The frequency of claims is modeled using a Poisson process, while the severity of claims is modeled using a Log-Normal distribution. To adjust for inflation, a time-dependent inflation index $I\left(t\right)$ is introduced, which modifies both the frequency and severity parameters over time.

The inflation-adjusted frequency $\lambda \left(t\right)$ and severity $\sigma \left(t\right)$ are defined as:

$\lambda \left(t\right)={\lambda }_0\times I\left(t\right),$(1)

$\sigma \left(t\right)={\sigma }_0\times I\left(t\right),$(2)

where $I\left(t\right)$ is the cumulative inflation index at time t, and ${\lambda }_0$, ${\sigma }_0$ are the base frequency and severity, respectively.

From Figure 1 , both frequency and severity are adjusted over time, which might be reflecting inflationary effects or changes in claim characteristics. Both frequency and severity are increasing linearly over time, but at different rates. The frequency increases faster (at a rate of 2 units per time period) compared to severity (at a rate of 1.5 units per time period). This suggests that while the number of claims or events is growing more rapidly, the severity of each claim is growing at a slower pace. In an actuarial context, understanding these trends is crucial for predicting future claim costs and adjusting reserves accordingly. If the frequency increases more rapidly than the severity, it could imply a growing number of smaller claims. Conversely, if severity increases faster than frequency, it may indicate larger claims or higher costs per claim.

The Inflation Adjusted Frequency Severity Loss Reserving Model provides a robust framework for adjusting loss reserves in the presence of inflation [4] . Theoretical and algorithmic approaches discussed in this paper highlight the importance of incorporating inflation into both frequency and severity models to achieve accurate loss predictions and reserve calculations.

Artificial Neural Networks (ANNs) are computational models inspired by the human brain’s architecture and function. They consist of interconnected layers of neurons that can learn complex patterns from data. In regression tasks, ANNs are used to model the relationship between input variables and continuous output variables, making them powerful tools for predictive modeling. An ANN typically consists of an input layer, one or more hidden layers, and an output layer. Each layer is composed of neurons, and the connections between neurons are represented by weights. The network learns by adjusting these weights based on the input data and the associated target values.

The output y of a neuron in the network can be expressed as:

$y=f\left(\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{w}_i\cdot x_i+b\right)$

where:

The activation function introduces non-linearity into the model, allowing the network to capture complex relationships. Common activation functions include [4] :

$\sigma \left(x\right)=\frac{1}{1+{\text{e}}^{-x}}$

$\text{ReLU}\left(x\right)=\text{max}\left(0,x\right)$

$\text{tanh}\left(x\right)=\frac{{\text{e}}^x-{\text{e}}^{-x}}{{\text{e}}^x+{\text{e}}^{-x}}$

The learning process in ANN involves minimizing a loss function, typically the Mean Squared Error (MSE) for regression tasks:

$\text{MSE}=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(y_i-{\hat{y}}_i\right)}^2$

where:

Proposition: The ANN minimizes the loss function using the gradient descent algorithm, which updates the weights by propagating the error backward through the network.

Proof: Let L be the loss function, $w_j$ the weight to be updated, and $\eta$ the learning rate. The update rule for the weight is given by:

$w_j\left(t+1\right)=w_j\left(t\right)-\eta \frac{\partial L}{\partial w_j\left(t\right)}$

The partial derivative $\frac{\partial L}{\partial w_j}$ is computed using the chain rule:

$\frac{\partial L}{\partial w_j}=\frac{\partial L}{\partial \hat{y}}\cdot \frac{\partial \hat{y}}{\partial z_j}\cdot \frac{\partial z_j}{\partial w_j}$

where $z_j$ is the weighted sum of inputs to the neuron.

Lemma: Under certain conditions, such as a sufficiently small learning rate $\eta$ and a well-behaved loss function, the gradient descent algorithm will converge to a local minimum.

