Introduction

Since the financial crisis that began in 2007–2008, the European Union (EU) has undertaken an ambitious scheme for supervisory and regulatory reform. That crisis spurred a severe worldwide economic crisis that marked the next decade and the first quarter of the twenty-first century. Until the coronavirus (COVID) pandemic broke out in 2020, it was probably the most serious financial crisis since the Great Depression.

The 2008 bankruptcy of Lehman Brothers sparked an international banking crisis. The European debt crisis began with a deficit in Greece in late 2009. The 2008–2011 Icelandic financial crisis involved the failure of all three of that country’s major banks. Relative to the size of its economy, the Icelandic crisis was the largest economic collapse suffered by any country in history.

Regulatory authorities reacted. The United States (U.S.) enacted the Dodd-Frank Wall Street Reform and Consumer Protection Act of 2010 in order to "promote the financial stability of the United States" (U.S. Library of Congress, 2010, PL 111–203). In 2009, countries around the world adopted the Basel III capital and liquidity standards. Because regulation is a complex interaction between politicians, civil servants, industry, interested groups, regulatory bodies, and consumers, its true impact demands close scrutiny.

The European Insurance and Occupational Pensions Authority (EIOPA, 2009a, b) introduced new solvency capital requirements as a system of governance and a mechanism for cooperation and coordination between supervisory authorities. Called Solvency II, this scheme took effect on January 1st, 2016 and applies to all European insurance and reinsurance companies (EIOPA, 2009a, b, 2015a, b, 2016, 2018). It attempts to create a level playing field for the European insurance sector. In addition to prescribing rules for the governance of insurance companies, Solvency II emphasizes the capital required to cover the assumed risks and safeguard the solvency of insurers.

Solvency II seeks to reduce the risk of failure of an insurance company to cover the claims of the insured and protect policyholders from losses due to such events. In addition, Solvency II sets rules for more detailed, public information included in the Solvency and Financial Condition Report (SFCR) and not just in the Report to Supervisors. Increased transparency should boost confidence in all types of insurance: life insurance, non-life insurance, and reinsurance. As risk taking is the primary component of the business of insurance, an insurer's risk management process is laid out in its own risk and solvency assessment (ORSA) (EIOPA, 2015a, b). ORSA includes a risk-based assessment of the insurer’s solvency needs based on its business profile and own risk appetite. It must be considered in running the business.

This study advances the existing literature in two ways. First, it identifies the most important variables affecting the solvency capital requirement (SCR) ratio, which is vital for the viability of European insurance companies. Second, this study sets a benchmark for monitoring and forecasting the effectiveness of the risk management process that insurers implement. Such forecasting ability is of the utmost importance for the insurance sector, since solvency enables insurers to deliver benefits promised to policyholders and fulfill their social obligations.

Literature Review

Financial services, banking and insurance have benefited from real-world applications of machine learning. Examples include customer/market segmentation, portfolio optimization, tracking and prevention of money laundering and other illegal financial activities, implementation of smarter and more effective risk management and regulatory compliance in finance and accounting. These capabilities enable organizations to achieve and maintain a long-term competitive advantage (Paltrinieri et al., 2019; Sen & Mehtab, 2021; Lei et al., 2020; Dornadula & Geetha, 2019; Eling et al., 2022; Yu et al., 2021; Leo et al., 2019; Gu et al., 2020; Ye & Zhang, 2019; Zand et al., 2020). The existing literature addresses a wide range of machine learning applications in insurance. These include the prediction of insolvency, fraud detection (in property and casualty insurance), claims (in export credit insurance), customer-risk level, and losses (in property and casualty insurance), claims analysis (in health and travel insurance), lapse-risk management, portfolio insurance strategies, and motor insurance analysis (Table 1).

