Advancing loss reserving: A hybrid neural network approach for individual claim development prediction

2.1 Loss reserving: An economic perspective

Loss reserving is a crucial and economically significant task within every insurance company (Radtke et al., 2016). As depicted in Figure 1, the loss reserving process can be described as follows2: First, a claim occurs. After a certain delay, the policy holder reports the claim to the insurance company. Based on the knowledge of the company at the time of reporting, parts of the claim might be settled immediately or partially until the claim is closed. Before the final closure, the claim may be reopened, and further payments and recoveries may happen. In nonlife insurance, actuaries go beyond analyzing raw claim payments by focusing on incurred losses, which are defined as the sum of raw claim payments and case reserves. Case reserves are usually set by experts and represent their current estimate of the outstanding loss on individual claims. The process involving payments, case reserves, and recoveries upon closure or final closure is referred to as incurred loss adjustments.

Incurred losses change dynamically over time. Payments and case reserves are not necessarily adjusted simultaneously. However, the incurred losses remain unaffected in scenarios where case estimates automatically adjust with claim payments while keeping the overall incurred loss constant. Additionally, for industrial insurance, incurred loss amounts may not change in every period, especially for complex claims characterized by long settlement horizons. Ultimately, the total paid amount aligns with the ultimate incurred loss amount, ensuring a balance in the final assessment. Our subsequent analyses considers these specifics of industrial insurance, and our machine learning methods are tailored to address these characteristics.

At the evaluation date, denoted by $t^{*}$ and representing the starting point of our prediction task, insurance companies distinguish between RBNS and IBNR claims. An IBNR claim was occurred before the evaluation date but is reported and settled afterwards. For an RBNS claim, the claim's occurrence and reporting before the evaluation date, while the settlement takes place afterward. Our model is specifically designed to leverage individual claim characteristics. As IBNR claims' characteristics are not observable at the time of evaluation, we exclusively focus on predicting RBNS claims.

We consider a portfolio comprising $K$ reported claims, indexed by $k=1,\mathrm{\dots },K$. Each claim is reported after a specified time point. We use discrete points in time with development periods of equal length; that is, each development period spans 1 year. We also assume that each claim in the portfolio is fully settled within a fixed number of $n$ development periods. Therefore, for each claim, we have development periods indexed by $j=0,\mathrm{\dots },n$. Our objective is to predict the incurred losses for the unknown periods $t^{*}+1,\mathrm{\dots },n$, for each claim $k$. Thus, at the evaluation date $t^{*}$, each claim is in its specific development period. Our task is to estimate the remaining development periods until settlement. Each claim's development period at $t^{*}$ is defined by the number of discrete time points before $t^{*}$.

Theoretically, there is a modeling equivalence between cumulative and incremental losses, making both approaches valid. However, incremental losses are more commonly predicted in the literature. By exploring both approaches, our model provides a robust and versatile tool for reserve estimation.

2.2 Loss reserving: A methodological perspective

Traditional loss reserving methods, such as the chain ladder method, rely on aggregating claims into a homogeneous portfolio, structured in a triangular format that captures the development of claims over time. These methods have been extensively employed and studied. For example, Mack (1993) introduces a distribution-free stochastic framework for the chain ladder method to quantify uncertainty in reserve estimates. Additionally, Verrall (2000) explores various stochastic models that align with chain ladder reserve estimates. Moreover, England and Verrall (2002), Pinheiro et al. (2003), and England and Verrall (2006) introduce bootstrapping techniques, which help in integrating expert judgments within a general linear model framework. For a comprehensive overview of traditional methods, see Wüthrich (2008).

New advances in data collection and increasing dataset complexity have led to extensions of traditional approaches, evolving the univariate chain ladder method into multivariate extensions (Merz & Wüthrich, 2008; Pröhl & Schmidt, 2005; Shi, 2017; Zhang, 2010). The multivariate extensions can aid in developing a portfolio of several correlated sub-portfolios by accounting for both contemporaneous correlations and structural relationships. However, the methods require strong structural assumptions on the relation between sub-portfolios and the aggregate portfolio, which are often rather ad hoc. We emphasize that this presents a huge advantage of machine learning methods, whereby nonlinear dependencies are captured without strong ad hoc assumptions. While using aggregated data for the chain ladder method is still the most popular approach, it has some limitations (Antonio & Plat, 2014): the inadequacy in handling outliers (Verdonck et al., 2009), parameter overfitting (Wright, 1990), and the presence of the chain ladder bias (Taylor, 2003). Antonio and Plat (2014) provide a comprehensive overview of these limitations. They argue that the existence of these problems, along with extensive literature on the topic, suggests that using aggregate data with the chain ladder method is not always suitable, especially when individual claims data are available.

