Reinsurance as a Mechanism for Optimizing Capital Requirements Under the Solvency II Regime: An Empirical Analysis of a Ten-Year Insurance Portfolio

1. Introduction

The modern insurance market operates in an environment of growing uncertainty stemming from a combination of the stochastic nature of insurance events, macroeconomic volatility, climate change, and the increasing frequency of catastrophic risks. In this context, the Solvency II prudential framework, introduced by Directive 2009/138/EC of the European Parliament and of the Council of November 25, 2009, places at the center of regulation a risk-oriented and economically integrated approach to assessing the capital adequacy of insurance and reinsurance companies. The Solvency Capital Requirement (Solvency Capital Requirement, SCR) is defined as the value-at-risk of own funds at a 99.5% confidence level over a one-year horizon, which means that capital must be sufficient to cover potential losses under plausible but extremely adverse scenarios (Doff 2008; Eling et al. 2007).

Within this framework, reinsurance has established itself as the most important instrument for vertical risk transfer and, therefore, as a key risk mitigation technique within the meaning of Articles 101 and 308a of Directive 2009/138/EC. At the theoretical level, reinsurance fulfills four main functions: (i) limiting exposure to catastrophic and large individual losses; (ii) stabilizing financial results by reducing the volatility of the net combined ratio; (iii) optimizing regulatory capital by reducing net technical provisions and the SCR; and (iv) expanding underwriting capacity, particularly for rapidly growing portfolios (Cummins and Trainar 2009; Albrecher et al. 2017). However, these functions are not realized automatically: their recognition in the SCR requires a demonstrable and sustainable reduction in risk under stress conditions, which raises a non-trivial question regarding how exactly different types of reinsurance contracts transform the distribution of net losses and, consequently, regulatory capital.

The literature from the past two decades offers a significant body of theoretical models for optimal reinsurance (Centeno and Simões 2009; Albrecher et al. 2017), but empirical and quasi-empirical analyses of the impact of specific types of reinsurance contracts on the SCR under Solvency II remain relatively limited, especially for markets in Southeast Europe. The Bulgarian insurance market is characterized by a relatively high degree of risk cession and a significant reliance on reinsurance transfers for coverage of catastrophic events, making it a particularly relevant subject for applying such an analytical approach.

The purpose of this study is to quantitatively assess the impact of five main types of reinsurance contracts quota share, surplus, quota-surplus, excess-of-loss (XoL), and stop-loss on the distribution of net losses and the Solvency Capital Requirement (SCR) under the Solvency II regulatory framework. The analysis is structured around the following research questions: (1) What is the magnitude and persistence of the capital relief generated through reinsurance over a ten-year time horizon? (2) How do different forms of reinsurance contracts transform the individual quantiles of the loss distribution, and in particular the behavior of the right tail of the distribution? (3) What is the relative contribution of key methodological components including model scope, data smoothing procedures, risk calibration, and diversification effects to the observed differences between the standard formula and the internal model? (4) To what extent does the effect of reinsurance protection remain robust when applying stress scenarios, including scenarios characterized by the presence of so-called “wrong-way risk”?

This study contributes to the existing academic literature in three main areas. First, it provides an integrated quantitative framework that combines the waterfall decomposition of the SCR (from the standard formula through the internal model to the actual business profile) with a detailed comparative analysis across different types of reinsurance contracts. The analysis is based on a consistent and methodologically homogeneous dataset, calibrated using publicly available sources from the Financial Supervision Commission (FSC) and the European Insurance and Occupational Pensions Authority (EIOPA). Second, the study conducts an explicit quantitative assessment of the relative contribution of reinsurance within the overall economic model (the standard formula) compared to the purely methodological effect resulting from the application of an internal model. This achieves a clear analytical distinction between the effect of the “standard formula” and the effect of “real business” a conceptual distinction that often remains implicit or insufficiently formalized in regulatory literature and supervisory practice.

Third, by including a scenario characterized by the presence of wrong-way risk, as well as a hypothetical failure of a leading reinsurer, the study provides an empirical assessment of the limits of reinsurance’s capital efficiency. In this context, it explains why EIOPA’s regulatory approach exhibits systematic caution regarding the automatic recognition of capital reliefs resulting from reinsurance risk transfer.

The remainder of the study is organized as follows. Section 2 presents the theoretical framework of the Solvency II regime and the taxonomy of reinsurance contracts. Section 3 describes the data used, the model assumptions, and the methodology of the empirical analysis. Section 4 presents the results, organized into six analytical areas. Section 5 examines the regulatory and practical implications of the results, as well as the study’s limitations. Section 6 formulates the main findings and conclusions.

2. Theoretical Framework and Literature Review

2.1. The Architecture of Solvency II

Directive 2009/138/EC introduces a harmonized regulatory regime built on three interrelated pillars: (i) quantitative requirements centered on the Solvency Capital Requirement (SCR), the Minimum Capital Requirement (MCR), and the economic valuation of technical provisions; (ii) a system of corporate governance and the Own Risk and Solvency Assessment (ORSA); and (iii) disclosure requirements and market discipline (Eling et al. 2007; Doff 2008).

The structure of the three pillars corresponds functionally to the Basel II/III banking regulations in the banking sector but is specifically adapted to the actuarial nature of insurance risk.

Under the quantitative pillar, technical provisions are calculated as the sum of the best estimate of future cash flows and a risk margin, determined using the cost-of-capital approach and reflecting the cost of maintaining capital for unhedged risks. The Solvency Capital Requirement (SCR) can be interpreted as the Value-at-Risk of a loss in Basic Own Funds over a one-year horizon at a confidence level of 99.5%: formally, SCR = VaR_0.995(ΔBoF), where ΔBoF is the change in Basic Own Funds over a one-year period (BoF₀– BoF₁). The SCR is aggregated through several key risk modules: market risk, underwriting risk (life, general insurance, and health insurance), counterparty default risk (CDR), intangible assets, and operational risk, using correlation matrices embedded in the standard formula or stochastic techniques for modeling dependencies including copula structures and Monte Carlo simulations in internal models. (EIOPA 2014; Christiansen and Niemeyer 2014). The MCR is determined based on a linear function of technical provisions and premium income, subject to an absolute minimum threshold, with the final value constrained within a regulatory corridor between 25% and 45% of the SCR.

The choice between the standard formula and an internal model is a central methodological issue within the Solvency II regime. The standard formula is calibrated to an average risk profile and ensures a high degree of comparability among insurance companies in the market, but it has limited sensitivity to the specific characteristics of an individual insurance portfolio. Internal models whether full or partial allow for a more precise reflection of the insurer’s actual risk profile, but require prior approval from the national supervisory authority and ongoing validation (EIOPA 2015). Empirical studies show that the differences between the two approaches can be substantial and depend to a significant extent on the portfolio structure, risk characteristics, assumptions regarding interdependence, and the degree of diversification. In many cases, internal models result in lower capital requirements due to a more precise accounting for diversification effects and the actual risk profile, whereas for exposures with pronounced tail dependencies, low diversification, or asymmetric risks, higher SCR values may also be observed

2.2. Reinsurance: Definition and Taxonomy

Pursuant to Article 13(7) of the Directive, reinsurance is the activity of assuming, under a reinsurance contract, all or part of the risks assumed by an insurer or reinsurer, in exchange for a reinsurance premium. From an economic perspective, reinsurance represents a vertical (secondary) equalization of risk outside the insurance pool (Gabrovski 2006). Reinsurance contracts are classified according to three main criteria (Albrecher et al. 2017; Cummins and Trainar 2009).

Based on the mandatory nature of risk cession, reinsurance contracts are classified as: facultative, facultative-obligatory, obligatory-facultative, and bilateral-obligatory. This taxonomy reflects the degree of commitment of the parties regarding the automatic acceptance and cession of risk. Based on the risk transfer mechanism, reinsurance contracts are divided into: proportional and non-proportional. In proportional reinsurance, the insurer and the reinsurer share the risk, premium income, and claims in a predetermined proportion, with the risk distributed on a pro rata basis between the parties. The two main types are: the quota-share contract, in which a fixed percentage α of each policy is ceded to the reinsurer, and the surplus contract, in which only the portion exceeding a predetermined retention is ceded, structured through “lines.”

Non-proportional contracts, on the other hand, do not allocate amounts but rather losses exceeding a certain threshold. In Excess-of-Loss (XoL) reinsurance, the reinsurer covers the portion of the loss exceeding an agreed retention (“priority”) up to a specified coverage limit, and the contract may be divided into multiple layers. In stop-loss reinsurance, coverage is triggered when the aggregate annual losses exceed a specified loss level, defined by the loss ratio or by an absolute threshold for accumulated losses.

Mathematically, if X denotes the random loss, then the net exposure for the insurer takes the following forms: for a quota share contract with a reinsurance percentage α, X_net = (1 − α)·X; for an excess-of-loss contract with a retention R and an unlimited limit, X_net = min(X, R); for a stop-loss contract with an aggregate retention A and a limit L, X_Re = max(min(X, A + L) − A, 0). These functional relationships lead to significantly different effects on the variance, skewness, and tail behavior of the net loss distribution, thereby directly impacting capital requirements and risk efficiency under the Solvency II regime.

2.3. Reinsurance as a Risk Mitigation Technique Under Article 101

Pursuant to Article 101(5) of the Directive and Delegated Regulation (EU) 2015/35, reinsurance is recognized as a risk mitigation technique in the calculation of the SCR, provided that: (i) the risk transfer is genuine and legally valid; the reinsurer is a licensed entity with sufficient credit quality; the contract specifically covers the scenarios that determine the SCR; and the effect of reinsurance can be clearly identified and measured (Christiansen and Niemeyer 2014; EIOPA 2014).

However, this effect of regulatory recognition of reinsurance introduces an additional risk component counterparty default risk (CDR) which is assessed within a separate module of the standard SCR formula. As a result, the net effect of reinsurance on the capital requirement is the net result of the reduction in underwriting and catastrophe risk, on the one hand, and the increase in exposure to CDR, on the other. Under certain structural conditions including a high concentration in the reinsurance program, a lower credit rating of reinsurance counterparties, or the presence of wrong-way risk the increase in CDR may partially or completely offset the capital relief resulting from the transfer of insurance risk.

This study operationalizes these theoretical relationships by constructing an empirically calibrated stochastic model, which assesses the relative impact of the five main types of reinsurance contracts on the quantiles of the loss distribution, as well as on the individual risk modules of the SCR under Solvency II.

3. Data and Methodology

3.1. Study Design

The study is designed as a controlled computational experiment based on a calibrated ten-year insurance portfolio covering the period 2016–2025. The experimental design integrates three complementary analytical levels: (i) the portfolio level, encompassing the dynamics of gross and net risk indicators over time; (ii) the level of the SCR risk modules and the relative contribution of the main submodules in the standard formula; (iii) the reinsurance contract level, at which a stratified comparison of the capital effect is performed by type of reinsurance contract and by confidence levels of the loss distribution. This three-tiered stratification is methodologically necessary, as the effects of the various types of reinsurance contracts are, by their nature, nonlinear and non-additive. In particular, the interactions between underwriting risk, catastrophe risk, and counterparty default risk cannot be adequately identified using aggregated indicators, as they manifest differently across the individual quantiles of the loss distribution and under different risk structures.

3.2. Data Sources and Calibration

The data are structured into four interrelated blocks, implemented in spreadsheets and calibrated against publicly available reference values.

The first section (Portfolio) includes annual data on gross and net premium income, ceded premiums, cession ratio, number of policies, gross and net claims, gross and net losses from catastrophic risks, gross and net combined ratio, and total technical reserves. Gross premium income ranges from 45,000 thousand EUR (2016) to 72,000 thousand EUR (2025), while the cession ratio increases from 0.32 to 0.40, which is comparable to the trends observed among medium-sized insurance companies in the region. The projections are based on aggregated public data from the Financial Supervision Commission (FSC) for general insurance and correspond to a medium-to-large company with a diversified portfolio (Property, Comprehensive Auto, Liability, Agricultural, Liability, Health, etc.). The gross combined ratio ranges from 86% (2022) and 105% (2020), with the trend reflecting the observed cyclicality of loss experience, including the effects of the pandemic shock in 2020 and increased catastrophe activity in 2018, 2020, and 2023. The second section (SCR_Modules) contains the annual SCR values for the main risk modules within the standard formula premium risk, reserve risk, catastrophe risk, market risk, counterparty default risk, and operational risk as well as the effect of diversification, distinguishing between gross and net values. The correlation matrices and shocks in the standard formula correspond to the latest available calibration published by EIOPA. The third block (Waterfall_SCR) performs two sequential decompositions: (i) a transition from the standard formula to an internal model structured through four methodological stages model scope, data harmonization, calibration of risk parameters and diversification effects; and (ii) a transition from an internal model to a full economic model by incorporating the effects of reinsurance, managerial actions, and the behavioral responses of policyholders. The fourth section (Comparison of Contracts) contains comparative assessments of the SCR with and without the inclusion of the five types of reinsurance contracts, as well as when combining a multi-tiered reinsurance program, analyzed against various quantiles of the loss distribution (50%, 95%, 99%, 99.5%, 99.9%).

The supplementary sections (“Counterparty Risk,” “Business Lines,” and “Stress Tests”) expand the analysis by incorporating specific risk factors and scenario experiments.

The data are quasi-empirical in nature, in that they are calibrated based on public sources (FSC, EIOPA, Insurance Europe), without directly replicating the financial statements of a specific insurer. All values are aggregated and rounded to preserve the structural comparability and replicability of the analysis. With regard to the internal model, calibrations provided for research purposes by an insurance company that is developing and testing an internal model but has not yet received regulatory approval were used. According to the information provided, the model is used for internal analytical purposes and ongoing refinement, and the estimated capital requirement is approximately 20% lower than the results obtained using the standard formula.

