Journal of Islamic Monetary Economics and Finance, Vol. 9, No. 3 (2023), pp. 511 - 532
p-ISSN: 2460-6146, e-ISSN: 2460-6618

RETAKAFUL CONTRIBUTIONS MODEL USING MACHINE
LEARNING TECHNIQUES
Kouach Yassine1, EL Attar Abderrahim2 and EL Hachloufi Mostafa3
1

University Hassan II of Casablanca, Morocco, yassine.kouach-etu@etu.univh2c.ma
2
University Hassan II of Casablanca, Morocco, abderrahimelattar@univh2c.ma
3
University Hassan II of Casablanca, Morocco, mostafaelhachloufi@univh2c.ma

ABSTRACT
Driven by the need to manage risk by the newly created Moroccan Takaful operators,
the Moroccan Insurance and Social Welfare Control Authority has authorized the
Central Reinsurance Company to create a ReTakaful window for the purpose of
reinsuring Takaful operations. Nevertheless, the main challenge is determining the
appropriate ReTakaful model for the Moroccan Islamic insurance sector by ensuring
compliance with Shariah. With this in mind, this article aims to determine the optimal
ReTakaful contributions model for the Moroccan Takaful industry via Machine
Learning algorithms. We select the best model by comparing the performance of
each algorithm. The achieved results of this study demonstrate the potential of using
Machine Learning algorithms to compute ReTakaful contributions that are more
suitable for Takaful operators and more optimal for the ReTakaful operator.
Keywords: ReTakaful, Takaful, Reinsurance, Treaty, Machine learning, Probability of ruin.
JEL classification: G22; C51; C52; C53.

Article history:
Received
: January 22, 2023
Revised		
: July 28, 2023
Accepted
: August 31, 2023
Available online : September 29, 2023
https://doi.org/10.21098/jimf.v9i3.1681

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I. INTRODUCTION
Re-Takaful, which is an alternative Islamic reinsurance to a conventional
reinsurance, is a process whereby a Re-Takaful company agrees to bear all or part
of the loss of primary Takaful companies (Billah, 2019a) if the risks insured are
above the normal underwriting or claim (Arbouna, 2000).
Indeed, ReTakaful helps Takaful operators to manage the risk of having
insufficient financial resources when multiple unwanted damages occur at the
same time. For this reason, ReTakaful aims to protect the Takaful operators from
the threat of insolvency and to consolidate the financial stability of the Takaful
operators (Arbouna, 2000; Billah, 2019a).
Re-Takaful principal is similar to conventional reinsurance principal in the
way that the ReTakaful operator (insurer) accepts to insure a part of the defined
risk in a specified category, if it exceeds prudent underwriting limits (Hassan &
Mollah, 2018), in the Takaful policy transferred by the Takaful operator (ceding
company or the cedent). In return, the Takaful operator transfers also a part of
the premium as contributions depending on the part of the risk covered by the
ReTakaful operator.
However, there are some differences between conventional reinsurance and
ReTakaful operations. The central difference is that ReTakaful transactions should
take into consideration the Shariah principles in the same way as Takaful does
(Arbouna, 2000). It should avoid Riba (interest), Gharar (uncertainty), and maysir
(gambling) (Billah, 2019a; IFSB-18, 2016). In addition, The Re-Takaful contracts
must be based on profit and loss sharing. If there is a profit, the Takaful operators
will get a share of the profit based on a predetermined ratio (Billah, 2019a; Budd,
2016).
Moreover, the ReTakaful operators’ income is limited to the fee (wakalah) and/
or profit sharing (mudarabah) (Archer et al., 2011). In addition, in the event of a
deficit in the participants’ fund, the ReTakaful company will make a free loan as
in Takaful insurance (Budd, 2016). Thus, ReTakaful is generally divided into two
categories, family ReTakaful and general ReTakaful (IFSB-18, 2016).
A ReTakaful operator, or ‘‘Takaful of Takaful’’ (Archer et al., 2011), has the
same role as the Takaful operator, that is, reducing the risks of loss faced by the
insured (Ali & Markom, 2021).
With that in mind, the newly created Takaful operators in Morocco are invited
to cover their risks and to place a certain part of their risk with another operator.
Though, they should not obtain coverage from conventional reinsurance operators
as their practice is contrary to Shariah principles (Billah et al., 2019). This explains
the need for Islamic reinsurance operators.
In this view and in order to ensure the proper progress of Islamic insurance in
Morocco, Insurance and Social Welfare Control Authority (ACAPS) has authorized
SCR, the Central Reinsurance Company, to open a reinsurance window (ReTakaful
window (Budd, 2016)) for Takaful operators based on the principles of Shariah, in
this case, ReTakaful. The challenge that remains is to determine the appropriate
ReTakaful model that fits the Islamic insurance industry in Morocco by ensuring
compliance with Shariah.
According to the literature review, we find that most works related to
ReTakaful or Takaful are theoretical in nature. There are limited works that deal
with the practical and actuarial aspects of Takaful and ReTakaful.

