudharabah and PL-sharing musharakah contracts.

To describe more explicit our contribution, consider a PL sharing-musharakah contract involving $d$ partners. Suppose that the partners of the PL sharing-musharakah contract contributed amounts $L_1, \dots, L_d$, to the capital $L = L_1 + \dots + L_d$. Let $\kappa_\ell = L_\ell/L$ be the equity stake of partners $\ell$, for every $\ell \in \{1, \dots, d\}$ and let $\gamma_1, \dots, \gamma_d$, with $\gamma_1 + \dots + \gamma_d = 1$, be the respective profit sharing ratios of the partners. As previously, $R_T$ denotes the net income process of the investment at the maturity $T$ of the contract. Then, the expected payoff of each partner $\ell$ at the maturity of the contract is given by

$$ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) = \gamma_\ell \mathbb{E}(R_T - L)^+ - \kappa_\ell \mathbb{E}(L - R_T)^+. \quad (2) $$
The fair profit sharing ratios $\gamma_\ell, \ell = 1, \dots, d$ is determined by solving the system of equations

$$ \text{Pay}_1(\gamma_1, \kappa_1) = \dots = \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) = \dots = \text{Pay}_d(\gamma_d, \kappa_d). $$

If all the partners contributed equally to the management of the project, we will see that this makes sense and leads to the solution $\gamma_\ell = \kappa_\ell, \ell = 1, \dots, d$. However, we know that this is not usually the case, so it is important to understand how to take into account the additional management or labour contribution of some partners, who may only be compensated for their extra contribution through their profit-sharing ratios. To illustrate this, let us consider a PL-sharing musharakah contract between two partners, whose contributions to the capital are denoted by $\kappa_1$ and $\kappa_2$. If the partners manage the project together with an equal contribution to management, the difference in their overall project contributions lies in their capital contributions, $\kappa_1$ and $\kappa_2$, respectively. It is the reasonable to assume that the fair profit-sharing ratios are $\gamma_1 = \kappa_1$ and $\gamma_2 = \kappa_2$. In fact, in this case, we have

$$ \frac{\text{Pay}_1(\gamma_1, \kappa_1)}{\kappa_1} = \frac{\text{Pay}_2(\gamma_2, \kappa_2)}{\kappa_2} \\ \text{or, equivalently, } \text{Pay}_1(\gamma_1, \kappa_1) = c \frac{\text{Pay}_2(\gamma_2, \kappa_2)}{\kappa_2}, \text{ with } c = \kappa_1/\kappa_2. $$

Now suppose that the two partners (Partner 1 and Partner 2) contribute the same amount of capital ($\kappa_1 = \kappa_2$), but, participate to different degrees to the management. They may then agree to weight their respective contributions to the project's overall success using the coefficients $c_1$ and $c_2$ in order to determine the profit-sharing ratios that reflect their respective payoffs. So, our aim is to find $\gamma_1$ and $\gamma_2$ such that $\text{Pay}_1(\gamma_1, \kappa_1) / c_1 = \text{Pay}_2(\gamma_2, \kappa_2) / c_2$. We will show (see Proposition 4.1) that, for any $c_1, c_2 > 0$,

$$ \gamma_1 = \frac{c_1}{c_1 + c_2} (1 - \rho(R_T, L)) + \kappa_1 \rho(R_T, L), \quad (3) $$

$$ \gamma_2 = \frac{c_2}{c_1 + c_2} (1 - \rho(R_T, L)) + \kappa_2 \rho(R_T, L), \quad (4) $$

where $\rho(R_T, L) = \mathbb{E}(L - R_T)^+ / \mathbb{E}(R_T - L)^+$ (observe that $\mathbb{E}(R_T - L)^+ > 0$ if $R_T \neq L$) is seen as the investment risk whereas $1 - \rho(R_T, L)$ is seen as the investment opportunity. It is very important to remark that the viability assumption (1) guarantees $\rho(R_T, L) \in [0, 1]$ and then, $\gamma_\ell \in [0, 1]$, for $\ell \in \{1, 2\}$. Also note that $\gamma_1 + \gamma_2 = 1$.

Furthermore, setting $\varpi_\ell = c_\ell / (c_1 + c_2)$, $\ell = 1, 2$, we show that the expected payoff of each partner $\ell \in \{1, 2\}$ is given by

$$ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) = \varpi_\ell \Delta(R_T, L), \quad \ell = 1, 2, \quad (5) $$

that is, the c-fair profit sharing ratios leads the two partners to share the expected investment profit $\Delta(R_T, L)$ according the respective weights $\varpi_\ell, \ell = 1, 2$.
It is also worth noting that a PL-sharing *mudharabah* contract appears to be a special case of a PL-sharing *musharakah* contract where the manager's (or partner 2's) share of the capital is zero. In fact, setting $κ_1 = 1$ and $κ_2 = 0$ in Equations (3) and (4) i yields the c-fair profit-sharing ratios obtained for a PL-sharing *mudharabah* contract:

$$ \gamma_1 = \frac{c_1}{c_1 + c_2} (1 - \rho(R_T, L)) + \rho(R_T, L), \quad (6) $$

$$ \gamma_2 = \frac{c_2}{c_1 + c_2} (1 - \rho(R_T, L)). \quad (7) $$

You can compare the previous result with that of [8] by setting $c_1 = c_2$ (see Remark 3.1 for more details).

In addition to the specific application of Equations (3) and (4) to the PL sharing - *mud-harabah* contract, it is important to understand the following remarks. Indeed, we can decompose it, for every $\ell \in \{1, 2\}$, as follows:

$$ \gamma_{\ell} = \underbrace{\varpi_{\ell} (1 - \rho(R_T, L))}_{\text{reward of labour}} + \underbrace{\kappa_{\ell} \rho(R_T, L)}_{\text{reward of capital}}. \quad (8) $$

The previous equation shows that the profit-sharing ratio is:

* a function of investment asset (or investment area), either through investment risk $\rho(R_T, L)$ or investment opportunity $1 - \rho(R_T, L)$;

* is proportional to the investment risk $\rho(R_T, L)$ and the invested capital;

* is proportional to the investment opportunity $1 - \rho(R_T, L)$ and the contribution to labour (or to the management).

We can still deduce from Equation (8) that a moderately risky investment (i.e. when the investment opportunity is greater than the investment risk, or equivalently, when $\rho(R_T, L) \le 1/2$) rewards a contribution to labour more than a contribution to capital, whereas a highly risky investment (i.e. when $\rho(R_T, L) \ge 1/2$) rewards a contribution to capital more. In other words, if a firm is looking to raise capital to finance its business, it is in its best interest to find a highly profitable (or low-risk) activity that will increase its share of the profits.

We also consider the case where the two partners of a PL sharing-musharakah contract appoint a manager (Partner 3) who is not one of them, and where they are bound to him by a PL sharing-mudharabah contract or a wakalah contract. In the former case where management is carried out via a PL-sharing mudharabah contract, the c-fair profit-sharing ratios are similar to those obtained for a PL-sharing musharakah contract with three partners, with the manager's capital contribution set to zero. When management is arranged via a wakalah contract with fixed remuneration (an amount $p$ paid $k$ times up to maturity $T$), and
if partners 1 and 2 and manager 3 agree to rate their respective payoffs by coefficients $c_1, c_2$ and $c_3$, then, the $(c_1, c_2, c_3)$-fair profit sharing ratios and periodic remuneration $p$ satisfy:

$$ \theta(r, T, k) p = \bar{\omega}_3 \Delta(R_T, L) \quad (9) $$

$$ \gamma_{\ell} = \left( \frac{\bar{\omega}_3}{2} + \bar{\omega}_{\ell} \right) (1 - \rho(R_T, L)) + \kappa_{\ell} \rho(R_T, L), \quad \ell \in \{1, 2\} \quad (10) $$

where the coefficient $\theta(r, T, k)$ comes from the updating value of the expected payoff of the manager. In fact, we may consider that their is a (T-period) discounted rate $r$ of return that could be arned from the investment on some good (in fact, when their is no actuatization, $\theta(r, T, k) = k$) and is given by $\theta(r, T, k) = ((1+r)^{\frac{T}{k}} - 1) / ((1+r)^T - 1)$. The weights $(\bar{\omega}_{\ell})_{\ell \in \{1,2,3\}}$ still be defined by

$$ \bar{\omega}_{\ell} = \frac{c_{\ell}}{c_1 + c_2 + c_3}, \quad \text{for all } \ell \in \{1, 2, 3\}. \quad (11) $$

Observe that $\gamma_1 + \gamma_2 = 1$. Furthermore, the expected payoff of each partner $\ell$ (even Partner 3 who has a fixed $k$-periodic remuneration up to maturity) is given by

$$ \text{Pay}_{\ell}(\gamma_{\ell}, \kappa_{\ell}) = \bar{\omega}_{\ell} \Delta(R_T, L), \quad \ell = 1, 2, 3. \quad (12) $$

Hence, the c-fair profit sharing ratios leads the three partners to share the *expected investment profit* $\Delta(R_T, L)$ with the respective weights $\bar{\omega}_{\ell}, \ell \in \{1, 2, 3\}$.

A typical application of this result is a situation involving three partners: an investor (Partner 1), who provides all the capital ($\kappa_1 = 1$); a bank (Partner 2), which acts as the investor's agent through a *wakalah* contract, arranging a fixed remuneration $p$, paid $k$ times up to maturity $T$; and a company (Partner 3), which requires funding and is bound to the bank by a *PL-mudharabah* contract ($\kappa_3 = 0$). This case is studied in Example 5.