This approach has two main advantages: (i) it allows for full replicability of the analysis while maintaining a realistic calibration of the parameters; and (ii) it ensures consistency with observed market orders of magnitude for the key risk parameters. The limitation of this approach is that the results should be interpreted as valid for a hypothetical, yet realistically calibrated, medium-to-large European insurer, rather than as a specific characteristic of a particular market participant.

3.3. Specification of the Standard Formula for SCR

Underwriting risk in general insurance is modeled in accordance with the modular structure of the standard formula under the Solvency II Directive, with the aggregate capital requirement for underwriting risk calculated using quadratic correlation aggregation between the premium, reserve, and catastrophe submodules:

SCR_nl = √(Σ ρ_ij · SCR_i · SCR_j)

where ρ denotes the regulatory correlation coefficients between the individual risk components, and SCRi and SCRj represent the respective capital requirements for premium, reserve, and catastrophe risk. This approach allows for the consideration of diversification effects among the various sources of underwriting risk, while preserving the conservative nature of the regulatory framework.

Premium and reserve risks are assessed using the standard volatility-based specification, in which the capital requirement is determined as a function of the exposure (volume) measure V and the corresponding volatility parameter σ, calibrated to the historical variability of losses in the respective lines of business. This approach ensures that the model is sensitive to both the size of the insurance portfolio and its risk profile.

Catastrophic risk is modeled using the scenario-based mechanism of the NatCat submodule, which simulates extreme but plausible natural events with low frequency and high loss severity. This ensures that tail risk which is critical for determining regulatory capital at a 99.5% confidence level is adequately captured.

Counterparty default risk for Type 1 reinsurers is assessed using the standard approach for Type 1 exposures, in which the capital requirement is determined as the product of the risk adjustment factor k and the loss given default (LGD), or

SCR_def,1 = k·LGD

The parameter k is a function of the degree of concentration and variation in the portfolio’s exposure, as well as the probability of default (PD) associated with the credit quality of the individual reinsurance counterparties. In turn, LGD is calculated in accordance with regulatory requirements as 50% of the value of recoverable amounts and recognized collateral. This specification reflects the prudent nature of the Solvency II framework by integrating both the reinsurer’s credit risk and the potential uncertainty regarding the effective recovery of reinsurance receivables under stress conditions.

3.4. Waterfall Decomposition: From the Standard Formula to Real Business

To identify and quantify the individual factors driving the difference between the capital requirement calculated using the Standard Formula (SF) and that obtained through an Internal Model (IM), a sequential waterfall decomposition approach was applied. The methodology is based on a step-by-step transformation from SCRSF to SCRIM, in which each step isolates a specific source of methodological or economic deviation

In the first stage, four consecutive adjustments were applied to achieve comparability between the two risk assessment approaches, namely: (i) scope alignment was performed by eliminating risk components included in the standard formula but not covered by the internal model; (ii) data alignment was performed, involving the harmonization of the reserves, premiums, and exposures used, in order to eliminate differences arising solely from the data set; (iii) risk calibration has been applied, whereby regulator-specified shocks and parameters have been replaced with proprietary stochastic estimates based on internal data and models. This step involves refining volatility, the shape of distributions, and tail risk characteristics; (iv) A transition has been made to a more realistic structure of dependencies and diversification, based on copula dependencies and nonlinear dependencies in the tails, which allow for more adequate modeling of the joint behavior of risk factors during extreme events.

Conceptually, this sequence can be represented as:

SCR_SF → Scope Alignment → Data Alignment → Risk Calibration → Dependency & Diversification Adjustment → SCR_IM

In the second stage of the waterfall decomposition, the analysis is expanded to include three additional business effects: (i) the effect of reinsurance, (ii) the effect of management actions, and (iii) the effect of behavioral responses by policyholders. By integrating these components, a capital requirement based on a full economic model is derived within the framework of the Own Risk and Solvency Assessment (ORSA) process.

This two-tier decomposition structure is of significant methodological and regulatory importance, as it allows for a clear distinction between the effects arising from more precise risk measurement and modeling (stages i–iv), and the effects resulting from actual risk management and transfer mechanisms (stages v–vii).

Such a distinction is particularly important in the context of the supervisory assessment under the Solvency II Directive, as regulatory authorities, including the European Insurance and Occupational Pensions Authority (EIOPA), require evidence that the observed capital reliefs stem from sustainable and legally enforceable risk-reduction mechanisms, rather than from purely methodological choices or modeling assumptions. In this regard, the proposed waterfall decomposition provides a transparent analytical framework for assessing the economic soundness and regulatory reliability of internal models.

3.5. Stress Tests and Scenario Analysis

To assess the resilience and effectiveness of the capital relief generated through reinsurance, a multi-scenario stress-testing framework has been developed, comprising eight sequentially escalating scenarios with varying intensities and nature of risk. The approach is consistent with the ORSA principles and EIOPA’s supervisory expectations regarding the forward-looking assessment of solvency under extreme but plausible conditions.

The first scenario represents the baseline configuration without stress and serves as a reference point for comparative analysis.

The second scenario simulates an event with a frequency of approximately once every ten years and includes a moderate shock, resulting in a 10% increase in the amount of claims.

The third scenario models a 1-in-50-year event characterized by a severe storm and a 25% increase in loss ratio.

The fourth scenario simulates an extremely catastrophic flood combined with an adverse market shock, corresponding approximately to a 1-in-100-year event.

The fifth scenario represents an extreme multifactorial configuration with a 1-in-200-year frequency, equivalent to the regulatory 99.5% VaR confidence level under the Solvency II Directive. This scenario combines severe underwriting, market, and catastrophe shocks and serves as the primary stress test for capital adequacy under Solvency II.

The sixth scenario analyzes credit risk in the reinsurance market through a hypothetical default of Munich Re as a leading reinsurance counterparty, resulting in a loss of approximately 30% of reinsurance coverage. This assesses the company’s sensitivity to concentration risk and potential inefficiencies in risk transfer in the event of a counterparty default.

The seventh scenario models a combined macroeconomic and insurance shock caused by a pandemic crisis and subsequent recession. Under this scenario, health claims increase approximately threefold, while the company’s market share decreases by 40%, which simultaneously worsens both the underwriting result and the ability to generate future premium income.

The eighth scenario represents a complex wrong-way risk scenario in which the occurrence of a large-scale catastrophic event coincides with the insolvency of two key reinsurers. This configuration aims to capture the nonlinear interactions between catastrophe risk and counterparty risk, as well as the potential nullification of reinsurance protection precisely under the conditions in which it is most needed.

For each scenario, the following are calculated: (i) the SCR before and after the effect of the current reinsurance program; (ii) the amount of eligible own funds; (iii) the solvency ratio; and (iv) the company’s position relative to the Minimum Capital Requirement (MCR). This analytical framework allows for a comprehensive assessment not only of the immediate effect of risk transfer on regulatory capital but also of the resilience of the capital position under extreme and systemically interconnected stress conditions.

4. Results

4.1. Portfolio Structure and Dynamics

The ten-year insurance portfolio presented in Table 1 demonstrates relatively stable business growth in the property insurance segment, characterized by moderate but consistent growth in gross premium income. For the period 2016–2025, gross written premiums increase from 45.0 million EUR to 72.0 million EUR, corresponding to an average annual growth rate of approximately 5.4%. At the same time, there is a gradual increase in the cession ratio from 32% in 2016 to 40% in 2025 which indicates a growing reliance on reinsurance risk transfer as a tool for stabilizing results and optimizing the capital position.

The claims experience reveals two distinct catastrophe years. In 2018, the portfolio was affected by severe weather events, including a storm and hail, resulting in gross catastrophe losses of EUR 8.5 million. Even greater stress was observed in 2020, when the combined impact of pandemic-related health claims and adverse weather events resulted in gross catastrophe losses of approximately EUR 15.0 million. These two years provide key insights for assessing the resilience of the reinsurance program under extreme but realistic shocks.

Analysis of the combined ratio shows a significant stabilizing effect of reinsurance on the technical result. The gross combined ratio ranges from 86% to 105%, with a standard deviation of 6.25 percentage points, indicating significant sensitivity to fluctuations in loss ratio and catastrophic events. After accounting for the effect of reinsurance protection, the net combined ratio exhibits significantly lower volatility, ranging from 92% to 115% with a standard deviation of just 2.94 percentage points. This corresponds to an approximate 53% reduction in volatility relative to the gross position and provides empirical evidence of the role of reinsurance as a mechanism for smoothing results and limiting tail risk.

The trends during the crisis years of 2018 and 2020 are particularly telling. Despite the significant increase in the gross loss ratio and the deterioration of the gross combined ratio to 99% and 105%, respectively, the net result remained relatively stable thanks to reinsurance coverage. This confirms that the increase in the cession ratio during the period under review does not represent merely a mechanical transfer of risk, but rather a strategic tool for reducing volatility in results and limiting capital fluctuations amid the increasing frequency and severity of catastrophic events.

From the perspective of the Solvency II Directive, the observed reduction in volatility has a direct impact on the determination of the solvency capital requirement, as the reduction in the variability of net losses leads to lower extreme quantiles in the distribution and, consequently, to a lower SCR. In this sense, the empirical results in Table 1 support the argument that reinsurance serves both as a risk management tool and as a mechanism for capital optimization.

4.2. Aggregate Impact of Reinsurance on the SCR

The dynamics of the aggregate Solvency Capital Requirement (SCR), presented in Table 2, show a clear relationship between the expansion of reinsurance coverage and the reduction in regulatory capital. Both the gross and net SCR increase over the period under review as a result of the expansion of the insurance portfolio, the increase in exposures, and the growing volume of risk assumed. However, the rate of growth in the net SCR remains significantly lower, leading to a gradual acceleration of the capital relief generated through the transfer of risk to the reinsurance market.

Empirical results show that the relative reduction in the aggregate SCR increases from 18.4% in 2016 to 23.4% in 2025. This trend follows an almost linear relationship with the increase in the cession ratio, suggesting the existence of an approximately proportional scaling mechanism between the degree of risk transferred and the magnitude of the capital relief. Given the relatively stable structure of the reinsurance program, the observed effect confirms that reinsurance functions as a predictable tool for capital optimization within the framework of the Solvency II Directive.

The most pronounced effect is observed in the catastrophe risk submodule. In 2025, the gross SCR for catastrophe risk amounts to 10,873 thousand EUR, while the net value after the reinsurance effect decreases to 6,089 thousand EUR, corresponding to a reduction of 44.0%. This effect is approximately twice as strong as the average reduction for the aggregate SCR and reflects the high effectiveness of catastrophe reinsurance coverage in mitigating tail risk and extreme losses. The result is consistent with the theoretical characteristics of non-proportional reinsurance structures, in which protection is activated precisely in the highest quantiles of the loss distribution.

At the same time, the analysis reveals a gradual increase in the capital requirement for counterparty default risk (CDR). The SCR for counterparty risk rises from 963 thousand EUR in 2016 to 1,656 thousand EUR in 2025, representing an increase of approximately 71.9%. This trend reflects the structural increase in the company’s dependence on the reinsurance sector due to rising recoverable amounts and a higher degree of risk transfer.

Despite the significant absolute growth in CDR, its relative weight within the total net SCR remains stable in the range of approximately 6.6–7.7%. This observation has important economic implications, as it shows that, in the presence of reinsurers with high credit quality, the increase in counterparty risk is not sufficient to offset the primary effect of the reduction in underwriting and catastrophe risk. In other words, the benefits of risk transfer continue to outweigh the additional credit risk generated by dependence on reinsurance counterparties.

From a methodological standpoint, the results in Table 2 provide two key empirical conclusions. First, the capital relief exhibits a monotonic and nearly proportional relationship with the cession ratio, implying an elasticity coefficient close to one. Second, although an increase in reinsurance exposure inevitably leads to higher counterparty risk, this effect remains secondary in the presence of a diversified and credit-stable reinsurance panel structure. In this sense, the results confirm that the effectiveness of reinsurance as a capital management mechanism depends not only on the volume of risk transfer but also on the quality and stability of the reinsurance counterparties used.

4.3. Comparative Analysis of the Main Types of Reinsurance Contracts

The stratified comparative analysis presented in Table 3 shows that different types of reinsurance contracts generate significantly different impact profiles on the solvency capital requirement and its submodules. The results confirm that the effectiveness of reinsurance cannot be assessed solely by the magnitude of the aggregate reduction in the SCR, as individual contract structures affect the frequency, severity, and risk behavior of the loss distribution in different ways.

A quota share contract with a 40% cession quota results in a 30.0% reduction in the total SCR, with the effect being practically symmetrical across the main underwriting submodules premium, reserve, and catastrophe risk each of which is reduced by approximately 40%. This proportionality is fully consistent with the theoretical characteristics of quota-share structures, in which risk transfer occurs through a linear scaling of exposure. As a result, the quota share treaty reduces both the average level of losses and their variability, without a significant change in the shape of the distribution.

The surplus treaty with a capacity of 15 lines demonstrates a more moderate overall effect on the SCR (−20.0%), with the impact concentrated primarily on large individual exposures. The reduction in premium and reserve risk remains relatively limited (−24% and −25%), while catastrophe risk decreases by −29%. This indicates that surplus structures function more as a tool for managing concentration risk than as a universal mechanism for reducing aggregate risk.