Early contributions towards granular modeling on loss reserving are based on parametric models (Buhlmann et al., 1980; Norberg, 1986). Arjas (1989) and Norberg (1993, 1999) model individual claims development to compute reserves based on Position Dependent Marked Poisson Processes (PDMPP). Larsen (2007) refines their approach by decomposing the complex stochastic process of claim developments into independent segments corresponding to calendar years. By treating these as independent segments, Larsen's method allows the overall likelihood function to be separated into individual components, which can be maximized separately. This decomposition simplifies the stochastic reserving model's estimation process. Zhao and Zhou (2010) extend this idea by accounting for the dependence of claim event times and covariates. Furthermore, Huang et al. (2015, 2016) develop a stochastic model based on individual claim data that considers factors such as each claim's occurrence, reporting, and settlement times. Their model demonstrates a significant reduction in the mean squared error loss compared to classical models based on aggregated data. Thus, their findings support the use of models that leverage individual claim data.

In their review article, Wüthrich (2018) summarizes the benefits of machine learning in individual claims reserving. Pigeon and Duval (2019) explore these ideas by using different tree-based approaches, like the XGBoost algorithm, to predict the total paid amount of individual claims. Meanwhile, Baudry and Robert (2019) use another tree-based ensemble technique, ExtraTrees, to compute individual-level IBNR and RBNS claims reserves. A survey of recently developed claims reserving techniques can be found in Taylor (2019), emphasizing that research in this area remains highly relevant.

While loss reserving models have significantly evolved, including the adoption of machine learning techniques, the limited availability of publicly accessible individual claims data remains a notable challenge. To address this, Gabrielli and Wüthrich (2018) develop a stochastic simulation machine based on neural networks to synthesize claims data and provide back-testing capabilities. Using this stochastic simulation machine, Gabrielli (2020) and Gabrielli et al. (2020) integrate classical loss-reserving models into neural networks. They initially align the network with a traditional model, such as the over-dispersed Poisson, and then refine it through training to reduce prediction errors, effectively leveraging a boosting-like method. Gabrielli (2021) presents an individual claims reserving model for reported claims, which uses summarized past information to predict expected future payments. Instead, Kuo (2020) again focuses on the aggregated losses but considers the underlying time series nature of the data. The author's framework facilitates the joint modeling of paid losses and claims outstanding, adapting to incorporate various data types. The author then tests their model on aggregated datasets and demonstrates the potential for broader application with more detailed data. Furthermore, Kuo (2020) introduces an individual claims model for RBNS claims incorporating an encoder LSTM for past payments and a decoder LSTM for generating paid loss distribution, augmented by a Bayesian neural network for uncertainty quantification. However, none of these proposed models consistently and systematically outperform the classical chain ladder method for the used datasets. Chaoubi et al. (2023) suggest a different model architecture using a LSTM network followed by two fully connected layers to jointly predict the probability of a payment or recovery and the corresponding amount. Their model focuses on predicting incremental payments and treats static features as dynamic within the network. While their model seemingly has an advantage over the chain ladder method, an extensive set of benchmark models and large samples of actual data are missing.

3.1 Data description

Our data comprise proprietary claims data provided by a large industrial insurance company covering both short-tail and long-tail LoBs, which are the property and liability lines.3 The scarcity of publicly available individual claims data in empirical studies of insurance loss reserving means that simulated claims datasets are often used to enable the back-testing of the proposed models. For instance, Baudry and Robert (2019), Kuo (2020), and Gabrielli (2021) all use synthetically generated data on individual claims for their analysis. Moreover, when real claim data are utilized, the focus tends to be narrowly on a single LoB, predominantly examining general liability insurance for private individuals.4 Thus, the use of real-world, individual claim data from two distinct LoBs within the industrial insurance domain is particularly interesting for analyzing the use of machine learning methods for insurance loss reserving.

