September 30, 2026
Comparison between models for predicting premiums in the Brazilian Automobile insurance market
Comparison between Models for Premium Prediction in the Brazilian Automobile Insurance Market
Angélica Tidori Takiguchi; Igor Pinheiro de Araújo Costa
DOI: 10.22167/2675-6528-202602675
Article derived from a Course Conclusion Work (TCC), with content based on the student’s original work and adapted to the editorial format of the E&S Magazine with the support of the ResumeAI tool, an artificial intelligence solution developed by Instituto Pecege for textual synthesis and organization.
Summary
The insurance market in Brazil, with the automotive branch representing a significant portion of GDP in 2024, demands methodologies that provide greater reliability to premium forecasts. In this context, the application of time series models constitutes a strategic tool for insurance companies. The study comparatively evaluated the predictive performance of a simpler Moving Averages model and more robust models, SARIMA and ETS, with the objective of determining which of them presented greater accuracy in monthly projections. For the analysis, monthly data from the Brazilian automotive insurance market, covering the period from January 2019 to December 2024, were used. The models’ performance was measured by the Mean Absolute Percentage Error (MAPE) for predictive accuracy, and by the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for statistical fit comparison. The results indicated that, despite the ETS and SARIMA models demonstrating better statistical adherence to historical data, the Moving Averages model achieved the best predictive performance for the analyzed scenario. This occurred because the actual values in the test period showed a stability that diverged from the oscillations and seasonalities captured by the more robust models in the training base. The Moving Averages model proved to be an effective and low-complexity tool for supporting decision-making.
Keywords: ETS; Moving averages; SARIMA; Time series.
1. Introduction
The insurance market in Brazil is a multifaceted sector, encompassing general insurance, open supplementary pension plans, supplementary health, and capitalization. The inspection, regulation, and supervision of a large part of these segments, specifically general insurance, open supplementary pension plans, and capitalization, are the responsibilities of the Superintendence of Private Insurance (SUSEP). According to the SUSEP’s Monthly Report of the Supervised Market (2024), the insurance market showed a growth of 12.2% compared to the previous year, with total revenues reaching R$ 435.56 billion. This amount represented 3.72% of Brazil’s Gross Domestic Product (GDP) in 2024.
Within this panorama, the general insurance sector stood out, contributing 3.29% of GDP, equivalent to R$ 385.35 billion (SUSEP, 2024). Fundación MAPFRE (2025) clarifies that the general insurance sector is composed of damage and personal insurance. While personal insurance aims to protect human beings and guarantee compensation related to their health, integrity, and existence, damage insurance focuses on property assets, ensuring the payment of compensation for losses or damages affecting the policyholder’s assets. The Automotive segment, one of the most relevant in damage insurance, collected R$ 57.66 billion in December 2024, corresponding to 42.5% of the total premiums in the damage insurance market (SUSEP, 2024). These data highlight the significant economic activity of the insurance market in Brazil, with a substantial portion coming from the Automotive group.
The importance of insurance transcends mere individual protection, playing a crucial role in the economy and society. According to Contador (2023), insurance acts in mitigating losses from unexpected events, promotes efficient risk management, and offers financial stability for individuals and companies. This security encourages institutions to undertake long-term projects and families to acquire durable goods, driving business and commercial growth. Borelli et al. (2015) complement this by stating that insurers contribute to economic growth and large-scale income generation, minimizing uncertainties and encouraging investments, innovations, and competition. In this context, the premium, which is the amount paid to the insurer, is fundamental for the coverage of all risks foreseen in each policy.
Given the relevance of insurance for the economy and society, it becomes imperative to adopt methodologies that enhance the reliability of premium forecasts. The application of time series models, which consist of describing the historical behavior of data, identifying seasonal patterns, and making predictions (Morettin and Toloi, 2018), can offer a strategic tool for insurance companies. This approach contributes to insurance companies performing their functions with greater analytical security, in addition to providing significant theoretical gain for the insurance market, especially in the Automobile sector, by supporting analyses and monitoring of pricing, conversion rates, and portfolio renewal.
The need for accurate forecasts and the importance of the Automobile insurance market justify this study. Thus, the objective of this work is to comparatively analyze the performance between a simpler Moving Averages model and more robust SARIMA and ETS models, in order to identify which model presents the best predictive results for the estimation of monthly Automobile insurance premiums in Brazil in 2024.
