Modeling Age-to-Age Development Factors in Auto Insurance Through Principal Component Analysis and Temporal Clustering

Abstract

The estimation of age-to-age development factors is fundamental to loss reserving, with direct implications for risk management and regulatory compliance in the auto insurance sector. The precise and robust estimation of these factors underpins the credibility of case reserves and the effective management of future claim liabilities. This study investigates the underlying structure and sources of variability in development factor estimates by applying multivariate statistical techniques to the analysis of development triangles. Departing from conventional univariate summaries (e.g., mean or median), we introduce a comprehensive framework that incorporates temporal clustering of development factors and addresses associated modeling complexities, including high dimensionality and temporal dependency. The proposed methodology enhances interpretability and captures latent structures in the data, thereby improving the reliability of reserve estimates. Our findings contribute to the advancement of reserving practices by offering a more nuanced understanding of development factor behavior under uncertainty.

1. Introduction

Age-to-age loss development factors (LDFs) are fundamental tools in property and casualty insurance, enabling the projection of future claim amounts. These projections are derived from observed claims organized in loss development triangles, with the primary goal of estimating future development. When applied to claim counts, these factors are referred to as count development factors. The accurate estimation of development factors (DFs) is crucial for effective risk assessment and the management of future financial liabilities. As highlighted in Berquist and Sherman (1977), DFs are integral to forecasting ultimate losses or claim counts. Furthermore, they offer insights into claim payment patterns (Pittarello et al. 2022; Shi and Hartman 2016), supporting financial planning in auto insurance.

Despite notable advancements in DF estimation, much of the literature remains anchored in traditional methods, each with distinct advantages and limitations (Berquist and Sherman 1977; D’Arcy 1987; Okyere Dwebeng 2016). Among these, the Chain Ladder method (Verrall 2000) is widely used for its simplicity, estimating DFs by tracking the progression of claims over time. However, it often performs poorly under volatile development conditions. The Bornhuetter–Ferguson method (Schmidt and Zocher 2007), a hybrid of the Chain Ladder and Expected Loss Ratio approaches, combines historical data with prior estimates of ultimate losses (Sakthivel 2016). While more stable, it is sensitive to prior assumptions and introduces additional uncertainty through its reliance on paid loss information. The Cape Cod method, an extension of Bornhuetter–Ferguson, can yield more accurate estimates but requires extensive internal and external data and greater computational resources (Radtke 2016). The Mack model (Kloek 1998), a stochastic extension of the Chain Ladder, assumes development factors follow specific distributions (e.g., log-normal), thereby allowing for stochastic reserving via Generalized Linear Models. This enhances estimation flexibility and enables the quantification of reserve uncertainty.

Our research shifts the focus from traditional age-to-age DF estimation to an in-depth analysis of the variability and structural patterns of DFs. Recent studies have explored refined estimation techniques for DFs to improve reserving accuracy (Han and Gau 2008; Jeong and Dey 2020; Jeong et al. 2021; Korn 2015; Verdier and Klinger 2005), reflecting a broader commitment to predictive analytics and financial stability in the insurance sector. However, DF estimates are subject to uncertainty stemming from model assumptions, small sample sizes, evolving claim patterns, and data quality issues (Feng and Robbin 2022; Mack 1995; Nugraha and Qoyyimi 2022; Raço et al. 2022). This inherent variability challenges the reliability of case reserve forecasts. Stochastic reserving models have thus gained traction as alternatives to deterministic approaches, incorporating statistical tools such as regression and time-series analysis, often complemented by actuarial judgment (Al-Mudafer et al. 2022; England and Verrall 2002; Sriram and Shi 2021; Taylor 2012).

To address these challenges, we propose further analysis of Development Rectangles (DRs), which aggregate age-to-age DFs across accident half-years. DRs help identify longitudinal and cross-sectional trends in loss development, which are often obscured in raw triangle data. Detecting such trends can improve reserving accuracy and inform regulatory pricing strategies (Frees et al. 2014). Moreover, DRs facilitate the identification of outliers—extreme values that, if uncorrected, may skew DF estimates and misrepresent future liabilities (Zhao et al. 2021). To mitigate this, de-noising techniques tailored to DRs have been proposed (Barlak et al. 2022; Neuhaus 2023), emphasizing the need for context-specific methods.

In this study, we analyze DRs derived from various regulatory datasets, focusing on incurred loss, paid loss, claim counts, and paid counts. Using the Mack model (Mack 1999), we construct DRs across different reporting periods and apply Principal Component Analysis (PCA) to uncover latent patterns of variation. Rather than developing new reserving methodologies, our contribution lies in promoting the use of multivariate statistical techniques for pattern discovery and uncertainty quantification in DFs. This approach enhances our understanding of variation across accident half-years and improves risk assessment under regulatory frameworks.

A key innovation in this study is the introduction of a time-constrained, weighted K-Means algorithm, which extends the classical K-Means by restricting cluster formation to temporally adjacent accident half-years. This constraint increases the interpretability and relevance of clustering results in the context of loss reserving, enabling the detection of evolving structural changes in development factors. Moreover, the exploration of DF patterns using industry-level data adds a unique and novel dimension to this study.

Ultimately, our approach advances the use of predictive analytics in the actuarial domain, with implications for pricing strategy and reserve adequacy. The remainder of the paper is structured as follows: Section 2 describes the data and methodology. Section 3 presents and interprets empirical findings based on regulatory datasets. Section 5 concludes the paper and outlines directions for future research.