The property claim dataset contains a total of 66,208 claims arising from 16,713 distinct policies reported from January 1, 2011 to December 31, 2016, where observations are available until the end of 2021. The long-tailed liability claims dataset includes 403,461 claims from 24,084 distinct policies, reported from January 1, 2000, to December 31, 2011, where observations are available until the end of 2022. Thus, our data ensure a comprehensive analysis framework with at least six and twelve discrete annual observation periods for property and liability claims, respectively. This timeframe aligns with the assumption that claims are fully settled within the specified periods—6 years for property and twelve years for liability—corresponding to the internal reserving practices of the insurance company. This extensive dataset allows to thoroughly analyze each claim's development trajectory, providing a critical advantage for back-testing our model against real-world outcomes.

The data can be dissected into two types of data. The first type includes static features, which are fixed and do not change over time. The second type of data is dynamic; that is, these features may vary over time.

The property dataset contains the following static features: the Contract category, which identifies whether an insurance policy is part of an international program or a local policy; Contract type, which is differentiated into standard, primary, layer, and master policies; Business type, segmented into sole, lead, coinsurance, and indirect business, the Policy share, which captures the share of the overall insured risk (ranging from 0% to 100%); Risk category, which classifies each claim into one of 12 distinct risk categories5; Risk class, with values ranging from 1 to 10, with 10 signifying a higher risk; Coverage, which is detailed with three different coverages; Covered perils, which include three classes; Claim type, differentiated into attritional losses, large losses, and natural catastrophes; Notification duration of the claim, measured in days; and Branch, indicating which of the 12 branches the corresponding policy belongs to. The dynamic features are the German real GDP (base year 2011) and an internal Inflation mixture indices, created internally by the insurance company to reflect inflationary trends relevant for the specific LoB.

The liability dataset comprises the following static features: the Contract category, Business type, Policy share, the Risk category containing seven categories, the Coverage detailed by 12 different coverages, the Notification duration, and the Branch. The dynamic features are the same as those for the property dataset. The features used in our model for the two datasets are summarized in Appendix A.

We remove claims categorized at the reporting date as large losses. At the point of notification, a claims handler classifies each claim as either an attritional loss, a large loss, or a natural catastrophe. The classification is based on whether the expected ultimate loss amount exceeds a predetermined threshold specific to the LoB. Large losses, especially in the industrial insurance, can further increase the inherent volatility of the data. This may distort the predictions of the model aimed at more typical claim behavior. Therefore, it is usually analyzed and modeled separately (Denuit & Trufin, 2018; Riegel, 2014). In practice, reserving for large losses often involves a significant amount of expert judgment and adjustments. Due to this complexity and the unique handling required for these claims, we have excluded them from our analysis. Excluding large losses narrows our focus to attritional patterns, enhancing model precision for typical claims. Moreover, it highlights the necessity of separately addressing large losses for a holistic loss reserving approach. Notably, some claims may ultimately develop into large losses, although they were initially categorized as attritional losses. Hence, they remain a part of our datasets.

3.2 Training- and testing-setup

Grouping claims not only by their development period $j$ but also by their reporting year, indexed by $i=1,\mathrm{\dots },n$, facilitates the structural organization of the data such that it aligns with traditional loss reserving methods based on loss triangles. This structure enables us to define the cumulative incurred loss amount for claim $k$, in reporting year $i$, and in development period $j$ as $S_{i,j}^{\left(k\right)}$.

Figure 2 appropriately illustrates how the property dataset is organized in the form of a common loss triangle. Here, each cell represents $S_{i,j}$, which is the sum of the incurred losses of all claims that were reported in year $i$ and are in their development period $j$. This is helpful for visualizing the comparison between our individual loss reserving approach and the traditional aggregated methods. Moreover, we can visualize how we split our data into training, validation, and test sets.

As our datasets contain the complete development trajectory of each claim up to their final development period $n$, we first split the data into training and testing sets by setting the evaluation date $t^{*}$ to December 31, 2016, and December 31, 2011, for the property and liability datasets, respectively.6 The cumulative incurred loss amounts $S_{i,j}^{\left(k\right)}$ for $i+j\le n$ are observable Up to these dates for each claim $k$ and serve as our training data (represented in light gray in Figure 2). The incurred loss amounts beyond this valuation date for $i+j\ge n+1$, which are not observable as of $t^{*}$, form the test set (represented in the dark gray in Figure 2).

That is, $t^{*}$ is the time point up to which data are considered known and used for training. Meanwhile, $n$ represents the final development period for any claim. Data observed up to $t^{*}$ are used for training, whereas those beyond $t^{*}$ are reserved for testing.