We also generalize the previous results to contracts involving multiple partners. However, it is important to note that computing the profit-sharing ratios and related quantities in all previous results only requires the computation of the quantities $\rho(R_T, L)$ and $\Delta(R_T, L)$. These quantities can be explicitly calculated in certain models, such as when the stochastic process ($R_t$) is assumed to evolve according to the well-known Black-Scholes model, which assumes, among other things, that the distribution of $R_T$ is log-normal. Nevertheless, even if the values obtained from the Black-Scholes model can be considered as benchmarks (particularly when considering a lower volatility framework, in which case these benchmark values will be close to those expected from the use of more general models), it is well known that most financial datasets are not normally distributed. One of the most challenging tasks is to use econometric models such as ARIMA or (G)ARCH, or more general stochastic diffusion models such as local volatility or stochastic volatility, to find the appropriate model for R given an asset dataset. This task will be addressed in a separate paper to reduce the scope of this one.
The paper is organised as follows. First, we introduce and discuss the viability assumption and its implications. We then study c-fair profit-sharing ratios for a PL-sharing mudharabah contract and a PL-sharing musharakah contract. In the latter case, we first consider the matter of two and three partners to help the reader better understand the quantities at stake before generalising.

## 2 Viable investment

To introduce what we mean here by a viable investment, consider an investor with initial capital $L$ (the value of the investment may be a cash or its equivalent labour value) whom invests on an asset $R$ and expects the income $R_T$ at a future time $T > 0$. Two situations may happen:

* even the operation has succeeded and the income $R_T$ at time $T$ is greater than $L$, in which case its profit is $(R_T - L)^+$,

* or, it failed because $R_T \le L$. In this case its loss is $-(L - R_T)^+$.

Let us define the expected profit of such a given investor as $\mathbb{E}(R_T - L)^+$, and its expected loss as $\mathbb{E}(L - R_T)^+$. We also define the *expected investment profit* as $\Delta(R_T, L) = \mathbb{E}(R_T - L)^+ - \mathbb{E}(L - R_T)^+$, the *investment risk* (supposing that $R_T \neq L$, in which case $\mathbb{E}(R_T - L)^+ > 0$) as

$$ \rho(R_T, L) = \frac{\mathbb{E}(L - R_T)^+}{\mathbb{E}(R_T - L)^+} \quad (13) $$

and the *expected investment return* (or the *investment opportunity*) as

$$ \beta(R_T, L) := \frac{\Delta(R_T, L)}{\mathbb{E}(R_T - L)^+} = 1 - \rho(R_T, L). $$

We will say that the $T$-time maturity investment on the asset $R$ is viable if the $T$-term expected profit of the investor is greater than its expected loss: that is, $\Delta(R_T, L) > 0$, or, equivalently,

$$ \beta(R_T, L) > 0 \quad \text{or} \quad \rho(R_T, L) \in [0, 1]. \quad (14) $$

It then appears that any $T$-term and $R$-asset based investment with capital $L$ for which $\rho(R_T, L) > 1$ is not viable because its expected loss is greater than the expected profit. So, a $T$-term and $R$-asset based investment (or contract) will be considered to be viable if the statement (14) holds. Consequently $\rho(R_T, L)$ seems to be inherent to any productive activity like the trading activities or manual labour. The quantity $L$ may be seen in general as the production costs (or its equivalence in cash as in a manual labour, for example). Finally, as mentioned previously, $\rho(R_T, L)$ may also be seen as the *investment risk* associated to an activity which productive costs is $L$ and that generates a random income $R_T$ at time $T$.
**Definition 2.1.** A T-term and R-asset based investment (or contract) of the capital L is said to be viable if

$$ \Delta(R_T, L) > 0 \quad \text{or, equivalently, if} \quad \rho(R_T, L) \in [0.1[. \qquad (15) $$

We have the following easy result which characterises a viable investment (or contract).

**Proposition 2.1.** A T-term and R-asset based investment (or contract) of the capital L is viable if and only if

$$ \mathbb{E}(R_T) \geq L. \qquad (16) $$

**Proof.** We have

$$ \begin{aligned} \Delta(R_T, L) &= \mathbb{E}(R_T - L)^+ - \mathbb{E}(L - R_T)^+ \\ &= \mathbb{E}((R_T - L)1_{\{R_T \le L\}} + (L - R_T)1_{\{R_T > L\}}) \\ &= \mathbb{E}((R_T - L)(1 - 1_{\{R_T \le L\}}) + (L - R_T)1_{\{R_T > L\}}) \\ &= \mathbb{E}(R_T - L). \end{aligned} $$

Thus the result.

□

In the next sections, we discuss the notion of c-fair profit sharing ratios for the PL sharing - mudharabah. For ease of understanding, we will study the fair profit sharing ratios case before introducing the c-fair one. We start by recalling the definition of a PL sharing - mudharabah contract.

# 3 The PL sharing - mudharabah contract

We start by describing a mudharabah contract before studying fair profit sharing ratios of these contract.

## 3.1 What is a mudharabah contract?

A mudharabah contract between two entities (which may be, for example, a bank and a company) is a partnership contract where one partner provide the capital (the owner of the capital is called the rabb al-maal) and the other partner (the mudarib) brings his labour or his expertise to manage the funds in order to make a profit. The profit, if any, is shared between the partners according to an allocation key that must be known at the conclusion of the contract. Each partner may terminate the contract at any time as long as a maturity is not fixed or before the contract execution. After contract completion and when the maturity
is contractual, the contract will still be irreversible up to execution of book value of cash assets or up to the maturity. However, with a mutual agreement between the partners, the profit sharing ratios may be updated at any time before the maturity.

It is very important to notice that this contract is a participative contract on the fact that since the investor does not participate to the work, the counterpart of his capital raising is the labour or the expertise of the worker. Then, as a basic principle, the capital owner is compensated only on the profit (in the same way as the manager) of the investment and has, in the same way as the manager who loses the reward of his labour in case of loss, to support the losses, if any, except in the event of mismanagement (a risk that the funding owner may cover by requiring guarantees from the worker).

Finally, the *mudharabah* may be absolute or restrictive. It is restrictive when the funding owner makes restrictions on the domain of activity, for example, or on any other aspect he consider appropriated. Otherwise, the *mudharabah* is said to be absolute.

Our aim in the next section is to determine a fair (in the sense that equalise the payoff of the partners at the maturity of the contract) profit sharing ratios of the partners of a *mudharabah* contract.

## 3.2 Fair profit sharing ratio

Consider a PL sharing - mudharabah contrat between a owner of the capital or the rabb al-maal (which may be a bank) and a worker or a mudarib (a firm for example). We denote by $(\gamma_1, \gamma_2)$, with $\gamma_1 + \gamma_2 = 1$, the profit sharing ratios between the partners (the funding owner and the worker). We want to determine the fair profit sharing between the funding owner and the worker. We denote by $R_T$ the net income process of the investment at the maturity $T$ of the contract. So, at the maturity of the contract.

◇ the payoff of the funding owner is

$$ \gamma_1 (R_T - L)^+ - (L - R_T)^+, \qquad (17) $$

◇ and the payoff of the worker is

$$ \gamma_2 (R_T - L)^+. \qquad (18) $$

**Definition 3.1.** A T-term viable PL sharing - mudharabah contract on an asset $R$ is said to be fair if the expected maturity payoff of the funding owner is equal to the expected payoff of the worker. This means,

$$ \gamma_1 \mathbb{E}[(R_T - L)^+] - \mathbb{E}[(L - R_T)^+] = \gamma_2 \mathbb{E}[(R_T - L)^+]. \qquad (19) $$
Remark that we may also consider the actualised values of the expected payoffs in Definition 3.1 by considering that there is a discounted rate r of return that could be earned from the investment on some good of the expected payoffs. However, this is not necessary since Equation (19) will still hold true by simplifying the discounted factor on both sides of the equality. In the sequel we will consider actualised values of the payoff when it is necessary.

The following proposition follows directly from Definition 3.1.

**Proposition 3.1.** In a fair PL sharing - mudharabah contract, the funding owner and the worker ratios are given by

$$ \gamma_1 = \frac{1}{2}(1 + \rho(R_T, L)) \quad (20) $$

$$ = \frac{1}{2}(1 - \rho(R_T, L)) + \rho(R_T, L), \quad (21) $$

$$ \gamma_2 = \frac{1}{2}(1 - \rho(R_T, L)), \quad (22) $$

and their common expected payoffs at maturity is given by

$$ \frac{1}{2}\Delta(R_T, L), $$

meaning that they share equally the expected investment profit $\Delta(R_T, L) = \mathbb{E}(R_T) - L > 0$.

The proof is in fact easy and is given after the following remark that discusses on the comparison with the results of [8].

**Remark 3.1.** Define the success probability of the project to be $\beta = \mathbb{P}(R_T > L)$. Observing that $\mathbb{E}((R_T - L)^+) = \beta \mathbb{E}(R_T - L|\{R_T > L\})$, $\mathbb{E}((L - R_T)^+) = (1 - \beta) \mathbb{E}(L - R_T|\{R_T \le L\})$ and setting

$$ R^{-}(L) := \mathbb{E}(R_{T}|\{L \ge R_{T}\}) \quad \text{and} \quad R^{+}(L) := \mathbb{E}(R_{T}|\{R_{T} > L\}), \quad (23) $$

we obtain

$$ \gamma_1 = \frac{1}{2} \left( 1 + \frac{(1 - \beta) (L - R^{-}(L))}{\beta (R^{+}(L) - L)} \right). \quad (24) $$

This must be compared with the result of Proposition 3.2.4. in [8] which states that if an investor invests an amount L in a company that may generate revenue $R^+(L) > L$ with probability $\beta$, or revenue $R^-(L) < L$ with a probability $1 - \beta$, then the fair profit-sharing ratio is

$$ \gamma_1 = \frac{R^+(L) - R^-(L)}{2(R^+(L) - L)} + \frac{R^-(L) - L}{2\beta(R^+(L) - L)} = \frac{1}{2} \left( 1 + \frac{1 - \beta}{\beta} \frac{L - R^-(L)}{R^+(L) - L} \right). $$
This shows that our model matches with that of [8] if the generated revenue $R^-(L)$ and $R^+(L)$ considered in [8] are defined as in (23).