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In regards to the Takaful, Khouaja (2015) propose generalized linear models in
pricing the contribution of a Takaful automobile insurance. Derkaoui & Halourou
Souley (2016) design in their thesis an Islamic borrower insurance product adapted
to a Mourabaha model taking into account the Moroccan regulations on Takaful.
In the recent work related to Takaful pricing, Kouach et al. (2022) seek in their
article to price Takaful automobile insurance using Machine Learning algorithms
and by interpreting the Takaful insurance model introduced in Law 59.13 and in
compliance with Shariah.
Within the context of the ReTakaful, Ahmad et al. (2015) attempt to examine the
Shariah issues related to ReTakaful and whether the risks in ReTakaful are shared
by Takaful operators and ReTakaful funds or simply transferred to ReTakaful
operators.
Alshammari & Altwijry (2023) introduce new perspectives to review the
ReTakaful industry’s legitimacy by exploring the views of six Shariah scholars on
whether the fatwā on the permissibly of using conventional reinsurance based on
the precept of ḍarūrah is still applicable (Alshammari & Altwijry, 2023).
However, with the exception of the work by El Attar & El Hachloufi (2022),
these works are limited to the theoretical aspect of ReTakaful through qualitative
research without developing a pricing method adopted for ReTakaful products.
El Attar & El Hachloufi (2022) design an actuarial model of Takaful insurance
contributions associated with Mourabaha real-estate contracts that meets Shariah
standards by introducing ReTakaful as a tool of security and technical support. In
addition, they seek to find the optimal structure of ReTakaful, which maximizes
the technical surplus under certain constraints. El Attar & El Hachloufi (2022)
develop an optimization program that allows not only to calculate the individual
Takaful contributions but also to determine the appropriate form of ReTakaful.
However, their work deals only with the family ReTakaful.
In this view, this article aims to propose pricing ReTakaful contributions based
on Machine Learning in the context of general ReTakaful especially ReTakaful of
an automobile Takaful product. We model ReTakaful treaty using Decision tree,
Random Forests, and Neural Networks. As well, we seek to choose the best model
to price ReTakaful contributions by taking into consideration two ReTakaful
methods, namely the “quota share” treaty and the “excess of loss” treaty. At
the same time, we propose a model of ReTakaful in conformity with Moroccan
regulations and in compliance with Shariah.
In addition, the purpose of this article is to contribute to the improvement
of the actuarial ReTakaful literature and to encourage practitioners to integrate
Machine Learning algorithms into the Islamic finance industry, especifically the
Takaful and ReTakaful industry.
The rest of the paper is structured as follows. The next section introduces
the theoretical framework of ReTakaful by highlighting ReTakaful methods
and ReTakaful models. In section III, we proceed to present the methodology,
Machine Learning algorithms adopted, and the criteria for an optimal ReTakaful
notably the probability of ruin minimization. The results of the application of our
proposed model for the ReTakaful of the damage and collision guarantee of the
motor insurance branch are presented section IV. And finally, section V presents
the conclusions and recommendations.

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II. THEORETICAL FRAMEWORK OF RETAKAFUL
2.1. ReTakaful Methods
The methods used by ReTakaful operators are the same as conventional
Reinsurance methods (Figure 1). There are two main categories: Treaty ReTakaful
and Facultative ReTakaful. The scope of this article is limited to the Treaty
ReTakaful.