Table 1 Literature review on the applications of ML in the insurance industry
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This study addresses a gap in the literature, the identification of the most important factors affecting SCR ratios. This paper studies the internal (firm-related) factors that allowed insurance companies to maintain SCR ratios that ensure solvency. These factors relate to premiums generated, insurers’ reserves, effectiveness in reinvesting in profitable assets, cash or cash equivalents held, long-term investments, losses and expenses (e.g., management, administrative), size, and income generated by each insurer’s total activity.

Data and Variables

The dataset consists of 29 insurance groups that operate across the EU, with a country of origin within the EU, from 2016 to 2020. The proxy employed for solvency is the SCR ratio, which is the sum of eligible own funds divided by the SCR, calculated on a consolidated basis. The SCR is the amount of assets that insurance and reinsurance companies are required to hold in order to attain 99.5% confidence that they will be able to meet the claims of policyholders under extreme expected losses. The SCR accounts for life insurance, health insurance, market, credit, operational and counterparty risk and must be recalculated at least once a year.

Eligible own funds are the component of actual own funds that qualify for coverage of the SCR and the minimum capital requirement (MCR), the minimum safety net of capital adequacy over one year. Eligibility is decided by the regulator, includes restrictions on the amount of each tier of capital an insurer can use to cover its SCR and MCR, and must be over 100% (EIOPA, 2020a, b).

The variables were constructed from annual financial statements retrieved from (Thomson Reuters) DataStream (2012). SCR ratios were obtained manually from the SFCR of each group. The variables are defined in Table 2.

Table 2 Brief definition and role of the model variables
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Methods

Traditional econometric approaches typically specify a model to be fitted. The model is usually based on economic theory and specifies a fixed functional form that includes a dependent variable and one or more independent variables. The ordinary least squares (OLS) procedure is the most common method in general and seeks to minimize the sum of the squared residuals. Given a regression line through the data, the sum of the squared residuals is estimated as the sum of the squares of the distances of the data points from the regression line. In contrast, machine-learning (ML) approaches capture data patterns and apply them to a wide range of problems. ML techniques are efficient and accurate in prediction and classification (Berry & Linoff, 2004; Kudyba, 2014; Sarker, 2021; Thompson, 2014). ML is primarily concerned with prediction: producing the best predictions of y given available data X. Informally, “machine learning belongs in the part of the toolbox marked \(\widehat{y}\) rather than in the more familiar \(\widehat{\beta }\) component” (Mullainathan & Spiess, 2017, p. 88). ML methods attempt to find generalizable patterns in the available data and exploit those patterns to make accurate predictions.

Since the objective of ML is to make accurate predictions, ML methods must be evaluated differently than econometric methods. The latter are commonly evaluated using metrics that are calculated using in-sample tests (e.g. R2, p-values) and out-of-sample tests (e.g., bias, accuracy). As in econometrics, ML methods typically partition the data into training and testing data (in-sample and out-of-sample, respectively). Holdout testing data are used to evaluate the model that has been fitted using the training data. ML methods typically employ cross-validation to train a model. This analytical framework follows the approach introduced by Chen (2021).

Findings

Summary statistics-Correlation analysis

Summary statistics of the key variables (Table 3) report negative values in the annual change of insurance premiums and reserves, as well as the reinvestment ratio and net investment income. Furthermore, some companies exhibit zero long-term debt. There is large variation in the exposure to long-term investments from 14.34% to 90.62% of assets. Expenses also exhibit great variation. Losses, benefits and adjustments expenses ranged from 22.46% to 131.26%. Selling, general and administrative expenses ranged from 0.04% to 38.13% of total revenue. In terms of solvency, one company in 2020 fell below the 100% security threshold and posted a SCR ratio of 66%.

Table 3 Summary statistics of the model variables for 2016–2020
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The SCR ratio has a positive correlation with the annual change in insurance premiums and reserves, the reinvestment ratio, net investment income, long-term debt and expenses in losses, benefits, and adjustments. In contrast, it has a negative correlation with cash and equivalents, premium and selling, and general and administrative expenses (Table 4).

