Pricing Life Insurance Research Articles

COVID-19 and Excess Mortality: An Actuarial Study

The study of mortality is an ever-active field of research, and new methods or combinations of methods are constantly being developed. In the actuarial domain, the study of phenomena disrupting mortality and leading to excess mortality, as in the case of COVID-19, is of great interest. Therefore, it is relevant to investigate the extent to which an epidemiological model can be integrated into an actuarial approach in the context of mortality. The aim of this project is to establish a method for the study of excess mortality due to an epidemic and to quantify these effects in the context of the insurance world to anticipate certain possible financial instabilities. We consider a case study caused by SARS-CoV-2 in Belgium during the year 2020. We propose an approach that develops an epidemiological model simulating excess mortality, and we incorporate this model into a classical approach to pricing life insurance products.

Time-series forecasting of mortality rates using transformer

Predicting mortality rates is a crucial issue in life insurance pricing and demographic statistics. Traditional approaches, such as the Lee-Carter model and its variants, predict the trends of mortality rates using factor models, which explain the variations of mortality rates from the perspective of ages, gender, regions, and other factors. Recently, deep learning techniques have achieved great success in various tasks and shown strong potential for time-series forecasting. In this paper, we propose a modified Transformer architecture for predicting mortality rates in major countries around the world. Through the multi-head attention mechanism and positional encoding, the proposed Transformer model extracts key features effectively and thus achieves better performance in time-series forecasting. By using empirical data from the Human Mortality Database, we demonstrate that our Transformer model has higher prediction accuracy of mortality rates than the Lee-Carter model and other classic neural networks. Our model provides a powerful forecasting tool for insurance companies and policy makers.

Surrender contagion in life insurance

Modern life insurance policies embed a surrender option that entitles policyholders to terminate their policy early for its cash value. Mass surrenders–spiking surrenders of life insurance policies within a short period–are widely documented prior to many life insurer failures, but their impacts on insurers are rarely quantified. This paper builds a model of mass surrenders in a heterogeneous pool consisting of a minority of financially guided policyholders who make rational surrender decisions and a majority of individual policyholders who are subject to herd. By revising the self-excitation in the Hawkes process to a contingent event-accumulation-excitation, we introduce a history-dependent surrender process for the individuals whose contagious behavior can be triggered contingent on past surrenders becoming salient. We then incorporate it into a life insurance pricing framework and study mass surrenders’ impacts on contract valuation and default risk of the insurer, allowing for its structural default under regulatory solvency intervention. Our numerical results show that while contagion aligns individual policyholders’ surrender behavior with the optimal surrender of financially guided ones, it jeopardizes their financial positions in favor of equity holders. Surprisingly, surrender contagion only mildly enlarges an insurer’s bankruptcy rate, which is not driven by strict solvency regulation. Despite its public perception of posing a significant risk to insurers as similar to a bank run, surrender contagion does not critically threaten insurers’ solvency: To prevent insurers’ bankruptcies, their investment riskiness and leverage pose more of a concern to regulators.

Impact of rough stochastic volatility models on long-term life insurance pricing.

The Rough Fractional Stochastic Volatility (RFSV) model of Gatheral et al. (Quant Financ 18(6):933–949, 2014) is remarkably consistent with financial time series of past volatility data as well as with the observed implied volatility surface. Two tractable implementations are derived from the RFSV with the rBergomi model of Bayer et al. (Quant Financ 16(6):887–904, 2016) and the rough Heston model of El Euch et al. (Risk 84–89, 2019). We now show practically how to expand these two rough volatility models at larger time scales, we analyze their implications for the pricing of long-term life insurance contracts and we explain why they provide a more accurate fair value of such long-term contacts. In particular, we highlight and study the long-term properties of these two rough volatility models and compare them with standard stochastic volatility models such as the Heston and Bates models. For the rough Heston, we manage to build a highly consistent calibration and pricing methodology based on a stable regime for the volatility at large maturity. This ensures a reasonable behavior of the model in the long run. Concerning the rBergomi, we show that this model does not exhibit a realistic long-term volatility with extremely large swings at large time scales. We also show that this rBergomi is not fast enough for calibration purposes, unlike the rough Heston which is highly tractable. Compared to standard stochastic volatility models, the rough Heston hence provides efficiently a more accurate fair value of long-term life insurance contracts embedding path-dependent options while being highly consistent with historical and risk-neutral data.

