On some effects of dependencies on an insurer’s risk exposure, probability of ruin, and optimal premium loading
Abstract: We study how the presence of dependencies between risks in a population of prospective insurance customers translates into risk exposure for an insurance company, depending on the company’s market share on the various risks. It turns out that the dependency structure in the insurer’s portfolio may differ significantly from the dependency structure of those risks in the general population, even when policyholders for different risks are selected independently. We obtain an upper bound for the difference between… Show more
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Presentation of EAJ Issue 13/1 - May 22nd
The seminar is chaired by Tim Boonen and Julien Trufin.Optimal portfolios with sustainable assets: aspects for life insurersSince August 2022, customers have to be asked if they are interested in sustainable investment when entering a pension contract. Hence, the provider has to be prepared to offer suitable investment opportunities. Further, the provider has to manage the new risks and chances of those assets in the whole portfolio. We therefore especially look at possible consequences for optimal portfolio decisions of a life insurer and suggest modeling approaches for the evolution of the demand and the sustainability ratings for sustainable assets. We will solve various portfolio problems under sustainability constraints explicitly and suggest further research topics. As a special feature for a life insurer, we particularly look at the role of the actuarial reserve fund and the annual declaration of its return.Micro-level prediction of outstanding claim counts based on novel mixture models and neural networksPredicting the number of outstanding claims (IBNR) is a central problem in actuarial loss reserving. Classical approaches like the Chain Ladder method rely on aggregating the available data in form of loss triangles, thereby wasting potentially useful additional claims information. A new approach based on a micro-level model for reporting delays involving neural networks is proposed. It is shown by extensive simulation experiments and an application to a large-scale real data set involving motor legal insurance claims that the new approach provides more accurate predictions in case of non-homogeneous portfolios.An incremental loss ratio method using prior information on calendar year effectsIn a run-off triangle external factors can have a similar influence on all incremental losses of the same calendar year. This can distort the triangle such that reserving methods like chain ladder or the loss ratio method do not work properly. A very recent example of such an external factor is the Covid-19 pandemic. In many countries, the insurance industry is in the process of establishing market knowledge about the impact of the pandemic on premiums and losses. We extend the additive claims reserving model to allow for calendar year effects and develop a variant of the incremental loss ratio method (also known as additive method) that can make use of such market knowledge. We derive formulas for the mean squared error of prediction and provide a detailed numerical example.Generalized PELVE and applications to risk measuresThe continuing evolution of insurance and banking regulation has raised interest in the calibration of different risk measures associated with suitable confidence levels. In particular, Li and Wang (2019) have introduced a probability equivalent level (called PELVE) for the replacement of Value at Risk (VaR) with Conditional Value at Risk (CVaR). Extending their work, we propose two generalizations of PELVE that combine useful theoretical properties with empirical benefits in risk analysis. The former, termed d-PELVE, establishes a correspondence between VaR and suitably parameterized distortion risk measures. The latter, termed g-PELVE, iterates the construction of CVaR starting from VaR to a general coherent risk measure. We state conditions for the existence and uniqueness of the proposed measures and derive additional properties for specific classes of underlying risk functionals. A study of Generalized Pareto Distributions reveals an interesting correspondence between PELVE and g-PELVE, and explores their relationship with the tail index. An empirical application illustrates the usefulness of (g-)PELVE in characterizing tail behavior not only for individual asset returns, but also for possible portfolio combinations.On some effects of dependencies on an insurer’s risk exposure, probability of ruin, and optimal premium loadingWe study how the presence of dependencies between risks in a population of prospective insurance customers translates into risk exposure for an insurance company, depending on the company’s market share on the various risks. It turns out that the dependency structure in the insurer’s portfolio may differ significantly from the dependency structure of those risks in the general population, even when policyholders for different risks are selected independently. We obtain an upper bound for the difference between the ruin probability and its estimate based on the company’s portfolio marginal distributions. Under certain conditions, dependencies between risks in the portfolio of a company with small market shares are mild. We characterize the optimal loadings and market shares, assuming generic demand functions for the different risks.Long-Term Stability of a Life Insurer's Balance SheetIn this paper, we devise a stochastic asset-liability management (ALM) model for a life insurance company and analyze its influence on the balance sheet within a low-interest rate environment. In particular, a flexible procedure for the generation of