*Proof.* The proof follows directly from Equation (19). On the other hand, replacing in Equation (19) the ratios $\gamma_1$ and $\gamma_2$ by their values from (20), we see that in an $T$-term viable investment of amount $L$ on the asset $R$, the common expected payoff at maturity of both the owner of the capital and the worker is

$$ \frac{1}{2} \Delta(R_T, L). $$

This ends the proof.

□

From Equation (20), we can see that in a fair profit-sharing *mudharabah* contract, the investor's profit ratio $\gamma_1$ is at least 50%, increased by a quantity that is a non-decreasing function of the investment risk$\rho(R_T, L)$. This implies that, in such a contract, the manager's ratio increases with the expected profit of the investment (or decreases with the expected loss). Furthermore, the larger the expected profit compared to the expected loss, the greater the worker's ratio will be. If the project is risky to the extent that the expected profit of the investment is almost equal to the expected loss, $\gamma_1$, will be almost equal to 100%, meaning the manager must agree to work without expecting a paycheck. This may occur in the case of a risky project in the short term, but with long-term profitability, where the *mudharabah* contract is structured over different periods and the manager would accept a paycheck over a longer term.

Now, consider the situation in which we wish to determine the profit-sharing ratios $\gamma_1$ and $\gamma_2$ such that the expected payoff for the capital owner at maturity is $c$ times (with $c > 0$) the payoff for the worker. Denote by

$$ \text{Pay}_1(\gamma_1) := \gamma_1 \mathbb{E}[(R_T - L)^+] - \mathbb{E}[(L - R_T)^+] \\ \text{and} \quad \text{Pay}_2(\gamma_2) := \gamma_2 \mathbb{E}[(R_T - L)^+], $$

the respective expected payoffs of the capital owner and the labourer (or the manager). We suppose that

$$ \text{Pay}_1(\gamma_1) = c \text{Pay}_2(\gamma_2). $$

Setting $c = c_1/c_2$, this boils down to

$$ \text{Pay}_1(\gamma_1) / c_1 = \text{Pay}_2(\gamma_2) / c_2. $$

This motivates the introduction of a $(c_1, c_2)$-fair PL sharing - *mudharabah* contract, where $c_\ell$ is a real number for $\ell = 1, 2$. In the notation of the $(c_1, c_2)$-fair contract, the weighting coefficient $c_\ell$ is associated with the payoff $\text{Pay}_\ell$. This weighting accounts for
each partner's contribution to the project's overall success. As will be seen in Equation (28), these weighting coefficients determine the weight assigned to each partner in the sharing of the expected common investment profit $\Delta(R_T, L)$.

**Definition 3.2.** A T-term viable PL sharing - mudharabah contract on an asset R is said to be $(c_1, c_2)$-fair if

$$ \text{Pay}_1(\gamma_1) / c_1 = \text{Pay}_2(\gamma_2) / c_2. \qquad (25) $$

**Proposition 3.2.** In a $(c_1, c_2)$-fair PL sharing - mudharabah contract, the funding owner and the worker profit ratios are given by

$$ \begin{aligned} \gamma_1 &= \frac{1}{c_1 + c_2} (c_1 + \rho(R_T, L) c_2) \\ &= \frac{c_1}{c_1 + c_2} (1 - \rho(R_T, L)) + \rho(R_T, L), \end{aligned} \qquad (26) $$

$$ \gamma_2 = \frac{c_2}{c_1 + c_2} (1 - \rho(R_T, L)). \qquad (27) $$

Moreover, their respective expected payoff are given by

$$ \frac{c_1}{c_1 + c_2} \Delta(R_T, L) \quad \text{and} \quad \frac{c_2}{c_1 + c_2} \Delta(R_T, L), \qquad (28) $$

meaning that they will share the expected investment profit $\Delta(R_T, L)$ with the respective weights $\varpi_1 = c_1 / (c_1 + c_2)$ and $\varpi_2 = c_2 / (c_1 + c_2)$.

*Proof.* Keeping in mind that $\gamma_1 + \gamma_2 = 1$, we get (26)-(27) by solving Equation (25). The expected payoff $\text{Pay}_1(\gamma_1)$ and $\text{Pay}_2(\gamma_2)$ of the capital owner and the manager are obtained by replacing $\gamma_1$ and $\gamma_2$ by their values in (26)-(27). $\square$

**Example 1.** In this example we fix a value of $\rho(R_T, L) \in \{\frac{1}{4}, \frac{1}{2}\}$ and aim to find the $(c_1, c_2)$-fair profit sharing ratios in a PL sharing - mudharabah contract (between a bank and a firm, for example).

◇ Suppose we want to choose the profit sharing ratios such that the expected payoff at maturity pays off the bank $\frac{3}{2}$ times than it pays off the company, so that we may set $c_1 = 3$ and $c_2 = 2$ (for the firm). In this case, the $(c_1, c_2)$-fair profit sharing ratios are given as follows:

$$ \text{when } \rho(R_T, L) = \frac{1}{4}, \text{ then, } \gamma_1 = 70\% \text{ and } \gamma_2 = 30\%, $$

$$ \text{and when } \rho(R_T, L) = \frac{1}{2}, \text{ then, } \gamma_1 = 80\% \text{ and } \gamma_2 = 20\%. $$

With the previous choice of the ratios $\gamma_1$ and $\gamma_2$, we expect a sharing of the anticipated profit $\Delta(R_T, L)$ with the respective weighted shares $(c_1 / (c_1 + c_2), c_2 / (c_1 + c_2)) = (3/5, 2/5)$.
◇ Now, for a fair (when $(c_1, c_2) = (1, 1)$) profit sharing, we get

$$ 
\begin{array}{l}
\text{when } \rho(R_T, L) = \frac{1}{4}, \text{ then, } \gamma_1 = 62.5\% \text{ and } \gamma_2 = 37.5\%, \\
\text{and when } \rho(R_T, L) = \frac{1}{2}, \text{ then, } \gamma_1 = 75\% \text{ and } \gamma_2 = 25\%.
\end{array}
$$

This leads to a equal sharing of the anticipated expected profit $\Delta(R_T, L)$.

◇ Finally, when we want, because of several extra services developed by the company to ease the management and the tracking of the project for example, to choose the profit sharing ratios such that the bank may expect to be paid off at maturity 2/3 times than the company may be expecting (we may set in this case $c_1 = 2$ and $c_2 = 3$) then

$$ 
\begin{array}{l}
\text{when } \rho(R_T, L) = \frac{1}{4}, \text{ then, } \gamma_1 = 55\% \text{ and } \gamma_2 = 45\%, \\
\text{and when } \rho(R_T, L) = \frac{1}{2}, \text{ then, } \gamma_1 = 70\% \text{ and } \gamma_2 = 30\%.
\end{array}
$$

Choosing these profit sharing ratios ($\gamma_1$ still be for the bank and $\gamma_2$ for the company) leads to a sharing of the anticipated expected profit $\Delta(R_T, L)$ with the respective weighted shares $(c_1/(c_1+c_2), c_2/(c_1+c_2)) = (2/5, 3/5)$.

As discussed previously, we can see from this example that the profit-sharing ratio depends on exposure to risk $\rho(R_T, L)$. Recall that the investor is more exposed to risk than the funding manager because he will have to cover any losses. This exposure to investment risk is compensated by his profit ratio share: the riskier the investment, the higher the investor profit ratio.

# 4 The PL sharing - musharakah contract

We first define a musharakah contract before dealing with the mixed contracts that combines a musharakah contract with mudharabah and wakalah contracts.

## 4.1 Definition of a musharakah contract

A musharakah contract involves two or more partners combining their assets, labour or liabilities to generate profits. Each partner's share of the capital must be determined in advance. The profit-sharing ratios are agreed upon by the partners and can be amended at any time before maturity. However, losses are absorbed by all partners in proportion to their capital contribution. In this section, we aim to determine fair profit-sharing ratios between partners in a PL-sharing musharakah contract.
## 4.2 Fair profit sharing ratio of mixed contracts

Suppose we have two partners who have contributed amounts $L1$ and $L2$ to the capital $L$, such that their respective contributions to the capital are given by $\kappa_1 = L1/L$ and $\kappa_2 = L2/L$. Denote by $Pay_1(\gamma_1, \kappa_1)$ and by $Pay_2(\gamma_2, \kappa_2)$ the expected payoff of the partners. Then,

$$ \text{Pay}_1(\gamma_1, \kappa_1) = \gamma_1 \mathbb{E}(R_T - L)^+ - \kappa_1 \mathbb{E}(L - R_T)^+ \\ \text{and} \\ \text{Pay}_2(\gamma_2, \kappa_2) = \gamma_2 \mathbb{E}(R_T - L)^+ - \kappa_2 \mathbb{E}(L - R_T)^+ $$

where $\gamma_1$ and $\gamma_2$ stand respectively for the profit ratios of partner 1 and partner 2.

We want to determine the c-fair profit-sharing ratios, $\gamma_1$ and $\gamma_2$, and possibly the project manager's remuneration share, depending on the type of contract binding him to the entrepreneurs.

We first consider the situation where the partners of the *musharakah* contract decide to manage the project themselves.