Further, we split the claims inside the training data randomly into the final training and validation sets.7 To be more precise, 80% of these data are used for model training, while the remaining 20% are used for model validation to evaluate the model's performance and generalizability.

4.1 Model architecture

The model architecture which is most similar to ours is introduced by Chaoubi et al. (2023). While inspired by this architecture, we introduce significant changes: We treat static and dynamic features individually within the network, include an attention mechanism besides our sequential model to further enhance the network's ability to focus on the input sequence's relevant parts, and use a decision rule for updating the predicted claim amounts.

Essentially, our model adopts a two-part approach: it separately processes static and dynamic features before integrating them to produce two task outputs. Consequently, the model can capture the unique characteristics of each type of feature. Furthermore, the final prediction is based on a simple decision rule that incorporates the two outputs: if the probability of a change in the cumulative incurred loss amount exceeds a predefined threshold, the model updates its forecast using the output of the regression task; otherwise, the current estimate is retained. The specific threshold is determined post-training by analyzing the validation set, ensuring that the forecast over the next period depends on a substantial probability of change.9 This approach mitigates the risk of over-adjustment during periods of claim stability, thereby implicitly optimizing operational efficiency. This strategy contrasts with the method proposed by Chaoubi et al. (2023), wherein the output of the regression task is weighted by the predicted probability through the multiplication of these two quantities.

In summary, the model operates under the following assumptions: (1) Claim development patterns are assumed to be consistent across reporting years. However, time dependence is introduced through dynamic factors such as GDP and inflation, which change over time and impact the claims process. These factors ensure that the model adapts to varying economic conditions while maintaining the assumption of underlying structural consistency in claim development. (2) Claims are fully settled within predefined periods (six and twelve for property and liability claims, respectively). (3) The final output is produced by combining the results of the regression and classification task, utilizing a decision rule to update predictions based on the probability of change in the incurred loss amount.

In our network, the first step is processing the static features, which contain both quantitative and categorical types. All static quantitative features are standardized. Each static categorical feature is initially indexed and then processed by embedding layers, as introduced by Bengio et al. (2000). According to Kuo (2020) and Gabrielli (2021), we embed categorical variables into two-dimensional vectors. The processed static features are concatenated into a single vector, denoted by ${\tilde{\mathbf{F}}}_0^{\left(k\right)}$ and fed into a fully connected (FC) layer10 with a rectified linear unit (ReLU) activation function. This yields the output vector ${\tilde{\mathbf{F}}}_0^{\prime \left(k\right)}$.

Dynamic features, denoted by ${\mathbf{D}}_j^{\left(k\right)}$, are processed within our model by a LSTM layer, introduced by Hochreiter and Schmidhuber (1997). This architecture is designed to handle sequential data by maintaining a memory of past information. This is particularly useful for modeling the time-dependent behavior in the claims reserving process. The LSTM operates on a sequence of dynamic features, updating two crucial components at every time step: the cell state ${\mathit{C}}_j$ and hidden state ${\mathit{h}}_j$. These states integrate new information provided by the current input, and are essential for the model to remember and forget information through a process involving three gates. Specifically, the forget gate controls the extent to which the previous cell state is retained. The input gate updates the cell state by adding new information. The output gate decides which information from the cell state will be used to generate the output hidden state, which is further used for predicting the next development period. These operations allow the LSTM to maintain and manipulate its internal state over time, providing the ability to remember information across sequences of variable lengths. We refer to Figure 3 for a visual representation of a single LSTM cell, which delineates the workflow through its various gates. As the LSTM processes the sequence of dynamic features up to the last observed period, it outputs a series of hidden states. The last hidden state is particularly important as it encapsulates the information from the entire input sequence. This state is used for making the one-step-ahead prediction for the coming unknown period.

To further refine the focus of the model on relevant temporal patterns, we introduce an attention mechanism besides the set of hidden states. Attention mechanisms, initially popularized in the context of neural machine translation (Bahdanau et al., 2014; Luong et al., 2015), allow a model to dynamically focus on different parts of the input sequence to generate the output sequence's each element. They work by assigning weights to different input elements, indicating their relative importance for the task at hand. This approach can significantly improve the performance of time series tasks, as demonstrated by Lai et al. (2018), who empirically show improved performance across various time series domains. It is also effective in financial time series predictions, where attention mechanisms have been successfully used in conjunction with LSTMs (Kim & Kang, 2019; Zhang et al., 2019). This demonstrates this approach's adaptability and efficiency in diverse applications. Our dot product attention mechanism is identical to that used by Lai et al. (2018) and is designed such that the hidden states can only be influenced by their predecessors without affecting them in return. The mechanism also maintains the temporal ordering of the sequences.