Table 4 Correlation matrix of the model explanatory variables for 2016–2020
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Results

The estimation results of all regression and supervised ML models showed that our model is weakly predictive, Pooled OLS, random forest, extra trees, eXtreme gradient boosting (XGBoost), gradient boosting, AdaBoost, support vector regression (SVR) and multi-layer perceptron (MLP), struggled to find just the right combination of independent variables to make good predictions. Traditional linear regression did not exceed 0.30 in R2 (Table 3). The metrics used to analyze regression models are R2 and the root mean squared error (RMSE) (Table 5).

Table 5 Outcome of the eight regression models for the period 2016–2020
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Implementation of Regularized linear models-LASSO

A modification of linear regression is the least absolute shrinkage and selection operator (LASSO). The loss function in LASSO is changed to reduce the model's complexity by limiting the sum of the absolute values of the model coefficients:

$$Loss function=OLS+a*summation$$
(1)

where the summation is the absolute value of the magnitude coefficients.

The default value of the regularization parameter in LASSO regression (given by α) is 1, where \(a\) is the parameter that balances the amount of emphasis given to minimizing the residual sum of squares (RSS) versus minimizing the sum of squares of coefficients. At α = 0, LASSO is equivalent to OLS. RSS is the sum of the squared errors between the predicted and actual values in the training data set. The larger the value of \(\alpha\), the more aggressive the penalization. The LASSO hyperparameter \(a\) reached its optimal value at 0.1453 (Fig. 1).

Fig. 1
figure 1

LASSO hyperparameter α reaches its optimal value at 0.1452999999999999998. Source: Authors’ estimates using Python and annual financial statements from the Thomson Reuters DataStream (2012) database provided by Refinitiv for 2016–2020. The SCR ratios were obtained manually from the SFCR of each insurance firm

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LASSO has the effect of reducing coefficients to zero if they do not contribute significant predictive value. The sparsity induced by LASSO indicates significance, akin to the role of p-values informal statistics.

LASSO model selection using an information criterion: AIC or BIC

The Akaike information criterion (AIC) or the Bayes information criterion (BIC) was used to select the optimal value of the regularization parameter \(a\). Before fitting the model, the data were standardized. The AIC and BIC values can be plotted for different values of \(a\). The vertical lines in the plot correspond to the \(a\) chosen for each criterion. The selected \(a\)(Fig. 2) corresponds to the minimum of the AIC and BIC criterion (Pedregosa et al., 2011).

Fig. 2
figure 2

Selecting the LASSO hyperparameter α via AIC and BIC. Source: Authors’ estimates using Python and annual financial statements from the Thomson Reuters DataStream (2012) database provided by Refinitiv for 2016–2020. The SCR ratios were obtained manually from the SFCR of each insurance firm. Due to tiny values, they are plotted on a negative base-10 logarithmic scale

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As α increases toward its optimized value of 0.1453, LASSO turns more coefficients into zero. The reinvestment rate, cash and equivalents, long-term investment, losses-benefits-and-adjustments expenses were selected from the LASSO regression. LASSO results are best understood through a comparison with the results of conventional OLS regression, which indicates that only the losses-benefits-and-adjustments expenses variable is statistically significant (Table 6). The coefficients of all four variables post the same sign in both regressions though. The other six explanatory variables were assigned zero coefficients at a relatively aggressive value of the LASSO complexity parameter α. In effect, LASSO regression reduced the dimensionality of the model from 10 to 4.

Table 6 LASSO regression results compared with OLS results for the period 2016–2020
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LASSO regression based on standardized data allows the resulting beta coefficients to be read directly. Although Table 6 reports a corresponding vector of p-values, the sparsity induced by this regression method is unequivocally clear and decisive. The zero-coefficient trick replaces or complements the more conventional removal of variables with a high (non-significant) p-value.