### 4.2.1 Musharakah contract between two partners

If the partners manage the project together with an equal contribution to management, the difference in their overall contributions to the project lies in their respective capital contributions, $\kappa_1$ and $\kappa_2$. In this case, it would be reasonable to choose fair profit-sharing ratios of $\gamma_1 = \kappa_1$ and $\gamma_2 = \kappa_2$. So, we have:

$$ \text{Pay}_1(\gamma_1, \kappa_1) / \kappa_1 = \text{Pay}_2(\gamma_2, \kappa_2) / \kappa_2 \\ \text{or, equivalently,} \\ \text{Pay}_1(\gamma_1, \kappa_1) = c \text{Pay}_2(\gamma_2, \kappa_2), \text{ with } c = \kappa_1 / \kappa_2. $$

Now suppose that the two partners in the *musharakah* contract agree to appoint one of them (Partner 2, for example) to manage the project. In this case, it is not permitted to specify fixed remuneration for the manager, who is also one of the *musharakah* contract partners. However, it is possible to reward their additional contribution to the management of the project by increasing their profit-sharing ratio. As before, we must determine the $(c_1, c_2)$-fair profit-sharing ratios $(\gamma_1$ and $\gamma_2)$ by selecting the weighting constants $c_1$ and $c_2$ for their payoffs in a way that offsets Partner 2's additional responsibilities. That is:

$$ \text{Pay}_1(\gamma_1, \kappa_1) / c_1 = \text{Pay}_2(\gamma_2, \kappa_2) / c_2 \\ \text{or, equivalently,} \\ \text{Pay}_1(\gamma_1, \kappa_1) = c \text{Pay}_2(\gamma_2, \kappa_2), \text{ with } c = c_2 / c_1. $$

So, we have to solve the system of equations

$$ \begin{cases} \text{Pay}_1(\gamma_1, \kappa_1) / c_1 = \text{Pay}_2(\gamma_2, \kappa_2) / c_2 \\ \gamma_1 + \gamma_2 = 1, \quad \kappa_1 + \kappa_2 = 1, \end{cases} \qquad (29) $$
which solutions reads

$$ \gamma_{\ell} = \frac{c_{\ell}}{c_1 + c_2} - \left( \frac{c_{\ell}}{c_1 + c_2} - \kappa_{\ell} \right) \rho(R_T, L), \quad \ell \in \{1, 2\}. \qquad (30) $$

Next, we consider the situation in which the two partners of the musharakah contract appoint an external manager to oversee the project. There may be several ways to compensate him, but here we consider two types of contract that could bind the manager to the entrepreneurs: a wakalah contract, and a mudharabah contract.

### 4.2.2 When the entreprise is managed through a wakalah contract

Here, we suppose that the two partners of the musharakah contract are bound to the manager by a wakalah contract that provides for fixed remuneration (an amount $p$, paid out $k$ times up to maturity $T$), which will be included in their expenses. In this case, if there is a discounted rate of return $r$ that could be earned from investing in a good over a $T$-period, the actualised expected payoffs of the partners in the profit-sharing musharakah contract, $Pay_1(\gamma_1, \kappa_1, p)$ and $Pay_2(\gamma_2, \kappa_2, p)$, satisfy the following, respectively:

$$ \text{Pay}_1(\gamma_1, \kappa_1, p) = (1+r)^{-T} \left( \gamma_1 \mathbb{E}(R_T - L)^+ - \kappa_1 \mathbb{E}(L - R_T)^+ \right) - \frac{\text{Pay}_3(p, k)}{2} $$

$$ \text{Pay}_2(\gamma_2, \kappa_2, p) = (1+r)^{-T} \left( \gamma_2 \mathbb{E}(R_T - L)^+ - \kappa_2 \mathbb{E}(L - R_T)^+ \right) - \frac{\text{Pay}_3(p, k)}{2}. $$

In the previous equations, $Pay_3(p, k)$ is the (actualised expected) payoff of Partner 3 (which is supposed to be the manager). The quantity $Pay_3(p, k)$ is the present value of the sum of the future cash flow $p$ up to maturity $T$ of the contract. So, we have

$$ \text{Pay}_3(p, k) = \sum_{i=1}^{k} p (1+r)^{-\frac{T}{k}i} = p (1+r)^{-T} \frac{(1+r)^T - 1}{(1+r)^{\frac{T}{k}} - 1}. $$

As in previous cases, we aim to determine the $(c_1, c_2, c_3)$-fair profit-sharing ratios in a viable PL-sharing musharakah contract. In other words, we want to calculate the periodic remuneration $p$ up to maturity and the profit-sharing ratios $\gamma_1$ and $\gamma_2$ satisfying:

$$ \left\{ \begin{array}{l} \text{Pay}_1(\gamma_1, \kappa_1, p) / c_1 = \text{Pay}_2(\gamma_2, \kappa_2, p) / c_2 = \text{Pay}_3(p, k) / c_3 \\ \gamma_1 + \gamma_2 = 1. \end{array} \right. \qquad (31) $$

If we want to determine first $p$, it is suitable to rewrite the system (31) as a system of three equations with three unknowns:

$$ \left\{ \begin{array}{l} \text{Pay}_1(\gamma_1, \kappa_1, p) / c_1 c_2 = \text{Pay}_3(p, k) / c_2 c_3 \\ \text{Pay}_2(\gamma_2, \kappa_2, p) / c_1 c_2 = \text{Pay}_3(p, k) / c_1 c_3 \\ \gamma_1 + \gamma_2 = 1. \end{array} \right. \qquad (32) $$
Solving the system (32) leads to (setting $\theta(r, T, k) = ((1+r)^{\frac{T}{k}} - 1) / ((1+r)^T - 1)$)

$$ \theta(r, T, k) p = \bar{\omega}_3 \Delta(R_T, L), \quad \text{with} \quad \bar{\omega}_3 = \frac{c_3}{c_1 + c_2 + c_3}. \qquad (33) $$

Then, replacing $p$ by its value in the two first equations gives, for $\ell \in \{1, 2\}$,

$$ \gamma_{\ell} = \left( \frac{\bar{\omega}_3}{2} + \bar{\omega}_{\ell} \right) (1 - \rho(R_T, L)) + \kappa_{\ell} \rho(R_T, L), \qquad (34) $$

with

$$ \bar{\omega}_{\ell} = \frac{c_{\ell}}{c_1 c_2 + c_1 c_3 + c_2 c_3}. \qquad (35) $$

### 4.2.3 When the entreprise is managed through a mudharabah contract

We suppose here that the two partners of the musharakah contract are bound to the manager (the partner 3) by a PL-sharing mudharabah contract. In this case, the expected payoffs $Pay_1(\gamma_1, \kappa_1)$, $Pay_2(\gamma_2, \kappa_2)$ and $Pay_3(\gamma_3)$ of the partners are given respectively by:

$$ \text{Pay}_1(\gamma_1, \kappa_1) = \gamma_1 \mathbb{E}(R_T - L)^+ - \kappa_1 \mathbb{E}(L - R_T)^+ $$

$$ \text{Pay}_2(\gamma_2, \kappa_2) = \gamma_2 \mathbb{E}(R_T - L)^+ - \kappa_2 \mathbb{E}(L - R_T)^+ $$

$$ \text{Pay}_3(\gamma_3) = \gamma_3 \mathbb{E}(R_T - L)^+. $$

Finding the $(c_1, c_2, c_3)$-fair profit sharing ratio amounts to solve the system

$$ \left\{ \begin{array}{l} \text{Pay}_1(\gamma_1, \kappa_1) / c_1 = \text{Pay}_2(\gamma_2, \kappa_2) / c_2 = \text{Pay}_1(\gamma_3, \kappa_3) / c_3 \\ \gamma_1 + \gamma_2 + \gamma_3 = 1. \end{array} \right. \qquad (36) $$

To compute $\gamma_1$, we consider the following equivalent system:

$$ \left\{ \begin{array}{l} \text{Pay}_1(\gamma_1, \kappa_1) / c_1 c_2 c_3 = \text{Pay}_1(\gamma_1, \kappa_1) / c_1 c_2 c_3 \\ \text{Pay}_1(\gamma_1, \kappa_1) / c_1^2 c_3 = \text{Pay}_2(\gamma_2, \kappa_2) / c_1 c_2 c_3 \\ \text{Pay}_1(\gamma_1, \kappa_1) / c_1^2 c_2 = \text{Pay}_3(\gamma_3) / c_1 c_2 c_3 \\ \gamma_1 + \gamma_2 + \gamma_3 = 1. \end{array} \right. $$

Now, adding the left hand side and then, the right hand side of the three first equations of the previous system gives

$$ \left( \frac{1}{c_2 c_3} + \frac{1}{c_1 c_3} + \frac{1}{c_1 c_2} \right) \text{Pay}_1(\gamma_1, \kappa_1) = \frac{1}{c_2 c_3} \left( \mathbb{E}(R_T - L)^+ - \mathbb{E}(L - R_T)^+ \right). $$

It follows that

$$ \gamma_1 = \bar{\omega}_1 - (\bar{\omega}_1 - \kappa_1) \rho(R_T, L), \quad \text{avec} \quad \bar{\omega}_1 = \frac{c_1}{c_1 + c_2 + c_3}. \qquad (37) $$
The ratio $\gamma_2$ is obtained by solving the equivalent system

$$ 
\left\{ 
\begin{array}{l} 
\text{Pay}_2(\gamma_2, \kappa_2) / c_2^2 c_3 = \text{Pay}_1(\gamma_1, \kappa_1) / c_1 c_2 c_3 \\ 
\text{Pay}_2(\gamma_1, \kappa_2) / c_1 c_2 c_3 = \text{Pay}_2(\gamma_2, \kappa_2) / c_1 c_2 c_3 \\ 
\text{Pay}_2(\gamma_2, \kappa_2) / c_2^2 c_2 = \text{Pay}_3(\gamma_3) / c_1 c_2 c_3 \\ 
\gamma_1 + \gamma_2 + \gamma_3 = 1, 
\end{array} 
\right. 
$$

which solution is given by

$$ \gamma_2 = \frac{c_2}{c_1 + c_2 + c_3} - \left( \frac{c_2}{c_1 + c_2 + c_3} - \kappa_2 \right) \rho(R_T, L). \quad (38) $$

Using the condition $\gamma_1 + \gamma_2 + \gamma_3 = 1$ yields

$$ \gamma_3 = \frac{c_3}{c_1 + c_2 + c_3} (1 - \rho(R_T, L)). \quad (39) $$

All the results of the previous discussions is summarised in the following proposition.