Finally, the processed static feature vector is replicated and concatenated with the attended hidden states to form a comprehensive feature set for each period. The resulting feature vector is then fed into another FC layer using the ReLU activation function, followed by two separate pathways to produce the one-step ahead prediction. One pathway involves an FC layer with the identity activation function to produce the regression output for the cumulative incurred loss amount of the next period, defined as ${\hat{Y}}_{j+1}^{\left(k\right)}$. The second pathway utilizes an FC layer with a sigmoid activation function to get the probability of a change in the cumulative incurred loss amount in the next period ${\hat{p}}_{j+1}^{\left(k\right)}$. The proposed architecture is illustrated in Figure 4.

To optimize the hyperparameters of the network, we perform a two-staged approach. Initially, Random Search is utilized to broadly explore the hyperparameter space, followed by a refinement process using Bayesian Search. Our training methodology and hyperparameter tuning strategy, including the two-staged approach's specifics, are described in detail in Appendix B. Notably, the training process of our model, particularly during hyperparameter tuning, does demand considerable computational resources (e.g., up to 9.25 hours for Random Search and 11.07 h for Bayesian Search on an Amazon-EC2-G5.8xlarge instance). However, such computational demands should not be a limiting factor for modern insurance companies. The potential for more accurate and flexible predictions, which advanced methods like ours aim to achieve, can justify these resources when the model outperforms classical methods.

6.1 Model performance

We first examine each model's overall performance, quantifying this via the estimated reserves' percentage error, shown in Table 1.

We find a significant heterogeneity in performance across the models. Our neural network models demonstrate superior accuracy, with the model based on cumulative data yielding the lowest percentage errors: −1.37% for property and −1.18% for liability. This is better than the chain ladder method, which exhibits larger negative errors, indicating an underestimation of the reserves. Interestingly, the neural network model applied to incremental data shows a notable deviation from the cumulative data-based network. It exhibits a positive percentage error of 7.59% for property, suggesting overestimation and comparable underestimation to the chain ladder method for liability with a −16.28% error.

The C-LSTM model, following the methodology proposed by Chaoubi et al. (2023), exhibits a similar pattern to our models but with larger percentage errors. With an 11.34% error for property, it slightly outperforms the chain ladder model but still overestimates the underlying claims portfolio. Conversely, for the liability line, the C-LSTM model shows an −18.58% error, indicating greater underestimation than traditional approaches. Clearly, our modifications in the model architecture and reserve estimation strategy are beneficial.

Lastly, the performance of the linear regression model highlights its limitations in handling claim data complexity, significantly underestimating property by −20.73% and liability by −61.37%. Thus, the simplicity of the model is inadequate for capturing the complex, possibly nonlinear patterns present in the data.

To provide deeper insights into the predictive performance of our model at the individual claim level, we present each development period's NMAE, NRMSE, and balanced accuracy. These metrics are detailed in Tables 2–4 for property and Tables 5–7 for liability. These tables show the final prediction errors of the neural network and isolate the NMAE and NRMSE for the regression task alone (${\text{NN}}_{\text{Reg}},{\text{NN-incr}}_{\text{Reg}}$, and ${\text{C-LSTM}}_{\text{Reg}}$). This allows to distinguish the classification task's impact on prediction accuracy.

Importantly, for the C-LSTM model, the final prediction is the product of the predicted claim amount and predicted probability of a non-zero amount. This approach inherently differs from ours, as it directly integrates the probability into the regression output to refine the prediction. This distinction is crucial as it highlights the significance of the classification task in enhancing the performance of the model, particularly when it accurately predicts periods without a change in the incurred claim amount. This accuracy effectively reduces the error for those instances, which may even go to zero if the last state is known.14

For the property line, the NMAE and NRMSE for all models, except for the linear regression, decline over the development periods. This indicates the increasing accuracy of the models as they gain more information about the trajectories of the claims over time. Notably, the neural network models' (NN, NN-incr, and C-LSTM) final predictions consistently outperform the regression-only predictions (${\text{NN}}_{\text{Reg}},{\text{NN-incr}}_{\text{Reg}}$, and ${\text{C-LSTM}}_{\text{Reg}}$). This highlights the significance of the classification task in the prediction process, as it enables the model to identify periods where the incurred claim amounts remain unchanged, potentially reducing the NMAE and NRMSE in those instances.

