Frailty mixture model with application in insurance industry

Introduction

The concept of frailty modeling is based on mixture distributions and survival analysis.

Varying parameter and unknown covariates

Johnson et al. (2005, p.345) [11] , describes a mixture distribution as a distribution that arises when the density function of a random variable depends on a parameter. Consider a random variable 𝑥 depending on its parameter 𝜃 then the conditional probability density function can be written as:

𝑓(𝑥) = ∫ 𝑓(𝑥|𝜃)𝑔(𝜃) 𝑑𝜃

Thus instead of considering a parameter 𝜃 , we consider an unknown covariate.

Survival analysis

Survival analysis is a branch of statistics which deals with time to the occurrence of a given event of interest. For insurers this event could be time to death, ill health or retirement. It's different from other fields of statistics in that we are observing something that develops dynamically over time and takes censoring into consideration which is partial information about the variable of interest. Three important functions of time to the event are:  The survival function, 𝑆(𝑡), describes the probability that an individual survives longer than time 𝑡.  The probability density function, 𝑓(𝑡)  The hazard function, ℎ(𝑡), describes the instantaneous death rate Estimating the survival function using non-parametric methods such as the Kaplan-Meier technique leads to obtaining the median time to the event under investigation. Determining factors affecting the hazard, the Cox PH model has been widely used.

Estimating the survival function

Existing literature on estimating the survival function includes the parametric, semi-parametric and non-parametric methods. Frailty models are extension of Cox-proportional hazard model where the relative risk is replaced with a random variable called the 'frailty term'.

Parametric Methods

For parametric inference, it is necessary to make assumptions about the distribution of failure times. Parametric approaches such as Weibull, lognormal, exponential, etc can be used to estimate the survival function for homogeneous populations. Basically, any distribution of non-negative random variables can be used.

Non-Parametric Methods

Non-parametric approaches such as Aalen-Nelson (1978) [2] can be used to estimate the survival function when assumption of the failure time distribution is to be avoided. An advantage of non-parametric models is their good fit and their ability to deal with any distribution without any additional assumptions.

Semi-Parametric Methods Cox proportional hazard model (1972)

A Cox model is a technique for exploring the relationship between the survival time of an individual and several explanatory variables. The hazard function for each individual is proportional to the baseline hazard ℎ 𝑜 (𝑡) and thus the hazard is fully determined by the covariate vector. The hazard function for individual 𝑖 at time (age) t is written as: ℎ 𝑖 (𝑡) = ℎ 𝑜 (𝑡) 𝑒 𝛽 1 𝑋 1 +𝛽 2 𝑋 2 +⋯+𝛽 𝑘 𝑋 𝑘 ℎ 𝑜 (𝑡) is the baseline hazard function and corresponds to the probability of dying (or reaching an event) when all the explanatory variables are zero. In this model it is left unspecified. 𝑒𝑥𝑝(𝛽 ′ 𝑋 𝑖 ) is the relative risk of individual i, where 𝑥 𝑖 = (𝑥 1 , … , 𝑥 𝑘 ) are vector of covariates and 𝛽 ′ = (𝛽 1 , … , 𝛽 𝑘 ) are vector of regression coefficients that give the proportional change that can be expected in the hazard, related to changes in the explanatory variables.

Frailty Models

Vaupel et al. (1979) [17] introduced the term frailty and used it in individual survival models. Clayton (1978) [6] promoted the model by its application to multivariate situations. Ordinary life table methods implicitly assume that the population under study is homogenous. This means that all individuals in that study are subject under the same risk (e.g., risk of death, risk of accident). Basic observation of medical statistics shows that individuals differ greatly. Thus, the study population cannot be assumed to be homogeneous but must be considered as a heterogeneous sample. A random effect model takes into account the effects of unobserved or unobservable heterogeneity, i.e. an individual's attitude towards health or some genotypic personal characteristics. Thus, the role of "frailty" is to include all unobservable factors acting on the individual mortality. The random effect denoted by Z is the term that describes the individual heterogeneity.