2. Materials and Methods

2.1. Data

The dataset analyzed in this study comprises incurred loss development, incurred claim count development, paid claim count development, and paid loss development for accident half-years spanning from 2003-1 to 2017-2 as reported at six-month intervals through 31 December 2017. For subsequent accident half-years from 2004-1 to 2018-2, development is reported as of year-end 2018. This structure applies across valuation years from 2017 through 2022, which form the basis of our analysis. Both incurred and paid loss amounts include associated settlement expenses, thereby reflecting the full cost of claim resolution. To enhance the homogeneity of development patterns, development triangles are constructed separately for major coverage types—medical, bodily injury (BI), and collision. These triangles are generated using a custom-built program in R software package, and the corresponding development rectangles are derived using the Mack Chain Ladder model, a widely adopted method in actuarial practice for case reserving.

For illustration purposes,

Table 1. Illustration of a portion of age-to-age incurred loss development factors rectangle obtained from Chain Ladder under the Mack model approach for the Bodily Injury coverage. The development age is every six months, and each row represents the development of incurred losses for each accident half year (HY).

Accident HY	6 to 12	12 to 18	18 to 24	24 to 30	30 to 36	36 to 42	42 to 48	48 to 54	54 to 60	60 to 66	66 to 72
200301	2.979	1.590	1.234	1.178	1.194	1.158	1.216	1.145	1.132	1.104	1.067
200302	3.482	1.504	1.375	1.318	1.200	1.181	1.208	1.169	1.094	1.101	1.062
200401	3.352	1.432	1.296	1.285	1.244	1.117	1.194	1.151	1.169	1.069	1.077
200402	3.557	1.497	1.489	1.274	1.219	1.167	1.153	1.191	1.148	1.096	1.102
200501	3.493	1.536	1.399	1.346	1.323	1.148	1.124	1.131	1.119	1.038	1.052
200502	4.219	1.649	1.462	1.244	1.146	1.191	1.210	1.149	1.113	1.095	1.075
200601	3.814	1.543	1.338	1.228	1.191	1.165	1.207	1.168	1.145	1.086	1.074
200602	4.204	1.513	1.317	1.286	1.212	1.239	1.145	1.114	1.168	1.059	1.057
200701	3.506	1.464	1.326	1.255	1.188	1.168	1.157	1.095	1.080	1.145	1.076
200702	3.977	1.540	1.286	1.231	1.212	1.202	1.207	1.093	1.101	1.066	1.075
200801	3.779	1.493	1.330	1.299	1.188	1.180	1.129	1.079	1.102	1.088	1.063
200802	4.331	1.540	1.248	1.205	1.148	1.146	1.121	1.134	1.096	1.070	1.062
200901	3.653	1.490	1.313	1.179	1.119	1.108	1.104	1.119	1.128	1.090	1.066
200902	4.478	1.560	1.277	1.151	1.124	1.137	1.106	1.127	1.080	1.101	1.040
201001	3.964	1.580	1.253	1.163	1.120	1.118	1.125	1.084	1.062	1.046	1.062
201002	3.069	1.489	1.296	1.238	1.160	1.226	1.138	1.124	1.118	1.108	1.095
201101	4.035	1.802	1.658	1.306	1.379	1.219	1.243	1.168	1.139	1.104	1.080
201102	4.351	1.787	1.485	1.404	1.259	1.364	1.221	1.239	1.181	1.117	1.116
201201	4.541	1.772	1.723	1.497	1.347	1.345	1.182	1.213	1.162	1.102	1.073
201202	4.607	1.860	1.380	1.429	1.389	1.381	1.213	1.256	1.135	1.105	1.052
201301	4.602	1.915	1.672	1.205	1.329	1.283	1.265	1.178	1.140	1.064	1.058
201302	4.005	2.045	1.366	1.465	1.286	1.370	1.298	1.217	1.129	1.082	1.057
201401	4.170	1.802	1.690	1.378	1.490	1.371	1.299	1.190	1.165	1.075	1.059
201402	5.042	1.887	1.575	1.576	1.300	1.307	1.360	1.196	1.150	1.123	1.090
201501	5.016	1.931	1.676	1.230	1.534	1.300	1.207	1.241	1.176	1.240	1.093
201502	7.001	1.868	1.477	1.516	1.293	1.361	1.228	1.226	1.167	1.115	1.074
201601	5.494	1.958	1.582	1.485	1.399	1.285	1.320	1.204	1.123	1.147	1.085
201602	5.419	2.434	1.459	1.420	1.250	1.318	1.216	1.163	1.130	1.105	1.083
201701	5.239	1.997	1.516	1.311	1.324	1.272	1.250	1.161	1.183	1.112	1.091
201702	4.874	1.977	1.642	1.391	1.296	1.261	1.249	1.167	1.136	1.128	1.072
201801	5.873	2.053	1.698	1.315	1.333	1.172	1.187	1.178	1.234	1.099	1.072
201802	5.325	2.059	1.576	1.228	1.405	1.287	1.139	1.237	1.130	1.099	1.072
201901	5.878	1.805	1.636	1.335	1.302	1.327	1.254	1.161	1.130	1.099	1.072
201902	5.073	1.957	1.530	1.404	1.282	1.295	1.193	1.161	1.130	1.099	1.072
202001	5.064	1.945	1.482	1.371	1.259	1.230	1.193	1.161	1.130	1.099	1.072
202002	6.043	1.948	1.717	1.323	1.241	1.230	1.193	1.161	1.130	1.099	1.072
202101	6.006	1.861	1.584	1.287	1.241	1.230	1.193	1.161	1.130	1.099	1.072
202102	7.513	2.086	1.403	1.287	1.241	1.230	1.193	1.161	1.130	1.099	1.072
202201	6.138	1.667	1.403	1.287	1.241	1.230	1.193	1.161	1.130	1.099	1.072
202202	4.160	1.667	1.403	1.287	1.241	1.230	1.193	1.161	1.130	1.099	1.072