A COMPARISON OF PARAMETRIC MORTALITY MODELS TO GRADUATE URBAN AND RURAL MORTALITY

This study examine twenty-nine parametric mortality models and assess their suitability for graduating mortality rates of urban and rural areas in Pakistan. Grouped age specific mortality rates of rural and urban populations for the year 2019 are used. The data is collected from the website of National Institute of Population Studies which conduct Maternal Mortality Survey in Pakistan on regular basis. The parametric mortality models were applied to rural and urban mortality data. We used R software to estimate the model’s parameters and assess their suitability for urban and rural populations. The suitability of these models was assessed by using 3 different loss functions. Our analyses found that the fourth type of Heligman-Polard’s model with loss function 3 provides reliable results for graduating the mortality of rural population while second type of Carriere model with loss function 3 produce best results for graduating the urban mortality of Pakistan. Based on two models, mortality rates of urban and rural population have been graduated over age range 0-85. We suggest the use the graduated mortality rates of urban and rural areas for pricing life insurance products in rural and urban areas respectively. In addition, graduated mortality rates are also suggested for use in calculation of life insurance liabilities.

Country factor behavior for integration improvement of European life insurance markets

This paper provides evidence of the role that financial market development and institutional quality play in the integration of European Union (EU) life insurance markets. It analyzes ten EU life insurance markets over a seventeen-year sample period. The meta-technology cost/revenue efficiency ratios, estimated under the meta-frontier DEA framework, are used as a measure of integration and the analysis is conducted by applying Tobit panel regression models. We find that national stock market development and institutional quality enhance cost performance and integration of EU life insurance markets. Results also show that in countries where bancassurance is the main life insurance distribution channel, banking sector development contributes to integration in terms of revenue efficiency. However, we find that better outcomes in national institutional quality decrease the meta-technology revenue efficiency ratio, suggesting that life insurance prices are lower in countries with better institutions.

Statistical features of persistence and long memory in mortality data

Abstract Understanding core statistical properties and data features in mortality data are fundamental to the development of machine learning methods for demographic and actuarial applications of mortality projection. The study of statistical features in such data forms the basis for classification, regression and forecasting tasks. In particular, the understanding of key statistical structure in such data can aid in improving accuracy in undertaking mortality projection and forecasting when constructing life tables. The ability to accurately forecast mortality is a critical aspect for the study of demography, life insurance product design and pricing, pension planning and insurance-based decision risk management. Though many stylised facts of mortality data have been discussed in the literature, we provide evidence for a novel statistical feature that is pervasive in mortality data at a national level that is as yet unexplored. In this regard, we demonstrate in this work a strong evidence for the existence of long memory features in mortality data, and second that such long memory structures display multifractality as a statistical feature that can act as a discriminator of mortality dynamics by age, gender and country. To achieve this, we first outline the way in which we choose to represent the persistence of long memory from an estimator perspective. We make a natural link between a class of long memory features and an attribute of stochastic processes based on fractional Brownian motion. This allows us to use well established estimators for the Hurst exponent to then robustly and accurately study the long memory features of mortality data. We then introduce to mortality analysis the notion from data science known as multifractality. This allows us to study the long memory persistence features of mortality data on different timescales. We demonstrate its accuracy for sample sizes commensurate with national-level age term structure historical mortality records. A series of synthetic studies as well a comprehensive analysis of real mortality death count data are studied in order to demonstrate the pervasiveness of long memory structures in mortality data, both mono-fractal and multifractal functional features are verified to be present as stylised facts of national-level mortality data for most countries and most age groups by gender. We conclude by demonstrating how such features can be used in kernel clustering and mortality model forecasting to improve these actuarial applications.

Surrender Contagion in Life Insurance: Modeling and Valuation

Mass surrenders—spiking surrenders of life insurance policies within a short period— are widely documented prior to many life insurers failures, but their impacts on insurers are rarely quantified. This paper builds a model of mass surrenders in a heterogeneous pool consisting of a minority of financially guided policyholders and a majority of individual policyholders with contagious behavior that can be triggered contingent on past surrenders becoming salient. By incorporating it into a life insurance pricing framework, we study mass surrenders’ impacts on contract valuation and early bankruptcy of the insurer, allowing for its structural default under regulatory solvency intervention. Our numerical results show that while contagion aligns individual policyholders’ surrender behavior with the optimal surrender of financially guided ones, it jeopardizes their financial positions in favor of equity holders. Surprisingly, contagion at worst only marginally enlarges the insurer’s bankruptcy rate, which is not driven by strict solvency regulation. Insurers have incentives to encourage contagion. Despite its public perception of endangering market stability as similar to a bank run, surrender contagion does not threaten insurers’ solvency, but can improve it instead with the assets left behind by individual policyholders.