Moreover, for the C-LSTM model, the differences between the final predictions and regression outputs are smaller than those observed in our proposed models. This may be attributable to the C-LSTM's approach, wherein the classification task is primarily used to scale the regression results subtly rather than drastically altering them, which improves the prediction.

The balanced accuracy results, calculated for our neural networks (NN and NN-incr) and presented in Table 4, follow a similar pattern. Accuracy generally increases over time, indicating the improving capability of the model to correctly classify changes in claim amounts as claims progress. Notably, not all claims follow the same settlement pattern. While our models assume a fixed number of periods for full settlement, some claims may settle earlier. As the model adapts to these settlement patterns over time, the classification task potentially becomes simpler with each passing period.

As the model progresses through successive periods, it becomes more proficient at identifying patterns and anomalies within the claims data, refining predictions, and thus, likely resulting in the observed decrease in NMAE and NRMSE. This trend reflects the growing adeptness of the model in predicting claim closure timings and adapting to the data's evolving nature over the development periods.

When examining the liability line, we observe patterns similar to those seen in the property line. The models reveal a consistent decrease in NMAE over the development periods, as shown in Table 5. This trend is corroborated by increasing balanced accuracy scores, provided in Table 7. The NRMSE, detailed in Table 6, also diminishes over time, albeit with less uniformity, hinting at the influence of outliers in the reserving process. In industrial insurance, outliers are generally more prevalent in the liability than in the property line. Liability claims often involve complex legal issues and long-tail liabilities. This increases the likelihood of large, unpredictable claims that result in outliers. Despite initially excluding claims marked as large losses, some claims evolve into large losses during their lifecycle. This evolution can partially explain the observed variability in NRMSE values, revealing areas where our model's performance can be improved in predicting these evolving large-loss claims within the liability line.

Next, we focus on a distinct advantage of our neural network model: its ability to perform runoff analysis at various granular levels, such as the branch level. This is a particularly relevant aspect for practical actuarial applications. Table 8 presents the percentage error in estimated reserves for the property line across branches. The neural network models (NN, NN-incr, and C-LSTM) generally exhibit improved accuracy than the chain ladder method, with notably precise estimates in certain branches. Interestingly, for branch 300, the neural networks based on incremental data (NN-incr and C-LSTM) show a lower percentage error than the cumulative model. This may be attributed to the tendency of the former model to overpredict; in the context of large claims that disproportionately affect reserve estimates, this may result in closer alignment with actual incurred losses. The cumulative model's underprediction of −10% suggests that it may not be effectively capturing the volatility associated with such large claims. This demonstrates that different models may be appropriate for large claims and can lead to improved performance. Overall, the performance across branches is heterogeneous. Despite this variability, the neural network model based on cumulative data generally delivers better performance than its incremental counterpart.

Table 9 presents the percentage error of the estimated reserves by branch for the liability dataset. The liability line results exhibit a similar pattern to the property lines, as the neural network models generally demonstrate more accurate performance than the chain ladder and linear regression methods across several branches. Particularly, the incremental data-based model (NN-Incr) shows a lower absolute percentage error in branches 430 and 650, indicating more precise estimates in these cases. However, the difference in performance compared to the cumulative model (NN) is not notably large, suggesting that both models possess their strengths in handling the data for these specific branches.

Clearly, our neural network model, particularly when employing cumulative data, has consistently outperformed traditional actuarial methods, such as the chain ladder, and simple econometric models, like linear regression, across both property and liability lines. Although the performance across branches demonstrates some variability, the overall trend suggests that the neural network's sophisticated architecture is more adept at capturing the claims data's complexities than the benchmark models. The incremental neural network model also shows promise in certain segments, indicating its potential applicability under specific conditions. The deviations observed in the cumulative data model can be attributed to the incremental claims data's inherent variability. Incremental data, which captures changes in claims from period to period, may exhibit a more erratic trajectory than the cumulative data's smoother progression. This complexity can lead to increased prediction errors. These results highlight the neural network's versatility and superior predictive capabilities, affirming its value as a powerful tool for actuarial forecasting and reserve estimation.