presents an example of BI incurred loss development factor rectangles for development ages up to 72 months. This table is based on data from the 2022 reporting year. As a result, the age-to-age development factor for the first 6 months of accident half-year 2022-02 is an estimated value, while the corresponding factor for 2022-01 is a calculated value derived from observed incurred losses. In this context, the values in the last row of Table 1 are entirely estimated. Estimation begins from the second-last row and proceeds backward, except for the portion on the left side of the table, where the accident half-year plus development period does not exceed the reporting year. For example, the accident half-year 2021-02 plus an 18-month development period extends to the end of the 2022 calendar year, the reporting period. This means that the values 7.513 and 2.086 in that row are based on observed data, while the remaining values in the same row are estimates.

In this study, the primary focus is to analyze these projected development rectangles using multivariate statistical techniques, with the aim of better understanding the variability inherent in age-to-age development factors. As such, we do not delve into the full forecasting process used to construct the development rectangles but rather emphasize the investigation of variability patterns in the obtained rectangles. Additionally, we propose a Generalized Linear Model (GLM)-based approach to estimate the mean development factors at each development age. This framework allows us to explain variations in DFs by incorporating relevant covariates, thereby enhancing interpretability and predictive accuracy. We assume that input data are available in the form of a development triangle. Additionally, we assume that a forecasting method is in place and ready to be applied for predicting the unobserved values in the development rectangle.

2.2. Estimating Age-to-Age Development Factors via Principal Component Analysis

Principal Component Analysis (PCA) is a multivariate statistical technique widely used to explore complex patterns and dependencies in high-dimensional datasets. It achieves dimensionality reduction by identifying orthogonal directions, principal components, which capture the maximum variance in the data. By reconstructing the original data using only a subset of these components, PCA effectively filters out noise, leading to a de-noising effect. In the context of DRs, we treat each age-to-age DF as a distinct random variable, each with its own underlying expected value representing the true DF. Observations across different accident half years are interpreted as replicates of these random variables. Given the high dimensionality and complexity of DRs—each encoding a wide array of age-to-age development patterns—PCA provides a powerful framework for uncovering underlying structures and reducing dimensionality while preserving the majority of the variation.

By isolating the most influential DFs that contribute significantly to the variation in loss development, PCA enhances the interpretability, stability, and precision of models used for LDF estimation and future claims forecasting. Conceptually, the DR is viewed as comprising both signal and noise components. PCA enables the separation of these components by approximating the original data structure using only the leading principal components. Furthermore, PCA facilitates the quantification of uncertainty associated with DF estimates through the analysis of principal component scores. The integration of PCA-derived insights into the modeling process can significantly improve the robustness and accuracy of loss reserving methods, supporting insurers in risk management and assisting regulators in making data-informed pricing and policy decisions.

PCA can be implemented using two standard approaches: eigenvalue decomposition of the covariance matrix and singular value decomposition (SVD) of the data matrix. For completeness and clarity, we briefly outline the eigenvalue decomposition method for performing PCA in the following section.

2.3. Eigenvalue Decomposition of DR

Given a DR with n accident half years and p development periods—each considered a random variable—we denote the observed data as an $n\times p$ matrix L. The first step in PCA is to standardize the data by subtracting the mean and dividing by the standard deviation of each variable. This results in a standardized matrix Z. The sample mean of the p-dimensional random vector provides an empirical estimate of the DF at each development age, while the sample standard deviation captures the uncertainty contributed by each accident year’s development data and by the method used to project unobserved points in the DR. The sample covariance matrix of Z, denoted by C, is a $p\times p$ matrix given by

$$C={\displaystyle \frac{1}{n-1}}{Z}^{\top }Z.$$

(1)

This covariance matrix reflects the uncertainty arising from accident half year variation, the correlation among development ages, and the variability introduced by the projection methodology. The concentration of variations across development ages helps assess the temporal robustness of DF patterns. To further examine this structure, we perform an eigenvalue decomposition of the covariance matrix C, resulting in eigenvectors and eigenvalues. The eigenvectors represent the principal components of the DFs, while the eigenvalues quantify the amount of variance captured by each component. These eigenvectors are ordered by decreasing eigenvalues.

We select the top k principal components based on the desired level of variance explained, denoted by q. For instance, if $q=95\%$, we choose the smallest k such that the sum of the first k eigenvalues is at least 95% of the total variance. In this work, we investigate how many components are necessary to capture the dominant fluctuations in age-to-age development factors across accident years. Let $V^{\left(k\right)}\in {\mathbb{R}}^{p\times k}$ be the matrix containing the first k eigenvectors. The matrix of principal component scores, $T^{\left(k\right)}\in {\mathbb{R}}^{n\times k}$, is computed as

$$T^{\left(k\right)}=Z{V}^{\left(k\right)}.$$

(2)

Each row of $T^{\left(k\right)}$ corresponds to the projection of an accident year onto the reduced k-dimensional space defined by the principal components. We then approximate the original data matrix L using the selected k components, yielding

$$L^{\left(k\right)}=T^{\left(k\right)}{V^{\left(k\right)}}^{\top }+\overline{L},$$

(3)

where $\overline{L}$ is the row vector of column means of the original DR data, repeated across all rows. We denote the approximations using the first and first two components as $L^{\left(1\right)}$ and $L^{\left(2\right)}$, respectively. The first and second principal component vectors are $V_1^{\left(k\right)}$ and $V_2^{\left(k\right)}$, and their associated score vectors are $T_1^{\left(k\right)}$ and $T_2^{\left(k\right)}$.