Different Shades of Risk: Mortality Trends Implied by Term Insurance Prices

To infer forward-looking, market-based mortality trends, we estimate a flexible affine stochastic mortality model based on a set of U.S. term life insurance prices using a generalized method of moments approach. We find that neither mortality shocks nor stochasticity in the aggregate trend seem to affect the prices. In contrast, allowing for heterogeneity in the mortality rates across carriers is crucial. We conclude that for life insurance, rather than aggregate mortality risk, the key risks emanate from the composition of the portfolio of policyholders. These findings have consequences for mortality risk management and emphasize important directions for mortality-related actuarial research.

Evidence for Persistence and Long Memory Features in Mortality Data

It is important to understand the statistical features of mortality data if one is to accurately undertake mortality projection and forecasting when constructing life tables. The ability to accurately forecast mortality is a critical aspect for the study of demography, life insurance product design and pricing, pension planning and insurance based decision risk management. Though many stylised facts of mortality data have been discussed in the literature, we provide evidence for a novel statistical feature that is pervasive in mortality data at a national level that is as yet unexplored. In this regard we demonstrate in this work strong evidence for the existence of long memory features in mortality data. We argue that it is important to consider as incorporate of such features in models will improve the understanding of mortality and the accuracy of forecasts. To achieve this we first outline the way in which we choose to represent persistence of long memory from a estimator perspective. To achieve this, we make a natural link between a class of long memory feature and an attribute of stochastic processes based on fractional Brownian motion. This allows us to use well established estimators for the Hurst exponent to then robustly and accurately study the long memory features of mortality data. A series of synthetic studies are implemented to evaluate the performance of three different estimators under different data lengths, different long memory strengths, different missing value settings, different aggregation type and different quantization. All of which are common transformations used in studying national level mortality data. Then the dynamic of the long memory across genders, age groups, countries and time periods is further analysed using real data from a range of different countries to demonstrate overwhelming evidence for this statistical property of mortality data.

The Impact of Longevity Risk on the Price of Life Insurance with the Accidental Option (Type AI and ADI)

Regardless of the type of insurance the basic part of the insurance premium is the pure premium, often referred to as net premium, which is based on the risk of death of the insured, averaged over the entire insurance period. The net premium is therefore the fund amount that the insured should pay in order for the insurer to possess sufficient funds to cover all benefits during the contract period. In accordance with the principles of actuarial mathematics the valuation of standard life insurance is based on the equivalence principle and takes into account the risk of death and also the change in time value of money, i.e. actuarial risk, which is estimated by the insurer at the time of signing the contract. Such method of calculating the net premium does not ensure the protection of collected funds against aggregated and individual longevity risk, which may negatively affect long-term financial stability of insurers as well as the level of financial security of the insured. Hence it is necessary to, on the one hand precisely valuate the trend of further life expectancy, and on the other there is a need to modify the methods of calculating premium rates and their adjustment in the duration of the contract. Therefore the article includes premium calculations that take into account the extended actuarial risk that encompasses longevity risk and its effect on their amount. This problem is the aim of the analysis carried out in the article. That is to say, taking into account the extended actuarial risk that encompasses longevity risk, a fair premium formula including the need for a risk adjustment was presented, followed by a premium calculation based on the example of life insurance with the AI and ADI option, as well as a look into the effect of change in longevity risk on their amount.

From insurance risk to credit portfolio management: a new approach to pricing CDOs

We present a new approach for pricing collateralized debt obligations (CDOs) which takes into account the issue of the market incompleteness. In particular, we develop a suitable extension of the actuarial framework proposed by Bayraktar et al. [Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities. J. Econ. Dyn. Control, 2009, 33, 676–691], Milevsky et al. [Financial valuation of mortality risk via the instantaneous Sharpe-ratio: Applications to pricing pure endowments. Working Paper, 2007. Available at: http://arxiv.org/abs/0705.1302], Young [Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio: Theorems and proofs. Technical Report, 2007. Available at: http://arxiv.org/abs/0705.1297] and Young [Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio. Insurance: Math. Econ., 2008, 42, 691–703], which is based on the so-called instantaneous Sharpe ratio. Such a procedure allows us to incorporate the attitude of investors towards risk in a direct and rational way and, in addition, is also suitable for dealing with the often illiquid CDO market. Numerical experiments are presented which reveal that the market incompleteness can have a strong effect on the pricing of CDOs, and allows us to explain the high bid-ask spreads that are frequently observed in the markets.