insurers' compressed contract portfolios that respects the given biometric structure is presented, extending the existing literature on stochastic ALM modeling. The introduced balance sheet model is in line with the principles of double-entry bookkeeping as required in accounting. We further focus on the incorporation of new business, i.e. the addition of newly concluded contracts and thus of insured in each period. Efficient simulations are retained by integrating new policies into existing cohorts according to contract-related criteria. We provide new results on the consistency of the balance sheet equations. In extensive simulation studies for different scenarios regarding the business form of today's life insurers, we utilize these to analyze the long-term behavior and the stability of the components of the balance sheet for different asset-liability approaches. Finally, we investigate the robustness of two prominent investment strategies against crashes in the capital markets, which lead to extreme liquidity shocks and thus threaten the insurer's financial health.Smooth projection of mortality improvement rates: A Bayesian two-dimensional spline approachThis paper proposes a spline mortality model for generating smooth projections of mortality improvement rates. In particular, we follow the two-dimensional cubic B-spline approach developed by Currie et al. (2004), and adopt the Bayesian estimation and LASSO penalty to overcome the limitations of spline models in forecasting mortality rates. The resulting Bayesian spline model not only provides measures of stochastic and parameter uncertainties, but also allows external opinions on future mortality to be consistently incorporated. The mortality improvement rates projected by the proposed model are smoothly transitioned from the historical values with short-term trends shown in recent observations to the long-term terminal rates suggested by external opinions. Our technical work is complemented by numerical illustrations that use real mortality data and external rates to showcase the features of the proposed model.Neural networks meet least squares Monte Carlo at internal model dataIn August 2020 we published “Comprehensive Internal Model Data for Three Portfolios” as an outcome of our work for the committee “Actuarial Data Science” of the German Actuarial Association. The data sets include realistic cash-flow models outputs used for proxy modelling of life and health insurers. Using these data, we implement the hitherto most promising model in proxy modeling consisting of ensembles of feed-forward neural networks and compare the results with the least squares Monte Carlo (LSMC) polynomial regression. To date, the latter represents—to our best knowledge—the most accurate proxy function productively in use by insurance companies. An additional goal of this publication is a more precise description of “Comprehensive Internal Model Data for Three Portfolios” for other researchers, practitioners and regulators interested in developing solvency capital requirement (SCR) proxy models.A public micro pension programme in Brazil: heterogeneity among states and setting up of a benefit age adjustment Brazil is the 5th largest country in the world, despite having a “High Human Development”, it is the 9th most unequal country. The existing Brazilian micro pension programme is one of the safety nets for poor people. To become eligible for this benefit, each individual must have an income that is less than a quarter of the Brazilian minimum wage and be either over 65 or considered disabled. That minimum income corresponds to approximately US 2 per day. This manuscript analyses quantitatively some aspects of this programme in the Public Pension System of Brazil. We look for the impact of some particular economic variables on the number of people receiving the benefit, and seek if that impact significantly differs among the 27 Brazilian Federal Units (UF). We search for heterogeneity. We perform a regression and spatial cluster analysis for detection of geographical grouping. We use a database that includes the entire population receiving the benefit. Afterwards, we calculate the amount that the system spends with the beneficiaries, estimate values per capita and the weight of each UF, searching for heterogeneity reflected on the amount spent per capita. In this latter calculation we use a more comprehensive database, by individual, that includes all people that started receiving a benefit under the programme between January and April 2018. We compute the expected discounted benefit and confirm a high heterogeneity among UF’s as well as by gender. We propose looking for a more equitable system by introducing “age adjusting factors” to change the benefit age.Does autocalibration improve goodness of lift?Autocalibration is a desirable property since it ensures that the information contained in a candidate premium is used without any bias. It turns out to be intimately related to the method of marginal totals that predates modern risk classification methods. The present note aims to assess the impact of autocalibration on the goodness of lift. It is shown on a case study that autocalibration does not only restore global and local balances but also improve lift.Model selection with Gini indices under auto-calibrationThe Gini index does not give a strictly consistent scoring function. Therefore, simply maximizing the Gini index may lead to a wrong model choice. The main issue is that the Gini index is a rank-based score that is not calibration-sensitive. We show that the Gini index allows for strictly consistent scoring if we restrict it to the class of auto-calibrated regression models. That is, on the class of auto-calibrated models we know that the true model maximizes the Gini index.