Figure 1.
ReTakaful MEthods

⁻⁻

Facultative ReTakaful:
Facultative ReTakaful allows the reinsurance of risks individually based on
each risk (case by case) (Gunardi et al., 2013). Indeed, the ReTakaful operator has
the ability to accept or reject any risk transferred by the Takaful operator. In other
words, the ReTakaful operator considers each risk separately and it retains the
‘faculty’ to accept or reject each risk offered (Billah, 2019b).
⁻⁻ Treaty ReTakaful:
Takaful operator can transfer during a specified period the risks wanted to
be reinsured but the ReTakaful Operator, contrary to the Facultative ReTakaful,
agrees to accept all risks in the scope of the treaty with the Takaful operator (Ali &
Markom, 2021; Hassan & Mollah, 2018).
For Treaty ReTakaful contract, the losses (Xi)i∊[1,…,N] are shared between the
Takaful operator and the ReTakaful operator as follows:
(1)
Where (XiT)i∊[1,…,N] the risk portion retained by Takaful operator, (XiR)i∊[1,…,N] the
risk portion retained by ReTakaful operator, and N represents the number of the
claims.
Treaty ReTakaful includes two methods (Figure 1), in occurrence, the
proportional method and the non-proportional method. According to the
proportional method, the ReTakaful can be expressed on a “quota-share” or
“surplus-share” agreement. By contrast, the non-proportional method can be
expressed on an “excess of loss” or “stop loss” agreement.

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2.1.1. Proportional Method:
When the ReTakaful operator opts for a proportional method, the ReTakaful
operator undertakes to reinsure part or a percentage of the risk transferred by the
Takaful operator. In return, the latter transfers the same part of the premiums to
the ReTakaful operator.
⁻⁻ Quota share treaty:
A quota share treaty is a pro-rata ReTakaful contract, whereby the Takaful
operator and ReTakaful operator accept to share premiums and losses of each
and every risk within a defined category of business (Billah, 2019b) according to a
fixed percentage (Essadic Amine & Mouhssine Yassine, 2018). In other words, each
disaster, regardless of its weight, is shared in the same proportion between Takaful
and ReTakaful operators (Arbouna, 2000).
In the quota share treaty, the ReTakaful operator limits its liability to a fixed
proportion of the risk α called the proportionality factor or the cession rate. In this
case, the proportions retained by the ceding company and the ReTakaful operator
are expressed as follows:
and

(2)

⁻⁻

Surplus treaty:
The cession rate in the case of surplus treaty is not fixed. Indeed, it varies
according to the maximum amount of claim and the full retention which
corresponds to a fixed amount retained by the ceding.
(3)
(4)
Where (mi > 0)i∊[1,…,N] represents the maximum amount of claim and (ai > 0)i∊[1,…,N]
represents the fixed amount retained by the ceding.
2.1.2. Non-proportional Method:
Non-proportional reinsurance only responds if the loss suffered by the insurer
exceeds a certain amount, which is called the “retention” or “priority” (Hasan,
2011). In fact, non-proportional ReTakaful allows the Takaful operator to limit its
liability to an amount L>0 called retention limit.
⁻⁻ Excess of loss treaty:
This treaty is a form of ReTakaful under which recoveries are available when
a given loss exceeds the retention limit as defined in the agreement (Billah, 2019b).
The retention limit is applied to each claim such as the Takaful operator portion
and the ReTakaful operator portion are expressed as follow:
(5)

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(6)
⁻⁻

Stop loss Treaty:
Unlike the excess of loss treaty, the retention limit is not applied to each claim.
Indeed, the retention limit is applied to the total claims.
(7)
(8)
Stop loss protects against catastrophic losses or large shock claims by protecting
reserves after a certain threshold is reached, as well as protecting the integrity of
the organization and its cash flow (Essadic Amine & Mouhssine Yassine, 2018).
2.2. ReTakaful models
ReTakaful operators have the option to operate using organizational structures that
are similar to those used in the Takaful insurance industry, such as Mudharabah,
Wakala, and hybrid models.
2.2.1. Mudharabah Model
Mudharabah ReTakaful model is operating in the same way as the Mudharabah
Takaful model. The ReTakaful operator acts as an entrepreneur (Mudharib)
who undertakes the ReTakaful business activities. Meanwhile, the participants,
comprised of different Takaful operators acting as capital providers (Rab al mal)
who give ReTakaful contributions to the ReTakaful Risk Fund for the ReTakaful
operator. These contributions are made based on tabarru (Htay et al., 2014). In
accordance with the Mudharabah principle, both the ReTakaful operator and
Takaful operator will agree to share any profits and losses at a predetermined rate
(Alkhan & Hassan, 2020; Bensed & Fasly, 2020).