In this study, the absolute value of the long-term investment coefficient exceeds the sum of the absolute value of the other three non-zero coefficients. Therefore, that variable commanded an overwhelming share of coefficient importance (defined as the absolute value of each non-zero coefficient divided by the sum of the absolute value of all non-zero coefficients). Therefore, LASSO reports the subset of predictive variables within the ℓ0 quasi-norm of variables with non-zero coefficients.

Finally, the negative sign attached to the coefficient for cash and equivalents should be highlighted. This is the only negative variable in the new ℓ0 vector of coefficients. Cash is not a risk-free asset, especially with respect to the solvency of financial institutions. Cash earns so little return that it undermines preparedness for future crises (Danielsson et al., 2016).

Discussion

The LASSO regression showed that the reinvestment rate, cash and equivalents, long term investment, and losses-benefits-and-adjustments expenses can predict the solvency of insurance companies during the period under investigation. Insurance companies operated under a low-interest rate environment and continue to earn less investment income. Annual investment returns are reinvested to generate additional future returns.

However, reinvestment at lower yields has a measurable impact on an insurer’s future financial health. Older, higher-yielding, maturing securities and cash are reinvested at current (lower) market rates, leading to reduced investment income. As a result, insurance companies must either hold more assets in the future to earn the same investment income, or else hold riskier assets to achieve better returns. The reinvestment rate can be considered as a tool for risk management, which discourages insurers from investing in risky portfolios and endangering their solvency ratio. Risk-averse insurers want to avoid losses from risky investments, even though they may benefit in the short-term.

Insurance companies are long-term investors. They invest premiums paid by policyholders. Due to the long-term nature of many products (such as annuities and life insurance policies), insurers invest in long-term assets to match their long-term liabilities. However, under Solvency II, assets and liabilities are valued mark-to-market. Consequently, short-term market movements pose a risk that must be managed. Mark-to-market valuation ensures that the SCR ratios reflect an insurer’s true economic position. Therefore, mark-to-market valuation is an instrument for risk management and policyholder protection, even though it does not fully capture the investments’ long-term horizon.

The solvency capital requirements motivate insurers to match the duration of their assets and liabilities. The better the duration match, the lower the solvency capital requirement is. SCR ratios increase insurers' appetite for long-term assets. Insurers are free to make prudent investments, and capital requirements will depend on the actual risk associated with those investments.

Cash is not a risk-free asset. There are differences between the SCR needed to cover cash deposits at a bank and other cash equivalents. Solvency II assumes that the loss (given default) for cash at a bank is 100%. The EIOPA estimates that approximately €190 billion in cash and cash equivalents were held on the balance sheets of European insurers at the end of the second quarter of 2020. A Euribor of -0.6% implies that roughly €1 billion of this cash will be lost through negative yields over the next 12 months (EIOPA, 2020a, b).

Losses-benefits-and-adjustments expenses reflect the cost of investigating and settling insurance claims, relative to an insurance company’s gross revenue. Investigations are necessary to prevent fraud and reduce exaggerated claims; in essence, to verify the amount of the loss. The business of insurance requires fair and prompt payment of valid claims. When an insurance company refuses claims without adequate investigation and fails to pay promptly and fairly when liability is clear, many insureds may sue to recover underpayments. If an insurance company loses many underpayment lawsuits, such defeats indicate that the insurance company is routinely underpaying claims. Therefore, losses-benefits-and-adjustments expenses can provide early warning of systematic under payment relative to gross revenue.

Conclusion

Five years after the implementation of the Solvency II Directive, this study makes two primary contributions. First, it identifies the most important factors in predicting SCR ratios and evaluates the impact of these factors on solvency. The reinvestment rate, cash and equivalents, and long-term investments (as part of total assets) and losses-benefits-and-adjustments expenses (as part of total revenue) can be used as benchmarks for monitoring and forecasting SCR ratios. Second, this study attains these results through computational extensions of OLS regression. It makes particularly illuminating use of LASSO.