6.2 Bootstrap analysis

Here, we provide a more comprehensive statistical overview of the forecast performance of our neural network and benchmark methods, including their capacity to estimate the impact of the uncertainty related to the initial conditions. We utilize a bootstrapping technique to assess the quality and reliability of our predictions. This method helps us generate a distribution over the predictions rather than providing a single value for the estimated reserve (Efron & Tibshirani, 1994). The following outlines the key steps and presents the corresponding results.

We first generate 100 bootstrapped datasets from our original training and validation dataset. Each bootstrapped dataset is created by randomly sampling with replacement claim sequences; that is, a time series of claim developments. Thus, each bootstrapped dataset retains the same size (i.e., the same number of claims) as the original dataset. Subsequently, we divide each bootstrapped dataset into training and validation sets using the same ratio as described in Section 3.2.15

Each neural network model (NN, NN-Incr, and C-LSTM) is trained on all 100 bootstrapped datasets using the best hyperparameters identified by Random Search. Initially, we used the final optimal hyperparameters obtained by Bayesian Search. However, this extended approach allows to further test the robustness and sensitivity of the model to different hyperparameter configurations, providing a comprehensive evaluation of their performance. Meanwhile, the linear regression model used the set of predictors identified from the original training dataset. After training each model on the bootstrapped datasets, we obtain an ensemble of reserve predictors, enabling us to provide insights into model confidence.

For the chain ladder method, we calculate predictors directly from each of the 100 bootstrapped samples. The chain ladder method cannot be applied in the same manner as the neural network models, which can use the trained models to make predictions based on the original dataset. This is because the chain ladder method calculates development factors specific to the data it is applied to. These factors cannot be directly used to predict another dataset. Instead, for each bootstrapped sample, we use the reserve estimates generated by the chain ladder method, which is similar to the bootstrapped chain ladder approach suggested by England and Verrall (2002).

Note that we could have also predicted the reserves for the neural networks based on the 100 bootstrapped samples. Ultimately, our methodology introduces an additional layer of uncertainty into the neural networks' predictions, making it more challenging for them to outperform the chain ladder method.

Next, we discuss the reliability and performance of the different models based on the percentage error of the ensemble point estimates across the 100 bootstrapped samples; that is, the point estimates are obtained as the average reserve prediction across all individual predictions. This bootstrap aggregation (bagging) technique improves the stability and accuracy of the classification and regression task, reducing the variance and avoiding overfitting. Additionally, we use box plots to illustrate the distribution of each model's percentage errors' distribution.16 The generated point predictions are presented in Table 10 for the property and liability LoBs, revealing similar patterns to those observed in the original estimates.

Crucially, the neural network models maintain their superior accuracy, with the cumulative neural network model (NN) showing a slight increase in percentage error for property claims but still outperforming all other models. The neural network models based on incremental data (NN-Incr and C-LSTM) continue to overestimate (underestimate) property (liability) data, with a slight increase (decrease) in percentage error. Both the chain ladder and linear regression models exhibit slight improvements for both datasets, indicating increased stability.

Figures 5 and 6 show box plots illustrating the error distributions, highlighting the median, quartiles, and potential outliers in the predictions.

For both property and liability LoBs, the cumulative neural network model (NN) demonstrates the smallest interquartile range (IQR) and least variability, indicating a high level of consistency and accuracy across all model predictions. Consistently, for both LoBs, the neural network models based on incremental data (NN-Incr and C-LSTM) exhibit greater variability, reflecting the higher dispersion of percentage errors observed in the bootstrap analysis. Overall, the error distribution is higher for these models compared to the cumulative model.

For the property line, the chain ladder method shows moderate variability but tends to underestimate reserves, resulting in an overall negative distribution of errors. Conversely, for the liability line, the chain ladder method shows substantially higher variability. The linear regression model provides generally inaccurate predictions, highlighting its limitations in accurately predicting reserves and leading to a negative distribution of percentage errors.

Overall, the bootstrapping analysis offers valuable insights into the practical implications of utilizing different models for loss reserving in the insurance sector. The neural network models, particularly the cumulative data ones (NN), demonstrate superior accuracy and consistency. This makes them highly suitable for practical applications. The low variability and narrow IQR indicate reliable performance, which is crucial for making informed decisions regarding reserve estimation.