Figure 2.
Mudharabah ReTakaful Model

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2.2.2. Wakala Model
Under Wakala model, the ReTakaful operator acts as an agent (Wakeel) who
manages the ReTakaful Risk Fund. On the other hand, the Takaful operators act as
principals (Muwakil) (Htay et al., 2014). In return for the service rendered by the
ReTakaful Operator as Wakeel, Wakeel receives an agreed management fee called
the Wakala fee, which is usually a percentage of the contributions paid by the
participants (IFSB-18, 2016). However, the profits and losses belong entirely to the
participants (Htay et al., 2014).

Figure 3.
Wakala ReTakaful Model

2.2.3. Hybrid Model
The hybrid or mixed model combines the best sides of Mudharabah and Wakala
operating models (Mohamed Yusuf, 2011). In ReTakaful’s hybrid model, the
operator fulfills both the roles of Wakeel and Mudharib. As a Wakeel, the
ReTakaful operator manages the ReTakaful Risk Fund, while as a Mudharib, the
operator manages the investment activities. The ReTakaful operator receives a
Wakala fee and a pre-determined percentage share of the investment proceeds as
remuneration (IFSB-18, 2016).

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Figure 4.
Hybrid ReTakaful Model

III. METHODOLOGY
3.1. Context of the Study
Since 2014, the monetary authorities in Morocco have undertaken an ambitious
strategy to promote Islamic finance in the country. This strategy involves
establishing an Islamic financial system that operates in parallel with the existing
conventional system, with the aim of catering to the financial needs of individuals
who hold religious convictions.
In 2018, a significant milestone was achieved when the central bank granted
consent to participating banks to offer Islamic banking products. This has allowed
them to introduce Islamic banking products and the issuance of Sukuk into the
market, which is considered a crucial step in expanding the scope of the financial
landscape.
Despite these advancements, the Islamic financial ecosystem in Morocco
is still evolving. The emergence of an Islamic insurance market alongside the
conventional insurance sector has evolved from being a mere option to a pressing
necessity to meet market demand and ensure financial inclusivity.
The demand for Islamic insurance products becomes even more evident,
especially in situations such as real estate transactions. Participating banks offer
Murabaha (real estate) products but without an insurance policy to cover the
death of borrowers. This absence not only hampers the expansion of real estate
loans but also introduces complications after the borrower’s death. The need for a
comprehensive solution is apparent for both financial institutions and borrowers.
Recognizing these gaps, Morocco introduced Takaful Insurance in 2016
through Law 59.13 by amending and supplementing the existing Insurance
Code outlined in Law 17.99. However, it was not until the end of 2021 that the
Ministry of Finance’s decree pertaining to Takaful insurance contracts came into
effect. This substantial progress was accentuated by the subsequent actions of the
Insurance and Social Welfare Supervisory Authority (ACAPS), which published

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the application circular for the law and collaborated closely with the esteemed
Superior Council of the Ulemas (Religious Scholars) to grant approval to three
operators, thus inaugurating Takaful insurance operations in the country.
However, the transition to a complete Islamic insurance environment has
highlighted the need to institute a risk management strategy that aligns with
Shariah principles.
In effect, these newly created Takaful operators in Morocco need to cover their
risks and to place a certain part of their risks with another operator. However, these
operations must be in accordance with Shariah principles and hence the need for
Islamic reinsurance operators. In view of this fact, ACAPS authorized SCR, Central
Reinsurance Company, to open a reinsurance window or ReTakaful window. This
innovative mechanism enables Takaful operators to access reinsurance services
based on the Shariah principles.
Nevertheless, ReTakaful in Morocco does not yet have the fundamental
theoretical and practical framework for the pricing of ReTakaful contributions.
Accordingly, the challenge that remains is to determine a suitable framework
for pricing ReTakaful contributions in Morocco. Furthermore, selecting an
appropriate ReTakaful model that adapts to the Moroccan Islamic insurance sector
while ensuring compliance with Shariah principles is key to the development of
the ReTakaful framework.
Aiming to explore and provide innovative solutions, this paper proposes
the use of Machine Learning algorithms in computing ReTakaful contributions
to improve accuracy and efficiency in determining risk-sharing arrangements in
accordance with Shariah principles.
3.2. Research Approach
Our research objective is to develop an appropriate Moroccan ReTakaful model for
pricing ReTakaful contributions aligned with Shariah principles through machine
learning algorithms for accurate and efficient risk-sharing arrangements. For this
reason, we adopt the following steps in our research approach.
Firstly, our focus lies on the interpretation of the ReTakaful model as proposed
by Law 59.13 amending and supplementing law 17.99 on the insurance code. This
involves a detailed examination of its nuances to better understand the ReTakaful
structure allowed in Morocco.
Afterward, we proceed to the calculation of ReTakaful contributions using
two distinct ReTakaful methods, namely, a method of the proportional category,
specifically the quota share treaty, and a method of the non-proportional category,
specifically the excess of loss treaty. In order to model the contributions, we exploit
the power of Machine Learning algorithms, namely Decision trees, Random
Forests, and Neural Networks.
In the next step, we embark on the task of selecting the suitable Machine
Learning algorithm for each ReTakaful treaty method. This selection is motivated
by an evaluation of the performance measures based on minimizing the root mean
square error.
Finally, we engage in a particular assessment to determine the most appropriate
ReTakaful treaty method. This evaluation is guided by an essential criterion,
namely the minimization of the probability of ruin.