The quota-surplus combination achieves a more balanced protection profile, reducing the total SCR by 32.5%. This result stems from the combination of the proportional reduction characteristic of the quota-share component with the targeted protection against large individual exposures provided by the surplus layer. As a result, catastrophe risk is reduced by −44%, which exceeds the effect of either of the two contracts considered individually.

Among the individual contract structures, the catastrophe excess-of-loss (Cat XoL) contract demonstrates the strongest impact, reducing the total SCR by 35.5%. However, its impact is highly asymmetric: the catastrophe submodule decreases by −65%, while premium and reserve risk decrease by only approximately −8%. This asymmetry reflects the fundamental functional characteristic of XoL contracts their ability to limit extreme losses by “trimming” the right tail of the distribution without significantly altering the central tendency of losses. From the perspective of the Solvency II Directive, this is particularly significant, as the SCR is determined precisely by the extreme quantiles of the distribution.

The working XoL contract (working excess-of-loss) demonstrates a more moderate but broader-ranging effect, reducing the total SCR by 25.8%. Unlike Cat XoL, the protection here is triggered at lower loss levels, which allows for the mitigation of both frequent large claims and a portion of the catastrophe exposure. However, the effect on catastrophe risk remains significantly weaker (−35%) compared to the specialized Cat XoL layer.

The stop-loss contract with a loss ratio threshold of 85% exhibits a fundamentally different risk profile. Its impact is concentrated primarily on premium risk (−42%), while the effect on catastrophe risk is practically zero. This is explained by the aggregate nature of the protection: the stop-loss contract is triggered only when the total combined ratio exceeds a predefined threshold; therefore, the mechanism functions as a tool for stabilizing the technical result rather than as direct protection against extreme catastrophic events.

The most significant capital relief is observed in the combined multilayer program, which includes quota share, surplus, Cat XoL, and stop-loss components. This structure reduces the total SCR by 48.8%, while simultaneously lowering premium risk by −50%, reserve risk by −45%, and catastrophe risk by −70%. The results clearly show that the various types of reinsurance contracts function as mutually complementary mechanisms rather than as perfect substitutes. Proportional contracts reduce the overall level of exposure, XoL structures limit tail risk, and stop-loss protection stabilizes the aggregate technical result. It is precisely this complementarity that accounts for the significantly higher effectiveness of multi-layered programs compared to any single contract considered in isolation.

Additionally, it should be noted that all the structures examined lead to an increase in counterparty default risk (CDR), with the highest increase observed in the combined program (+35%). This is a natural consequence of the increased reliance on reinsurance recoveries. Nevertheless, even with the most complex structures, the positive effect of transferring underwriting and catastrophe risk remains significantly greater than the additional counterparty risk, which confirms the economic rationality of multi-layered reinsurance programs as a tool for capital optimization and solvency stabilization.

4.4. Impact of Reinsurance on the Quantiles of the Loss Distribution

To explain the varying degrees of capital efficiency among different types of reinsurance contracts, it is necessary to analyze not only the aggregate change in the SCR but also the way in which the respective contractual structures transform the loss distribution across different confidence levels. Table 4 illustrates precisely this transformation by comparing loss amounts at different quantiles of the distribution for five alternative configurations of the reinsurance program.

The results reveal the existence of three clearly distinguishable patterns of impact on the shape and risk behavior of the loss distribution. The first pattern is characteristic of quota-share contracts, in which a nearly perfectly proportional scaling of the entire distribution is observed. Each quantile is reduced by approximately 40%, regardless of its confidence level. For example, the median loss decreases from 30,000 thousand EUR to 18,000 thousand EUR, and the value at the 99.5% confidence level corresponding to the regulatory SCR decreases from 200,000 thousand EUR to 120,000 thousand EUR. This result confirms that proportional contracts reduce the magnitude of risk without significantly altering the shape, asymmetry, or tail thickness of the distribution.

Excess-of-loss (XoL) contracts exhibit a fundamentally different profile, with a pronounced nonlinearity in the effect. In the central portions of the distribution, the impact is limited: the median loss decreases by only 6.7% (from 30,000 thousand EUR to 28,000 thousand EUR). However, as the confidence level increases, the effect grows exponentially. At the 99th percentile, the reduction reaches approximately 43.3%, while at the regulatory level of 99.5%, the reduction amounts to 52.5% (from 200,000 thousand EUR to 95,000 thousand EUR). An even more significant effect is observed at the extreme tail of the distribution: at the 99.9% quantile, losses are reduced by 68.6%, from 350,000 thousand EUR to 110,000 thousand EUR.

This nonlinear transformation is a fundamental characteristic of XoL structures and can conceptually be interpreted as “trimming” the right tail of the loss distribution. Unlike proportional contracts, XoL coverage has virtually no effect on frequent and moderate losses, but is directed almost entirely toward limiting extreme events with low frequency and high severity. It is precisely this characteristic that accounts for the high capital efficiency of XoL contracts under the Solvency II Directive framework, in which the SCR is defined by the 99.5% Value-at-Risk.

Stop-loss contracts demonstrate a third, conceptually different impact model. In these contracts, the protection operates at the aggregate portfolio level and is triggered only after a predefined loss ratio threshold is exceeded. As a result, a moderate reduction is observed in the higher quantiles for example, 20.0% at a 99% confidence level but a significantly weaker effect in the far tail of the distribution. At the 99.5% quantile, the reduction reaches only 27.5%, as the stop-loss coverage limit is exhausted before the most extreme losses are realized. This indicates that stop-loss contracts are more effective as a tool for stabilizing technical performance and limiting medium-term volatility than as a means of reducing extreme tail risk.

The combined multilayer reinsurance program has the strongest impact on the loss distribution. It results in a significant reduction across all quantiles, with the effect becoming more pronounced as one moves toward the right tail of the distribution. At the 99.5% level, losses are reduced by 72.5%, and at the 99.9% level, by approximately 80.0%, which effectively leads to an almost complete mitigation of catastrophic risk. This result confirms the existence of strong complementary effects among the different types of contracts: proportional structures reduce overall exposure, XoL contracts limit extreme losses, and stop-loss mechanisms stabilize the aggregate technical result.

The results obtained are of significant importance for understanding the capital efficiency of reinsurance. Since the SCR is defined by the 99.5% VaR, contracts that specifically target the extreme tail of the distribution primarily XoL structures provide maximum capital relief per unit of ceded premium. In contrast, quota-share contracts are more effective as a tool for stabilizing financial results, improving the net combined ratio, and reducing operational volatility, but they are relatively less optimized for the specific logic of regulatory capital under Solvency II.

4.5. Empirical Results of the Analysis

The empirical analysis is based on an integrated database covering the company’s financial, reinsurance, and capital characteristics for the period 2016–2025. The dataset used includes: (i) ten-year time series for premiums, claims, technical reserves, own funds, capital requirements, and cession ratios; (ii) information on 50 reinsurance contracts, structured by five main contract types for each year of the period; (iii) panel data on counterparty risk, including probability of default, exposure, and credit ratings for five reinsurers over the ten-year horizon; and (iv) cross-sectional data by line of business according to the Solvency II Directive classification, including Liability Insurance (MTPL), Other Motor Insurance, Fire and Other Property Damage Insurance, Marine, Aviation, and Transport Insurance, General Liability Insurance, Credit and Surety Insurance, and Medical Expense Insurance.

4.5.1. Methodological Framework of Regression Analysis

Regression analysis answers the question: “How does one variable change when another variable changes?” Regression seeks a quantitative relationship between two or more variables. Regression analysis was used to identify the quantitative relationships between key financial and capital indicators and to assess the economic significance of the observed relationships.

The analysis was conducted using three main empirical frameworks: (i) time-series regressions on ten-year series (n=10); (ii) panel regressions for counterparty risk (n=50); and (iii) cross-sectional regressions by business line (n=8). The format of the results is tailored to the practical requirements of SFCR and ORSA reporting.

When analyzing financial time series, a fundamental econometric problem arises related to the presence of sustained upward trends in the key variables. Indicators such as premium income, SCR, technical reserves, and cession ratios demonstrate systematic growth over time, which creates a risk of so-called spurious regression a statistically significant relationship between two variables arising not from economic causality but from their shared trend component.

To mitigate this effect, the analysis was conducted using two complementary specifications.

The first specification uses classical linear regressions on the levels of the variables.

The second specification includes a set of transformations designed to eliminate the common trend and isolate the actual economic relationship: (i) first differences (Δ); (ii) log differences (Δlog), allowing the coefficients to be interpreted as elasticities; and (iii) detrending using the Frisch–Waugh–Lovell (FWL) approach, in which the linear time trend is removed before estimating the relationships.

This two-step approach allows for a clear distinction between structurally stable relationships and relationships that are artifacts of the joint growth of the variables.

4.5.1.1. Results and Conclusions from the First Specification of the Regression Analysis:

The results of the first specification are presented in Table 5

The data show an extremely strong linear relationship between net SCR and gross premium revenue. The model: SCRnet = α + β ⋅ Premiumgross + ε yields a coefficient of determination R² = 0.9963, suggesting a practically perfect linear relationship between the size of the business and the capital requirement. The estimated slope indicates that every additional 1,000 EUR in gross premium income generates approximately 0.26,000 EUR in additional SCR, which attests to the portfolio’s high capital intensity.

When the cession ratio is included as an additional explanatory variable, it is found that every 1 percentage point increase in the cession ratio reduces the net SCR by approximately 63,000 EUR, ceteris paribus. Furthermore, the regression between the reduction in SCR and the cession ratio shows R² = 0.9773, which initially suggests nearly perfect elasticity between the degree of reinsurance and the capital relief.

The results from R4 show no statistically significant relationship between the combined ratio and the solvency ratio. The low R² and high p-value indicate that the solvency ratio does not follow technical profitability in a linear fashion, but is determined primarily by capital policy, reinsurance structure, and management decisions. Similarly, the regression between the gross catastrophe SCR and realized catastrophe losses proves to be completely insignificant (p=0.90), confirming that the standard formula is calibrated to hypothetical 1-in-200 scenarios rather than to historically observed losses. This result provides a strong empirical basis for considering partial or internal models in catastrophe risk.

A particularly strong result is observed in the panel model for counterparty risk, where LGD and credit rating explain over 91% of the variation in the SCR for counterparty risk. This provides a quantitative basis for the strategic importance of credit quality in the selection of reinsurance partners.

Reinsurance exhibits stable and economically significant capital leverage. The results from R3 show that a 1 percentage point increase in the cession ratio leads to an approximate 0.65 percentage point additional reduction in the SCR. This implies an almost proportional, though not entirely linear, transfer of risk. From a practical perspective, increasing the cession ratio from 40% to 50% would result in approximately 6.5 percentage points of additional capital relief, confirming the significant role of reinsurance as a tool for capital optimization. Discrepancy between SF and reality in catastrophe risk. R5 is completely insignificant (p=0.90). The gross catastrophe SCR does not track realized catastrophe losses at all the standard formula is calibrated to a 1-in-200 scenario, not to observed history. This is a strong argument for an internal/partial model in the catastrophe submodule.

Counterparty risk is highly predictable and structurally determined. The panel regression R6 (n = 50) shows high explanatory power (R² = 0.91), with the combination of exposure and credit rating explaining the majority of the variation in the SCR for counterparty risk. Quantitatively, a downgrade in credit quality from AA− to A s leads to an increase in SCR of approximately 914,000 EUR, ceteris paribus. This provides a clear empirical basis for credit-based selection of reinsurance partners and for managing counterparty risk concentration.

Capital intensity is homogeneous across the various lines of business. The results show an approximately equal ratio of SCR to net premiums (around 0.24 thousand EUR SCR per 1 thousand EUR of premium) across all lines considered. This suggests that the observed diversification in the portfolio does not stem from significant differences in the capital “weight” of the individual lines, but rather from differences in the risk structure and the correlation relationships among them. Consequently, the diversification effect is primarily structural, rather than a result of heterogeneity in the individual capital requirements.

[Conclusions from the First Specification of the Regression Analysis]

The analysis allows us to draw several key conclusions.

First, the capital requirement exhibits a stable and nearly unit elasticity with respect to the volume of business, which confirms the linear nature of the standard formula with respect to premium exposure.

Second, reinsurance does indeed generate capital relief, but its effect is weaker and less certain after adjusting for time trends.

Third, solvency is not a mechanical reflection of current technical profitability but rather the result of targeted capital and reinsurance strategies.

Fourth, the standard formula for catastrophe risk turns out to be practically independent of actually observed catastrophe losses, which raises the question of its limited sensitivity to the company’s specific risk profile.

Fifth, credit rating and LGD are the dominant factors in determining the SCR for counterparty risk, which underscores the critical importance of the quality of reinsurance counterparties.

Finally, the lines of business exhibit relatively similar capital intensity, suggesting that the diversification effect stems more from differences in risk profiles and interdependencies across lines than from significant differences in individual capital ratios.

4.5.1.2. Results and Conclusions from the Second Specification of the Regression Analysis:

When estimating regression models on time series in levels: Yt = α + β . Xt + εt, there is a significant risk of obtaining spurious relationships when both the dependent and explanatory variables contain a common deterministic or stochastic trend. In such cases, artificially high values of the coefficient of determination R² and statistically significant coefficients β are commonly observed, even in the absence of an economic causal relationship.

A classic example in the literature involves correlations between unrelated processes that nevertheless share a common trend component (e.g., demographic and environmental variables influenced by third factors such as urbanization). Similarly, in the present study, premium income, SCR, and the cession ratio exhibit a joint upward trend driven by the growth in the size of the insurance portfolio, which creates conditions for spurious relationships.