Differential Mortality Models

The Multiplicative approach. Vaupel et al. (1979) [17] described the model as ℎ(𝑥|𝑧) = 𝑧ℎ 𝑜 (𝑥) ℎ 𝑜 (𝑥) is the baseline mortality considered to be a known function of x that is to be specified. The frailty, Z is meant to quantify uncertainty associated with the hazard rate which acts in a multiplicative manner The Aalen Additive Model. Aalen (1978) [2] described a nonparametric additive hazard model given by

This model is useful in dealing with right censored survival data, especially in the presence of time-varying covariates.

Age Shifting Model ℎ'(𝑥) = ℎ(𝑥 + 𝑧)

The age shift model was proposed by G Humphreys (1874). Who argued that the mortality experience of a group of impaired lives accepted for life insurance should have an increased premium rating determined by assuming that the insured's age is higher than the real current age, hence adopting the "age shift".

Types of Frailty Models

Frailty models without observed covariates. This model is used when only survival data is available for the analysis, or when additional information is of no interest.

ℎ(𝑡, 𝑍) = 𝑍ℎ 𝑜 (𝑡)

The non-negative random variable 𝑍 is called frailty and ℎ 𝑜 (𝑡) is the baseline hazard. This model is non-identifiable from survival data, since different combinations of ℎ 𝑜 (𝑡) and frailty distributions may produce the same marginal hazard rate ℎ(𝑡). The model becomes identifiable when the parametric structure of ℎ 𝑜 (𝑡) is fixed and 𝑍 is assumed to belong to some parametric distribution family.

Univariate Frailty Models

This model describes the influence of unobserved covariates in a proportional hazards model for independent lifetimes. The variability can be split into a part that depends on observable risk factors, and is therefore theoretically predictable, and a part that is theoretically unpredictable, even when all relevant information is known. This model has been used by Hougaard (1984) [10] to show that these two sources of variability can explain some unexpected results or gives an alternative explanation of some results.

Literature Review Frailty Distributions

First, neither theory nor data typically provides much guidance for choosing a specific distribution from which to draw the frailty, thus any distribution with positive support and finite mean is suitable to represent the frailty distribution. However, for tractability reasons the choice of distribution is limited to those that provide a closed form expression for the frailty survivor function, density and hazard functions. The choice of parametric distributions for Z is often a matter of computational convenience and it should be strictly positive support, since negative frailty leads to negative mortality rates. Some of the distributions considered in this study are:  Gamma distribution Vaupel et al. (1979) [17]  Inverse-Gaussian distribution Manton et al. (1986) [12]  Non Central Gamma distribution

Baseline Hazard Distributions

Two different approaches are possible. In the parametric case the baseline hazard is chosen in the class of parametric lifetime distributions. The model also works without any specification of the baseline hazard function. However, there has been no study or survival experiment, which restricts estimates for the parametric form of the baseline hazard.

The baseline hazard considered in this study is the Weibull distribution (Manton and Stellard, 1986) [12] Applications in Life Insurance Frailty models are used in life insurance to represent heterogeneity in a population due to unobservable risk factors. Heterogeneity due to observable risk factors is addressed at policy issue during the underwriting process to ensure that each contract is assigned premium consistent with the insured risk. Neglecting such factors may lead to biased valuation of insurance products. Actuaries have developed models for valuing life insurance that only consider observable risk factors. However, in general insurance models accounting for unobservable risk has been developed to explain overall claim frequency i.e. the Negative Binomial~Pareto Model 18 .

Life Insurance

Life insurance contracts with benefits contingent on the lifetime of an individual and whose benefit is stated in advance is considered. Heterogeneity can be classified as emerging from observable risk factors (at issue) i.e. age, sex, health status, profession, smoke habits, sport activities, and so on. Or unobservable risk factors like an individual's attitude towards risk. For immediate annuities, the relation of premium and annuity depends on the health of the insured at the time the contract is taken out. However, in deferred annuities (pension schemes), the insurer has to perform some kind of underwriting at the end of the deferment period.

The underwriting process

The purpose of underwriting is to assign each insured a frailty factor 𝑍 ̃ as an estimate of 𝑍 to determine the pricing mortality rates. These underwriting factors are observable characteristics, such as smoking status, that explain mortality heterogeneity. Underwriting is done to ensure that premiums and benefits are fairly priced.