This low-rank approximation helps reduce the influence of noise and potential outliers in the DFs. By focusing on the major components, the reconstruction preserves the underlying development patterns while filtering out irregular fluctuations, thereby enhancing the interpretability and robustness of the loss development analysis.

2.4. K-Means Clustering of Principal Component Scores with Contiguity Constraints in Time

Principal component scores serve as key features for exploring the characteristics of DFs within a reduced, low-dimensional feature space. These scores capture both the magnitude of each principal component and the variability of features across accident years. When analyzing DFs over time, it is natural to consider temporal contiguity, i.e., the idea that neighboring accident half years are likely to exhibit similar development patterns due to shared economic conditions, inflationary trends, and safety-related advancements that influence incurred losses or claim counts. This notion parallels spatial clustering in auto insurance pricing, where geographical proximity is used to ensure spatial continuity in risk segmentation. Similarly, temporal contiguity should be preserved when clustering accident half years based on their development profiles.

To incorporate this insight, we propose a novel clustering algorithm that assigns a weight to each feature prior to applying the K-Means clustering method. The optimal weight value is selected by minimizing a metric that quantifies cluster contiguity—specifically, the total number of temporally disjoint observations within clusters. The primary objective of this algorithm is to balance clustering performance with the requirement that accident half years grouped within a cluster exhibit temporal continuity. This emphasis on contiguity is grounded in the assumption that adjacent accident years should demonstrate shared characteristics in their DF trajectories. The algorithmic procedure is outlined in Algorithm 1, where two key innovations are introduced to the traditional K-Means approach:

Algorithm 1 Procedure for K-Means clustering of principal component scores with contiguity constraints in time.

1: Perform the PCA on the covariance matrix of the data, and compute the variance (${\lambda }_i$) for the ith PC. Select the smallest number of components k such that $${\displaystyle \frac{{\sum }_{i=1}^k{\lambda }_i}{{\sum }_{i=1}^{N-1}{\lambda }_i}}>q,$$ where $0<q<1$, and N is the total number of Accident Half-Year. By default, $q=0.95$. 2: Construct dataset including two components: PC scores for the first kth PCs, and an additional column representing Accident Half-Year, ranging from 1 to N. Scale this data and denote it by D, where D can be written as $(P{S}_1,\dots ,P{S}_k,t)$. 3: For each pre-specified scalar value of w, a: Create new dataset $D_w^{\ast }$, where $D_w^{\ast }$ = ${(P{S}_1\ast w,...,P{S}_k\ast w,t\ast (1-w\ast k))}_{n\ast k}$. b: Apply K-Means clustering methods on $D_w^{\ast }$, and extract the cluster number that each observation belongs to. Reorganize the cluster numbers in ascending order, denoted by ${\mathbf{c}}_w$, where ${\mathbf{c}}_w=(c_{1w},...,c_{nw})$ and n is the number of observations. c: Use $C{T}_w$ to quantify the contiguity of the clusters, $$C{T}_w={\displaystyle \frac{\# \mathrm{of} (c_{(i+1)w}-c_{iw})<0}{n}}$$ for $i=1,\dots ,n-1$. 4: Minimize $C{T}_w$ with respect to w, then denote the corresponding membership of accident year as ${\mathbf{C}}_{argmi{n}_w}=(c_1,...,c_n)$. 5: Make the final adjustment for the membership assignment. Let $\mathbf{M}=(m_1,\dots ,m_n)$ represent the optimal membership of the accident year. a: $m_1\leftarrow c_1$ b: For $i=2,\dots ,n-1$, c: if $m_{i-1}=c_{i+1}$ then $m_i\leftarrow m_{i-1}$ d: else $m_i\leftarrow c_i$

• A weight is incorporated into the objective function to control the influence of each feature during clustering.
• Post-clustering adjustments are made to ensure that clusters maintain full temporal contiguity, and no cluster consists of a single, isolated accident half year.

These enhancements enable the clustering results to better reflect underlying temporal structures in the DR. The resulting clusters are subsequently used in downstream modeling, where age-to-age development factors are estimated at the cluster level. This approach generalizes the traditional method of estimating a single DF for each development age by incorporating group-level effects that account for shared characteristics among similar accident years.

2.5. Modeling Development Factors by Generalized Linear Models

In addition to using Principal Component Analysis to decompose the development rectangle and approximate the development pattern using a subset of principal components, we propose applying a Generalized Linear Model (GLM) to directly model the development factors within the DR. The GLM-based approach offers several advantages, including improved interpretability of loss or count development patterns and enhanced uncertainty quantification. By fitting a GLM to the DR data, one can obtain estimates of the age-to-age DFs along with their associated standard errors. These standard errors provide a direct measure of the uncertainty associated with the DF estimates. Moreover, the estimated residual standard deviation from the GLM model can be used to assess the overall reliability of the model and the stability of the DF estimates. This GLM-based approach contrasts with traditional methods, which typically rely on summary statistics—such as the sample mean or median across all accident years—to estimate each age-to-age DF. By leveraging regression modeling, our approach introduces flexibility to incorporate covariates that explain variation in DFs across time and groups.