To minimize this effect and identify stable structural relationships, an extended regression framework (Specification 2) is applied, featuring three alternative, mutually complementary transformations aimed at eliminating common trend components and isolating structural relationships: first differences, log differences, and detrending using the Frisch–Waugh–Lovell approach. This approach allows for a distinction between (i) a common trend component and (ii) incremental economic relationships relevant to the capital structure.

• First differences (Δ):

The first approach is based on the transformation:

ΔYt = Yt − Yt−1 , ΔXt = Xt − Xt−1 .

and the model estimation:

ΔYt = α + β . ΔXt + εt

In the presence of first-order integrated processes I(1), first differences eliminate the overall stochastic trend and allow for the identification of short-term dynamic relationships. A major drawback is the loss of observations and the potential increase in standard errors.

• Logarithmic differences (Δlog):

The second specification uses the transformation:

Δlog Y ≈ ΔYt / Yt

and the regression:

Δ logYt = β · Δlog Xt + εt

The coefficient β can be directly interpreted as elasticity, i.e., the percentage change in Y induced by a 1% change in X. This approach further stabilizes the variation in variables with increasing scale and partially reduces heteroscedasticity.

• Detrending using the Frisch–Waugh–Lovell (FWL) theorem:

The third approach uses orthogonalization with respect to the time trend. First, the following are estimated:

Yt = γ0 + γ1 t + ut , Xt = δ0 + δ1 t + vt

and the residuals Ȳt = ut , X̃t = vt are calculated, after which the following is estimated:

Ȳt = β . X̃t + εt

According to the Frisch–Waugh–Lovell theorem, the resulting coefficient is equivalent to that from a multiple regression including a linear trend. This approach isolates the relationships between deviations from the overall time trend and is particularly suitable for trend-stationary processes.

The main objective of the three specifications is to decompose the observed correlation between financial and capital variables into: (i) a structural (economically driven) component; (ii) a trend-induced (spurious) component.

This is particularly important in the present context, as the key variables premiums, SCR, and cession ratio exhibit synchronous growth driven by the expansion of the insurance portfolio. As a result, high R² values in level regressions cannot be directly interpreted as evidence of causality.

Applying the three alternative specifications allows us to identify those relationships that are robust to transformations of the time structure and to distinguish them from artifacts arising from a common trend.

Table 6. Detrending Using the Frisch–Waugh–Lovell (FWL) Theorem.

#	Specification	β (slope)	SE	p-value	R²	DW	n	Note
R1 — Net Total SCR ~ Gross Premiums
	Niva (original)	0.2630	0.0057	0.0000	0.9963	2.9174	10	Suspected autocorrelation of the residuals
	First differences (Δ)	0.3074	0.0469	0.0003	0.8599	2.8838	9	Removes I(1) general trend
	Δ through zero	0.2713	0.0239	0.0000	0.9414	3.1432	9	No constant (theoretically correct if ΔX=0⇒ΔY=0)
	Logarithmic differences (elasticity)	0.9602	0.1564	0.0005	0.8434	2.7551	9	β = elasticity (% change in Y per 1% change in X)
	Trend-adjusted (FWL)	0.2777	0.0301	0.0000	0.9140	2.8211	10	Linear trend removed (FWL)
	✓ ROBUST. The relationship holds across all specifications (p<0.001 in Δ and in Δlog). The true elasticity is ~0.96 (close to unity). The high R² at the level is not an artifact.
R3 — SCR reduction from over-insurance ~ Cession ratio
	Niva (original)	0.6478	0.0349	0.0000	0.9773	2.55	10	Suspected autocorrelation of the residuals
	First differences (Δ)	0.7003	0.5033	0.2067	0.2167	2.51	9	Removes I(1) general trend
	Δ through zero	0.6259	0.1572	0.0041	0.6646	2.53	9	No constant (theoretically correct if ΔX=0⇒ΔY=0)
	Trend-adjusted (FWL)	0.7345	0.3173	0.0493	0.4011	2.58	10	Linear trend removed (FWL)
	⚠ PARTIALLY DRIVEN BY THE TREND. R² drops dramatically from 0.98 to 0.22 in Δ (p=0.21). After passing through zero and detrending, the relationship persists, but with 4–9× wider standard errors. The true effect is present, but it is weaker than the level suggested.
R4 — Solvency ratio ~ Gross combined ratio
	Niva (original)	0.2488	0.3808	0.5319	0.0507	1.79	10	Base model
	First differences (Δ)	0.2568	0.3281	0.4595	0.0805	2.52	9	Removes I(1) general trend
	Detrended (FWL)	0.3141	0.4057	0.4611	0.0697	1.82	10	Linear trend removed (FWL)
	⊖ ZERO IN ALL SPECIFICATIONS. p > 0.45 everywhere. The solvency ratio does not actually track the combined ratio it is driven by capital decisions.
R5 — Gross Cat. SCR ~ Gross Catastrophic Loss
	Field (original)	-0.0132	0.1014	0.8995	0.0021	0.23	10	Suspected autocorrelation of the residuals
	First differences (Δ)	-0.0124	0.0233	0.6110	0.0389	2.38	9	Removes I(1) general trend
	Log differences	-0.0024	0.0109	0.8319	0.0069	2.36	9	β = elasticity (% change in Y per 1% change in X)
	Trend-adjusted (FWL)	-0.0204	0.0237	0.4149	0.0846	2.37	10	Linear trend removed (FWL)
	⊖ ZERO IN ALL SPECIFICATIONS. p > 0.41 everywhere. The standard formula for catastrophic SCR is completely disconnected from realized losses. DW = 0.23 at the level confirms: the model misses all the information in the data.
Legend (p-value colors):
		p < 0.05 (significant at the 5% level)
		0.05 ≤ p < 0.10 (marginal)
		p ≥ 0.10 (not significant)

Table 6. Detrending Using the Frisch–Waugh–Lovell (FWL) Theorem.

#	Specification	β (slope)	SE	p-value	R²	DW	n	Note
R1 — Net Total SCR ~ Gross Premiums
	Niva (original)	0.2630	0.0057	0.0000	0.9963	2.9174	10	Suspected autocorrelation of the residuals
	First differences (Δ)	0.3074	0.0469	0.0003	0.8599	2.8838	9	Removes I(1) general trend
	Δ through zero	0.2713	0.0239	0.0000	0.9414	3.1432	9	No constant (theoretically correct if ΔX=0⇒ΔY=0)
	Logarithmic differences (elasticity)	0.9602	0.1564	0.0005	0.8434	2.7551	9	β = elasticity (% change in Y per 1% change in X)
	Trend-adjusted (FWL)	0.2777	0.0301	0.0000	0.9140	2.8211	10	Linear trend removed (FWL)
	✓ ROBUST. The relationship holds across all specifications (p<0.001 in Δ and in Δlog). The true elasticity is ~0.96 (close to unity). The high R² at the level is not an artifact.
R3 — SCR reduction from over-insurance ~ Cession ratio
	Niva (original)	0.6478	0.0349	0.0000	0.9773	2.55	10	Suspected autocorrelation of the residuals
	First differences (Δ)	0.7003	0.5033	0.2067	0.2167	2.51	9	Removes I(1) general trend
	Δ through zero	0.6259	0.1572	0.0041	0.6646	2.53	9	No constant (theoretically correct if ΔX=0⇒ΔY=0)
	Trend-adjusted (FWL)	0.7345	0.3173	0.0493	0.4011	2.58	10	Linear trend removed (FWL)
	⚠ PARTIALLY DRIVEN BY THE TREND. R² drops dramatically from 0.98 to 0.22 in Δ (p=0.21). After passing through zero and detrending, the relationship persists, but with 4–9× wider standard errors. The true effect is present, but it is weaker than the level suggested.
R4 — Solvency ratio ~ Gross combined ratio
	Niva (original)	0.2488	0.3808	0.5319	0.0507	1.79	10	Base model
	First differences (Δ)	0.2568	0.3281	0.4595	0.0805	2.52	9	Removes I(1) general trend
	Detrended (FWL)	0.3141	0.4057	0.4611	0.0697	1.82	10	Linear trend removed (FWL)
	⊖ ZERO IN ALL SPECIFICATIONS. p > 0.45 everywhere. The solvency ratio does not actually track the combined ratio it is driven by capital decisions.
R5 — Gross Cat. SCR ~ Gross Catastrophic Loss
	Field (original)	-0.0132	0.1014	0.8995	0.0021	0.23	10	Suspected autocorrelation of the residuals
	First differences (Δ)	-0.0124	0.0233	0.6110	0.0389	2.38	9	Removes I(1) general trend
	Log differences	-0.0024	0.0109	0.8319	0.0069	2.36	9	β = elasticity (% change in Y per 1% change in X)
	Trend-adjusted (FWL)	-0.0204	0.0237	0.4149	0.0846	2.37	10	Linear trend removed (FWL)
	⊖ ZERO IN ALL SPECIFICATIONS. p > 0.41 everywhere. The standard formula for catastrophic SCR is completely disconnected from realized losses. DW = 0.23 at the level confirms: the model misses all the information in the data.
Legend (p-value colors):
		p < 0.05 (significant at the 5% level)
		0.05 ≤ p < 0.10 (marginal)
		p ≥ 0.10 (not significant)

[Conclusions from the second specification of the regression analysis:]

• R1: Net SCR ~ gross premiums

The relationship between net SCR and gross premiums holds across all specifications, including first differences and logarithmic transformations. The elasticity, estimated using log differences, is approximately 0.96, suggesting an almost proportional scaling of the capital requirement relative to the volume of business. A 1% increase in gross premiums leads to a 0.96% increase in the net SCR. The stability of the estimates across different transformations and the persistence of significance indicate that the observed effect is not an artifact of a common trend, but rather a structural property of the standard SCR formula.

• R3: Effect of the cession ratio on the reduction in SCR

The relationship between the cession ratio and the reduction in SCR is sensitive to the model specification. While a high explanatory power is observed in level data, the transition to first differences and detrended data leads to a significant decrease in R², which falls from 0.98 to 0.22 (with a constant in Δ) or 0.40 (detrended). This indicates that part of the observed effect in the level data is due to the joint temporal dynamics of increasing variables.

However, the correlation does not disappear entirely under alternative specifications, suggesting the presence of a real but less pronounced structural effect. Therefore, reinsurance remains a relevant factor for capital reduction, but its effect is likely overestimated when analyzed solely in terms of levels.

• R4: Solvency ratio ~ combined ratio

In none of the specifications is a statistically significant relationship found between the solvency ratio and the combined ratio. The lack of a relationship is stable and robust across various data transformations, indicating that capital adequacy is determined primarily by capital and structural decisions, rather than by short-term operational profitability.

• R5: Catastrophe SCR ~ realized catastrophe losses

The relationship between the gross catastrophe SCR and observed catastrophe losses is statistically insignificant in all specifications. Furthermore, the diagnostics of the residuals in the level specification (low Durbin–Watson value) suggest omitted dynamics, which, however, are not recovered through differentiation or detrending.

This indicates that the standard formula for catastrophic SCR is fundamentally calibrated to theoretical scenarios (99.5% VaR / 1-in-200-year) rather than to empirically observed losses. The result provides empirical support for the limited explanatory power of the standard approach in assessing actually realized catastrophic risk and serves as a strong quantitative argument in favor of an internal model.

The following table summarizes the information from the two regression analyses and the results obtained:

Table 7. Summary of the two regression analyses.

What We Tested	Result	Brief explanation
Is the capital requirement dependent on the volume of premiums?	✓ YES, confirmed	There is a statistically significant and approximately proportional relationship between the SCR and gross premiums, with an elasticity close to one.
Does the cession ratio reduce the SCR?	✓ YES, confirmed (moderate)	The effect of reinsurance on the SCR is statistically significant but sensitive to the model’s specification and is partly driven by joint trend movements.
Does solvency follow operating profitability?	✗ NO, rejected	No statistically significant relationship is found between the solvency ratio and underwriting profitability indicators. It is driven by capital decisions, not by current results.
Does the catastrophe SCR reflect realized catastrophe losses?	✗ NO, rejected	There is no statistically significant relationship, which is consistent with the regulatory calibration of the model to theoretical scenarios rather than to the empirical frequency of events.
Does the credit rating determine the SCR for counterparty risk?	✓ YES, confirmed	LGD and the credit rating demonstrate high explanatory power (≈91%), which confirms the structural validity of the model.
Is capital intensity homogeneous across business lines?	≈ YES, partially confirmed	An approximately constant capital intensity (~0.24 SCR units per unit of net premiums) is observed, suggesting limited variation across lines at the aggregate level.

Table 7. Summary of the two regression analyses.

What We Tested	Result	Brief explanation
Is the capital requirement dependent on the volume of premiums?	✓ YES, confirmed	There is a statistically significant and approximately proportional relationship between the SCR and gross premiums, with an elasticity close to one.
Does the cession ratio reduce the SCR?	✓ YES, confirmed (moderate)	The effect of reinsurance on the SCR is statistically significant but sensitive to the model’s specification and is partly driven by joint trend movements.
Does solvency follow operating profitability?	✗ NO, rejected	No statistically significant relationship is found between the solvency ratio and underwriting profitability indicators. It is driven by capital decisions, not by current results.
Does the catastrophe SCR reflect realized catastrophe losses?	✗ NO, rejected	There is no statistically significant relationship, which is consistent with the regulatory calibration of the model to theoretical scenarios rather than to the empirical frequency of events.
Does the credit rating determine the SCR for counterparty risk?	✓ YES, confirmed	LGD and the credit rating demonstrate high explanatory power (≈91%), which confirms the structural validity of the model.
Is capital intensity homogeneous across business lines?	≈ YES, partially confirmed	An approximately constant capital intensity (~0.24 SCR units per unit of net premiums) is observed, suggesting limited variation across lines at the aggregate level.