Presentation of EAJ Issue 13/1 - May 22nd
The seminar is chaired by Tim Boonen and Julien Trufin.Optimal portfolios with sustainable assets: aspects for life insurersSince August 2022, customers have to be asked if they are interested in sustainable investment when entering a pension contract. Hence, the provider has to be prepared to offer suitable investment opportunities. Further, the provider has to manage the new risks and chances of those assets in the whole portfolio. We therefore especially look at possible consequences for optimal portfolio decisions of a life insurer and suggest modeling approaches for the evolution of the demand and the sustainability ratings for sustainable assets. We will solve various portfolio problems under sustainability constraints explicitly and suggest further research topics. As a special feature for a life insurer, we particularly look at the role of the actuarial reserve fund and the annual declaration of its return.Micro-level prediction of outstanding claim counts based on novel mixture models and neural networksPredicting the number of outstanding claims (IBNR) is a central problem in actuarial loss reserving. Classical approaches like the Chain Ladder method rely on aggregating the available data in form of loss triangles, thereby wasting potentially useful additional claims information. A new approach based on a micro-level model for reporting delays involving neural networks is proposed. It is shown by extensive simulation experiments and an application to a large-scale real data set involving motor legal insurance claims that the new approach provides more accurate predictions in case of non-homogeneous portfolios.An incremental loss ratio method using prior information on calendar year effectsIn a run-off triangle external factors can have a similar influence on all incremental losses of the same calendar year. This can distort the triangle such that reserving methods like chain ladder or the loss ratio method do not work properly. A very recent example of such an external factor is the Covid-19 pandemic. In many countries, the insurance industry is in the process of establishing market knowledge about the impact of the pandemic on premiums and losses. We extend the additive claims reserving model to allow for calendar year effects and develop a variant of the incremental loss ratio method (also known as additive method) that can make use of such market knowledge. We derive formulas for the mean squared error of prediction and provide a detailed numerical example.Generalized PELVE and applications to risk measuresThe continuing evolution of insurance and banking regulation has raised interest in the calibration of different risk measures associated with suitable confidence levels. In particular, Li and Wang (2019) have introduced a probability equivalent level (called PELVE) for the replacement of Value at Risk (VaR) with Conditional Value at Risk (CVaR). Extending their work, we propose two generalizations of PELVE that combine useful theoretical properties with empirical benefits in risk analysis. The former, termed d-PELVE, establishes a correspondence between VaR and suitably parameterized distortion risk measures. The latter, termed g-PELVE, iterates the construction of CVaR starting from VaR to a general coherent risk measure. We state conditions for the existence and uniqueness of the proposed measures and derive additional properties for specific classes of underlying risk functionals. A study of Generalized Pareto Distributions reveals an interesting correspondence between PELVE and g-PELVE, and explores their relationship with the tail index. An empirical application illustrates the usefulness of (g-)PELVE in characterizing tail behavior not only for individual asset returns, but also for possible portfolio combinations.On some effects of dependencies on an insurer’s risk exposure, probability of ruin, and optimal premium loadingWe study how the presence of dependencies between risks in a population of prospective insurance customers translates into risk exposure for an insurance company, depending on the company’s market share on the various risks. It turns out that the dependency structure in the insurer’s portfolio may differ significantly from the dependency structure of those risks in the general population, even when policyholders for different risks are selected independently. We obtain an upper bound for the difference between the ruin probability and its estimate based on the company’s portfolio marginal distributions. Under certain conditions, dependencies between risks in the portfolio of a company with small market shares are mild. We characterize the optimal loadings and market shares, assuming generic demand functions for the different risks.Long-Term Stability of a Life Insurer's Balance SheetIn this paper, we devise a stochastic asset-liability management (ALM) model for a life insurance company and analyze its influence on the balance sheet within a low-interest rate environment. In particular, a flexible procedure for the generation of insurers' compressed contract portfolios that respects the given biometric structure is presented, extending the existing literature on stochastic ALM modeling. The introduced balance sheet model is in line with the principles of double-entry bookkeeping as required in accounting. We further focus on the incorporation of new business, i.e. the addition of newly concluded contracts and