In this section, we briefly describe the application of our GLM-based modeling for estimating age-to-age DFs. Let $Y_{i,j}$ represent the incurred losses (or counts) for the jth accident half year at the ith development period. The corresponding age-to-age development factor, denoted by $d_{ij}$, is defined as the ratio of the $(i+1)$th to the ith period losses (or counts):

$$d_{ij}={\displaystyle \frac{Y_{i+1,j}}{Y_{i,j}}}.$$

The response variable $d_{ij}$ is assumed to follow a distribution from the exponential family, characterized by the general form

$$f(d\mid \theta ,\varphi )=exp\left\{{\displaystyle \frac{d\theta -b\left(\theta \right)}{a\left(\varphi \right)}}+c(d,\varphi )\right\},$$

(4)

where $\theta$ is the canonical parameter, $\varphi$ is the dispersion parameter, and the functions $a(·)$, $b(·)$, and $c(·)$ depend on the specific distribution family. The variance of d is given by

$$\mathrm{Var}\left(d\right)=\varphi V\left(\mu \right),$$

(5)

where $V\left(\mu \right)$ is the variance function that describes how the variance changes with the mean $\mu$. For example, in a normal distribution, $V\left(\mu \right)=1$, while in a Poisson distribution, $V\left(\mu \right)=\mu$.

Given that DFs are closely associated with the development period in which they are observed, we include ‘Period’ as a key covariate in our model. Additionally, to investigate whether clusters of accident years affect DF patterns, we introduce a dummy variable ‘Group’, with the earliest accident year group serving as the reference category. The predictor associated with GLM can thus be expressed as

$$g\left(E\left(d_{ij}\right)\right)={\beta }_0+{\beta }_1·{\mathrm{Group}}_{i,j}+{\beta }_2·{\mathrm{Period}}_j,$$

(6)

where $g(·)$ is a link function used to linked mean response $E\left(d_{ij}\right)$ to the linear combination of covariates. Given that both Group and Period are categorical covariates, the estimated effects for each level represent the mean difference from the respective base level. The model is applied separately to data from each reporting year (e.g., 2020, 2021, and 2022) to capture time heterogeneity in the estimates. Parameter estimates in the GLM are obtained via iteratively reweighted least squares (IRWLS). For implementation, we use the glm function in R. When the response follows a normal distribution, the estimated DF for the ith development period can be calculated as ${\widehat{\beta }}_0+{\widehat{\beta }}_{i2}$, where the estimate also depends on the cluster to which the accident year belongs, through the coefficient ${\widehat{\beta }}_1$.

2.6. Implications of Proposed Methods

Traditional risk management in auto insurance has long relied on understanding the development of future liabilities through the projection of unobserved losses in loss development triangles or claim counts in count development triangles. A more detailed exploration of age-to-age development factors (DFs) in the reserving process offers significant potential to enhance both risk management and financial forecasting.

This study introduces a novel multivariate analysis of forecasted development factors, structured in rectangular form rather than conventional triangular models. This shift enables a better understanding of DF behavior, moving beyond summary statistics to examine patterns, variability, and dependencies. Such deeper insights not only refine the accuracy of case reserve estimates but also illuminate the inherent uncertainty and variability surrounding DF predictions. These improvements are critical for insurers aiming to make informed reserving decisions, optimize capital allocation, and more effectively manage portfolio risk. In doing so, this work contributes to more resilient and financially sound insurance operations.

Conventional approaches typically assume the independence of DFs across accident years—a simplification that fails to capture the evolving nature of insurance systems. In reality, development factors are influenced by a range of time-dependent variables, including improvements in vehicle safety technology, shifts in medical treatment standards, and changes in infrastructure or regional transportation systems. To account for these dynamics, our study advocates for the temporal clustering of development factors. By organizing DFs according to their position in time, rather than treating them as interchangeable across years, we uncover meaningful patterns that reflect systemic change. This temporal perspective allows for more accurate modeling of development behavior and ultimately leads to better-informed forecasting.

Our research highlights the practical advantages of this approach in modern auto insurance, offering a more adaptive framework that aligns with the complex, evolving realities of claims development. Temporal clustering not only strengthens predictive accuracy but also supports more flexible and responsive risk management strategies, an essential advancement for navigating today’s dynamic insurance landscape.

3. Results

3.1. Results on Principal Component Analysis and Weighted K-Means of Development Factors

Principal Component Analysis is conducted to examine whether a small number of principal components (PCs) can effectively reduce noise while retaining the primary variability in LDFs. Figure 1, Figure 2, Figure 3 and Figure 4 present the LDFs for incurred and paid counts and losses across all development ages, for both bodily injury (BI) and medical coverages. These factors are estimated using either one or two PCs.

A consistent pattern emerges: LDFs in the first development half-year tend to exhibit higher magnitudes and greater variability, followed by a decline in both as the development age increases. This trend may be attributed to the resolution of uncertainty, where larger claims are more likely to have been settled or reserved earlier in the development period. Specifically, in medical coverage, the LDFs for incurred counts and losses initially decrease, then exhibit a temporary increase, and eventually stabilize. This mid-development increase may be due to delayed diagnosis or the late reporting of certain injuries or illnesses. However, as development progresses, the LDFs tend to converge toward one, reflecting stabilization in claim settlements and reserve development. Moreover, the LDFs estimated using either the first PC alone or the first two PCs show minimal differences and closely resemble those derived from the original data. This suggests that the first principal component captures most of the essential variation in the data and is sufficient for accurate DF estimation. The percentage of variation explained by the first and second principal components, along with their corresponding standard deviations, is reported in

Table 2. Standard deviation, proportion of variance and cumulative proportion of first two principle components for loss development factor of incurred and paid loss, for body injured, collision and medical coverages for the 2022 reporting year.