**Proposition 4.1.** Consider a viable T-term and R-asset based PL - musharakah contract of 2 partners. Then:

1. If they manage themselves the project and agree, according to their contribution to the funding and the management of the project, to rate their respective payoff by the coefficients $c_1$ and $c_2$, then, the $(c_1, c_2)$-fair profit sharing ratios are given by

$$ \gamma_{\ell} = \varpi_{\ell} (1 - \rho(R_{T}, L)) + \kappa_{\ell} \rho(R_{T}, L), \quad (40) $$

$$ = \varpi_{\ell} \beta(R_{T}, L) + \kappa_{\ell} \rho(R_{T}, L), \quad \ell = 1, 2, \quad (41) $$

where, for every $\ell \in \{1, 2\}$, $\varpi_{\ell} = c_{\bar{\ell}} / (c_1 + c_2)$. Furthermore, the expected payoff of each partner $\ell$ is given by

$$ \text{Pay}_{\ell}(\gamma_{\ell}, \kappa_{\ell}) = \varpi_{\ell} \Delta(R_{T}, L), \quad \ell = 1, 2. \quad (42) $$

2. If they appoint a manager outside of them and enter into a wakalah contract with him, whereby the manager receives fixed remuneration (an amount $p$ paid out $k$ times up to maturity $T$), and if Partner 1, Partner 2 and the manager agree to rate their respective payoffs using the coefficients $1/c_1$, $1/c_2$ and $1/c_3$, then the fair profit-sharing ratios and periodic remuneration $p$ satisfy the following:

$$ \theta(r, T, k) p = \varpi_3 \Delta(R_T, L) \quad (43) $$

$$ \gamma_{\ell} = \left( \frac{\varpi_3}{2} + \varpi_{\ell} \right) (1 - \rho(R_T, L)) + \kappa_{\ell} \rho(R_T, L), \quad \ell \in \{1, 2\}. \quad (44) $$
The updating coefficient $\theta(r, T, k) = ((1+r)^{\frac{T}{k}} - 1) / ((1+r)^T - 1)$ and the weights $(\bar{\omega}_\ell)_{\ell \in \{1,2,3\}}$ are given by $\bar{\omega}_\ell = c_\ell / (c_1 + c_2 + c_3)$, for all $\ell \in \{1, 2, 3\}$. Moreover, the expected payoff of each partner $\ell$ (even the partner 3 who has a fixed k-periodic remuneration up to maturity) is given by

$$ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) = \bar{\omega}_\ell \Delta(R_T, L), \quad \ell \in \{1, 2, 3\}. \qquad (45) $$

3. If they appoint a manager (Partner 3) outside of them and are bound to him by a PL-sharing mudharabah contract, the $(c_1, c_2, c_3)$-fair profit sharing ratios are given by (keeping in mind that $\kappa_3 = 0$)

$$ \gamma_\ell = \bar{\omega}_\ell (1 - \rho(R_T, L)) + \kappa_\ell \rho(R_T, L), \quad \text{for all } \ell \in \{1, 2, 3\}, \qquad (46) $$

with $\bar{\omega}_\ell = c_\ell / (c_1 + c_2 + c_3)$, for all $\ell \in \{1, 2, 3\}$.

Still, the expected payoff of each partner $\ell$ is given by

$$ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) = \bar{\omega}_\ell \Delta(R_T, L), \quad \ell \in \{1, 2, 3\}. \qquad (47) $$

**Remark 4.1.** It is important to remark that putting $\kappa_1 = 1$ and $\kappa_2 = 0$ in Equation (41) leads to the $(c_1, c_2)$-fair profit sharing ratios obtained in Equation (26) and (27) for a PL-sharing mudharabah contract between two partners. Consequently, a PL-sharing mudharabah contract between two partners is a PL-sharing musharakah contract where of the owner of the capital (the partner 1) brings the whole capital (so that $\kappa_1 = 1$) and the other, the labour ($\kappa_2 = 0$). Additional discussions on the results of Proposition 4.1 are made in the discussions that follow the proposition 4.2.

Our aim now is to generalize Proposition 4.1 to mixed contracts with several partners.

## 4.3 Fair profit sharing ratio: the general framework

We consider a PL sharing - musharakah contrat between $d$ partners: Partner 1, ..., and Partner $d$ (when $d = 2$, the partnership may be for example between a bank and a company, between two companies, between two banks, etc), who contributed amounts to the capital are $L_1, \dots$, and $L_d$, respectively. So, let $L = L_1 + \dots + L_d$ be the initial capital, and, define $\kappa_\ell = L_\ell/L$, as the equity stake of partners $\ell$, for every $\ell \in \{1, \dots, d\}$. Finally, let us denote by $(\gamma_1, \dots, \gamma_d)$, with $\gamma_1 + \dots + \gamma_d = 1$, the profit sharing ratios of the partners. We want to determine the fair profit sharing between the partners.

As previously, $R_T$ denotes the net income process of the investment at the maturity $T$ of the contract. Then, at the maturity of the contract, the expected payoff of any Partner $\ell$ is given by

$$ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) = \gamma_\ell \mathbb{E}(R_T - L)^+ - \kappa_\ell \mathbb{E}(L - R_T)^+. \qquad (48) $$

We next define a $(c_1, \dots, c_d)$-fair PL - musharakah contract.
**Definition 4.1.** A viable T-term and R-asset based PL - musharakah contract of d partners is said to be $(c_1, \dots, c_d)$-fair if

$$ \text{Pay}_1(\gamma_1, \kappa_1) / c_1 = \dots = \text{Pay}_d(\gamma_d, \kappa_d) / c_d. \qquad (49) $$

The following result gives the profit sharing ratio of each partner of a PL - musharakah contract.

**Proposition 4.2.** Consider a viable T-term and R-asset based PL - musharakah contract of d partners whom manage themselves the project and agree, according to their contribution to the funding and the management of the project, to rate their respective payoff by the coefficients $1/c_\ell$, $\ell \in \{1, \dots, d\}$, then, the $(c_1, \dots, c_d)$-fair profit sharing ratios are given by

$$ \gamma_{\ell} = \bar{\omega}_{\ell} \beta(R_{T}, L) + \kappa_{\ell} \rho(R_{T}, L) \qquad (50) $$

$$ = \bar{\omega}_{\ell} (1 - \rho(R_{T}, L)) + \kappa_{\ell} \rho(R_{T}, L), \quad \text{for any } \ell \in \{1, \dots, d\}, \qquad (51) $$

where the weights $(\bar{\omega}_{\ell})_{1 \le \ell \le d}$ are defined for every $\ell \in \{1, \dots, d\}$ by

$$ \bar{\omega}_{\ell} = c_{\ell} / \sum_{i=1}^{d} c_{i}. \qquad (52) $$

Furthermore, the expected payoff of each partner $\ell$ is given by

$$ \text{Pay}_{\ell}(\gamma_{\ell}, \kappa_{\ell}) = \bar{\omega}_{\ell} \Delta(R_{T}, L), \quad \ell = 1, \dots, d. \qquad (53) $$

This means that the partners will share the expected investment profit $\Delta(R_T, L)$ with the respective weights $\bar{\omega}_\ell$.

One interesting case is when $d-1$ partners of a PL-sharing musharakah contract appoint a manager (Partner $d$) who is not one of them, and are bound to him by a PL-sharing mudharabah contract. This appears to be a particular case of Proposition 4.2, which we state as a corollary.

**Corollary 4.3.** Consider a viable T-term and R-asset based musharakah contract of $d-1$ partners who appoint a manager (Partner $d$) outside of them and are bound to him by a PL-sharing mudharabah contract. If they agree to weight their respective payoff by the coefficients $1/c_\ell$, $\ell \in \{1, \dots, d\}$, according to their contribution to the funding and the management of the project, then, the $(c_1, \dots, c_d)$-fair profit sharing ratios are given by

$$ \gamma_{\ell} = \bar{\omega}_{\ell} (1 - \rho(R_{T}, L)) + \kappa_{\ell} \rho(R_{T}, L), \quad \ell = 1, \dots, d-1, \qquad (54) $$

$$ \gamma_d = \bar{\omega}_d (1 - \rho(R_T, L)), \qquad (55) $$
where for any $\ell \in \{1, \dots, d\}$, $\bar{\omega}_{\ell}$ is defined from (52). Furthermore, the expected payoff of each partner $\ell$ is given by

$$ \text{Pay}_{\ell}(\gamma_{\ell}, \kappa_{\ell}) = \bar{\omega}_{\ell} \Delta(R_{T}, L), \quad \ell = 1, \dots, d. \qquad (56) $$

Let us make the following remarks before giving the proof of Proposition 4.2.

**Remark 4.2.** Owing to Equation (51) (or to equations (54) and (55)) that may be decomposed as

$$ \gamma_{\ell} = \underbrace{\bar{\omega}_{\ell} (1 - \rho(R_{T}, L))}_{\text{reward for labour}} + \underbrace{\kappa_{\ell} \rho(R_{T}, L)}_{\text{reward of capital}}, $$

we see that the $(c_1, \dots, c_d)$-fair profit-sharing ratio $\gamma_{\ell}$ of each partner $\ell$ appears to be a reward for his contribution to the capital ($\kappa_{\ell} \rho(R_{T}, L)$) and a reward for his contribution to the management (or labour) of the project ($\bar{\omega}_{\ell} (1 - \rho(R_{T}, L))$). We can also deduce that a moderately risky investment (i.e. an investment opportunity greater than the investment risk, or equivalently, $\rho(R_{T}, L) \le 1/2$) rewards a contribution to project management more than capital participation, whereas a high-risk investment (i.e. $\rho(R_{T}, L) \ge 1/2$) rewards a contribution to capital more. In other words, if we (a firm, for example) are looking to raise capital to finance our business, it is in our best interests to find a highly profitable (or low-risk) activity that will increase our share of the profits.