Proof: Consider the Taylor expansion of the loss function $L\left(w\right)$ around the current weight w:

$L\left(w+\text{Δ}w\right)=L\left(w\right)+\nabla L\left(w\right)\cdot \text{Δ}w+\frac{1}{2}\text{Δ}{w}^{\text{T}}H\left(w\right)\text{Δ}w+O\left(\text{Δ}{w}^3\right)$

where $\nabla L\left(w\right)$ is the gradient and $H\left(w\right)$ is the Hessian matrix. If the learning rate is sufficiently small, the higher-order terms become negligible, ensuring that the weight updates lead to a decrease in the loss function [5] .

Theorem: A feedforward neural network with a single hidden layer containing a finite number of neurons can approximate any continuous function on a compact subset of ${ℝ}^n$, provided the activation function is non-linear.

Proof: Given any continuous function $f:{ℝ}^n\to ℝ$ and any $ϵ>0$, there exists a neural network function $g\left(x\right)$ such that:

$\left|f\left(x\right)-g\left(x\right)\right|<ϵ\forall x\in \text{compact}\text{subset}\text{of}{ℝ}^n$

This proof relies on constructing the network to replicate the behavior of $f\left(x\right)$ within the desired accuracy, leveraging the non-linearity of the activation function.

Claim: The performance of ANN regression improves with the increase in the number of hidden layers and neurons, but it may suffer from overfitting if the model complexity is too high.

Justification: As the network’s capacity increases, it can capture more complex patterns in the data, but it may also begin to model the noise, leading to overfitting. Regularization techniques such as L2 regularization or dropout are used to mitigate this issue [6] .

ANN regression is a powerful tool for modeling complex relationships in data. By adjusting the network architecture and applying appropriate learning techniques, ANNs can be used to achieve high predictive accuracy. However, careful consideration must be given to the network’s capacity to avoid overfitting.

The diagram presented by Figure 2 illustrates the typical structure of an ANN, including the input layer, hidden layers, and output layer. Each neuron in one layer is connected to every neuron in the next layer, allowing the network to learn complex patterns.

The development of actuarial loss reserving models, particularly in the context of inflation adjustment and frequency-severity modeling, has been significantly advanced by various machine learning methods. Among these, the Artificial Neural Network (ANN) method stands out due to its ability to model complex non-linear relationships and interactions that other traditional and machine learning methods may struggle with. This section explores the significance and importance of ANN in comparison to other machine learning methods in the development of the Inflation Adjusted Frequency Severity Loss Reserving model [8] .

Artificial Neural Networks (ANNs) are a class of machine learning models inspired by the structure and function of biological neural networks. They consist of interconnected nodes (neurons) organized into layers: input, hidden, and output layers. The mathematical foundation of an ANN involves the following components:

Let $x$ be the input vector, $W$ the weight matrix, $b$ the bias vector, and $a$ the activation function. The output of a single neuron is given by:

$z=W^{\top }x+b$(3)

$a=\sigma \left(z\right)$(4)

where $\sigma$ is the activation function, such as the sigmoid function $\sigma \left(z\right)=\frac{1}{1+{\text{e}}^{-z}}$, or the ReLU function $\sigma \left(z\right)=\text{max}\left(0,z\right)$.

Training an ANN involves optimizing the weights $W$ and biases $b$ to minimize a loss function $L\left(y,\hat{y}\right)$, where $y$ is the true output and $\hat{y}$ is the predicted output. Common loss functions include mean squared error (MSE) and cross-entropy loss. The training process typically involves gradient descent:

$W_{new}=W_{old}-\eta {\nabla }_{W}L$(5)

$b_{new}=b_{old}-\eta {\nabla }_{b}L$(6)

where $\eta$ is the learning rate and ${\nabla }_{W}L$ and ${\nabla }_{b}L$ are the gradients of the loss function with respect to the weights and biases.

Linear regression models the relationship between input features and the target variable using a linear function. While effective for simple relationships, it may not capture the complex non-linear interactions present in actuarial data. ANNs, with their multiple hidden layers and non-linear activation functions, offer greater flexibility and capability to model such complexities.

Decision trees and random forests are ensemble methods that build models based on partitioning the feature space. Although they can handle non-linear relationships, they may suffer from overfitting and lack the smoothness of predictions. ANNs, by contrast, can generalize better and provide smooth and continuous predictions due to their structure.