thus of insured in each period. Efficient simulations are retained by integrating new policies into existing cohorts according to contract-related criteria. We provide new results on the consistency of the balance sheet equations. In extensive simulation studies for different scenarios regarding the business form of today's life insurers, we utilize these to analyze the long-term behavior and the stability of the components of the balance sheet for different asset-liability approaches. Finally, we investigate the robustness of two prominent investment strategies against crashes in the capital markets, which lead to extreme liquidity shocks and thus threaten the insurer's financial health.Smooth projection of mortality improvement rates: A Bayesian two-dimensional spline approachThis paper proposes a spline mortality model for generating smooth projections of mortality improvement rates. In particular, we follow the two-dimensional cubic B-spline approach developed by Currie et al. (2004), and adopt the Bayesian estimation and LASSO penalty to overcome the limitations of spline models in forecasting mortality rates. The resulting Bayesian spline model not only provides measures of stochastic and parameter uncertainties, but also allows external opinions on future mortality to be consistently incorporated. The mortality improvement rates projected by the proposed model are smoothly transitioned from the historical values with short-term trends shown in recent observations to the long-term terminal rates suggested by external opinions. Our technical work is complemented by numerical illustrations that use real mortality data and external rates to showcase the features of the proposed model.Neural networks meet least squares Monte Carlo at internal model dataIn August 2020 we published “Comprehensive Internal Model Data for Three Portfolios” as an outcome of our work for the committee “Actuarial Data Science” of the German Actuarial Association. The data sets include realistic cash-flow models outputs used for proxy modelling of life and health insurers. Using these data, we implement the hitherto most promising model in proxy modeling consisting of ensembles of feed-forward neural networks and compare the results with the least squares Monte Carlo (LSMC) polynomial regression. To date, the latter represents—to our best knowledge—the most accurate proxy function productively in use by insurance companies. An additional goal of this publication is a more precise description of “Comprehensive Internal Model Data for Three Portfolios” for other researchers, practitioners and regulators interested in developing solvency capital requirement (SCR) proxy models.A public micro pension programme in Brazil: heterogeneity among states and setting up of a benefit age adjustment Brazil is the 5th largest country in the world, despite having a “High Human Development”, it is the 9th most unequal country. The existing Brazilian micro pension programme is one of the safety nets for poor people. To become eligible for this benefit, each individual must have an income that is less than a quarter of the Brazilian minimum wage and be either over 65 or considered disabled. That minimum income corresponds to approximately US 2 per day. This manuscript analyses quantitatively some aspects of this programme in the Public Pension System of Brazil. We look for the impact of some particular economic variables on the number of people receiving the benefit, and seek if that impact significantly differs among the 27 Brazilian Federal Units (UF). We search for heterogeneity. We perform a regression and spatial cluster analysis for detection of geographical grouping. We use a database that includes the entire population receiving the benefit. Afterwards, we calculate the amount that the system spends with the beneficiaries, estimate values per capita and the weight of each UF, searching for heterogeneity reflected on the amount spent per capita. In this latter calculation we use a more comprehensive database, by individual, that includes all people that started receiving a benefit under the programme between January and April 2018. We compute the expected discounted benefit and confirm a high heterogeneity among UF’s as well as by gender. We propose looking for a more equitable system by introducing “age adjusting factors” to change the benefit age.Does autocalibration improve goodness of lift?Autocalibration is a desirable property since it ensures that the information contained in a candidate premium is used without any bias. It turns out to be intimately related to the method of marginal totals that predates modern risk classification methods. The present note aims to assess the impact of autocalibration on the goodness of lift. It is shown on a case study that autocalibration does not only restore global and local balances but also improve lift.Model selection with Gini indices under auto-calibrationThe Gini index does not give a strictly consistent scoring function. Therefore, simply maximizing the Gini index may lead to a wrong model choice. The main issue is that the Gini index is a rank-based score that is not calibration-sensitive. We show that the Gini index allows for strictly consistent scoring if we restrict it to the class of auto-calibrated regression models. That is, on the class of auto-calibrated models we know that the true model maximizes the Gini index.
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