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By following this proposed research approach based on legal interpretation,
advanced modeling, and decision-making, we aim to contribute to the development
of a ReTakaful framework that not only meets regulatory and Shariah requirements
but also optimizes risk management practices.
3.3. Proposed Model of ReTakaful
Law 59.13 defines Takaful reinsurance as being “a reinsurance operation carried
out in accordance with the assent of the Superior Council of Ouléma having
as its objects the coverage of the risks provided for in the Takaful reinsurance
treaty by a Takaful reinsurance fund managed, in return for remuneration of
management, by an insurance and reinsurance company approved to carry out
Takaful reinsurance operations”. In addition, surplus Takaful Reinsurance Funds
shall be distributed among Takaful Operators in accordance with the Takaful
Reinsurance Fund Management Regulations, after deduction of Takaful advances,
if they exist. No part of the technical and financial surpluses may be granted to the
Takaful reinsurance company managing the Fund. The distribution of technical
and financial surpluses can only take place after the constitution of provisions and
reserves.
According to Law 59.13, we can conclude that the model proposed by the
cited law is similar to the Wakala ReTakaful model. Hence, Moroccan ReTakaful
operators will be paid by an agreed management fee as a Wakala fee.
Consequently, the contribution paid by the Takaful operators will be calculated
as follows:
(9)
Wakala fee represents the operating expenses and the margin Wakala (the
profit from the Wakala). The ReTakaful contribution can be expressed as follows:

(10)
We are interested in calculating the ReTakaful pure contribution, which is the
proportion of the expected cost of claims retained by the ReTakaful operator.
Under the quota share treaty, the ReTakaful pure contribution E[SR] is
calculated as follows:
(11)
Where α is the cession rate.
Under the excess of loss treaty, the ReTakaful pure contribution E[SR] is
expressed as follows:
(12)

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Where L is the retention limit.
We assume that the random variables X1R,…,XNR are independently and
identically distributed, and the costs of claims X1R,…,XNR and the number of claims
N are independent. In this case, the average total charge of claims is equal to the
average number multiplied by the average cost:
(13)
For the purpose of ReTakaful pure contribution calculation, we compute the
frequency of claims which is the same of Takaful pure contribution E[N], and the
average cost of claims E[XR] retained by the ReTakaful operator. We will apply
Machine Learning algorithms in order to determine the frequency and the average
cost of claims.
3.4. Machine Learning Algorithms
In this study, we seek to exploit the potential of non-parametric methods, in this
case, Machine Learning algorithms. Indeed, statistical learning methods allow
actuaries to go beyond the limits of traditional methods (Kouach et al., 2022).
Traditional methods such as linear regressions or generalized linear models
often do not provide satisfactory results due to assumptions of linearity,
assumptions of independence of explanatory variables, assumptions about
data distribution, difficulties in processing large datasets, and manual feature
engineering (Bellina, 2014; Fotia Santsa, 2018; Fox, 2022).
These limitations justify the use of Machine Learning algorithms as alternative
solutions. In fact, machine learning algorithms can adopt nonlinear relationships
between variables, capture complex interactions, handle high-dimensional data,
and perform automatic feature selection (Gotlieb et al., 2022; Xie et al., 2023).
In addition, Machine Learning algorithms make no assumptions about data
distribution (Sheetal et al., 2023). Statistical learning theory assumes just one
hypothesis, in occurrence, the data to predict (Bekkaye & Zari, 2023).
Machine Learning usually provides systems with the ability to learn and
enhance from experience automatically without being specifically programmed
(Saint-Cirgue, 2019; Sarker, 2021).
Machine Learning includes four learning algorithms categories depending
on the type of data, namely supervised learning, unsupervised learning, semisupervised learning, and reinforcement learning.
The category of Machine Learning algorithms used in this article is supervised
learning. This choice is explained by the type of data available, which is labeled.
Indeed, our model explains the response variable “ReTakaful charge” or “claim
frequency” according to the explanatory variables.
In order to determine the frequency of claims and the average cost of claims
retained by ReTakaful operator, we apply supervised learning algorithms
appropriate for regression, particularly, Decision tree, Random Forest, and Neural
Networks.

