Pricing implications of trends in population mortality and underwriting effectiveness

Pricing actuaries try to anticipate insured lives mortality rates for decades into the future by considering historic relationships between population and insured lives mortality and trends in population mortality. The degree to which underwriting might decrease insured lives mortality relative to population mortality is of particular importance. A comparison of trends in population and insured mortality is presented to illustrate historic relationships. Two theories for future life expectancy trends are: 1) no foreseeable limit to life expectancy, and 2) life expectancy limited by biological forces. Factors that may increase or decrease the future effectiveness of underwriting are reviewed.     

Infant mortality: an insured population perspective

Many insurers offer life coverage to individuals during the first year of life. The policies tend to have small face values, but frequently contain premium waiver or additional purchase options. General population mortality is significantly higher at this age relative to older children and even middle-aged adults. This article presents the mortality experience of an insured cohort in which death occurred under 1 year of age. In summary, the insured population's mortality rate was significantly lower and the leading causes of death were different than the general population.     

Developing Mortality Improvement Formulas

Abstract

Longevity improvements have contributed to widespread underfunding of pension plans and losses in insured annuity portfolios. Insurers might reasonably expect some upside from the effect of lower mortality on their life business. Although mortality improvement scales, such as the Society of Actuaries Scale AA, are widely employed in pension and annuity valuation, the derivation of these scales appears heuristic, leading to problems in deriving meaningful measures of uncertainty. We explore the evidence on mortality trends for the Canadian life insurance companies, data, using stochastic models. We use the more credible population data to benchmark the insured lives data. Finally, we derive a practical, model-based formula for actuaries to incorporate mortality improvement and the associated uncertainty into their calculations.     

A Model for Analyzing the Impact of Selective Lapsation on Mortality

Abstract

This paper presents a model for examining the effect of various relationships between mortality rates and lapse rates on the mortality experience of a cohort of insured lives. The approach is individual rather than the aggregate traditionally used in analyzing selective lapsation. The model assumes that insured lives are healthy at policy issue, but later may move to an impaired state from which the lapse rate is zero. Associated with each insured is an unobservable “risk level” random variable, which reflects the heterogeneity of the insured group. Individual mortality and lapse rates are functions of the risk level. A numerical illustration provides some interesting results obtained by using this model.     

Compound Poisson Model with Covariates

Abstract

Pet insurance in North America continues to be a growing industry. Unlike in Europe, where some countries have as much as 50% of the pet population insured, very few pets in North America are insured. Pricing practices in the past have relied on market share objectives more so than on actual experience. Pricing still continues to be performed on this basis with little consideration for actuarial principles and techniques. Developments of mortality and morbidity models to be used in the pricing model and new product development are essential for pet insurance. This paper examines insurance claims as experienced in the Canadian market. The time-to-event data are investigated using the Cox’s proportional hazards model. The claim number follows a nonhomogenous Poisson process with covariates. The claim size random variable is assumed to follow a lognormal distribution. These two models work well for aggregate claims with covariates. The first three central moments of the aggregate claims for one insured animal, as well as for a block of insured animals, are derived. We illustrate the models using data collected over an eight-year period.     

The mortality among industrial insured lives in Norway during the war

In the Norwegian life insurance company Fram a continuous mortality investigation takes place in connection with the yearly valuation of policies issued with weekly premiums. The investigation gives the aggregate mortality, the unit is the policy and the year of observation is the calendar year. A detailed account of the method used has been given by Fredrik Borch in his paper: “The mortality among industrial insured lives in Norway 1931–1940” in this journal 1943. The most important results of the investigation from the years 1940–1946 are rendered below.     

Claiming Strategies and Premium Levels for Bonus Malus Systems

In this paper we study a bonus malus system (bms) with deductibles. A bms is characterized by its premium levels and the transition rules among them. An insured is being moved among premium levels according to his/her claim record. Thus, an insured has to find an optimal strategy of submitting claims. Here optimal is in the sense of minimizing the total expected present value (epv) costs. Such strategies are found both for finite and infinite horizons. Furthermore, premium levels balancing the cost to the insured and the payoff of the insurer are given. The methods used to analyze the problem are from dynamic programming and Markov chains.     

Post-war mortality among industrial insured lives in Norway

Abstract

The results of mortality investigations among industrial insured lives with weekly premiums in the Norwegian life insurance company Fram for the period 1931/40 and for the period 1940/46 have earlier been published in this journal.l It is now possible to render the main results of the continued investigation for the post-war years 1946/50.     