2. Material and Methods
The present study was characterized by a quantitative approach, of a descriptive and predictive nature, with the objective of comparatively analyzing the performance of different time series models in forecasting monthly Automobile insurance premiums in Brazil. The research sought to identify which model would present the best predictive results for the year 2024, as outlined in the general objective of the work.
The database used, titled “Ses_seguros”, was obtained through the Statistics System (SES) of the Superintendence of Private Insurance (SUSEP). The data comprised monthly values referring to the automobile insurance market in Brazil, covering the period from January 2019 to December 2024.
For the analysis, the most frequent and relevant insurance branches in the Brazilian automotive market were considered: Personal Accident Passengers (code 0520), Automobile – Hull (code 0531), Assistance and Other Coverages (code 0542), and Facultative Civil Liability for Vehicles (code 0553). Observations from the five companies with the largest Direct Premium volumes in the analyzed period were selected, due to their relevance to the market and computational capacity for model generation.
The variables used in the calculations were the year and month of the information (damesano), the company code (coenti), and the Direct Premium (premio_direto). The Direct Premium variable was defined as the result of Issued Risk Premiums, subtracting the values of Cancellations and Refunds, according to the SUSEP FIP Filling Manual. The other variables present in the original database were excluded because they were not relevant to the study’s objectives.
The data processing and cleansing process was carried out using Microsoft Access software. All subsequent analyses and model implementation were executed using Python software. The time series analysis methodology, according to Morettin and Toloi (2018), consisted of describing the historical behavior of the data, identifying seasonal patterns, and making forecasts.
The time series models studied were Moving Average (MA), Seasonal Auto-Regressive Integrated Moving Average (SARIMA), and Error, Trend, and Seasonality (ETS). The Moving Average model, one of the components of ARIMA, represents the time series as a linear combination of past and present white noises, with its order identified through the sample autocorrelation function (ACF) and partial sample autocorrelation function (PACF).
The SARIMA model, an extension of ARIMA, incorporates a seasonal component to handle periodicities in the time series, being applicable in situations where the series is not stationary and requires differencing for stabilization (Morettin and Toloi, 2018). Its parameters are also identified with the aid of ACF and PACF functions. The ETS model, on the other hand, based on exponential smoothing (Hyndman and Athanasopoulos, 2018), decomposes the series into error, trend, and seasonality, which can be additive or multiplicative, including variations such as the Holt-Winters model.
To evaluate the statistical fit of the models, the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) were applied. Predictive accuracy was measured by the Mean Absolute Percentage Error (MAPE). Additionally, the stationarity of the time series was verified through the Augmented Dickey-Fuller (ADF) test, which assesses the presence of a unit root. The model residuals were subjected to the Ljung-Box test, to verify the absence of autocorrelation, and the ARCH-LM test, to assess homoscedasticity.
3. Results and Discussion
The initial data analysis, fundamental for understanding the behavior of time series, revealed distinct patterns of trend, seasonality, and residuals for each of the five insurance companies studied. For insurance company A, a growth trend was observed between the end of 2020 and the beginning of 2023, followed by a period of stability. This upward movement was attributed to factors such as component scarcity in the automotive industry during the pandemic, the appreciation of used vehicles, and an increase in theft and robbery rates, which directly impacted insurance prices. Seasonality for this insurer showed regular fluctuations, with notable production peaks in the middle and, mainly, at the end of each year, indicating predictable cyclical behavior.
For insurer B, the time series decomposition revealed a seasonality with considerable oscillations over the years, characterized by a sharp drop in the early months and a significant increase in the middle of the period. Insurer C also exhibited expressive seasonal variation, with marked reductions between the end and the beginning of the years, interspersed by growth peaks. These patterns highlight the complexity inherent in the automobile insurance market, where external factors and consumer behavior can influence demand and premium pricing at different times of the year, requiring models that can capture these nuances.
Insurer D presented the most volatile seasonality among the five analyzed, with prominent peaks at the beginning and middle of the years, followed by declines at the end. Additionally, the trend graph for this insurer showed an abrupt change between 2020 and 2021, which broke the smoothness of the series and generated a sudden slope. Despite this break, the overall view indicated an increasing trend between 2020 and 2023, aligned with the other insurers. In turn, insurer E exhibited seasonality similar to that of insurer A, with a considerable increase in premiums at the end of each year, reinforcing the presence of specific seasonal patterns that need to be considered in forecasts.