**Remark 4.3.** We still deduce from Equation (51) that for any $\ell, \ell' \in \mathcal{P}_d$,

$$ \gamma_{\ell} - \gamma_{\ell'} = (\bar{\omega}_{\ell} - \bar{\omega}_{\ell'}) (1 - \rho(R_{T}, L)) + (\kappa_{\ell} - \kappa_{\ell'}) \rho(R_{T}, L). \qquad (57) $$

Then, the difference in profit-sharing ratios between partners $\ell$ and $\ell'$ ($\gamma_{\ell} - \gamma_{\ell'}$) is balanced by the difference in their contributions to capital ($\kappa_{\ell} - \kappa_{\ell'}$) and the difference in their weighted contributions to management ($\bar{\omega}_{\ell} - \bar{\omega}_{\ell'}$). So, it follows from Equation (57) that:

1. If $\kappa_{\ell} \ge \kappa_{\ell'}$ and $\bar{\omega}_{\ell} \ge \bar{\omega}_{\ell'}$ (i.e., the partner $\ell$ contributes more than partner $\ell'$ in terms of both capital and labour) then, $\gamma_{\ell} \ge \gamma_{\ell'}$.

2. If $\kappa_{\ell} < \kappa_{\ell'}$ and $\bar{\omega}_{\ell} > \bar{\omega}_{\ell'}$, that is, partner $\ell$ contributes less capital than partner $\ell'$ but, more labour to the project's overall success, then,

$$ \gamma_{\ell} \ge \gamma_{\ell'} \iff \rho(R_{T}, L) \le \frac{\bar{\omega}_{\ell} - \bar{\omega}_{\ell'}}{(\bar{\omega}_{\ell} - \bar{\omega}_{\ell'}) - (\kappa_{\ell} - \kappa_{\ell'})} \in (0, 1). \qquad (9) $$

If in addition, $\rho(R_{T}, L) \le 1/2$, $\bar{\omega}_{\ell} - \bar{\omega}_{\ell'} > \kappa_{\ell'} - \kappa_{\ell}$ (Partner $\ell$ bridges the funding gap for capital through his management contribution), then, $\gamma_{\ell} \ge \gamma_{\ell'}$.
3. If $\bar{\omega}_\ell < \bar{\omega}_{\ell'}$ and $\kappa_\ell > \kappa_{\ell'}$ (partner $\ell$ contributes less in terms of management than partner $\ell'$, but more in terms of capital), then,

$$ \gamma_{\ell} \geq \gamma_{\ell'} \iff \rho(R_T, L) \geq \frac{\bar{\omega}_{\ell} - \bar{\omega}_{\ell'}}{(\bar{\omega}_{\ell} - \bar{\omega}_{\ell'}) - (\kappa_{\ell} - \kappa_{\ell'})} \in (0, 1). \quad (10) $$

If furthermore, $\rho(R_T, L) \geq 1/2$, $\kappa_\ell - \kappa_{\ell'} > \bar{\omega}_{\ell'} - c_\ell$ (Partner $\ell$ closes the gap in management contributions through his capital contribution), then, $\gamma_\ell \geq \gamma_{\ell'}$.

**Remark 4.4.** We may also remark that in the particular case where $c_i = 1$, for any $\ell \in \{1, \dots, d\}$, we have $\bar{\omega}_\ell = 1/d$ and Equation (51) becomes

$$ \gamma_{\ell} = \frac{1}{d} - \left( \frac{1}{d} - \kappa_{\ell} \right) \rho(R_T, L). \quad (58) $$

In this case, we have for any $\ell, \ell' \in \mathcal{P}_d$, $\gamma_\ell - \gamma_{\ell'} = (\kappa_\ell - \kappa_{\ell'})\rho(R_T, L)$, so that the difference in the profit-sharing ratio between two partners comes from the difference in their share of the capital contribution. If all have contributed the same amount of capital, i.e. $\kappa_\ell = \kappa_{\ell'}$, for any $\ell \neq \ell'$, then the fair profit-sharing ratio will be equal for all partners, regardless of the value of $\rho(R_T, L)$: $\gamma_\ell = \frac{1}{d}$, for any $\ell \in \{1, \dots, d\}$. This also suggests that the best way to absorb investment risk in a PL-musharakah contract is to encourage equal participation in capital and labour.

We depict in Figure 1 the $(c_1, c_2, c_3, c_4)$-fair profit sharing ratio as a function of the investment risk $\rho(R_T, L)$. We choose $(c_1, c_2, c_3, c_4) = (3, 5, 4, 2)$ and $(\kappa_1, \kappa_2, \kappa_3, \kappa_4) = (1/8, 5/8, 2/8, 0)$. We see that the profit sharing ratios may be a non-increasing as well as a non-decreasing function of $\rho(R_T, L)$, depending on the values taken by the $c_\ell$'s and the $\kappa_\ell$'s. This is in line with Remarks 4.2 and 4.3.

Let us prove now Proposition 4.2).

*Proof.* (of Proposition 4.2)] Let $\ell \in \{1, \dots, d\}$. Supposing that the contract is $(c_1, \dots, c_d)$-fair, we have in particular

$$ \left\{ \begin{array}{l} \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) / c_\ell = \text{Pay}_1(\gamma_1, \kappa_1) / c_1 \\ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) / c_\ell = \text{Pay}_2(\gamma_2, \kappa_2) / c_2 \\ \vdots \quad \vdots \quad \vdots \quad \vdots \quad \vdots \\ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) / c_\ell = \text{Pay}_d(\gamma_d, \kappa_d) / c_d \\ \gamma_1 + \dots + \gamma_d = 1. \end{array} \right. $$
<figure><img src="image_2.png" /><figcaption>Figure 1: We represent the $(c_1, c_2, c_3, c_4)$-fair profit sharing ratios $\gamma_\ell$ (in ordinate axis) as functions of the investment risk $\rho(R_T, L)$ (in abscissa axis): $(c_1, c_2, c_3, c_4) = (3, 5, 4, 2)$ and $(\kappa_1, \kappa_2, \kappa_3, \kappa_4) = (1/8, 5/8, 2/8, 0)$.</figcaption></figure>

It follows that

$$ 
\left\{ 
\begin{array}{l}
\text{Pay}_\ell(\gamma_\ell, \kappa_\ell) / (c_\ell \prod_{j \neq 1} c_j) = \text{Pay}_1(\gamma_1, \kappa_1) / (\prod_{j=1}^d c_j) \\
\text{Pay}_\ell(\gamma_\ell, \kappa_\ell) / (c_\ell \prod_{j \neq 2} c_j) = \text{Pay}_2(\gamma_2, \kappa_2) / (\prod_{j=1}^d c_j) \\
\vdots \quad \vdots \quad \vdots \quad \vdots \quad \vdots \\
\text{Pay}_\ell(\gamma_\ell, \kappa_\ell) / (c_\ell \prod_{j \neq d} c_j) = \text{Pay}_d(\gamma_d, \kappa_d) / (\prod_{j=1}^d c_j) \\
\gamma_1 + \dots + \gamma_d = 1.
\end{array}
\right.
$$

Adding the left and right hand sides of the $d$ first equations of the previous system gives (setting $E_1 := \mathbb{E}(R_T - L)^+$ and $E_2 := \mathbb{E}(L - R_T)^+$)

$$ (\gamma_{\ell} E_{1} - \kappa_{\ell} E_{2}) \frac{1}{c_{\ell}} \left( \sum_{i=1}^{d} \frac{1}{\prod_{j \neq i} c_{j}} \right) = (E_{1} - E_{2}) / \left( \prod_{i=1}^{d} c_{i} \right). $$

We deduce that

$$ (\gamma_{\ell} E_{1} - \kappa_{\ell} E_{2}) \frac{1}{c_{\ell}} \left( \sum_{i=1}^{d} \frac{c_{i}}{\prod_{j=1}^{d} c_{j}} \right) = (E_{1} - E_{2}) / \left( \prod_{i=1}^{d} c_{i} \right). $$
Consequently,

$$ (\gamma_{\ell} E_{1} - \kappa_{\ell} E_{2}) \frac{1}{c_{\ell}} \sum_{i=1}^{d} c_{i} = (E_{1} - E_{2}). $$

So,

$$ \gamma_{\ell} E_{1} - \kappa_{\ell} E_{2} = \bar{\omega}_{\ell} (E_{1} - E_{2}), $$

and this leads to Equation (51).

Now, the expected payoff $Pay_{\ell}(\gamma_{\ell}, \kappa_{\ell})$ of each partner $\ell$ is obtained by replacing in Equation (48), $\gamma_{\ell}$ by its value in (51). $\square$

**Example 2.** Consider a viable T-term and R-asset based PL - musharakah contract of 4 partners (say, a joint venture between 4 companies to develop a product) who agree, according to their contributions to the capital (of amount $L$) and the management, to rate their respective payoff by the coefficients $c_i$, $i = 1, \dots, 4$. We fix $\rho(R_T, L) \in \{\frac{1}{8}, \frac{3}{4}\}$, the rating coefficients $c_{\ell}$, and want to determine the $(c_1, \dots, c_4)$-fair profit sharing ratios.

Suppose that all the partners contributed to the same amount of the capital: $\kappa_{\ell} = 25\%$, for any $\ell \in \{1, 2, 3, 4\}$.