The tests carried out during the underwriting process are based on.  Biological and physiological factors, such as age, gender, genotype.  Features of the living environment; in particular: climate and pollution, nutrition standards population density, hygienic and sanitary conditions.  Occupation, in particular in relation to professional disabilities or exposure to injury, and educational attainment.  Individual lifestyle, in particular with regard to nutrition, alcohol and drug consumption, smoke, physical activities and pastimes.  Current health conditions, personal and/or family medical history, civil status, and so on. This assessment can be performed through proper questions in the application form and as to health conditions through a medical examination.

Modeling unobservable factors

In addition to observable factors, heterogeneity may be caused by unobservable individual-specific factors, referred to as frailty. Frailty models may provide an appropriate description of the age-specific mortality shape, as well as the estimate of parameters of the relevant models according to mortality observed within the portfolio. However, there is no data available that can be linked to the choice of the distribution of Z since it is unobserved The rest of this paper is organized as follows; Section 3 describes the frailty model construction and parameter estimation, Section 4 describes an application of frailty modeling in actuarial science and Section 5 is on discussions and recommendation for further research. Finally, Section 6 gives references used in the study

Methodology

Probability Tools Some of the common probability tools used in survival analysis, Bowers et al. (1997) [3] are described below. Let T x be the future lifetime variable i.e. the remaining duration of life of a person aged x, which is a positive real valued variable, having a continuous distribution with finite expectation. Several functions characterize the distribution of T x :

is the survival function, which is sometimes denoted with t 𝑃 𝑥  𝐹 𝑥 (𝑡) = 𝑃(𝑇 𝑥 ≤ 𝑡) Expresses the probability of dying within t years for a person age x and is denoted by

Laplace Transform

The Laplace transform is crucial in this study since it makes computations of the survival and hazard functions from the density function easy. The Laplace transform of a random variable 𝑍 with density function 𝑓(𝑧) is given by;

Frailty Models

The frailty approach aims to account for heterogeneity, caused by unmeasured covariates in the Cox-proportional model which is described by a mixture variable 𝑍 called frailty.

The Cox-proportional model is given by;

The hazard is modified to the frailty model by substituting the relative risk 𝑒𝑥𝑝(𝛽 ′ 𝑋 𝑖 ) by a random variable 𝑍 which represents the unobserved covariates 𝑋 𝑖 i.e.

ℎ(𝑡, 𝑧) = ℎ 𝑜 (𝑡) * 𝑍

The frailty 𝑍 is then assumed to follow some distribution with positive support and has a multiplicative effect on the baseline hazard function which is common to all individuals.

The Multiplicative Model

This model describes the population as a mixture and assumes that each individual correspond a frailty quantity 𝑍, describing the individual's relative risk. The non-negative quantity 𝑧 encompasses all other factors affecting mortality other than age. The hazard at age 𝑥 conditional on 𝑍 is assumed to be 𝑍 ℎ 𝑜 (𝑥) I.e. ℎ(𝑥|𝑧) = 𝑧 * ℎ 𝑜 (𝑥) where ℎ 𝑜 (𝑥) is the 'standard hazard function' corresponding to a 'standard individual', conventionally those with frailty 𝑧 = 1 Individuals with 𝑍 > 1 experience a force of mortality that is proportionally higher than ℎ(𝑥) at all ages. Individuals with 𝑍 < 1 experience proportionally lower mortality rates. Model Assumptions  The frailty Z has a multiplicative effect on the mortality rate of the individuals:  ℎ(𝑡; 𝑍) = 𝑍ℎ 𝑜 (𝑡)  The frailty 𝑍 𝑥 is stationary. i.e. the frailty of an individual keeps constant throughout the whole lifetime span (but the probability distribution does depend on the age, and this justifies the suffix x)  Z is distributed independent of age(x) or time(t)  Z has a strictly positive support since negative hazards are impossible.

Gamma Frailty Model

Vaupel et al. (1979) [17] suggest a Gamma distribution, due to its mathematical tractability. From a computational and analytical point of view, it fits well to failure data because it is easy to derive the closed form expressions of unconditional survival, cumulative density and hazard function. This is due to the simplicity of the Laplace transform.