6.3 Explainability

While we have examined the performance of the different reserving models, we have not yet addressed a crucial aspect of using machine learning methods in insurance applications: the driving features' transparency and explainability. For this, we rely on a state-of-the-art explainability technique used for applications like ours: SHAP values (Lundberg & Lee, 2017). SHAP values offer insights into each feature's contribution to a specific prediction, enabling us to evaluate the importance and impact of various variables in the dataset. To determine this impact, each feature is systematically omitted to observe the resulting change in the output of the model. This process is repeated across all possible feature combinations, ensuring that each feature's importance is evaluated within every possible feature set's context (Lundberg & Lee, 2017). We calculate SHAP values for a randomly sampled 10% of our data. This helps ensure that we capture a comprehensive picture of feature importance without overwhelming computational resources. After computing SHAP values, we aggregate them to summarize each feature's overall contribution to the predictions of the model. This aggregation helps us identify which features are consistently influential and which have less impact. As a further advantage this provides us with a clear understanding of the driving factors behind the behavior of the model.

Figures 7 and 8 display the mean absolute SHAP values for the top 10 features in the property and liability datasets, respectively, for both regression and classification tasks. The left panel of Figure 7 shows that for the regression task, Losses Incurred is the most important feature. This is unsurprising, as the incurred loss directly relates to the amount to be predicted. In particular, we consider incurred losses as payments plus individual case reserves. This expert information is critical because it reflects the claims handler's current estimate of the outstanding loss. Moreover, this information provides a robust basis for future loss predictions. Interestingly, all other features, such as Business Type, Risk Category, and Branch, only play a minor and do not significantly contribute to the regression task. Thus, although these features provide additional context, the incurred losses predominantly drive the model's predictive accuracy for estimating future claim amounts.

The right panel of Figure 7 illustrates the importance of different features for the classification task. Here, we observe a different picture than that for the regression task. Although Losses Incurred remains the most important feature, its relative importance is reduced compared with the regression task. Other features, such as Business Type, Risk Category, and Branch, also show considerable importance, highlighting their relevance in predicting changes in the cumulative incurred loss amounts. These features capture various dimensions of risk and policy characteristics that can influence the probability of changes in incurred losses. For instance, different business types and risk categories are likely to exhibit distinct patterns in claim development and settlement, thereby affecting the likelihood of adjustments in the incurred amounts.

For the liability dataset's regression task (Figure 8), insights from the SHAP values are nearly the same as for the property line. However, for the classification task, the magnitude and order of the features besides Losses Incurred differ compared to the property dataset. Other features, such as Branch, Contract Category, and Coverage, only show slightly higher importance. Thus, while these additional features contribute to predicting changes in the cumulative incurred loss amounts, their impact remains relatively minor for liability claims.

To further illustrate the benefits of our granular machine learning approach to reserve prediction, we provide SHAP values for the property dataset for each point in time (i.e., every development period).17 The left (right) panel of Figure 9 presents the regression (classification) task's 10 most important features at each development period. This temporal analysis offers a more detailed understanding of how features' importance evolves as additional information becomes available over the claim development periods.

In the regression task, Losses Incurred consistently emerges as the most important feature across all development periods, reaffirming its importance in predicting future claim amounts. This consistent prominence underscores the direct relationship between the incurred losses and the final amounts to be predicted. As the development periods progress, the importance of Losses Incurred from previous time points increases. Thus, earlier assessments of incurred losses accumulate valuable information, enhancing the ability of the model to make accurate predictions. As additional information about the incurred losses becomes available over time, initial estimates are continually updated and improved throughout the claim's lifecycle.

For the classification task, the picture is slightly different. While information about the loss amounts remains the primary driver, other variables such as Business Type, Risk Category, and Branch also play crucial roles, particularly in the earlier development periods. As the claim evolves, these features' relative importance changes, highlighting the dynamic nature of the factors influencing the likelihood of changes in incurred loss amounts.

For instance, Notification Duration and Contract Category exhibit varying importance across different stages. Thus, the timeliness of claim reporting and specific policy terms can significantly impact the classification of claim developments at different points in time.

This temporal analysis highlights the complex interplay of various features over time, deepening our understanding of how different factors influence reserve predictions throughout the claim lifecycle. By examining these dynamics, we can achieve greater transparency and accuracy in the reserving process, ensuring that predictions are informed by the most relevant and timely information.