4.5.2. Correlation Analysis

The purpose of correlation analysis is to identify the degree of co-movement among the key financial and risk indicators in the insurance portfolio under study. It answers the question, “Which variables move together?” Correlation measures the strength of the relationship between two variables on a scale from −1 (moving in opposite directions) to +1 (moving exactly in tandem). A value close to 0 indicates no relationship. Unlike regression analysis, correlation does not imply a cause-and-effect relationship; rather, it measures the intensity and direction of the linear or monotonic association between two variables.

The study uses Pearson’s and Spearman’s correlation coefficients, with estimates calculated in three specifications: (i) on the original levels of the variables, (ii) on logarithmic transformations, where applicable, and (iii) on first differences (Δ). This multi-specification approach is necessary due to the presence of a pronounced upward trend in the key variables premiums, reserves, SCR, and equity. Under such conditions, high correlations at the level may result from common temporal dynamics rather than an economically meaningful relationship. Therefore, comparing correlations in levels and in first differences allows for a distinction between stable structural relationships and spurious correlations arising from overall business growth.

One should also take into account the limitation associated with the small size of the time sample (n=10), which reduces the statistical power of the tests and increases the sensitivity of the results to individual outliers. Therefore, the results should be interpreted with caution and in the context of all the specifications used.

Table 8. Results of the correlation analysis.

Variable A	Variable B	r (levels)	p (levels)	r (Δ)	p (Δ)	Interpretation
✓ Robust correlations (strong in both levels and first differences)
Gross premiums	Technical reserves	0.99751	0.00000	0.89455	0.00113	Reserves automatically track premium volume
Gross premiums	Own funds	0.97744	0.00000	0.84129	0.00447	Capital grows with the volume of business
Gross premiums	SCR SF	0.98527	0.00000	0.81805	0.00705	SCR increases linearly with premiums — unit elasticity
Gross premiums	Total gross	0.99822	0.00000	0.92360	0.00038	Gross SCR directly follows premium volume
Gross premiums	Total net	0.99814	0.00000	0.92730	0.00032	Net SCR follows the premium volume on a linear basis
Gross cat loss	Gross CR	0.88114	0.00075	0.97509	0.00001	CAT losses DIRECTLY drive the combined ratio (r=0.98 in Δ!)
Technical reserves	SCR SF	0.98153	0.00000	0.69523	0.03761	The reserve SCR is a component of the total SCR
Technical Reserves	Total Gross	0.99499	0.00000	0.78102	0.01296	Reserves form the basis for the SCR reserve module
Technical reserves	Total net	0.99427	0.00000	0.75354	0.01904	Net SCR is also linked to reserves
Own Funds	SCR SF	0.97827	0.00000	0.78065	0.01303	Capital management aligns equity and SCR
Own Funds	Total Gross	0.97296	0.00000	0.79032	0.01124	Capital is maintained in accordance with requirements
Own Funds	Total net	0.97773	0.00000	0.82952	0.00568	The capital buffer increases with the net SCR
SCR SF	Total Net	0.98204	0.00000	0.69709	0.03688
Total Gross	Total net	0.99927	0.00000	0.98361	0.00000
⚠ Spurious correlations (strong at level, disappear in first differences)
SCR SF	Reduction in overinsurance	0.94628	0.00003	0.00687	0.98601	The reduction is better because both SCR and cession are increasing
Gross premiums	Renewal reduction	0.95206	0.00002	-0.02795	0.94311	Premiums are rising, and so is cession a joint trend, not a cause
Total Gross	Reinsurance reduction	0.95762	0.00001	0.07066	0.85666	Same as above: general trend, not a real correlation
Gross losses	Reduction in reinsurance	0.87779	0.00084	0.02197	0.95526	Damage and reduction increase in parallel, without a common cause
Total net	Reinsurance reduction	0.94692	0.00003	-0.10108	0.79582	Trend artifact — net SCR is also rising
Technical reserves	Decrease in reinsurance	0.94913	0.00003	0.10939	0.77937	Apparently both grow with volume
Assignment ratio	SCR SF	0.95397	0.00002	-0.12181	0.75490	Both increase over time; there is no causal relationship between them
Cession ratio	Technical reserves	0.96389	0.00001	0.17195	0.65823	Sporadic correlation it may be real, but is not confirmed in Δ
Gross losses	Own funds	0.88384	0.00069	0.10608	0.78592	Both increase with volume they are not correlated in Δ
Gross premiums	Cession ratio	0.96534	0.00001	-0.19528	0.61460	Both are increasing, but r_Δ = −0.20: when premiums ↑, the cession does NOT ↑
Equity	Decrease in reinsurance	0.90474	0.00032	-0.16834	0.66506	Trend artifact, not a real relationship
Assignment ratio	Total gross	0.96650	0.00001	-0.28142	0.46320	Expected Decline — Cession Ratios Are Rising Regardless of the Gross SCR
Gross claims	Total gross	0.88781	0.00060	-0.27797	0.46893	Losses do not affect the gross SCR on an annual basis
Cession ratio	Total net	0.95859	0.00001	-0.35425	0.34960	As R3: the actual correlation is weaker than the levels
Gross losses	Total net	0.88333	0.00070	-0.29153	0.44657	Losses do not affect the annual net SCR

Table 8. Results of the correlation analysis.

Variable A	Variable B	r (levels)	p (levels)	r (Δ)	p (Δ)	Interpretation
✓ Robust correlations (strong in both levels and first differences)
Gross premiums	Technical reserves	0.99751	0.00000	0.89455	0.00113	Reserves automatically track premium volume
Gross premiums	Own funds	0.97744	0.00000	0.84129	0.00447	Capital grows with the volume of business
Gross premiums	SCR SF	0.98527	0.00000	0.81805	0.00705	SCR increases linearly with premiums — unit elasticity
Gross premiums	Total gross	0.99822	0.00000	0.92360	0.00038	Gross SCR directly follows premium volume
Gross premiums	Total net	0.99814	0.00000	0.92730	0.00032	Net SCR follows the premium volume on a linear basis
Gross cat loss	Gross CR	0.88114	0.00075	0.97509	0.00001	CAT losses DIRECTLY drive the combined ratio (r=0.98 in Δ!)
Technical reserves	SCR SF	0.98153	0.00000	0.69523	0.03761	The reserve SCR is a component of the total SCR
Technical Reserves	Total Gross	0.99499	0.00000	0.78102	0.01296	Reserves form the basis for the SCR reserve module
Technical reserves	Total net	0.99427	0.00000	0.75354	0.01904	Net SCR is also linked to reserves
Own Funds	SCR SF	0.97827	0.00000	0.78065	0.01303	Capital management aligns equity and SCR
Own Funds	Total Gross	0.97296	0.00000	0.79032	0.01124	Capital is maintained in accordance with requirements
Own Funds	Total net	0.97773	0.00000	0.82952	0.00568	The capital buffer increases with the net SCR
SCR SF	Total Net	0.98204	0.00000	0.69709	0.03688
Total Gross	Total net	0.99927	0.00000	0.98361	0.00000
⚠ Spurious correlations (strong at level, disappear in first differences)
SCR SF	Reduction in overinsurance	0.94628	0.00003	0.00687	0.98601	The reduction is better because both SCR and cession are increasing
Gross premiums	Renewal reduction	0.95206	0.00002	-0.02795	0.94311	Premiums are rising, and so is cession a joint trend, not a cause
Total Gross	Reinsurance reduction	0.95762	0.00001	0.07066	0.85666	Same as above: general trend, not a real correlation
Gross losses	Reduction in reinsurance	0.87779	0.00084	0.02197	0.95526	Damage and reduction increase in parallel, without a common cause
Total net	Reinsurance reduction	0.94692	0.00003	-0.10108	0.79582	Trend artifact — net SCR is also rising
Technical reserves	Decrease in reinsurance	0.94913	0.00003	0.10939	0.77937	Apparently both grow with volume
Assignment ratio	SCR SF	0.95397	0.00002	-0.12181	0.75490	Both increase over time; there is no causal relationship between them
Cession ratio	Technical reserves	0.96389	0.00001	0.17195	0.65823	Sporadic correlation it may be real, but is not confirmed in Δ
Gross losses	Own funds	0.88384	0.00069	0.10608	0.78592	Both increase with volume they are not correlated in Δ
Gross premiums	Cession ratio	0.96534	0.00001	-0.19528	0.61460	Both are increasing, but r_Δ = −0.20: when premiums ↑, the cession does NOT ↑
Equity	Decrease in reinsurance	0.90474	0.00032	-0.16834	0.66506	Trend artifact, not a real relationship
Assignment ratio	Total gross	0.96650	0.00001	-0.28142	0.46320	Expected Decline — Cession Ratios Are Rising Regardless of the Gross SCR
Gross claims	Total gross	0.88781	0.00060	-0.27797	0.46893	Losses do not affect the gross SCR on an annual basis
Cession ratio	Total net	0.95859	0.00001	-0.35425	0.34960	As R3: the actual correlation is weaker than the levels
Gross losses	Total net	0.88333	0.00070	-0.29153	0.44657	Losses do not affect the annual net SCR

4.5.2.1. Robust Correlations

The analysis identifies a group of correlations that remain statistically significant in both levels and first differences, suggesting the presence of a robust structural relationship.

The strongest relationship is between gross premiums and technical reserves (r = 0.998 at the level; r = 0.895 in first differences), which is an economically expected result given the mechanical relationship between exposure and required reserves. Similarly, the relationship between gross premiums and the total SCR both gross and net remains high and statistically significant across all specifications, confirming that the capital requirement scales proportionally with the volume of business.

A key finding is the exceptionally strong correlation between catastrophe losses and the gross combined ratio. In first-differences analysis, the correlation coefficient reaches r = 0.975, indicating that catastrophe events have a direct and significant effect on the company’s operating results. This result is particularly important in the context of the previous regression analysis, according to which realized catastrophic losses do not explain the catastrophic SCR. Consequently, the empirical data reveal an asymmetry between the accounting treatment of catastrophic events (a strong effect on P&L) and their impact on regulatory capital (no effect on SF SCR).

The relationship between own funds and the SCR is also stable, suggesting the existence of an active capital management policy and the maintenance of a capital buffer in line with regulatory requirements.

4.5.2.2. Spurious Correlations

The second group of results includes correlations that are strong and statistically significant at the level but disappear when analyzed using first differences. This pattern is characteristic of spurious relationships induced by a joint upward trend.

This is most clearly observed in the relationships between the reduction in SCR due to reinsurance and various volume indicators gross premiums, gross claims, reserves, and total SCR. At the level of variables, the correlations exceed r = 0.90, but in first differences they drop to practically zero. This indicates that the observed association is primarily due to the fact that all variables increase simultaneously as the business expands.

A similar result is observed in the relationship between the cession ratio and premium volume. At the level, the correlation is extremely high (r = 0.965), but in first differences it becomes statistically insignificant and even negative. Consequently, an increase in the cession ratio cannot be interpreted as an automatic response to business growth, but rather as an independent strategic decision driven by risk and capital management policy.

A similar conclusion holds for the relationship between the cession ratio and the SCR. Although a high correlation is observed at the cross-sectional level, it is not confirmed in the dynamic specifications, which is consistent with the results of the regression analysis and suggests that the actual effect of reinsurance on the SCR is more moderate than the cross-sectional analysis alone would suggest.

4.5.2.3. Conclusions from the Correlation Analysis:

• The balance sheet demonstrates strong internal interdependence. Gross premiums, technical reserves, own funds, and the SCR show very high positive correlations both at the level (r > 0.97) and in first differences (r > 0.78). The persistence of this relationship even after removing the overall time trend suggests the presence of a stable economic relationship, rather than merely a trend artifact.
• Catastrophic losses have a significant impact on the technical result, but not on the capital requirement. The correlation between gross catastrophic losses and the combined ratio is high both in terms of levels (r = 0.88) and first differences (r = 0.975). This indicates that in years with elevated catastrophic losses, there is a significant deterioration in the technical result and the portfolio’s profitability. At the same time, the results of the regression analysis (R5) do not establish a statistically significant relationship between realized catastrophic losses and the catastrophic SCR under the standard formula. This suggests that the standard Solvency II formula is calibrated primarily to hypothetical “1-in-200-year” scenarios, rather than to the observed annual losses over the historical period. Consequently, realized catastrophic events affect the current profit and loss (P&L) statement but are not automatically reflected in the annual capital requirement under the standard formula.
• A significant portion of the correlations with “SCR Reduction from Reinsurance” is driven by the overall trend. Six of the eleven correlations examined with the variable “Reduction in SCR from Reinsurance” show very high values in first differences (r > 0.87), but practically disappear in first differences (r ≈ 0).

This suggests that a large portion of the observed relationships is due to the parallel growth of:

• premium volume,
• SCR,
• technical reserves,
• cession activity

during the period under review, rather than a direct economic relationship between the respective variables.

After adjusting for the overall time trend, only the cession ratio retains a relatively stable relationship with the decline in SCR, albeit significantly weaker than that observed in the regressions on the original levels. This result confirms the conclusions from the second specification of the regression analysis (R3).

• The cession ratio does not show a consistent relationship with the growth in premium volume. The correlation between gross premiums and the cession ratio is very high at the level (r = 0.965), but becomes negative and statistically insignificant in first differences (r = −0.20; p = 0.62). This suggests that the increase in the cession ratio during the period was not a mechanical response to the increase in premium volume, but rather the result of independent strategic decisions related to: (i) capital management; (ii) solvency optimization; (iii) catastrophe risk management; (iv) changes in reinsurance policy.