MARITAL STATUS AS A RISK FACTOR IN LIFE INSURANCE: AN EMPIRICAL STUDY IN TAIWAN

AbstractGender and age are the top two risk factors considered in pricing life insurance products. Although it is believed that mortality rates are also related to other factors (e.g. smoking, overweight, and especially marriage), data availability and marketing often limit the possibility of including them. Many studies have shown that married people (particularly men) benefit from the marriage, and generally have lower mortality rates than unmarried people. However, most of these studies used data from a population sample; their results might not apply to the whole population. In this study, we explore if mortality rates differ by marital status using mortality data (1975–2011) from the Taiwan Ministry of the Interior. In order to deal with the problem of small sample sizes in some marital status groups, we use graduation methods to reduce fluctuations in mortality rates. We also use a relational approach to model mortality rates by marital status, and then compare the proposed model with some popular stochastic mortality models. Based on computer simulation, we find that the proposed smoothing methods can reduce fluctuations in mortality estimates between ages, and the relational mortality model has smaller errors in predicting mortality rates by marital status. Analyses of the mortality data from Taiwan show that mortality rates differ significantly by marital status. In some age groups, the differences in mortality rates are larger between marital status groups than between smokers and non-smokers. For the issue of practical consideration, we suggest modifications to include marital status in pricing of life insurance products.

Optimizing insurance pricing by incorporating consumers’ perceptions of risk classification

Insurers primarily set premiums using cost-oriented pricing methods based on claims history. Customer-oriented pricing has yet to be widely applied. This article empirically tests consumers’ acceptance of (currently unused) risk characteristics in motor and term life insurance pricing in the United Kingdom, France, and Germany. We derive implications concerning how insurers’ can use the knowledge of consumer acceptance or rejection of specific price-determining factors when standardizing their pricing schemes. The results indicate that consumers highly approve commonly used risk-rating factors when their price-determining function is transparent. Furthermore, consumers are willing to provide insurance companies further personal information when such information is used for pricing. The provision of personal information allows insurance companies to conduct a critical review of attributes requested and used that have a low impact on premium amounts and lower consumer acceptance1.

Regression-based algorithms for life insurance contracts with surrender guarantees

We present a general framework for pricing life insurance contracts embedding a surrender option. The model allows for several sources of risk, such as uncertainty in mortality, interest rates and other financial factors. We describe and compare two numerical schemes based on the Least Squares Monte Carlo method, emphasizing underlying modeling assumptions and computational issues.

Life insurance and income growth: the case of Sweden 1830–1950

In this paper we provide an analysis of the life insurance market in Sweden from the early nineteenth century to the mid-twentieth century. We consider determinants put forward in the financial history literature to explain the growth of life insurance. The paper shows that income elasticity of demand gives a fairly good approximation of the development in the twentieth century, while the development of risk and insurance innovation among other things need to be taken into account to explain the growth of life insurance in the nineteenth century. The price of life insurance, measured as the overhead-to-premium-income ratio, remained fairly constant during the second half of the nineteenth century, while the risk, as indicated in terms of crude mortality rates and its volatility, did decline. This probably improved the return on life insurance savings and further helped the entry of new firms. The average premium size was reduced to enable the diffusion of life insurance to workers.

Pricing life insurance contracts with early exercise features

In this paper we describe an algorithm based on the Least Squares Monte Carlo method to price life insurance contracts embedding American options. We focus on equity-linked contracts with surrender options and terminal guarantees on benefits payable upon death, survival and surrender. The framework allows for randomness in mortality as well as stochastic volatility and jumps in financial risk factors. We provide numerical experiments demonstrating the performance of the algorithm in the context of multiple risk factors and exercise dates.

Pricing Life Insurance Contracts with Early Exercise Features

In this paper we describe an algorithm based on the Least Squares Monte Carlo method to price life insurance contracts embedding American options. We focus on equity-linked contracts with surrender options and terminal guarantees on benefits payable upon death, survival and surrender. The framework allows for randomness in mortality as well as stochastic volatility and jumps in financial risk factors. We provide numerical experiments demonstrating the performance of the algorithm in the context of multiple risk factors and exercise dates.

Regression-Based Algorithms for Life Insurance Contracts with Surrender Guarantees

We present a general framework for pricing life insurance contracts embedding a surrender option. The model allows for several sources of risk, such as uncertainty in mortality, interest rates and other financial factors. We describe and compare two numerical schemes based on the Least Squares Monte Carlo method, emphasizing underlying modeling assumptions and computational issues.

Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.