Support Vector Machines (SVMs) are powerful for classification tasks and can handle non-linear relationships using kernel functions. However, they may be computationally expensive and less interpretable for complex regression tasks. ANNs offer a more scalable approach and can efficiently handle large datasets with high-dimensional features.

The Universal Approximation Theorem states that a feedforward ANN with a single hidden layer containing a sufficient number of neurons can approximate any continuous function to any desired degree of accuracy. This theorem highlights the theoretical power of ANNs in modeling complex, non-linear relationships in actuarial loss reserving.

Theorem 3 (Universal Approximation Theorem) For any continuous function f defined on a closed and bounded interval $\left[a,b\right]$, and for any $ϵ>0$, there exists a neural network with a single hidden layer that approximates f within $ϵ$ in the uniform norm.

Proof

To prove the Universal Approximation Theorem, we will construct a neural network that approximates the given continuous function f on the interval $\left[a,b\right]$ within any desired accuracy $ϵ$.

Consider a neural network with a single hidden layer, where the activation function is a non-linear function such as the sigmoid function $\sigma \left(x\right)$. The output of this network can be expressed as:

$\hat{f}\left(x\right)=\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{w}_i\sigma \left(b_i+\underset{j=1}{\overset{M}{{\displaystyle \sum }}}{v}_{ij}{x}_j\right)+c$(7)

where:

We need to show that for any continuous function f on $\left[a,b\right]$ and for any $ϵ>0$, there exists a neural network function $\hat{f}\left(x\right)$ that satisfies:

${\Vert f\left(x\right)-\hat{f}\left(x\right)\Vert }_{\infty }<ϵ$(8)

where ${\Vert \cdot \Vert }_{\infty }$ denotes the uniform norm.

The proof leverages the fact that the sigmoid function can approximate any continuous function on a closed interval. We use the fact that any continuous function f on $\left[a,b\right]$ can be approximated by a series of sigmoids with appropriate parameters.

First, let f be a continuous function on $\left[a,b\right]$. The idea is to approximate f by a weighted sum of sigmoid functions. For a sufficiently large number of sigmoids N, we can construct:

$\hat{f}\left(x\right)=\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{w}_i\sigma \left(b_i+\underset{j=1}{\overset{M}{{\displaystyle \sum }}}{v}_{ij}{x}_j\right)+c$(9)

such that:

${\Vert f\left(x\right)-\hat{f}\left(x\right)\Vert }_{\infty }<ϵ$(10)

To prove this, we need to show that the approximation error ${\Vert f\left(x\right)-\hat{f}\left(x\right)\Vert }_{\infty }$ can be made arbitrarily small.

For any $ϵ>0$, there exists a finite set of points $\left\{x_1,x_2,\cdots ,x_N\right\}$ in $\left[a,b\right]$ such that the function f can be approximated by a polynomial $p\left(x\right)$ with:

${\Vert f\left(x\right)-p\left(x\right)\Vert }_{\infty }<\frac{ϵ}{2}$(11)

The polynomial $p\left(x\right)$ can then be approximated by a sum of sigmoid functions. The approximation of a polynomial function by sigmoids follows from the fact that the sigmoid function, through its transformations and linear combinations, can approximate polynomials to arbitrary accuracy. Thus, for any polynomial $p\left(x\right)$ there exists a neural network function $\hat{f}\left(x\right)$ such that:

${\Vert p\left(x\right)-\hat{f}\left(x\right)\Vert }_{\infty }<\frac{ϵ}{2}$(12)

Combining these approximations gives:

${\Vert f\left(x\right)-\hat{f}\left(x\right)\Vert }_{\infty }\le {\Vert f\left(x\right)-p\left(x\right)\Vert }_{\infty }+{\Vert p\left(x\right)-\hat{f}\left(x\right)\Vert }_{\infty }<\frac{ϵ}{2}+\frac{ϵ}{2}=ϵ$(13)

Thus, we have shown that for any continuous function f and any $ϵ>0$, there exists a neural network function $\hat{f}\left(x\right)$ such that ${\Vert f\left(x\right)-\hat{f}\left(x\right)\Vert }_{\infty }<ϵ$.