Consequently, in a given year, a significant increase in premium volume is possible without a substantial change in the cession ratio.

The following table summarizes the information from the correlation analysis and the results obtained:

Table 9. Summary of the Correlation Analysis Results.

Correlation between...	Type of relationship	What it means
Premiums, reserves, capital, and SCR	Stable	The main balance sheet items and the capital structure evolve in tandem
CAT losses and combined ratio	Sustainable and strong	Catastrophic events have a direct impact on the technical result
CAT losses and catastrophic SCR	No consistent relationship	The standard formula does not reflect actual annual losses
Cession quota and premium volume	Apparent	The high correlation in levels is mainly explained by a general time trend
Decrease in SCR and most financial indicators	Mostly spurious	A significant portion of the correlations disappear after removing the trend.
Assignment ratio and a decrease in SCR	Partially robust	The relationship persists but is weaker after trend adjustment

Table 9. Summary of the Correlation Analysis Results.

Correlation between...	Type of relationship	What it means
Premiums, reserves, capital, and SCR	Stable	The main balance sheet items and the capital structure evolve in tandem
CAT losses and combined ratio	Sustainable and strong	Catastrophic events have a direct impact on the technical result
CAT losses and catastrophic SCR	No consistent relationship	The standard formula does not reflect actual annual losses
Cession quota and premium volume	Apparent	The high correlation in levels is mainly explained by a general time trend
Decrease in SCR and most financial indicators	Mostly spurious	A significant portion of the correlations disappear after removing the trend.
Assignment ratio and a decrease in SCR	Partially robust	The relationship persists but is weaker after trend adjustment

4.5.3. Chi-Square Analysis

Chi-square analysis aims to determine whether the observed empirical distribution differs statistically significantly from the theoretically expected distribution. Unlike regression and correlation analysis, which examine relationships between continuous variables, the chi-square approach assesses deviations between observed and expected frequencies within categorical data. The method is particularly suitable for analyzing the correspondence between the empirical behavior of the insurance portfolio and the assumptions underlying the Standard Formula (SF), the structure of the reinsurance program, and the concentration of exposures to counterparties. For example: if the Standard Formula assumes that a catastrophe year occurs once every 10 years, but we observe 3 such years over a 10-year period, is this difference random, or is it a real deviation?

The chi-square statistic is defined as follows: χ² = Σ (O − E)² / E, where Oi denotes the observed frequencies and Ei denotes the expected frequencies under the null hypothesis H0. Higher values of the test statistic indicate more significant deviations between the observed and expected distributions and, accordingly, increase the probability of rejecting H0. A large χ² value means that the observed distribution differs significantly from the expected one, and H0 must be rejected.

Statistical significance is assessed using the p-value, which measures the probability that the observed deviation is due solely to chance under the null hypothesis. In accordance with standard statistical practice, results with p < 0.05 are interpreted as statistically significant.

As part of this study, five chi-square tests were conducted on the same source dataset used in the regression and correlation analyses. The tests cover: (i) the frequency of catastrophic years relative to the calibration of the Standard Formula; (ii) the relationship between the type of reinsurance contract and its activation; (iii) the concentration of exposures by credit rating; (iv) the correspondence between the portfolio structure and the market structure.

For each test, the null hypothesis, test statistic, p-value, and effect size (Cramér’s V, where applicable) are presented.

Table 10. Results of the chi-square analysis.

#	Test	Type	n	df	χ²	p-value	Effect	Conclusion
T1	Frequency of catastrophic years (threshold: 5 million EUR) based on the SF assumption of “1 in 10”	GOF	10	1	4.444	0.0350 *		Three catastrophic years were observed, compared to the expected one according to SF, suggesting a possible underestimation of moderate-severity catastrophic events.
T2	Contract Type × Activation (5×2)	Indep.	50	4	40.642	3.19e-08 ***	V=0.902	There is a strong correlation between the contract structure and its activation. Proportional contracts are activated systematically, while Stop Loss contracts were not activated during the period under review.
T2b	Proportional vs. Non-proportional Contracts × Activation (Fisher’s exact test)	Indep.	50	1		1.29e-05 ***	OR=∞	All 20 proportional ones are activated; compared to 13 out of 30 non-proportional ones.
T3	Rating × LGD category (3×3 tertiles)	Indep.	50	4	38.015	1.11e-07 ***	V=0.617	High rating ↔ high exposure. Concentration relative to AA-. Rating and exposure are related but the test does not indicate whether a high rating “attracts” exposure, or whether high exposure leads to a preference for high ratings. This describes the structure, not causality. The test finds an association, but NOT A CAUSE. A strong association is found between credit rating and the size of the exposure, indicating a concentration toward highly rated reinsurers.
T4a	Distribution of exposure by rating versus a uniform distribution	GOF	100	2	30,500	2.38e-07 ***		The observed proportions of 55% / 35% / 10% differ statistically significantly from the uniform distribution of 33% / 33% / 33%.
T4b	Distribution of exposure by rating versus proportional distribution by number of counterparties	GOF	100	2	84.375	4.77e-19 ***		The concentration of exposure significantly exceeds the level that would result from the numerical distribution of counterparties.
T5	Premiums by business line vs. market structure	GOF	100	7	7,682	0.3615		The portfolio structure does not differ statistically significantly from the market structure.
Color legend:
			Significant difference / correlation (p < 0.05)
			H0 not rejected – consistent with expectations
Significance markers:
*** p < 0.001 | ** p < 0.01 | * p < 0.05 | · p < 0.10

Table 10. Results of the chi-square analysis.

#	Test	Type	n	df	χ²	p-value	Effect	Conclusion
T1	Frequency of catastrophic years (threshold: 5 million EUR) based on the SF assumption of “1 in 10”	GOF	10	1	4.444	0.0350 *		Three catastrophic years were observed, compared to the expected one according to SF, suggesting a possible underestimation of moderate-severity catastrophic events.
T2	Contract Type × Activation (5×2)	Indep.	50	4	40.642	3.19e-08 ***	V=0.902	There is a strong correlation between the contract structure and its activation. Proportional contracts are activated systematically, while Stop Loss contracts were not activated during the period under review.
T2b	Proportional vs. Non-proportional Contracts × Activation (Fisher’s exact test)	Indep.	50	1		1.29e-05 ***	OR=∞	All 20 proportional ones are activated; compared to 13 out of 30 non-proportional ones.
T3	Rating × LGD category (3×3 tertiles)	Indep.	50	4	38.015	1.11e-07 ***	V=0.617	High rating ↔ high exposure. Concentration relative to AA-. Rating and exposure are related but the test does not indicate whether a high rating “attracts” exposure, or whether high exposure leads to a preference for high ratings. This describes the structure, not causality. The test finds an association, but NOT A CAUSE. A strong association is found between credit rating and the size of the exposure, indicating a concentration toward highly rated reinsurers.
T4a	Distribution of exposure by rating versus a uniform distribution	GOF	100	2	30,500	2.38e-07 ***		The observed proportions of 55% / 35% / 10% differ statistically significantly from the uniform distribution of 33% / 33% / 33%.
T4b	Distribution of exposure by rating versus proportional distribution by number of counterparties	GOF	100	2	84.375	4.77e-19 ***		The concentration of exposure significantly exceeds the level that would result from the numerical distribution of counterparties.
T5	Premiums by business line vs. market structure	GOF	100	7	7,682	0.3615		The portfolio structure does not differ statistically significantly from the market structure.
Color legend:
			Significant difference / correlation (p < 0.05)
			H0 not rejected – consistent with expectations
Significance markers:
*** p < 0.001 | ** p < 0.01 | * p < 0.05 | · p < 0.10

Interpretation of the results:

• The results indicate that the Standard Formula likely underestimates the frequency of moderately severe catastrophic events. Over the observed 10-year period, catastrophic losses exceeding 5 million EUR occurred three times, whereas the calibration of the Standard Formula suggests approximately one such event over the same time horizon (p=0.035). When the threshold is lowered to 3 million EUR, the discrepancy becomes even more pronounced (five observed events versus one expected, p < 0.001). These results are consistent with the conclusions of the regression analysis (R5), according to which the catastrophic SCR under the Standard Formula is weakly correlated with actual catastrophic claims. Taken together, the results suggest that the SF is calibrated primarily to extreme tail scenarios (of the “1-in-200” type) rather than to recurring, moderately severe catastrophic events observed empirically.

Recommendation: Consider using a proprietary catastrophe model or Volatility Adjustment in ORSA.

• The second significant finding relates to the activation of the reinsurance program. The Stop Loss contract was not activated even once during the entire period analyzed, despite the significant amount of ceded premiums. This can be interpreted in two ways: either the portfolio demonstrated a high degree of stability and the loss ratio never reached the levels required to trigger the coverage (the loss ratio never approached 85%), or the contract functioned as protection against extreme systemic events with a low probability of occurrence (Stop Loss was purchased as an expensive option that would pay out only in the event of a systemic crisis). The results show that over a 10-year period, the ceded Stop Loss premium amounted to approximately 9 million EUR without a single claim payment. From the perspective of risk management effectiveness, this result raises questions about the economic efficiency of this type of protection within the time horizon under consideration.
• The analysis also identifies a significant concentration of exposures to reinsurers with the highest credit ratings. Counterparties rated AA− (Munich Re and Swiss Re) account for approximately 55% of the total ceded exposure, even though they represent a significantly smaller share of the total number of counterparties (40%). Both goodness-of-fit tests decisively reject the hypothesis of a uniform or proportional distribution of exposures (χ² = 30.5 (p < 0.001)). From a prudential perspective, such a structure is economically justified, as a higher rating leads to lower capital requirements for counterparty risk under Solvency II. At the same time, however, concentration risk arises, which is not fully captured by the Standard Formula.
• Furthermore, the relationship between credit rating and exposure size is statistically strong (Cramér’s V = 0.617). This result should be interpreted as a structural association rather than a causal relationship. The analysis does not allow us to determine whether a high rating “attracts” larger exposures or whether the insurer deliberately directs larger exposures toward higher-rated counterparties as part of a conservative risk management policy. Regardless, the observed concentration suggests the presence of potential wrong-way risk, as a potential default by a highly rated reinsurer (AA-) would affect a disproportionately large portion of the ceded exposure (55%). In this context, the results support the need to introduce explicit concentration limits by rating and counterparty.
• Finally, the portfolio structure appears statistically consistent with the market structure. The null hypothesis of similarity is not rejected (p=0.362), suggesting the absence of excessive concentration in individual lines of business relative to the market. The most significant deviation is observed in “Civil Liability” insurance, where the company’s market share is lower than the market average (by ~9 percentage points). This can be interpreted as a strategic positioning aimed at limiting exposure to a highly regulated and relatively low-margin segment.

The following table summarizes the information from the chi-square analysis and the results obtained:

Table 11. Summary of the Chi-square Analysis.

What We Tested	Result	What it means
Frequency of catastrophic years relative to the calibration of the regulatory formula	There is a statistically significant difference	The empirically observed frequency of catastrophic years exceeds the frequency implicitly assumed in the Standard Formula, suggesting a possible underestimation of moderately severe catastrophic events.
Relationship Between the Type of Reinsurance Contract and Its Activation	A strong relationship has been established	The probability of activation is significantly determined by the structure of the contract. Proportional contracts are activated systematically, while Stop Loss coverages were not activated during the analyzed period.
Relationship between credit rating and exposure size	A strong association has been identified	Higher-rated reinsurers accumulate larger exposures, indicating a concentration toward counterparties with high credit ratings.
Distribution of exposure by counterparty rating	The distribution is not uniform	Exposure is disproportionately concentrated among a limited number of high-rated counterparties, with approximately 55% of the exposure directed toward 40% of the counterparties.
Alignment between the portfolio structure and the market structure	No statistically significant differences were found	The portfolio structure is consistent with the market structure and shows no signs of excessive concentration in individual business lines, which is an indicator of relatively balanced diversification.

Table 11. Summary of the Chi-square Analysis.

What We Tested	Result	What it means
Frequency of catastrophic years relative to the calibration of the regulatory formula	There is a statistically significant difference	The empirically observed frequency of catastrophic years exceeds the frequency implicitly assumed in the Standard Formula, suggesting a possible underestimation of moderately severe catastrophic events.
Relationship Between the Type of Reinsurance Contract and Its Activation	A strong relationship has been established	The probability of activation is significantly determined by the structure of the contract. Proportional contracts are activated systematically, while Stop Loss coverages were not activated during the analyzed period.
Relationship between credit rating and exposure size	A strong association has been identified	Higher-rated reinsurers accumulate larger exposures, indicating a concentration toward counterparties with high credit ratings.
Distribution of exposure by counterparty rating	The distribution is not uniform	Exposure is disproportionately concentrated among a limited number of high-rated counterparties, with approximately 55% of the exposure directed toward 40% of the counterparties.
Alignment between the portfolio structure and the market structure	No statistically significant differences were found	The portfolio structure is consistent with the market structure and shows no signs of excessive concentration in individual business lines, which is an indicator of relatively balanced diversification.

4.5.4. Summary of Conclusions from the Three Analyses

The integrated analysis of the results from the regression, correlation, and χ² analyses allows for the formulation of several key empirical conclusions regarding the adequacy of the standard Solvency II formula, the effectiveness of the reinsurance program, and the risk structure of the insurance company under study. The conclusions presented below are arranged according to their analytical and practical significance, rather than according to the sequence of the statistical methods applied.