A new parametric model for converting excess mortality from clinical studies to insured population

We propose a new parametric model – the generalized excess mortality (GEM) model – for converting excess mortality from clinical to insured population. The GEM model has been formulated as a generalization of the excess death rate (EDR) model in terms of a single adjustment parameter (m) that accounts for a partial elimination of a clinical study’s EDR due to the underwriting selection process. The suggested value of the parameter m depends only on the ratio of the impairment’s prevalence rate in the insured population to that in the clinical population. The model’s development has been implemented in two phases: the design phase and the validation phase. In the design phase, the data from the National Health and Nutrition Examination Survey I pertaining to three broad impairments (diabetes, coronary artery disease, and asthma) have been used. As a result, the following equation for the parameter m has been proposed: m_k?=?(P_i,k/P_c,k)ⁿ, where P_i,k, P_c,k are the prevalence rates of impairment k under study in the insured and the clinical populations, respectively, and n a single universal parameter with its value best approximated as n?=?0.5 (95% confidence interval 0.5–0.6). In the validation phase, several independent clinical studies of three other impairments (Crohn’s disease, epilepsy, and chronic obstructive pulmonary disease) were used. As it has been demonstrated in the validation phase, for a number of impairments, the GEM model can provide a better fit for observed insured population mortality than either one of the conventional EDR or mortality ratio models.     

Select mortality and other durational effects modelled by partially observed Markov Chains

Abstract

In a recent paper Norberg explained select mortality tables for insured lives by a simple Markov model where the lives are classified as active/disabled and insured/not insured, and where no return is possible to previously visited states. The present paper extends the set-up and its results to more complex state spaces and patterns of transition, the key tool being the Kolmogorov backward differential equations.     

Converting Clinical Literature to an Insured Population: A Comparison of Models Using Nhanes

Abstract

The use of clinical literature to set risk classification standards for life insurance underwriting stems from the need to set the most accurate standards using the best available information. A necessary hurdle in this process is converting any excess mortality observed in a clinical study to the appropriate rating for use in underwriting. A widely accepted model in the insurance industry, the Excess Death Rate model, treats the excess as additive to the conditional probability of death for an insurance company’s unimpaired class.

In this paper we test the validity of that model versus other common predictive models of excess mortality in an insured population. Applying these models to National Health and Nutrition Examination Survey (NHANES) data, we derive estimates for excess mortality from three commonly seen underwriting impairments in what could be considered a clinical population. These estimates are added to an estimate of an insurance company’s unimpaired mortality class and then used to predict deaths in an “insurable” subset of that clinical population.

The Excess Death Rate model performed the best of all models, having the smallest cumulative difference of actual to predicted deaths. The use of publicly available data, such as that in NHANES, could help bridge the gap between clinical literature and its application in insurance underwriting if insurable cohorts can be reliably identified from these generally healthy, ambulatory groups.     

Reference mortality K2004 of personal life insurance policies in Finland

This article presents the reference mortality model K2004 approved by the Actuarial Society of Finland and the technique that was implemented in developing it. Initially, I will present the historical development of individual mortality rates in Finland. Then, the requirements posed for a modern mortality modelling will be presented. Reference mortality model K2004 is based on total population mortality rates, which were adjusted to correspond with that portion of the population that has a life insurance policy. First, the model presents a margin of the observed life insurance mortality rate in the total population with a Lee-Carter method together with a forecast, where the downward trend in mortality rates is expected to continue at the rate illustrated since the 1960s. Then, the mortality rate has been adjusted into life insurance mortality per age so that it corresponds to the differences observed between total population and the portion of population that has a life insurance during 1991–2001. Finally, a cohort and gender-specific functional margin will be presented to obtained data.     

Deposit Insurance, Capital Adequacy Requirements and Interest Rate Dynamics

This paper examines the long run interaction among deposit insurance, bank deposit rates and capital adequacy requirements. Using analysis similar to the price discrimination model of Lott and Roberts (1991) we find that a competitive environment among banks would link the spread between insured and uninsured deposit rates to the size of the insurance premium. We also find that banks that choose to operate at the regulatory minimum capital level, would increase asset risk with increased capital requirements if (1) the implicit interest paid to insured and uninsured depositors is equally sensitive to changes in risk and capital adequacy and (2) the insurance premium is independent of the level of risk and capital adequacy. Under the present risk-based premium structure, asset risk has the potential to decline when the regulatory agency raises capital requirements. Finally, we examine the time series behavior of insured and uninsured interest rates to see if it is consistent with our theoretical model. We find that insured and uninsured rates, along with deposit insurance premiums, are cointegrated series as suggested by our model.     

Screening potential elderly preferred markers: exploratory analysis of Cardiovascular Health Study (CHS) data

The Cardiovascular Health Study (CHS) analyzes risk factors for coronary heart disease and stroke in people age 65 and older. Since CHS is designed to comprehensively study cardiovascular risk factors in an elderly population, it provides a unique opportunity to study the association of risk factors with mortality, as well as morbidity risk. With the growth of the elderly as population and life insurance market segments, the need to more precisely stratify mortality within a standard risk group of the elderly has grown as well. This exploratory analysis assesses medical factors that could be used to improve mortality risk stratification within a "standard" mortality population, using the CHS public use data set. Participants with a personal history of cardiovascular disease, diabetes, or major electrocardiographic abnormalities were excluded from the analysis in order to mimic a standard life insurance selection process. Then, Cox proportional hazards regression was used to study 10 medical risk factors. This model suggested that forced vital capacity >80% predicted, serum creatinine <1.5 mg/dL (133 mcmol/L), hemoglobin >11 g/dL (110 g/L), and serum albumin >3.5 mg/L (35 mmol/ L) are significantly associated (p = 0.05) with favorable mortality. C-reactive protein <1 mg/L is associated with favorable mortality at borderline significance levels (p = 0.09). On the other hand, a family history of cardiovascular disease (MI and/or stroke) and low BMI (<26 kg/m2) are associated with unfavorable mortality in the analysis. Total to HDL cholesterol ratio of <6, presence of supine systolic blood pressure < or = 140 mmHg, and the presence of minor rest electrocardiographic findings were not statistically significant factors in the multivariate model. Further assessment of the predictive value of the "significant" medical factors identified is required in insured lives.     