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3.4.1. Decision tree:
The Decision tree algorithm is chosen for its ability to handle both categorical and
numerical data (Saed & Jaharadak, 2022), which makes this algorithm suitable
for predictive modeling. The advantage of the Decision tree lies in its simplicity
of visual interpretation to explain the results of the model (Nanfack et al., 2022).
Additionally, the Decision tree has the ability to capture complex nonlinear
relationships within the data (McNamara et al., 2022).
The Decision tree algorithm consists of splitting the tree from node to leaf
containing the outputs based on criteria such as “Gini impurity” or “entropy” for
information gain.
3.4.2. Random Forest:
Random Forest is an ensemble supervised learning technique. This method uses
“parallel assembling” which fits several Decision trees in parallel, on different
dataset sub-samples, and uses majority voting or averages for the outcome or final
result (Sarker, 2021).
The Random Forest algorithm is used to solve overfitting and variance
problems often associated with individual Decision trees (Correa-Garcia et
al., 2022; Santhosh et al., 2022). By combining multiple Decision trees through
ensemble learning, Random Forest improves the generalization and accuracy of
predictions (Correa-Garcia et al., 2022; McNamara et al., 2022).
3.4.3. Neural Networks:
The computational architecture of Artificial Neural Networks combines three
processing layers, namely the Input layer, the Hidden layer, and the Output layer.
The input layer contains the explanatory variables that will be transmitted to
the hidden layer thanks to the combination function. In order to introduce the nonlinearity, an activation function will be applied to the linear combination output.
Neural Networks are used for their capacity to capture nonlinear relationships
within data (Paturi et al., 2022). Their hierarchical structure allows feature
extraction (Taye, 2023). The neural Networks are particularly useful for tasks
involving high-dimensional data (Gotlieb et al., 2022).
After we have chosen the Machine Learning algorithms, we divide the database
into two sub-databases, namely the training database to train each algorithm and
the testing database to evaluate the performance of each algorithm. In order to
optimize the parameters, we use cross-validation instead of having a third base
i.e. database validation.
3.4.4. Performance Measure:
In order to choose the best algorithm, we use the mean square error as a measure
of performance particularly suited to regression problems where the goal is to
predict continuous numerical values.

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(14)
Where yi the actual value of the observation i, ŷi the predicted value of the
observation i, and n the total number of observations.
The mean square error is easy to interpret. A value of zero indicates perfect
predictions, while higher values indicate
​​
non-negligible prediction errors.
3.5. Optimization Program
In this work, we consider a ReTakaful treaty pricing is optimal for the ReTakaful
operator when this treaty avoids the risk of having a total amount of contributions
lower than the total amount of claims charges. In other words, it avoids the ruin
or insolvency of the ReTakaful operator. For this purpose, we choose to use the
criterion of minimization of the probability of ruin to evaluate the two treaty
models of ReTakaful namely the “quota share” and “excess of losses”.
First, the probability of ruin is defined as the probability that the operator’s
reserve will become negative due to the excess of the total burden of claims over
the contributions collected.
The reserve of an operator with an initial capital k and who receives net
premiums Pnet with a view to damaging claims X1,X2,…., is written as follows:
(15)
(16)
With

(17)

We assume that the portion of risk transferred to the ReTakaful operator is
priced according to a pricing method based on the principle of mathematical
expectation. Hence, the net premium is expressed as follows:
(18)
(19)
Where ηR is the safety loading.
So
(20)
(21)
(22)
(23)

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And
(24)
Therefore, the ruin probability function is:
(25)
(26)
(27)
Adding E(Rt) and dividing the left and right sides of the inequality by
we find:

,
(28)
(29)
(30)
(31)

Note that the left side of the inequality converges to a normal law N(0,1). Using
the central limit theorem and a normal approximation, the probability of ruin is
given by:
(32)
Where:
FZ is the distribution function of the random variable

which follows

the normal law N(0,1).
By definition

represents the factor of safety.