This preliminary analysis was crucial for identifying the presence of seasonal patterns and trends in the insurance premium time series. To verify the stationarity of the series, the Augmented Dickey-Fuller (ADF) test was applied. The initial p-values obtained were above 0.05 for all insurers, indicating that the series were non-stationary and therefore exhibited a unit root. This result confirmed the need to apply a differencing process to the data to stabilize the series and make them suitable for time series modeling, as recommended by Morettin and Toloi (2018).
After applying differentiation, the final p-values of the ADF test for all insurers were below 0.05. This result demonstrated that the time series became stationary, meaning their statistical properties, such as mean and variance, became constant over time. Transforming the data into stationary series is a fundamental step for applying models like SARIMA and ETS, which assume stationarity to ensure the validity and reliability of forecasts. The suitability of the data after differentiation allowed us to proceed with the application and comparison of predictive models.
With the series properly treated and stationary, the Moving Average, SARIMA, and ETS models were applied to forecast the monthly automobile insurance premiums for the year 2024. The performance of each model was evaluated using three main metrics: the Mean Absolute Percentage Error (MAPE), to measure predictive accuracy, and the Akaike (AIC) and Bayesian (BIC) information criteria, to compare the statistical fit to historical data. The results revealed an interesting contrast between the statistical fit and the predictive capability of the models.
In general, the SARIMA models presented the lowest AIC and BIC values for most insurance companies, indicating a better statistical fit to historical data. This suggests that these more robust models were able to capture with greater precision the complexity of the autocorrelation and seasonality structures present in the training series. However, when evaluating the predictive accuracy for the test period, the Moving Average model consistently achieved the lowest MAPE values across all analyzed insurance companies, demonstrating superiority in its ability to forecast future values.
This divergence between the best statistical fit and the best predictive performance is a crucial finding. It suggests that a model that fits historical data very well, capturing all its oscillations and seasonalities, is not always the most accurate for predicting out-of-sample values. The simplicity of the Moving Average model, by focusing on the average of recent values, proved more effective in a scenario where the actual values of the test period showed a stability that did not reflect the more complex variations captured by the SARIMA and ETS models in the training phase.
The comparative analysis of the forecasts for insurer A clearly illustrated this dynamic. The actual values observed in 2024 remained at a more stable level compared to the historical series, and the Moving Averages model managed to follow this behavior effectively. In contrast, the projections from the SARIMA and ETS models, although capturing historical oscillations, diverged from the actual values of 2024. The SARIMA model tended to overestimate, while the ETS underestimated the actual values, highlighting the difficulty of more complex models in adapting to a change in the stability pattern.
For insurer B, the actual values for 2024 showed strong growth followed by stabilization at a high level. The Moving Averages model best aligned with this movement, remaining close to the actual data. On the other hand, the projections from the SARIMA and ETS models, although capturing the historical cyclical fluctuations, diverged significantly from the actual values observed in 2024, underestimating the series’ behavior. This reinforces the idea that the unexpected stability in the test period favored the simplicity of the Moving Averages model.
In insurer C, actual values maintained a stable seasonal oscillation during 2024, and the Moving Averages model estimates managed to accurately follow these movements. In contrast, the SARIMA and ETS models projected much deeper oscillations, based on the historical behavior of the series, diverging from the actual values. The SARIMA model tended to considerably underestimate the test series, while the ETS oscillated around the training mean, but with lower predictive accuracy. This scenario reiterated the effectiveness of the simpler model in capturing the observed seasonal stability.
The insurer D showed a growth peak followed by a drop in actual values in 2024, and the Moving Averages model followed these movements, staying close to the observed data. However, the projections from the SARIMA and ETS models showed great difficulty in reproducing the real behavior of the series for the test period. This limitation was attributed to a break in the series’ smoothness, caused by a sudden increase in production between 2020 and 2021. Both models stabilized on a horizontal line significantly below the actual values of 2024, underestimating the new level reached by the insurer.
For insurer E, actual values remained stable in 2024, after a strong upward trend in previous years. The Moving Averages model effectively captured this growth slowdown, adhering to the actual data. In contrast, the SARIMA and ETS models continued to project the strong oscillations from the training period, which distanced them from the actual values of 2024. The SARIMA model overestimated the series peaks in certain months, while the ETS tended to underestimate the actual values, confirming the preference for a more conservative and stable model for this context.