	BI, Incurred Loss		Collision, Incurred Loss		Medical, Incurred Loss
	PC1	PC2	PC1	PC2	PC1	PC2
Standard deviation	1.066	0.208	0.030	0.006	0.202	0.066
Proportion of Variance	0.938	0.036	0.960	0.034	0.754	0.081
Cumulative Proportion	0.938	0.974	0.960	0.995	0.754	0.835
	BI, Paid Loss		Collision, Paid Loss		Medical, Paid Loss
	PC1	PC2	PC1	PC2	PC1	PC2
Standard deviation	1.270	0.670	0.081	0.005	0.275	0.060
Proportion of Variance	0.754	0.210	0.995	0.004	0.878	0.042
Cumulative Proportion	0.754	0.964	0.995	1.000	0.878	0.920

and

Table 3. Standard deviation, proportion of variance and cumulative proportion of first two principle components for loss development factor of incurred and paid counts, for body injured, collision and medical coverages for the 2022 reporting year.

	BI, Incurred Counts		Collision, Incurred Counts		Medical, Incurred Counts
	PC1	PC2	PC1	PC2	PC1	PC2
Standard deviation	0.672	0.175	0.061	0.026	0.112	0.052
Proportion of Variance	0.916	0.062	0.706	0.125	0.562	0.123
Cumulative Proportion	0.916	0.977	0.706	0.831	0.562	0.685
	BI, Paid Counts		Collision, Paid Counts		Medical, Paid Counts
	PC1	PC2	PC1	PC2	PC1	PC2
Standard deviation	0.061	0.001	0.176	0.026	2.326	0.166
Proportion of Variance	1.000	0.000	0.964	0.021	0.993	0.005
Cumulative Proportion	1.000	1.000	0.964	0.984	0.993	0.998

for each major coverage, with respect to incurred loss and incurred counts, respectively.

In the plots of Figure 1, Figure 2, Figure 3 and Figure 4, the blue and grey shaded areas represent the variability of DFs across different accident years. Notably, PCA reduces the width of these intervals, particularly when using only the first PC, indicating enhanced precision in the DF estimates. This improved accuracy can support insurers in making more informed loss reserving decisions, contributing to better financial planning. From a practical standpoint, these findings suggest that PCA can effectively filter noise and highlight the dominant trend in the development patterns, which in turn enhances the reliability of reserve estimates. For insurers, this translates to more accurate and consistent reserving practices, ultimately supporting better capital management, pricing strategies, and regulatory compliance.

Figure 5 and Figure 6 display the first two PC scores of LDFs for incurred and paid counts and losses across medical, BI, and collision coverages. Colors represent the clusters determined using traditional K-Means clustering, while the numbers correspond to accident half-years. Interestingly, the clustering results for paid and incurred losses (Figure 5) show that cluster membership is not strictly determined by accident year. For instance, in BI coverage, the second halves of 2012, 2017, and 2020 form one cluster, while the first halves of 2013, 2016, and 2022 form another. Similar observations are made across other coverages. However, the development factors are inherently time sensitive. For example, LDFs from adjacent accident half-years may be expected to be similar, and DFs after 2019 may exhibit distinct characteristics due to the COVID-19 pandemic. To account for such temporal dependencies, we integrate accident year contiguity into our clustering approach as described in Algorithm 1.

Our proposed algorithm introduces a weight parameter w to control the influence of PC scores in clustering. We assess the contiguity of clusters (denoted $CT$) for various values of w, selecting the value that minimizes $CT$. Once the optimal w is identified, we refine cluster memberships to improve temporal continuity. For example, in the case of incurred loss DFs for BI coverage, the minimum $CT$ is achieved when $w=0.2$. However, with $w=0.2$, the accident half-years 201501, 201502, and 201601 are assigned to clusters 3, 4, and 3, respectively. To enhance temporal coherence, we reassign 201502 to cluster 3, aligning it with adjacent periods. This post-adjustment step ensures contiguous and practically meaningful clustering results.

3.2. Results on Development Factor Estimates

Based on the clustering results from the modified K-Means approach, we fit the data using GLM introduced in Section 2.5, for the purposes of identifying the variation contribution from different clusters and overall determining the DF estimates. The estimated DFs by group and period of incurred counts and loss, as well as paid counts and loss in three coverages, are reported in

Table 4. Loss development factor of incurred and paid loss, estimated by GLM model under Model 2: regression + group using first five periods, for body injured, collision and medical coverages from 2020 to 2022.