Once the ReTakaful model has been determined within the context of Law
59.13, presenting the Machine Learning algorithms used in this study and
considering the probability of ruin as a criterion for optimal ReTakaful pricing,
we proceed to formulate our optimization program in order to select the optimal
ReTakaful model.
The objective of our proposed program is to choose the optimal ReTakaful
treaty pricing that minimizes the objective function ψ(R). Our optimization
program then consists in finding the minimum probability of ruin ψ(R) among the
different ReTakaful methods and models adopted in this study.

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In this perspective, we have the following optimization program:

(33)

Where ηR is the safety loading.
The optimization program is written in the case of the quota share treaty, as
follows:

(34)

Where ηR is the safety loading and α is the cession rate.
Our objective then for this program is to find the optimal ηR and α parameters
that minimize the probability of ruin in the case of the quota share treaty.
Under the excess of loss treaty, the optimization program is formulated as
follows:

(35)

Where ηR is the safety loading and L is the retention limit.
The objective then for this program consists in finding the optimal parameter
ηR that minimizes the probability of ruin in the case of the excess of loss treaty.

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Figure 5.
Optimization Process

•
•
•
•
•

Our optimization process (Figure 5) takes place as follows:
Choose the best algorithm to model the frequency of claims and the average
cost of claims of ReTakaful under both methods, quota share and excess of
losses, based on the minimum mean squared error MSE.
Compute the ReTakaful contributions under both methods, quota share and
excess of losses.
Assess the objective function ψ(R) under defined constraints.
Select the minimum probability of ruin for both the quota share and excess of
losses scenarios.
Select the optimal ReTakaful method that has the lowest probability of ruin.

3.6. Data and Materials
In order to choose the best ReTakaful treaty and to determine the portion to be
transferred to the ReTakaful operator, we use two open-source databases. One
database contains policy information, and the other database contains the history
of claims charges. We are interested in the pricing of the portion of ReTakaful
operator for the damage and collision guarantee of the motor insurance branch.
To compute the portion of the risk accepted by the ReTakaful operator, we
need to determine the frequency and average claim of ReTakaful. The database
intended for frequency modeling contains 50,000 observations. For the modeling of
the average claim of ReTakaful, we use a database that contains 1735 observations.
For the ReTakaful “quota share” treaty, we will test the Machine Learning
algorithms according to different levels of cession rate α={5%,10%,….,95%}. For the

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ReTakaful “excess of losses” treaty, we retain L=5000 as retention limit. In addition,
different safety loading levels ηR={5%,10%,….,100%}, are used to determine the net
premium.
In our study, to calculate the probability of ruin or the reserve, we set the initial
capital to zero since the ReTakaful operator only manages the fund and in case of
insufficiency it lends the Takaful operators a Qard Hassan. In addition, the horizon
for calculating the probability of ruin is one year.
For ReTakaful fees, in general, the management fees can be between 10%
and 20%. We retain 20% in our case. Since the ReTakaful operator works in the
same economic context as the conventional reinsurer, we retain the average ratio
between the premiums and the operating expenses of conventional operators.
According to the annual report of ACAPS for the year 2021, the average operating
expense ratio is equal to 28%. Calculations are performed using Python.
IV. RESULTS AND ANALYSIS
The results illustrated in table 1, that outline the mean squared error on the test
dataset regarding the application of the Machine Learning algorithms to the
quota share ReTakaful method, show important outputs. Indeed, according to the
mean squared error, the Neural Network algorithm demonstrates an exceptional
ability to minimize prediction errors. The Neural Network algorithm achieves a
considerably low mean squared error of 0.00688669, a value that is almost zero.
This result significantly outperforms the error metrics associated with the other
algorithms. The Decision tree has the highest value of prediction error with 5398.90
followed by Random Forest with 608.1508.
For this reason, we choose the Neural Network algorithm to model the
average cost claims of ReTakaful under the quota shares ReTakaful method with
the specified cession rate of 10%.
Table 1.
Mean Squared Error of Machine Learning Algorithms Under Quota Share
Retakaful Method for the Average Claims Cost of ReTakaful
Algorithm
Neural Network
Random Forest
Decision tree

Cession rate

Mean squared error / test data

0.1
0.05
0.05

0.00688669
608.1508
5398.90

Regarding the application of the Machine Learning algorithms to the excess
of losses ReTakaful method, the overall outcomes are unsatisfactory. Notably,
the Random Forest is the algorithm that minimizes the mean squared error well
among the applied algorithms. The Random Forest algorithm has the ability to
produce a particularly low mean squared error of 3.17E + 06, that makes it possible
to estimate the average cost of ReTakaful under the excess of losses. In contrast,
the Neural Network algorithm is the worst algorithm that does not significantly
minimize the mean squared error with MSE equal to 3.37E+06.