论责任保险被保险人“怠于请求”行为之司法界定

2009年新修订的《保险法》第65条增加规定,赋予保险事故受害第三者在一定条件下对保险人的直接赔偿请求权,这是我国责任保险法律制度的进步。然而,《保险法》对被保险人“怠于请求”行为的认定并未作具体明确的规范,这必将使司法实践对此类案件的处理陷入困境。在目前条件下,为切实保证责任保险制度保护受害第三者利益立法目的之实现,...     

Markov Aging Process and Phase-Type Law of Mortality

Abstract

In this article, we propose a finite-state Markov process with one absorbing state to model human mortality. A health index called physiological age is introduced and modeled by the Markov process. Under this model the time of death follows a phase-type distribution. The model possesses many desirable analytical properties useful for mortality analysis. Closed-form expressions are available for many quantities of interest including the conditional survival probabilities of the time of death and the actuarial present values of the whole life insurance and annuity. The heterogeneity or frailty effect of a cohort can be expressed explicitly. The model is also able to explain some stylized facts of observed mortality data. We fit the model to some Swedish population cohort data and life tables compiled by the U.S. Social Security Administration. The fitting results are very satisfactory.     

The Metabolic Syndrome and All-Cause Mortality in an Insured Lives Population

Abstract

Metabolic syndrome and its association with mortality have not been studied in insured lives populations. The Swiss Re Study evaluated metabolic syndrome prevalence and associated mortality from all causes and circulatory disease in a cohort of 35,470 predominantly healthy individuals, aged 18–83 years, who were issued life insurance policies between 1986 and 1997. Metabolic syndrome was defined using the National Cholesterol Education Program (NCEP) Expert Panel Adult Treatment Panel (ATP) III guidelines. The NCEP obesity criteria were modified with a prediction equation using body mass index, gender, and age substituted for waist circumference. Adjustments also were made for nonfasting triglyceride and blood glucose values. Risk ratios for policyholders identified with metabolic syndrome were 1.16 (P = .156) for mortality from all causes and 1.45 (P = .080) for mortality from circulatory disease compared with individuals without the syndrome. Risk was proportional to the number of components, or score, of the metabolic syndrome present. Risk ratios for metabolic syndrome score were 1.14 (P < .001) for mortality from all causes and 1.38 (P < .001) for mortality from circulatory disease compared with individuals without metabolic syndrome factors. In both all-cause and circulatory death models, relative risk was highest for the blood pressure risk factor. Based on a modified NCEP definition, increased mortality risk is associated with metabolic syndrome in an insured lives cohort and has life insurance mortality pricing implications.     

The importance of socialepidemical research of premium calculation of life and pension insurances

Differences in mortality and morbidity depend not only on medical parameters but also on social determinants. Social epidemiology studies demonstrated the influence of general living conditions and social behavior on health status. This results should be taken into consideration by private insurance companies when calculating premiums in selected populations. It may be hazardous if only upper social classes are insured in pension schemes, for example. On the other hand, people in lower social classes have a higher risk of premature death and would also represent a risk for insurers with a portfolio including only this type of insured person.     

The Impact of the Secondary Market on Life Insurers' Surrender Profits

Life insurers often claim that the life settlement industry reduces their surrender profits and leads to an adverse shift in their portfolio of insured risks; that is, high risks remain in the portfolio instead of surrendering. In this article, we aim to quantify the effect of altered surrender behavior––subject to the health status of an insured––in a portfolio of life insurance contracts on the surrender profits of primary insurers. Our model includes mortality heterogeneity by applying a stochastic frailty factor to a mortality table. We additionally analyze the impact of the premium payment method by comparing results for annual and single premium payments.     

On systematic mortality risk and risk-minimization with survivor swaps

A new market for so-called mortality derivatives is now appearing with survivor swaps (also called mortality swaps), longevity bonds and other specialized solutions. The development of these new financial instruments is triggered by the increased focus on the systematic mortality risk inherent in life insurance contracts, and their main focus is thus to allow the life insurance companies to hedge their systematic mortality risk. At the same time, this new class of financial contract is interesting from an investor's point of view, since it increases the possibility for an investor to diversify the investment portfolio. The systematic mortality risk stems from the uncertainty related to the future development of the mortality intensities. Mathematically, this uncertainty is described by modeling the underlying mortality intensities via stochastic processes. We consider two different portfolios of insured lives, where the underlying mortality intensities are correlated, and study the combined financial and mortality risk inherent in a portfolio of general life insurance contracts. In order to hedge this risk, we allow for investments in survivor swaps and derive risk-minimizing strategies in markets where such contracts are available. The strategies are evaluated numerically.