Additionally, the model residuals were subjected to the Ljung-Box and ARCH-LM tests to verify their adequacy. The ARCH-LM test assessed the homoscedasticity of the residuals, and the p-values obtained did not indicate statistical evidence of the presence of heteroscedasticity. This means that there was no rejection of the null hypothesis of absence of ARCH effects, suggesting that the residual variance was approximately constant in the analyzed series. This result is favorable, as it indicates that the models did not present problems of non-constant variance in the prediction errors.
Regarding the Ljung-Box test, which assesses the absence of autocorrelation in the residuals, most models presented independent residuals, suggesting adequate treatment of temporal dependence. However, the ETS model for insurer A was an exception, with a p-value indicating significant autocorrelation in the residuals. This result suggests that the ETS model, for this specific insurer, did not fully capture the temporal structure of the series, even though its overall predictive performance might still be useful for comparative purposes with other models, according to Hyndman and Athanasopoulos (2018).
In summary, the research results indicated that, although the SARIMA and ETS models demonstrated a better statistical fit to the historical data, the Moving Average model stood out for its superiority in predictive accuracy for the analyzed scenario in 2024. This performance was largely due to the stability of the real values in the test period, which diverged from the more pronounced oscillations and seasonalities captured by the more robust models in the training base. Thus, the Moving Average model proved to be an effective and low-complexity tool to support decision-making in forecasting automobile insurance premiums, responding to the objective of identifying the model with the best predictive results.
4. Conclusion
This study comparatively analyzed the predictive performance of time series models for estimating monthly automobile insurance premiums in Brazil in 2024. Initially, the presence of trends and seasonality in the insurers’ premium series was verified, requiring data differencing to achieve stationarity, as indicated by the Augmented Dickey-Fuller test. After processing, the Moving Average, SARIMA, and ETS models were applied. The results indicated that, although the SARIMA and ETS models showed a better statistical fit to the historical data, with lower AIC and BIC values, the Moving Average model consistently achieved higher predictive accuracy for the test period, as measured by MAPE. This superiority of the simpler model occurred because the actual values for 2024 showed a stability that diverged from the more pronounced oscillations and seasonalities captured by the more robust models in the training set. Thus, the Moving Average model proved to be an effective and low-complexity tool for supporting decision-making in premium forecasting.
The main contribution of this work lies in demonstrating that the simplicity of a model can overcome the complexity of others in specific forecasting scenarios, offering a significant practical and managerial gain for insurance companies. However, the findings are limited to the automobile insurance portfolio. It is suggested for future studies to apply this comparative methodology to other insurance branches, such as Transport, Property, or Life. Such an expansion would allow verification of whether the superior performance of the Moving Averages model is maintained in different market contexts or if more robust models, such as ETS and SARIMA, would be more suitable for series with more pronounced and predictable seasonalities.
Bibliographic References
Borelli, E.; Chan, B. L.; Peris, R. W.; Silva, F. L. 2015. Evolução do Mercado Segurador e Crescimento Econômico no Brasil, Redeca, 2 (2): 21-36.
Contador, C. R. 2023. Seguros e Economia: Princípios e Aplicações. ENS. Rio de Janeiro, RJ, Brasil.
Fundación MAPFRE Brasil. Dicionário MAPFRE de Seguros. Disponível em: <https://www.fundacionmapfre.com.br/publicacoes/dicionario-mapfre-seguros/>. Acesso em: 21 mar. 2025.
Hyndman, R. J.; Athanasopoulos, G. 2018. Forecasting: principles and practice.
Morettin, P. A.; Toloi, C. М. С. 2018. Análise de Séries Temporais – Modelos Lineares Univariados. 3ed. Edgard Blücher Ltda, São Paulo, SP, Brasil.
Superintendência de Seguros Privados [SUSEP]. 2024. Seguro de Pessoas. Disponível em: <https://www.gov.br/susep/pt-br/copy_of_planos-e-produtos/seguros/seguro-de-pessoas>. Acesso em 16 mar. 2025.
Article originating from the Final Course Project of the Specialization in Data Science and Analytics of the MBA USP/Esalq
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