◇ If we suppose that they have an equal contribution to the management of the project so that they agree to set $(c_1, c_2, c_3, c_4) = (1, 1, 1, 1)$, then, the allocation key of the expected investment profit equals $(\bar{\omega}_1, \bar{\omega}_2, \bar{\omega}_3, \bar{\omega}_4) = (1/4, 1/4, 1/4, 1/4)$ and, regardless of the value taken by $\rho(R_T, L)$, the $(c_1, \dots, c_4)$-fair profit sharing ratios are equal (as discussed in Remark 4.4):

$$ \gamma_{\ell} = 25\%, \quad \text{for any partner } \ell \in \{1, \dots, 4\}. $$

◇ If we suppose now that partners 1 and 3 contributed more to the management of the project and that all partners agree to rate the maturity payoff as $(c_1, c_2, c_3, c_4) = (4, 2, 4, 1)$, then, the distribution key of the expected investment profit equals $(\bar{\omega}_1, \bar{\omega}_2, \bar{\omega}_3, \bar{\omega}_4) = \frac{1}{11} (4, 2, 4, 1)$. Furthermore,

$$ (\gamma_1, \gamma_2, \gamma_3, \gamma_4) = (35\%, 11\%, 35\%, 19\%) \quad \text{if } \rho(R_T, L) = 1/8 $$

$$ \text{and} \quad (\gamma_1, \gamma_2, \gamma_3, \gamma_4) \approx (29\%, 20\%, 29\%, 22\%) \quad \text{if } \rho(R_T, L) = 2/3. $$

Since $\kappa_{\ell} = \kappa_{\ell'}$, for any $\ell \neq \ell'$, $\ell, \ell' \in \{1, 2, 3, 4\}$, we deduce from Remark 4.3 that $\gamma_{\ell} \ge \gamma_{\ell'}$ whenever $\bar{\omega}_{\ell} \ge \bar{\omega}_{\ell'}$.
**Example 3.** The context still be the one of Example 2 where we suppose that $(\kappa_1, \kappa_2, \kappa_3, \kappa_4) = (1/8, 3/8, 1/8, 3/8) = (12.5\%, 37.5\%, 12.5\%, 37.5\%).$

◇ If $(c_1, c_2, c_3, c_4) = (1, 1, 1, 1)$, then, $(\varpi_1, \varpi_2, \varpi_3, \varpi_4) = (1/4, 1/4, 1/4, 1/4)$,

$$ (\gamma_1, \gamma_2, \gamma_3, \gamma_4) = (23\%, 27\%, 23\%, 27\%) \quad \text{if } \rho(R_T, L) = 1/8 $$

$$ \text{and} \quad (\gamma_1, \gamma_2, \gamma_3, \gamma_4) = (17\%, 33\%, 17\%, 33\%) \quad \text{if } \rho(R_T, L) = 2/3. $$

We see from this example that, since $c_{\ell, \bar{\ell}} = c_{\ell', \bar{\ell}'}$, for any $\ell \neq \ell'$, $\ell, \ell' \in \{1, 2, 3, 4\}$, then, $\gamma_{\ell} \ge \gamma_{\ell'}$ whenever $\kappa_{\ell} \ge \kappa_{\ell'}$ (referring to Remark 4.3).

◇ If $(c_1, c_2, c_3, c_4) = (1, 2, 1, 4)$, then, $(\varpi_1, \varpi_2, \varpi_3, \varpi_4) = \frac{1}{11}(4, 2, 4, 1)$,

$$ (\gamma_1, \gamma_2, \gamma_3, \gamma_4) = (33\%, 21\%, 33\%, 13\%) \quad \text{if } \rho(R_T, L) = 1/8 $$

$$ \text{and} \quad (\gamma_1, \gamma_2, \gamma_3, \gamma_4) \approx (20\%, 32\%, 20\%, 28\%) \quad \text{if } \rho(R_T, L) = 2/3. $$

We see from the foregoing that, when $\rho(R_T, L) = 1/8$ for example, then, $\gamma_1 \ge \gamma_4$ while $\kappa_1 \le \kappa_4$. This is because $\varpi_1 - \varpi_4 \ge \kappa_4 - \kappa_1$ (see Remark 4.3).

**Example 4.** Consider a viable $T$-term and $R$-asset based PL-musharakah contract of 3 partners that appoint a manager (still be, as in Example 2, a joint venture between 3 companies to develop a product). We suppose that they employ a manager outside of them: partner 4, and are bound with him by a PL-sharing mudharabah contract. Suppose that $(\kappa_1, \kappa_2, \kappa_3) = (1/3, 1/3, 1/3)$ and $\rho(R_T, L) = 1/2$.

◇ If $(c_1, c_2, c_3, c_4) = (1, 1, 1, 1)$, then, $(\varpi_1, \varpi_2, \varpi_3, \varpi_4) = (1/4, 1/4, 1/4, 1/4)$ and $(\gamma_1, \gamma_2, \gamma_3, \gamma_4) = (29\%, 29\%, 29\%, 13\%)$.

◇ If $(c_1, c_2, c_3, c_4) = (2, 2, 2, 3)$, then, $(\varpi_1, \varpi_2, \varpi_3, \varpi_4) = \frac{1}{9}(2, 2, 2, 3)$ and $(\gamma_1, \gamma_2, \gamma_3, \gamma_4) = (28\%, 28\%, 28\%, 16\%)$.

**Remark 4.5.** (About a diminishing musharakah contract) First, note that if we consider a viable $T$-term and $R$-asset-based musharakah contract, Equation (51) provides the profit-sharing ratios of the 'stockholders' of the musharakah contract at time $t = 0$. Now suppose that we have a secondary stock market in which stocks issued from a musharakah contract may be traded, or consider a diminishing musharakah contract in which some stockholders wish to sell a portion of their stocks at time $t$, where $0 < t < T$. In this situation, we need to price the stocks and evaluate the profit-sharing ratios at time $t$. This is a very challenging question for the islamic finance industry.

We next consider a generalisation of Proposition 4.1.2. to $d$ partners, where the $(d-1)$ partners of a PL sharing-musharakah contract appoint a manager (the $d$th partner) outside of them and are bound with him by a Wakalah (an agency) contract.
**Proposition 4.4.** Consider a viable T-term and R-asset based PL - musharakah contract of $d-1$ partners who appoint a manager outside of them (the d'th partner) and are bound with him by a Wakalah contract which arranges a fixed remuneration (a amount $p$ that is paid out $k$ times up to the maturity $T$). If the partners agree to rate their respective expected payoff by the coefficients $1/c_\ell$, $\ell \in \{1, \dots, d\}$, then, the $(c_1, \dots, c_d)$-fair profit sharing ratios and periodic remuneration $p$ satisfy

$$ \theta(r, T, k) p = \bar{\omega}_d \Delta(R_T, L) \quad (59) $$

$$ \gamma_\ell = \left( \frac{\bar{\omega}_d}{d-1} + \bar{\omega}_\ell \right) (1 - \rho(R_T, L)) + \kappa_\ell \rho(R_T, L), \quad \ell = 1, \dots, d-1 \quad (60) $$

where $\theta(r, T, k) = (1+r)^{\frac{T}{k}} - 1 / (1+r)^T - 1$, and the weights $(\bar{\omega}_\ell)_{1\le\ell\le d}$ are defined for every $\ell \in \{1, \dots, d\}$ by $\bar{\omega}_\ell = c_\ell / (c_1 + \dots + c_d)$.

Furthermore, the expected payoff of each partner $\ell$ (even the partner $d$ who has a fixed $k$-periodic remuneration up to maturity) is given by

$$ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell) = \bar{\omega}_\ell \Delta(R_T, L), \quad \ell = 1, \dots, d. \quad (61) $$

Before giving the proof, we may observe that $\gamma_1 + \dots + \gamma_{d-1} = 1$. We may also note that a typical example where Proposition 4.4 applies is the situation where we have three partners:

* an investor (partner 1) that brings all the capital (so that $\kappa_1 = 1$),

* a bank (partner 2) that acts as an agent for the investor via a *wakalah* contract and arranges a fixed remuneration $p$ that is paid out $k$ times up to the maturity $T$ and,

* a company (partner 3) that needs a funding and that is bound with the bank by a PL-mudharabah contract (so that $\kappa_3 = 0$).

*Proof.* The $(c_1, \dots, c_d)$-fair assumption implies for any $\ell = 1, \dots, d-1$,

$$ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell, p, k) / c_\ell = \text{Pay}_3(p, k) / c_d, \quad (62) $$

where for any $\ell = 1, \dots, d-1$,

$$ \text{Pay}_\ell(\gamma_\ell, \kappa_\ell, p, k) = (1+r)^{-T} \left( \gamma_\ell \mathbb{E}(R_T - L)^+ - \kappa_\ell \mathbb{E}(L - R_T)^+ \right) - \frac{\text{Pay}_3(p, k)}{d-1} \quad (63) $$

$$ \text{and } \text{Pay}_3(p, k) = p (1+r)^{-T} \theta(r, T, k). $$

It follows from (62) that for any $\ell = 1, \dots, d-1$,

$$ \left( \gamma_{\ell} \mathbb{E}(R_T - L)^{+} - \kappa_{\ell} \mathbb{E}(L - R_T)^{+} - \frac{p \theta(r, T, k)}{d-1} \right) / c_{\ell} = p \theta(r, T, k) / c_d. $$
Set, for all $\ell, d \in \{1, \dots, d\}$, $\mathcal{P}_d = \{1, \dots, d\}$ and $\bar{\mathcal{P}}_d(\ell) = \mathcal{P}_d\setminus\{\ell\}$. Multiplying both sides of the previous equation by $\prod_{i \in \bar{\mathcal{P}}_{d-1}(\ell)} (1/c_i)$ gives

$$ \left( \gamma_{\ell} \mathbb{E}(R_T - L)^+ - \kappa_{\ell} \mathbb{E}(L - R_T)^+ - \frac{p \theta(r, T, k)}{d-1} \right) \prod_{i=1}^{d-1} c_i^{-1} = p \theta(r, T, k) \prod_{i \in \bar{\mathcal{P}}_d(\ell)} c_i^{-1}. $$