	BI, Incurred Loss			Collision, Incurred Loss			Medical, Incurred Loss
Reported Year	2020	2021	2022	2020	2021	2022	2020	2021	2022
group2	0.021	0.004	0.018	−0.006 ***	0.007 ***	0.008 ***	0.077	0.024	0.054 ***
	(0.110)	(0.118)	(0.139)	(0.002)	(0.002)	(0.003)	(0.078)	(0.020)	(0.018)
group3	0.207 *	0.195 *	0.256 *	−0.007 ***	0.006 **	0.006 **	0.193 **	0.106 ***	0.083 ***
	(0.116)	(0.118)	(0.131)	(0.002)	(0.002)	(0.003)	(0.076)	(0.018)	(0.018)
group4	0.450 ***	0.437 ***	0.556 ***	0.00003	0.006 **	0.007 ***	0.132 *	0.099 ***	0.082 ***
	(0.110)	(0.108)	(0.131)	(0.002)	(0.002)	(0.003)	(0.076)	(0.018)	(0.018)
	Mean Age-to-Age DFs
Period12	3.917	3.973	4.008	1.010	1.069	1.071	1.688	1.489	1.518
Period18	2.422	2.428	2.448	1.002	0.999	0.999	0.995	1.026	1.041
Period24	1.834	1.832	1.806	1.003	0.995	0.995	0.929	0.984	0.997
Period30	1.592	1.594	1.549	1.003	0.995	0.995	0.942	1.004	1.014
Period36	1.381	1.376	1.324	1.003	0.995	0.995	0.923	1.024	1.035
	BI, Paid Loss			Collision, Paid Loss			Medical, Paid Loss
Reported Year	2020	2021	2022	2020	2021	2022	2020	2021	2022
group2	0.086	0.080	0.068	−0.0003	0.022 ***	0.019 ***	0.073 ***	0.073 ***	0.073 ***
	(0.071)	(0.077)	(0.091)	(0.006)	(0.007)	(0.007)	(0.018)	(0.018)	(0.018)
group3	0.460 ***	0.455 ***	0.458 ***	−0.003	0.004	0.003
	(0.071)	(0.077)	(0.089)	(0.006)	(0.007)	(0.007)
group4	0.469 ***	0.500 ***	0.536 ***	0.006	0.007	0.010
	(0.071)	(0.073)	(0.085)	(0.006)	(0.007)	(0.007)
	Mean Age-to-Age DFs
Period12	4.102	4.205	4.342	1.218	1.273	1.279	2.288	2.288	2.288
Period18	1.485	1.480	1.472	1.005	1.004	1.004	1.153	1.153	1.153
Period24	1.209	1.188	1.174	1.001	0.994	0.994	1.041	1.041	1.041
Period30	1.072	1.049	1.025	1.000	0.993	0.993	1.024	1.024	1.024
Period36	1.029	1.006	0.976	1.000	0.993	0.993	1.015	1.015	1.015

Note: * p < 0.1; ** p < 0.05; *** p < 0.01.

and

Table 5. Loss development factor of incurred and paid counts, estimated by GLM model under Model 2: regression + group using first five periods, for body injured, collision and medical coverages from 2020 to 2022.

	BI, Incurred Counts			Collision, Incurred Counts			Medical, Incurred Counts
Reported Year	2020	2021	2022	2020	2021	2022	2020	2021	2022
group2	−0.002	−0.005	−0.007	0.010 *	0.003	−0.003	−0.012	−0.010	0.002
	(0.005)	(0.004)	(0.005)	(0.005)	(0.006)	(0.006)	(0.008)	(0.009)	(0.013)
group3	−0.008	−0.005	−0.015	−0.003	−0.008	−0.009	0.078 ***	0.071 ***	−0.017
	(0.005)	(0.004)	(0.013)	(0.008)	(0.008)	(0.006)	(0.009)	(0.009)	(0.012)
group4	−0.003		0.0003	0.002	−0.002	−0.005	0.033 ***	0.035 ***	0.033 **
	(0.005)		(0.005)	(0.005)	(0.005)	(0.006)	(0.006)	(0.007)	(0.013)
	Mean Age-to-Age DFs
Period12	1.225	1.223	1.227	1.069	1.076	1.081	0.754	0.748	0.995
Period18	1.011	1.011	1.011	0.962	0.965	0.970	0.861	0.862	0.947
Period24	1.004	1.004	1.004	0.981	0.985	0.987	0.971	0.972	0.991
Period30	1.003	1.004	1.003	0.997	1.002	1.005	1.075	1.076	1.022
Period36	1.003	1.004	1.003	0.995	0.999	1.002	0.965	0.961	0.992
	BI, Paid Counts			Collision, Paid Counts			Medical, Paid Counts
Reported Year	2020	2021	2022	2020	2021	2022	2020	2021	2022
group2	0.229 ***	0.234 ***	0.229 ***	0.015	0.023 *	0.014	0.269 ***	0.248 ***	0.185
	(0.045)	(0.050)	(0.052)	(0.013)	(0.013)	(0.015)	(0.035)	(0.039)	(0.205)
group3	0.375 ***	0.370 ***	0.363 ***	0.024 *	0.035 ***	0.032 **	0.359 ***	0.379 ***	0.429 **
	(0.044)	(0.049)	(0.053)	(0.013)	(0.013)	(0.015)	(0.036)	(0.038)	(0.205)
group4	0.412 ***	0.449 ***	0.461 ***	0.045 ***	0.055 ***	0.065 ***			0.864 ***
	(0.046)	(0.052)	(0.051)	(0.013)	(0.013)	(0.016)			(0.200)
	Mean Age-to-Age DFs
Period12	3.196	3.309	3.383	1.668	1.672	1.688	3.095	3.180	3.925
Period18	1.420	1.449	1.476	1.061	1.059	1.061	1.464	1.507	1.212
Period24	1.162	1.178	1.205	1.013	1.010	1.012	1.233	1.262	0.881
Period30	1.085	1.097	1.115	1.000	0.996	0.997	1.154	1.178	0.778
Period36	0.973	0.975	0.984	0.993	0.988	0.989	1.031	1.041	0.696

Note: * p < 0.1; ** p < 0.05; *** p < 0.01.

. For illustration purposes, only the estimates for the first five periods are listed in the main body of the paper. The tables reveal a significant group effect on the incurred loss development factors within both BI and medical coverages. When examining the temporal aspect, disparities in group effects become evident. These findings underscore the inadequacy of relying solely on single statistical measures like mean or median for age-to-age DF analysis, as they fail to capture the variability and uncertainty inherent in data across various accident years. Grouping accident years mitigates this issue, facilitating more meaningful modeling of DFs. Moreover, the observed differences among these groups point to temporal trends and variability in age-to-age DFs. Their notable distinctions emphasize that they cannot be treated uniformly. Hence, employing multiple estimates of age-to-age DFs becomes imperative, offering a dynamic perspective that captures diverse trends in reaching the ultimate DF and final estimate. This approach is essential for a comprehensive understanding of the development process.