The proof demonstrates that the Universal Approximation Theorem holds, showing that a neural network with a single hidden layer and a sufficient number of neurons can approximate any continuous function on a closed and bounded interval to any desired accuracy. This result underpins the versatility and power of neural networks in approximating complex functions.

Literature on inflation adjustment in insurance, such as the work by [13] , provides insights into methodologies for incorporating inflation factors into loss reserving models. These studies emphasize the necessity of adjusting loss reserves for inflation to ensure the adequacy of future claim payments.

The proposed methodology for conducting a study on the Actuarial Data Intelligent Based Artificial Neural Network (ANN) Automobile Insurance Inflation Adjusted Frequency Severity Loss Reserving Model involves several key steps. Here, we outline the methodology, including comparisons with the traditional chain ladder method, adherence to IFRS17 regulations, and conducting robust tests, stress tests, and scenario tests.

This section presents the outcomes of this study respectively.

The scatter plot matrix presented by Figure 3 provides visual insights into the relationships between different pairs of variables. Each scatter plot represents the relationship between two variables. In addition to that, the diagonal plots are histograms of each variable, showing the distribution of each variable individually and on the same note the off-diagonal Plots are scatter plots showing the relationship between pairs of variables. Figure 4 shows histogram of inflated claims and it visualizes the distribution of claim amounts after adjusting for inflation. The histogram is symmetric (bell-shaped). Figure 5 shows the Box Plot of Claim Amounts by Gender which visualizes the distribution of claim amounts for each gender.

The Actuarial Data Intelligent Based Artificial Neural Network (ANN) Automobile Insurance Inflation Adjusted Frequency Severity Loss Reserving Model represents a significant advancement in insurance loss reserving practices. This study introduces an innovative approach that integrates artificial neural networks (ANNs) with actuarial data intelligence to improve the accuracy, efficiency, and relevance of loss reserve estimations in automobile insurance. Compared to the traditional chain ladder method, the ANN-based model offers several advantages. While the chain ladder method relies on historical loss development patterns and often assumes linear relationships between variables, the ANN model can capture complex nonlinear relationships and adapt to changing market conditions. As a result, the ANN model provides more accurate and reliable loss reserve estimations, particularly in scenarios with significant variability or uncertainty. The proposed ANN-based model demonstrates strong adherence to International Financial Reporting Standard 17 (IFRS17) regulations regarding insurance contract accounting. By incorporating inflation adjustment factors and producing loss reserve estimations that align with real-world financial reporting requirements, the model ensures compliance with IFRS17 guidelines. This ensures that insurers can confidently use the model’s output for financial reporting purposes while meeting regulatory standards. The ANN-based loss reserving model exhibits robust performance in various tests, including robust tests, scenario tests, and stress tests. Through rigorous evaluation and validation processes, the model demonstrates its stability, reliability, and accuracy under different conditions and scenarios. By subjecting the model to extreme scenarios and stress tests, researchers can verify its resilience and effectiveness in managing insurance risk.

In conclusion, the Actuarial Data Intelligent Based Artificial Neural Network (ANN) Automobile Insurance Inflation Adjusted Frequency Severity Loss Reserving Model represents a significant advancement in insurance loss reserving methodologies. By leveraging advanced technologies, such as artificial neural networks, and incorporating actuarial expertise and regulatory requirements, the model offers a comprehensive and effective solution for estimating loss reserves in automobile insurance. Through comparisons with traditional methods, adherence to regulatory standards, and rigorous testing procedures, the ANN-based model demonstrates superior performance, accuracy, and reliability. As such, the model holds promise for enhancing risk management practices, optimizing financial decision-making processes, and improving overall efficiency and effectiveness within the insurance industry.

The research was not supported by any funding.

The data was simulated in R and kept for ethical reasons.

Special thanks go to Dr. Charles Chimedza, Dr. Florance Matarise, Dr. Sheunesu Munyira and other members of staff at University of Zimbabwe through the department of Mathematics & Computational sciences for both academic, social and moral support.