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In light of this finding, we opt for Random Forest algorithm to model the
average cost claims in the context of the excess of losses ReTakaful method, using
a retention limit of 5000.
Table 2.
Mean Squared Error of Machine Learning Algorithms Under Excess of Losses
Retakaful Method for the Average Claims Cost of ReTakaful
Algorithm
Decision tree
Neural Network
Random Forest

Mean squared error / test data
3.23E+06
3.37E+06
3.17E+06

We model the number of claims for both methods at once because the frequency
of claims is the same for both treaty types.
As shown in Table 3, the results highlight that among the applied algorithms,
the Random Forest is the model that performs well. The Random Forest algorithm
achieves a notably low mean squared error of 0.036280 at the test database.
However, it is notable that the Neural Network algorithm does not present
the same level of performance in terms of error minimization, followed by the
Decision tree algorithm with 0.036298 and 0.036333 mean squared error values
respectively.
Random Forest algorithm is selected to model the frequency of claims in
ReTakaful operations considering its performance in minimizing mean squared
error.
Table 3.
Mean Squared Error of Machine Learning Algorithms for the Frequency Claims of
ReTakaful
Algorithm
Decision tree
Neural Network
Random Forest

Mean squared error / test data
0.036333
0.036298
0.036280

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Figure 6.
Probability of Ruin under Quota Share and Excess of Loss ReTakaful Treaty in
Function the Security Loading

After we have chosen the best algorithms to model the frequency of claims
and the average cost of claims of ReTakaful under both methods, quota share and
excess of losses, we compute the probability of ruin of both ReTakaful models.
This calculation is performed on different safety loading levels, as illustrated in
Figure 6. On one hand, we find that the probability of ruin is a decreasing function
of the safety loading. On the other hand, the probability of ruin is minimum in the
case of the ReTakaful quota share method when the safety loading is above 40%. In
this case, the probability of ruin is less than 5%. On the other hand, the probability
of ruin using the excess of loss method is less than 5% when the security load is
above 15%. In other words, there is a high probability that the ReTakaful operator
will be insolvent under the quota share method if the ReTakaful operator does
not increase the security loading. However, the probability of granting a Quard
Hassan to Takaful operators is lower with the excess of loss method.
According to this application, we recommend pricing the ReTakaful
contributions under the excess of loss method by utilizing the Random Forest
algorithm for modeling both the average cost of claims and the frequency of claims.
V. CONCLUSION AND RECOMMENDATION
Moroccan Insurance and Social Welfare Control Authority (ACAPS) authorized
SCR, Central Reinsurance Company, to open a ReTakaful window based on the
principles of Shariah in order to cover the risks transferred by Takaful operators.
However, the challenge that remains is to determine the appropriate ReTakaful
model for the Moroccan Islamic insurance sector. Unfortunately, the works
examined in the literature review that deal with the practical and actuarial aspects
of Takaful and ReTakaful, are rare. In this context, this article seeks to price
ReTakaful contributions based on Machine Learning in the context of general

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ReTakaful especially ReTakaful of an automobile Takaful product. At the same
time, this work proposes a model of ReTakaful in conformity with Moroccan
regulations and in compliance with Shariah.
We model the ReTakaful contributions using Decision tree, Random Forests,
and Neural Networks under two ReTakaful methods, namely the “quota share”
treaty and the “excess of loss” treaty. In order to choose the best Machine Learning
algorithm, we used the mean squared error. The selection of the best ReTakaful
treaty is based on the probability of ruin minimization criteria.
The results of this work demonstrate the advantage to price the ReTakaful
contributions under the excess of loss method that minimizes well the probability
of ruin. In addition, we suggest using Machine Learning algorithms, especially,
the Random Forest algorithm to model the average cost of claims and the Random
Forest algorithm to model the frequency of claims.
Through this work, we try to enrich the literature of ReTakaful from an
empirical angle. However, future empirical research should consider other
methods of ReTakaful and other prime principles that are beyond the scope of this
work and worth investigating.
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