Conclusion 1: The standard formula likely underestimates the frequency of moderate-severity catastrophic events

In summary: Analysis of historical data shows that, over the 10-year period under review, there were three years with catastrophic losses exceeding 5 million EUR (2018, 2020, and 2023), whereas the calibration of the standard formula suggests approximately one such year over the same time horizon. The χ² goodness-of-fit test rejects the hypothesis of a random deviation (p = 0.035), suggesting a systematic discrepancy between the empirically observed frequency and the regulatory calibration.

This result is consistently confirmed by the three independent statistical approaches:

• Regression analysis finds no statistically significant relationship between realized catastrophic losses and the capital requirement for catastrophic risk (Cat SCR). The slope remains statistically insignificant in all specifications, including after removing the trend and when analyzing first differences. In other words: when there is a large loss during the year, the capital requirement does not change in any way.
• Correlation analysis shows virtually zero correlation between catastrophic losses and the Cat SCR in both level and dynamic specifications.
• The chi-square analysis finds that the frequency of observed catastrophic years significantly exceeds (by about a factor of 3) the frequency expected according to the standard formula

The results suggest that the standard formula is primarily sensitive to extreme events with very low probability (1-in-200 scenarios), but does not adequately capture the frequency of recurring moderate-severity catastrophic events characteristic of the regional market profile. This represents a significant limitation of the regulatory framework, which is quantitatively evident in the empirical data.

Practical implication: In future ORSA assessments, it would be advisable for the Company to consider developing a partial internal model for the catastrophe submodule or introducing an additional management capital buffer for catastrophe risk.

Conclusion 2: Reinsurance effectively reduces the SCR, but the effect is weaker than that suggested by the level regressions

In summary: Reinsurance effectively reduces the capital requirement, but the actual effect is smaller than what the raw data suggest. The correlation appears very strong (close to 1.0) simply because all financial variables grow over time.

The initial analysis showed an extremely strong relationship between the cession ratio (the share of premiums transferred to reinsurers) and the reduction in the capital requirement (R² = 0.98, nearly perfect). At first glance, this suggests that reinsurance works exceptionally well.

However, when the analysis is conducted on annual changes (i.e., “by how much does cession change from year to year relative to the reduction in capital”), the correlation weakens significantly (R² drops to 0.22), and the standard errors of the coefficients increase manyfold. This result indicates that a significant portion of the initially observed correlation is driven by the joint growth of premium volume, SCR, and reinsurance activity over time. After removing the trend, the relationship between the cession ratio and the reduction in SCR remains positive but is statistically less robust.

Consequently, reinsurance does indeed fulfill its function of reducing capital requirements, but the quantitative effect is less certain than suggested by static level regressions.

Practical implication: When planning changes to the cession policy, one should not rely solely on historical linear relationships. It is necessary to combine statistical models with stress tests and scenario analyses.

Conclusion 3: The stop-loss contract is a costly protective mechanism that has not been empirically triggered

In summary: Over the 10-year period under review, approximately 9 million EUR in ceded premiums were paid under Stop Loss contracts, with no recoveries realized. The χ² test for dependence between the type of reinsurance contract and its activation shows an extremely strong statistical dependence (p < 0.00001).

The results show that: (i) proportional contracts are almost always involved in covering losses; (ii) Stop Loss contracts were not triggered even once during the entire period analyzed.

This suggests that the contract is structured to be triggered only under extreme systemic scenarios, which did not occur within the observed time horizon. It is possible that the activation threshold is significantly above the historically observed range of the combined ratio, meaning that in practice the contract would cover only extreme scenarios that have not occurred. A stop-loss contract is an expensive safeguard against a systemic crisis that simply has not materialized. This may be a reasonable decision, but it requires a careful assessment of the cost versus the benefit.

Practical implication: A detailed cost-benefit analysis of the stop-loss structure is needed, including an assessment of whether a lower trigger level and a lower limit would provide more effective protection under moderate stress scenarios. 9 million euros over 10 years without a single recovery is a significant expense that warrants analysis.

Conclusion 4: There is a concentration of reinsurance exposure to counterparties with the highest ratings

In summary: The two AA−-rated counterparties Munich Re and Swiss Re account for approximately 55% of the total ceded exposure, even though they represent only 40% of the total number of counterparties. This structure makes sense from a prudential risk management perspective, as it minimizes counterparty default capital charges. At the same time, however, it gives rise to concentration risk and the potential for wrong-way risk in the event of systemic market shocks

The χ² analysis reveals a statistically significant relationship between the credit ratings of reinsurers and the size of the exposure to them (Cramér’s V = 0.62).

The chi-square analysis revealed a strong structural relationship between the credit ratings of reinsurers and the size of the exposure to them. The higher the rating, the greater the exposure:

Table 12. Reinsurers’ Ratings and Share of Exposure.

Rating	Reinsurers	Number	Share of Exposure
AA- (highest)	Munich Re, Swiss Re	2	55%
A+	Hannover Re, SCOR	2	35%
A	PartnerRe	1	10%

Table 12. Reinsurers’ Ratings and Share of Exposure.

Rating	Reinsurers	Number	Share of Exposure
AA- (highest)	Munich Re, Swiss Re	2	55%
A+	Hannover Re, SCOR	2	35%
A	PartnerRe	1	10%

The main identified risks are:

• Wrong-way risk the risk of a simultaneous downgrade of the ratings of leading reinsurers during a global financial crisis.
• Excessive reliance on a limited number of counterparties for a key portion of the coverage. If one of the two leading reinsurers is unable to meet its obligations, the company loses coverage for approximately 30% of its premiums.

Practical implication: It is recommended to formalize explicit exposure limits both by rating category and per individual counterparty. For example: “no more than 60% per rating” and “no more than 30% per counterparty.” The current situation (Munich Re alone accounts for about 30%) is at the limit.

Conclusion 5: Solvency Does Not Directly Follow Profitability

In short: Neither regression nor correlation analysis establishes a statistically significant relationship between the combined ratio and the solvency ratio. This means that solvency trends are not directly determined by current underwriting profitability.

This may seem counterintuitive. The logical expectation is that better profitability (a low combined ratio) leads to better solvency. But the data show otherwise these two metrics do not move in tandem. This raises the question, “Why?” The answer is that solvency depends on capital, and capital is managed through decisions regarding profit distribution, reinsurance, and investment policy. The current annual result from underwriting activity is just one of the factors.

Practical implication: In SFCR and ORSA analyses, the explanation of changes in the solvency ratio should include capital decisions and reinsurance policy, not just the technical result.

Conclusion 6: The portfolio structure is statistically consistent with the market

In summary: A comparison of the distribution of premiums by line of business with the market structure reveals no extreme deviations. The company is reasonably diversified (p = 0.36 does not reject the null hypothesis of market conformity).

The χ² test for conformity between the premium portfolio structure and the market structure of the general insurance sector in the Republic of Bulgaria does not reject the null hypothesis (p = 0.36). This means that no statistically significant deviations from market diversification are observed. No line is overly dominant, and none is critically underrepresented

The most noticeable deviation is the relatively lower share of Motor Liability Insurance (about 9 percentage points below the market average), which likely reflects a strategic limitation of exposure to a low-margin and highly regulated segment. Property and industrial risks, conversely, account for a slightly higher share.

The results suggest that the company maintains a relatively balanced diversification across business lines, without excessive concentration in any single segment.

No statistical analysis provides 100% certain answers. In the following section, we outline the reliability of our conclusions that is, the extent to which we can rely on them and the limitations they impose. We have divided them into two groups:

• Conclusions with a high degree of reliability: (i) The lack of a correlation between Cat SCR and realized catastrophic losses; (ii) The concentration of exposure to highly rated reinsurers; (iii) The non-triggering of the stop-loss contract throughout the entire observation period.

These results are confirmed simultaneously by several independent statistical approaches and are characterized by high statistical significance.

• Conclusions with a moderate degree of reliability: (i) No correlation between solvency and profitability; (ii) The compatibility of the portfolio structure with the market structure.

However, the limited sample size does not allow for the definitive exclusion of weaker nonlinear or latent relationships.

The main limitation of the analysis stems from the length of the time series. The sample of 10 years of observations is sufficient to identify strong structural relationships and clearly defined trends, but remains limited for a precise assessment of moderate effects and complex dynamic relationships.

Using data at a higher frequency (e.g., quarterly observations) would significantly increase the statistical power of the tests, and the confidence intervals would narrow.

Based on the three analyses and their conclusions, we have made practical recommendations addressed to specific key functions within the insurance company

• To the management body: (i) Review and, if necessary, restructure the Stop Loss program; (ii) Formalize concentration limits by rating categories and individual counterparties; (iii) Expand the ORSA analysis to include additional catastrophic risk scenarios.
• To the management body of the insurance company: (i) Review the Stop Loss agreement, as 9 million euros over 10 years, without a single recovery, represents a significant expense that must be justified or restructured; (ii) Update the reinsurer concentration policy. Explicit limits on the share of exposure to a single rating and to a single counterparty; (iii) Inclusion of a catastrophe risk analysis in the ORSA. The data indicate that the standard formula may underestimate the frequency of medium-severity catastrophes.
• Regarding the actuarial function: (i) Use of higher-frequency data (quarterly instead of annual); (ii) Development of a partial internal model for the catastrophe submodule; (iii) Conducting scenario analyses for simultaneous downgrade risk of key reinsurers
• Regarding the risk management function: (i) A comprehensive cost-benefit analysis of the reinsurance program; (ii) Assessment of the optimal balance between proportional and non-proportional reinsurance; (iii) More detailed consideration of capital factors in the analysis of the solvency ratio in the SFCR documentation.

4.6. Decomposition of the Difference Between the Standard Formula and the Internal Model

The company under review has developed a partial internal model for assessing Solvency II capital requirements, which is currently in the internal validation and calibration phase. Although the model has not yet been officially approved by the supervisory authority, the company intends to initiate the regulatory approval process during the next reporting period. For the purposes of this study, historical results from the test version of the model were provided, allowing for a comparative analysis between the standard formula and the internal model.

A breakdown of the difference between the SCR calculated using the standard formula and the SCR estimated using the internal model reveals a clearly defined and exceptionally stable pattern throughout the entire ten-year period analyzed (Table 13). The results show that the standard formula systematically overestimates the capital requirement by approximately 20.5–21.3% relative to the internal model. More importantly, this difference remains virtually constant over time, with a standard deviation of less than 0.3 percentage points. This suggests that the observed gap is not the result of temporary stochastic fluctuations or cyclical factors, but stems from persistent methodological differences between the two approaches.

Table 13. Waterfall decomposition: transition from SCR based on the standard formula to SCR based on an internal model (thous. EUR).

Year	SCR standard formula	Scope	Data	Calibration	Diversification	SCR Internal Model	Difference
2016	12,473	−480	−311	−1,096	−726	9,860	−20.9%
2017	13,625	−565	−332	−1,208	−797	10,723	−21.3%
2018	15,155	−604	−388	−1,274	−906	11,983	−20.9%
2019	15,891	−626	−398	−1,320	−898	12,649	−20.4%
2020	15,063	−621	−388	−1,283	−894	11,877	−21.2%
2021	16,112	−625	−390	−1,420	−898	12,779	−20.7%
2022	17,011	−679	−444	−1,482	−981	13,425	−21.1%
2023	18,699	−725	−468	−1,609	−1,118	14,779	−21.0%
2,024	18,886	−729	−456	−1,645	−1,125	14,931	−20.9%
2,025	20,509	−819	−529	−1,696	−1,162	16,303	−20.5%

Table 13. Waterfall decomposition: transition from SCR based on the standard formula to SCR based on an internal model (thous. EUR).

Year	SCR standard formula	Scope	Data	Calibration	Diversification	SCR Internal Model	Difference
2016	12,473	−480	−311	−1,096	−726	9,860	−20.9%
2017	13,625	−565	−332	−1,208	−797	10,723	−21.3%
2018	15,155	−604	−388	−1,274	−906	11,983	−20.9%
2019	15,891	−626	−398	−1,320	−898	12,649	−20.4%
2020	15,063	−621	−388	−1,283	−894	11,877	−21.2%
2021	16,112	−625	−390	−1,420	−898	12,779	−20.7%
2022	17,011	−679	−444	−1,482	−981	13,425	−21.1%
2023	18,699	−725	−468	−1,609	−1,118	14,779	−21.0%
2,024	18,886	−729	−456	−1,645	−1,125	14,931	−20.9%
2,025	20,509	−819	−529	−1,696	−1,162	16,303	−20.5%

An analysis of the relative contribution of the individual components reveals a clear hierarchy of factors driving the difference between the two approaches.

The most significant methodological factor is risk calibration. On average over the period, this component reduces the SCR by approximately 1,403 thousand EUR, which corresponds to about 8.3% of the SCR calculated using the standard formula and approximately 41% of the total difference between the two methods. This effect stems primarily from replacing the regulator-specified shock parameters with empirically calibrated distributions, which, for the portfolio under consideration, exhibit lower observed volatility and skinnier tails compared to the conservative assumptions embedded in the standard formula.

The second most significant factor is the diversification effect, which reduces the SCR by an average of approximately 951,000 EUR (roughly 4.7% of the SCR calculated using the standard formula). This result reflects the use of more flexible relationships between risk modules via copula structures, which capture the empirical correlations in the portfolio more realistically. In contrast, the standard formula uses fixed regulatory correlation matrices, which are necessarily simplified and conservative.