Consistent trends emerged regarding paid losses. Significant disparities are noted in the estimations across accident year groups, highlighting the dynamic nature of age-to-age development factors. Moreover, the progression of losses for medical coverage appears temporally consistent, with fewer groups due to temporal clustering compared to other coverages. Another notable finding is the variation in group effect estimates across reporting years in our models. This suggests a dynamic pattern in loss development across different reporting years, with significant updates possibly occurring even among adjacent reporting periods. In Table 5, we observe that the claim development pattern is more time homogeneous for BI, incurred counts than the BI, paid counts. This may imply that higher variability of paid counts development pattern than the incurred counts for the BI coverage. Similar patterns are observed for the collision coverage. However, the paid counts development pattern associated with medical coverage are all volatile.

These findings have significant implications for auto insurance pricing and rate regulation practices. Understanding the group effects and temporal variability of DFs in the time domain is crucial for accurately assessing risk. By recognizing the impact of different accident years on DFs and the temporal trends therein, insurers can refine their pricing models to better reflect the evolving nature of risk. This approach allows for a more precise risk assessment of future liability, potentially leading to fairer premiums for policyholders. Moreover, regulatory bodies can utilize these insights to establish more effective rate regulations. By considering the variability and uncertainty inherent in age-to-age DFs, regulators can implement policies that ensure insurance rates remain both competitive and actuarially sound.

4. Some Discussions

This study builds upon the foundational Mack Chain Ladder model, using it to construct the development triangle that underlies our proposed analysis. From this, we form the development rectangle, a structure comprising both the observed upper triangle and a forecasted lower triangle, serving as the basis for our dimensionality reduction and clustering techniques. While the Mack model is used illustratively, our approach is model agnostic: the PCA framework can be readily applied to development rectangles generated from any reserving method, whether traditional (e.g., Bornhuetter–Ferguson) or modern (e.g., machine learning-based forecasts). By extracting dominant modes of variation across a set of projected development outcomes, PCA enables a systematic assessment of how development factors evolve, offering insights into the consistency, or disruption, of reserving assumptions over time.

Traditional chain ladder-type models typically assume that development patterns across accident half-years are identically distributed, simplifying estimation by treating these periods as replicates of a common process. However, this assumption is often violated in practice due to temporal heterogeneity induced by regulatory changes, inflation, operational shifts, or broader macroeconomic trends. To address this, we introduce a temporal clustering framework that groups accident periods based on similarity in their development trajectories. This data-driven approach relaxes the stationarity assumption and allows for the detection of structural changes in development behavior. In doing so, it supports more sophisticated reserve analysis that can adapt to evolving claims environments, thereby improving both interpretability and robustness of the estimation approaches.

Importantly, our focus is not on forecasting the unobserved lower triangle per se but rather on analyzing the variability and structural dynamics embedded within the completed development rectangle. The lower triangle is assumed to be generated by external reserving methods; we take it as given and use it as input for further analysis. Our contributions are thus exploratory and diagnostic rather than predictive in nature. Nevertheless, several limitations must be acknowledged. First, our analysis is contingent on the quality and reliability of the forecasted lower triangle; we do not assess or enhance its predictive validity. Second, the framework does not explicitly model forecast uncertainty, which may impact interpretation. Third, while the methods reveal latent structure and clustering within the data, they are not designed to produce reserve estimates or confidence intervals. Lastly, our approach presumes some level of comparability across development rectangles, an assumption that may not fully hold in the presence of significant shifts in claim handling practices or portfolio composition. Despite these limitations, our methodology offers a valuable toolkit for visualizing and interpreting complex development patterns, particularly in settings where traditional assumptions of reserving models may be under strain.

5. Conclusions

The ultimate goal of this investigation of development pattern is to estimate age-to-age development factors using observed development triangles. Our analysis employs Principal Component Analysis to investigate the variability and noise reduction of development factors. A general pattern emerges, wherein DFs exhibit higher variability and magnitude in the initial half-year, followed by a decrease as development ages progress. This trend suggests a stabilization in claim settlements and reserves over time. Moreover, the first PC proves to be sufficient in capturing the majority of information for DF estimation, with minimal differences observed when compared to DFs calculated from the original data. While clustering outcomes appear independent of accident years, the analysis underscores the necessity of considering accident years, especially in light of external factors such as the impact of COVID-19 pandemic.

Our study has shed light on the dynamic nature of age-to-age development factors in insurance claim data as evidenced by significant disparities across accident year groups and temporal variations. Through the application of principal component analysis, modified K-Means clustering and GLM, we have provided comprehensive estimates of DFs, revealing important patterns in claim development across different coverages and reporting periods. Our findings underscore the inadequacy of relying solely on single statistical measures for DF analysis and highlight the importance of using multiple estimates to capture the variability and uncertainty inherent in the loss data. These insights hold significant implications for insurance pricing and rate regulation practices, emphasizing the need for a dynamic perspective in assessing risk and ensuring fair premiums for policyholders. By recognizing and addressing the temporal trends and group effects in DFs, insurers and regulatory bodies can refine their models and policies to better reflect the evolving nature of insurance risk, ultimately promoting a more stable and equitable insurance market.