Since $\gamma_1 + \dots + \gamma_{d-1} = 1$ and $\kappa_1 + \dots + \kappa_{d-1} = 1$, taking the sum over the $\ell$'s (from 1 to $d-1$) yields:

$$ \left( \Delta(R_T, L) - p \theta(r, T, k) \right) \prod_{i \in \mathcal{P}_{d-1}} c_i^{-1} = p \theta(r, T, k) \sum_{\ell \in \mathcal{P}_{d-1}} \prod_{i \in \bar{\mathcal{P}}_d(\ell)} c_i^{-1}. $$

We deduce that

$$ \begin{aligned} \Delta(R_T, L) \prod_{i \in \mathcal{P}_{d-1}} c_i^{-1} &= p \theta(r, T, k) \left( \prod_{i \in \mathcal{P}_{d-1}} c_i^{-1} + \sum_{\ell \in \mathcal{P}_{d-1}} \prod_{i \in \bar{\mathcal{P}}_d(\ell)} c_i^{-1} \right) \\ &= p \theta(r, T, k) \left( \prod_{i \in \bar{\mathcal{P}}_d(d)} c_i^{-1} + \sum_{\ell \in \mathcal{P}_{d-1}} \prod_{i \in \bar{\mathcal{P}}_d(\ell)} c_i^{-1} \right) \\ &= p \theta(r, T, k) \left( \sum_{\ell \in \mathcal{P}_d} \prod_{i \in \bar{\mathcal{P}}_d(\ell)} c_i^{-1} \right), \end{aligned} $$

and, finally, that,

$$ \theta(r, T, k) p = \varpi_d \Delta(R_T, L). $$

Now, replacing $\theta(r, T, k) p$ by its value in Equation (62) gives

$$ c_{\ell} \left( \gamma_{\ell} \mathbb{E}(R_T - L)^+ - \kappa_{\ell} \mathbb{E}(L - R_T)^+ - \frac{\varpi_d \Delta(R_T, L)}{d-1} \right) = c_d p \theta(r, T, k). $$

This implies that

$$ \begin{aligned} \gamma_{\ell} &= \left( \frac{\varpi_d}{d-1} + \frac{c_{\ell}}{c_d \varpi_d} \right) (1 - \rho(R_T, L)) + \kappa_{\ell} \rho(R_T, L) \\ &= \left( \frac{\varpi_d}{d-1} + \varpi_{\ell} \right) (1 - \rho(R_T, L)) + \kappa_{\ell} \rho(R_T, L). \end{aligned} $$

Now, the expected payoff of each partner $\ell$ is obtained by replacing $\theta(r, T, k) p$ and $\gamma_{\ell}$ by their values in Equation (62) $\square$

**Example 5.** Consider the following situation involving three partners. The first partner is an investor who provides all the capital (so that $\kappa_1 = 1$). The second partner is a bank that
acts as an agent for the investor via a *wakalah* contract, arranging a fixed remuneration *p* that is paid *k* times up to maturity (*T*). The third partner is a company that requires funding and is bound to the bank by a *PL-mudharabah* contract ($\kappa_3 = 0$).

Suppose that they invest the amount *L* on the company asset *R* for a maturity *T*. When the vector of weighting coefficients *c* = (*c*₁, *c*₂, *c*₃) is given, the respective shares of the bank, the investor and the company are given by:

$$
\begin{align*}
\theta(r, T, k) p &= \varpi_3 \Delta(R_T, L), \\
\gamma_1 &= \left(\frac{\varpi_3}{2} + \varpi_1\right) (1 - \rho(R_T, L)) + \rho(R_T, L), \\
\gamma_2 &= \left(\frac{\varpi_3}{2} + \varpi_2\right).
\end{align*}
$$

If for example, $c = (1, 1, 1)$ and $\rho(R_T, L) = 1/2$, then,

$$
\theta(r, T, k)p = \frac{1}{3}\Delta(R_T, L), \quad \gamma_1 = \frac{3}{4}, \quad \gamma_2 = \frac{1}{4},
$$

that leads to an equal sharing of the expected investment profit between the three partners,
means, $\frac{1}{3}\Delta(R_T, L)$ for each partner.

**5 Modelisation perspectives**

First remark that, following our approach, the computation of the profit sharing ratios and all the also quantities of interest depend on both $\rho(R_T, L)$ and $\Delta(R_T, L)$. To compute these quantities in practice, we need to specify a adequate model (dependent on the observations) for the dynamics of the investment asset $R$. Among the possible models, we may enumerate the usual econometrics models like autoregressive (AR) models, moving average (MA) models, autoregressive integrated moving average (ARIMA) models, autoregressive conditional heteroskedasticity (ARCH) models and their generalizations like (G)ARCH models, etc. We may also use diffusion model ruled by stochastic differential equations like those use in finance. Among these possible models used in finance, we next consider the example of the famous Black-Scholes model for which all the quantities of interest may be computed explicitly.

**5.1 The Black-Scholes model**

**5.1.1 The model and the main result**

In the Black-Scholes model, we suppose that the investment asset $(R_t)_{t \in [0,T]}$ evolves following the SDE:

$$
dR_t = \mu R_t dt + \sigma R_t dW_t, \quad R_0 = x_0, \quad t \in [0, T], \qquad (64)
$$
where $\mu$ is the instantaneous return, $\sigma$ is the instantaneous volatility, and, the process $(W_t)_{t \in [0,T]}$ is a standard brownian motion defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
We have the following easy result.

**Proposition 5.1.** If the dynamics of R is that of Black-Scholes, then, an investment on the asset R is viable if and only if $\mu > 0$. Furthermore, we have

$$ \rho(R_T, L) = - \frac{\Phi(-\vartheta(\mu, \sigma)) - e^{\mu T} \Phi(-\vartheta(\mu, \sigma) - \sigma \sqrt{T})}{\Phi(\vartheta(\mu, \sigma)) - e^{\mu T} \Phi(\vartheta(\mu, \sigma) + \sigma \sqrt{T})} \quad (65) $$

$$ \text{and} \quad \Delta(R_T, L) = L(e^{\mu T} - 1), \quad (66) $$

where $\vartheta(\mu, \sigma) = (2\mu - \sigma^2)T / (2\sigma\sqrt{T})$ and $\Phi(\cdot)$ is the cdf of the standard gaussian distribution.

*Proof.* The viability result follows from (66). In fact

$$ \Delta(R_T, L) > 0 \iff \mu > 0. $$

For the rest, we just need to show that $\mathbb{E}((L-R_T)^+) = L(\Phi(-\vartheta(\mu, \sigma)) - e^{\mu T} \Phi(-\vartheta(\mu, \sigma) - \sigma\sqrt{T}))$ and $\mathbb{E}((R_T-L)^+) = L(\Phi(\vartheta(\mu, \sigma)) - e^{\mu T} \Phi(\vartheta(\mu, \sigma) + \sigma\sqrt{T}))$, and use the relation $\Delta(R_T, L) = \mathbb{E}((R_T-L)^+) - \mathbb{E}((L-R_T)^+)$. $\square$

### 5.1.2 The investment risk as functions of the volatility and the return

To understand how the investment risk $\rho(R_T, L)$ behaves with respect to volatility and return parameters $(\sigma$ and $\mu)$, Figure 2 shows how $\rho(R_T, L)$ varies with $\sigma$ and $\mu$: with $\sigma$ ranging from 5% to 90% in steps of 1%, and $\mu$ in $\{0.05, 0.07, 0.09, 0.11, 0.13\}$.

As expected, we see that

* for a fixed level of return, the investment risk is a nondecreasing function of the volatility: the greater the volatility of the investment asset, the higher is the investment risk;

* for a fixed level of volatility, the investment risk is a non-increasing function of the return: the greater the yield, the smaller it is.

### 5.1.3 Practical use of the previous results

Consider a bank that aims to invest in a specific sector of the economy (Fintechs, for example) through a mudharabah contract with a maturity period of $T$. Suppose the bank wants to capture 2/5 of the sector's development potential. Below are some steps the bank could take to achieve this.
<figure><img src="image_3.png" /><figcaption>Figure 2: Black-Scholes model: $\rho(R_T, L)$ as a function of the volatility $\sigma$, for different values of the return $\mu$.</figcaption></figure>

1. Suppose we have access to a dataset for the investment sector and that we can model the dynamics of the investment asset $R$ up to maturity $T$;

2. We must ensure that the investment is viable by checking (from the model) that $\Delta(R_T, L) > 0$. Also compute (or approximate) $\rho(R_T, L)$ from the model.

3. Since we want to capture 2/5 of the $\Delta(R_T, L)$, we may invest amount $L$ in a large number of (independent) projects (say, more than 30, having in mind the law of large number). Now, if for example,

$\rho(R_T, L) = 1/4$, we broker a cut of $\gamma_1 = 55\%$ of the profits for each contract.

$\rho(R_T, L) = 1/2$, we broker a cut of $\gamma_1 = 70\%$ of the profits for each contract.

Both previous strategies guarantee to hold 2/5 of $\Delta(R_T, L)$.

# 6 Conclusion

In this work, we introduce a new concept of c-fair profit-sharing ratio based on Islamic investment contracts, such as a musharakah or mudharabah contract, which may be combined with a wakalah contract. The c-fair concept allocates profits based on each partner's contribution to the overall success of the undertaking.
We show that the profit-sharing ratio is proportional to the investment risk and investment opportunity weighted by capital and labour contribution, respectively.

In order to calculate the quantities of interest, we must model the dynamics of the investment asset. This step depends, of course, on the nature of the available dataset and must be thoroughly investigated for any given dataset. Among the possible models used in finance and econometrics, we considered the famous Black-Scholes model, for which all the quantities of interest can be computed explicitly. This model also enables us to observe the expected behaviour of investment risk, which increases with volatility and decreases with return.

Finally, this work opens up new perspectives in understanding these Islamic contracts and in engineering and the industry of Islamic finance. It may also help Islamic finance institutions to value Islamic investment contracts and related financial instruments such as Sukuk instruments based on these Islamic investment contracts, as well as manage *Takaful* (Islamic insurance) via a *wakalah* model or a *wakalah-moudharabah* model.

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