The components related to the model’s scope and the data used have a significantly smaller impact. Scope alignment results in an average reduction of approximately 647,000 EUR, while data harmonization results in a reduction of approximately 410,000 EUR. The limited effect of these two steps indicates that the standard formula and the in-house model operate with virtually identical business scope and similar input data. Consequently, the main difference between the two approaches does not stem from a different information base, but from the way in which risk is modeled, calibrated, and aggregated.

When the actual business effects reinsurance, management actions, and policyholder behavior are added to this methodological decomposition, a complete economic model of risk is obtained. The results show that the overall deviation between the SCR calculated using the standard formula and the company’s economic SCR reaches approximately 30.7–31.5% during the period under review (Table 14).

Among the business effects, reinsurance makes the dominant contribution. In 2025, it reduces the capital requirement by approximately 1,367 thousand EUR, which represents about 69% of the total business effect. In second place are managerial actions (−340 thousand EUR; 17%), while the behavior of policyholders contributes an additional reduction of about 278 thousand EUR (14%).

These results have important conceptual significance. They show that the discrepancy between the standard formula and economic reality is not a single-factor effect, but rather a combination of: (i) methodological conservatism in the regulatory framework; (ii) limited sensitivity to actual diversification; (iii) the standard formula’s inability to account for managerial responses and adaptive business behavior; (iv) the actual effect of the reinsurance structure.

Consequently, while the standard formula fulfills its role as a unified prudential framework, for a specific company it systematically overestimates economic risk. This is precisely the main argument in favor of using an internal model: not capital minimization for its own sake, but a more precise and economically sound measurement of the company’s actual risk profile.

4.7. Structure of Counterparty Default Risk

A detailed analysis of the five reinsurers included in the program (Table 15) shows that, at the portfolio level, counterparty default risk is disproportionately concentrated among counterparties with average credit quality. Specifically, Munich Re and Swiss Re, rated AAby S&P Global Ratings, generate capital requirements under the counterparty default risk module (SCR) amounting to 22.2% and 21.7%, respectively, of the recoverable amounts, while PartnerRe, rated A by S&P Global Ratings, accounts for an SCR of 59.8%.

The results show that a one-notch downgrade in credit rating can increase the capital requirement per unit of recoverable exposure by approximately 2.5–3 times. This empirically confirms the theoretical framework set forth in Solvency II, according to which the creditworthiness of counterparties has a strongly nonlinear effect on the capital efficiency of the reinsurance program. Consequently, the potential benefits of diversification through the inclusion of lower-rated reinsurers may be partially or entirely offset by a significant increase in the regulatory capital requirement.

Furthermore, the portfolio structure shows that reinsurers with high credit ratings assume the largest relative share of the ceded exposure, while maintaining a significantly lower capital burden intensity. In contrast, counterparties with lower credit quality generate disproportionately high capital requirements despite their smaller share of the portfolio. This asymmetry underscores the strategic importance of optimizing the balance between cost-effectiveness, diversification, and capital relief when structuring reinsurance programs under a risk-based solvency framework.

4.8. Stress Tests and the Resilience of Capital Relief

The results of the stress scenarios (Table 16) show that the capital efficiency of reinsurance increases as extreme catastrophe scenarios intensify, but at the same time can deteriorate sharply in scenarios where insurance risk is accompanied by a significant risk of counterparty default. In the base case scenario, reinsurance leads to a 39.5% reduction in the Solvency Capital Requirement (SCR), while in a scenario with a 1-in-200-year frequency (VaR 99.5%), the reduction reaches 54.5%. This confirms that the capital protection effect of reinsurance is strongest precisely at the tails of the distribution, where extreme losses occur. This result is consistent with the structure of excess-loss coverage, in which the effect is concentrated in the most adverse loss scenarios.

The “wrong-way risk” scenario, which combines a catastrophic insurance event with the simultaneous failure of two of the five reinsurers, outlines the lower bound of the capital relief’s effectiveness. Under this scenario, the effect of reinsurance is significantly reduced (a 27.3% decrease in the SCR), while equity shrinks to 5,000 thousand EUR, and the solvency ratio falls to 12.5%, i.e., significantly below the minimum capital requirement (MCR). These results provide quantitative empirical support for EIOPA’s regulatory position regarding restrictions on the recognition of capital relief from reinsurance in the presence of concentration and counterparty risk. This justifies the need to apply concentration limits, strengthen monitoring of counterparties’ credit quality, and incorporate specific stress tests for wrong-way risk within the ORSA process.

The “pandemic and recession” scenario adds an additional dimension to the system’s vulnerability: although reinsurance reduces the SCR by 41.7%, the solvency ratio drops to 43%, a value close to the critical level. This is due to the combined impact of a market shock amounting to a 40% decline, which directly reduces the value of own funds, regardless of the effect of reinsurance on underwriting risk. This demonstrates that reinsurance is an effective tool for transferring underwriting and catastrophe risk, but cannot offset systemic market and liquidity shocks that affect the capital base regardless of the insurance structure

5. Discussion

5.1. Reinsurance and the SCR Architecture: An Integrated Interpretation

The results of this study confirm and expand upon the main premise in the literature on prudential regulation, namely that the solvency capital requirement is not solely a function of the volume of risk assumed, but rather the result of the interaction between the level of risk, the method of its measurement, the degree of diversification, and the risk transfer mechanisms (Doff 2008; Eling and Holzmüller 2008). Within the analyzed ten-year portfolio, reinsurance leads to a sustained reduction in the SCR in the range of 18.4–23.4%, with the effect increasing approximately in proportion to the cession quota. However, this aggregate result masks significant internal heterogeneity: different reinsurance structures affect different risk modules and different parts of the loss distribution, making the optimization of the contract mix a conceptually and empirically non-trivial problem.

The first key finding is that the capital efficiency of excess-of-loss (XoL) coverages stems directly from the mathematical definition of the SCR as a 99.5% quantile (VaR). Since the SCR is sensitive precisely to the tail of the distribution, contracts that transform extreme losses but leave the middle portion nearly unchanged achieve the maximum effect on the capital requirement with a minimal change in the expected value of losses. The empirical results (Table 4) confirm this mechanism: under median scenarios, the effect of XoL is limited (−6.7%), while at the 99.5% quantile the reduction reaches −52.5%, and at the 99.9% quantile, −68.6%. This indicates that XoL structures are a practically optimal tool for capital optimization under the Solvency II regime.

The second key conclusion is that quota-share reinsurance contracts, although less effective in terms of SCR, fulfill a different systemic function. They do not merely reduce the scale of risk but stabilize the overall dynamics of the results. The standard deviation of the net combined ratio (2.94 percentage points) is approximately half that of the gross ratio (6.25 percentage points), which is of significant importance for companies with limited capital or rapid growth. Consequently, quota share and XoL contracts are not interchangeable but rather complementary instruments within a multi-layered reinsurance architecture (Albrecher et al. 2017).

The third conclusion concerns stop-loss contracts. Although at first glance they appear similar to XoL structures, their aggregate nature results in a fundamentally different risk profile. Stop-loss is sensitive to the accumulation of multiple claims of moderate severity but does not respond to single catastrophic events. Empirically, it reduces premium risk by 42% but has no effect on catastrophe risk (0%). This makes it particularly suitable for lines of business with high frequency and low to medium loss severity (e.g., motor, health, and liability lines), whereas for low-frequency, high-severity risks (property and natural disasters), XoL remains the dominant tool.

5.2. Regulatory Implications: Why Is EIOPA’s Approach Structurally Conservative?

The results of the stress tests provide an empirical basis for EIOPA’s regulatory approach to limiting the excessive recognition of capital relief from reinsurance. In catastrophic scenarios (1-in-50 to 1-in-200 years), the effect of reinsurance increases from 45.3% to 54.5%, which is consistent with the expectation that protection should be most effective precisely at the tails of the distribution.

In a “wrong-way risk” scenario, however, involving a combination of a catastrophic event and the default of two key counterparties, the effect of reinsurance drops to 27.3%, and the solvency ratio falls to 12.5% significantly below the MCR. This clearly demonstrates the discrepancy between nominal and effective coverage under stress conditions.

The economic mechanism behind this discrepancy is clear: coverage that is valid under normal conditions may be reduced or disappear precisely when it is most needed due to: (i) the simultaneous insolvency of the reinsurer; (ii) contractual limitations or termination clauses; (iii) concentration of exposure; (iv) a positive correlation between the probability of loss and the probability of default (wrong-way risk) (Albrecher et al. 2017; Cummins and Trainar 2009).

From a practical standpoint, this leads to three regulatory conclusions. First, concentration limits must be structurally strict, as a 30% exposure to a single reinsurer already constitutes a systemic risk. Second, credit quality monitoring should be continuous, as rating dynamics during systemic crises are highly nonlinear. Third, the ORSA process must explicitly include wrong-way risk modeling, including the use of t-copulas with low degrees of freedom, since Gaussian copulas systematically underestimate dependencies in the tails.

5.3. The Difference Between “Model” and “Real Business”

The breakdown in Table 6 reveals a fundamental distinction within the Solvency II framework: the difference between the effects of more accurate risk measurement and the effects of actual risk management. The transition from the standard formula (20,509 thousand EUR) to an internal model (16,303 thousand EUR) results in a 20.5% reduction attributable to methodological precision. The additional transition to “actual business SCR” (EUR 13,836 thousand) adds a further 12.5% reduction resulting from management actions.

This second effect is distributed among three main sources: reinsurance (69%), management interventions (17%), and policyholder behavior (14%). This distinction is conceptually important from a regulatory perspective: the first reduction is subject to validation within the model, while the second represents a real transformation of risk and should be subjected to enhanced stress testing.

Empirically, the management effect turns out to be roughly comparable in scale to the methodological effect, which underscores that reinsurance has not only a financial but also a structurally defining effect on the insurer’s capital position.

5.4. Limitations of the Study

This study has three main limitations.

First, the data used are quasi-empirical, based on public aggregate sources (FSC, EIOPA, and Insurance Europe), which ensures replicability but limits the specificity of the results with respect to individual companies. Future studies using microdata from multiple insurers would allow for greater empirical identification.

Second, the model of dependencies and tails is simplified relative to the practices of large European insurers. More complex approaches including regime-switching models and dynamic copulas could improve the accuracy of estimating capital effects under extreme scenarios.

Third, the analysis focuses on property and casualty insurance and does not include life insurance, where the behavior of policyholders (lapsation, surrender options) plays a significant role. An extension to life insurance would require a different structural specification of the model and could lead to different quantitative results, although the underlying mechanisms would remain valid. A similar extension to life insurance would be a natural direction for future work.

6. Conclusion

Within the framework of the “Solvency II” prudential regime, reinsurance has established itself as a key instrument for managing the solvency capital requirement, with its effectiveness depending both on the structure of the contracts used and on the resilience of the risk transfer under extreme stress conditions. The empirical analysis, based on a ten-year insurance portfolio (2016–2025), shows that the aggregate reduction in the SCR due to reinsurance is stable within the range of 18.4–23.4%, with the effect increasing approximately linearly as the cession ratio rises.

A stratified analysis of the different types of contracts reveals significant structural heterogeneity in the way they affect the capital position. Quota-share contracts scale the entire loss distribution proportionally and contribute to the stabilization of results; excess contracts limit exposure to large individual losses; excess-of-loss (XoL) contracts reduce the right tail of the distribution and achieve the highest capital efficiency per unit of transferred risk (−35.5% compared to −30.0% for quota-shares and −20.0% for excess contracts). Stop-loss contracts, for their part, stabilize the aggregate annual result but have limited effectiveness against catastrophic events. The combined multi-layered structure (quota + excess + Cat XoL + stop-loss) provides the maximum observed relief of −48.8%, confirming that the various instruments are complementary rather than interchangeable.

A breakdown of the difference between the standard formula and the internal model shows that risk calibration (−8.3%) and the effects of diversification (−4.7%) are the main methodological factors driving the reduction in the SCR, while the additional 12.5% reduction stems from actual management mechanisms, including reinsurance, managerial actions, and policyholder behavior. This distinction empirically confirms the fundamental conceptual difference within the “Solvency II” framework between risk measurement and actual risk management, as well as the need for regulatory authorities to assess the sustainability of capital reliefs, rather than merely their formal validity.

The results of the stress tests highlight the limits of reinsurance effectiveness. In pure catastrophe scenarios (up to a 1-in-200-year event), the capital relief reaches 54.5%, confirming that reinsurance protection is most pronounced precisely at the tails of the distribution. In a “wrong-way risk” scenario, however, the effect is reduced to 27.3%, and the solvency ratio drops to 12.5%, i.e., significantly below the minimum capital requirement (MCR). This result provides a quantitative basis for EIOPA’s regulatory approach regarding the need for concentration limits, enhanced monitoring of credit quality, and explicit modeling of wrong-way risk within the ORSA process.

The practical implications of the analysis can be summarized in four main areas. First, the optimal reinsurance structure should combine proportional and non-proportional contracts in order to simultaneously stabilize results and provide protection against extreme events. Second, the choice between a standard formula and an internal model must take into account not only methodological precision but also the organizational capacity for continuous validation and regulatory maintenance of the model. Third, concentration in a limited number of reinsurers should be structurally limited, regardless of their credit quality, due to the amplified effect of wrong-way risk in systemic crises. Fourth, the ORSA process must function as an integrated strategic tool that unites capital planning, risk management, and business decisions within a single framework.

The regulatory conclusion is clear: under “Solvency II,” capital does not merely reflect the level of risk assumed, but rather the insurer’s combined ability to measure, diversify, and effectively transfer that risk. Reinsurance remains a central element in this architecture, but its capital recognition depends critically on the sustainability, legal enforceability, and behavior of the transfer under extreme stress.