Vaupel Approach (1979) [17] Construction

𝐿𝑒𝑡 𝑍 ~ 𝛤(𝑝, 𝑏)

With shape parameter 𝑝 and scale parameter 𝑏. The marginal density of 𝑍 is;

The Laplace transformation is given by;

This is required to integrate out the distribution of the unobserved frailty. Once the frailty is integrated out, accounting for unobserved heterogeneity is reduced to estimating the variance of the frailty term. The CV shows that 𝑝 plays the role of measuring, in relative terms, the level of heterogeneity in population. If 𝑝 → ∞, then 𝑐𝑣(𝑧) → 0, i.e. the population can be considered homogeneous; for small values of 𝑝, on the contrary, the value of 𝑐𝑣(𝑧) is high, expressing a wide dispersion, i.e. heterogeneity in the population.

However, the coefficient of variation is constant and does not change with age. This is a unique property of the gamma distributed frailty, since other assumed forms of frailty usually exhibit a decreasing coefficient of variation. i.e. Inverse Gaussian distributed frailty.

The marginal survival function is given by; 𝑆(𝑥) = 𝐿 𝑍 (

Inverse-Gaussian Frailty Model

Alternative to the Gamma distribution is the Inverse Gaussian as a frailty distribution introduced by Hougaard (1984) [10] . When the inverse Gaussian is used, the variability of Zx decreases with age which can be justified by the fact that those with low frailty keep on living.

Hougaard Approach (1984) [10] Construction

𝐿𝑒𝑡 𝑍 ~ 𝐼𝐺(𝜇, Ө)

The probability density function of Z is

The Laplace transform is given by;

For identifiability reasons the mean is normalized to one. i.e. 𝜇 = 1 Thus the variance 𝛿 2 = 𝛽 The Laplace transform becomes

The marginal survival function is given by;

Choice of 𝒉 𝒐 (𝒙) Weibull-Inverse Gaussian Frailty Model

Using Weibull distribution for ℎ 𝑜 (𝑡)

Non-Central Gamma Frailty Model

The probability density function for the non-central gamma distribution with Y being a mixing of the distributions of 𝑋 1 , 𝑋 2 , … , 𝑋 𝑁 Where 𝑋𝑖'𝑠 ~ 𝐺𝑎𝑚𝑚𝑎(𝑛, 1) 𝑎𝑛𝑑 𝑁~𝑝𝑜𝑖𝑠𝑠𝑜𝑛(𝜆)

Then the density function is a convolution with respective weights

Where 𝛤(𝑛) is the central complete gamma function with 𝑛 > 0 𝜆 > 0 𝑥 ≥ 0 The hazard function is a special case of the three parameter power variance function when 𝑟 = -1

Choice of 𝒉 𝒐 (𝒙) Weibull -Non central Gamma Frailty Model

Using Weibull distribution for ℎ 𝑜 (𝑡)

Parameter Estimation

Model parameters are fixed quantitative values that characterize the model believed to reflect the real world. They have to be estimated either by statistical inference from observations or by expert opinion.

Choice of Explanatory Variables

In order to make comparisons between the Gamma-Weibull model, the Inverse-Gaussian-Weibull model and the Non-central Gamma-Weibull model, it is necessary to estimate and fix the baseline model parameters using insurance based mortality rating. The baseline model has no underwriting. The weibull parameters are estimated using MCMC technique see Gamerman, D. (1997) [19] the R-CODE used in this study is shown below (appendix).

Choice of the Insureds Level of Heterogeneity

In Butt and Haberman (2004) [5] an insurance application of frailty-based survival model is proposed. In particular, the authors discuss various choices and fit some models to two sets of life insurance mortality data. The obtained results suggest that when life annuities are referred to 𝛿 2 = 1/𝑏 should fall in the range (0.025, 0.05).

Unless otherwise stated in this exercise the insured population will be considered to have heterogeneity level of 𝛿 2 = 0.05 We use of the Bayesian Inference with Gibbs sampling and Metropolis algorithm to estimate parameters of the proposed baseline distribution.

Bayesian Inference

Bayesian Inference is based on bayes theorem and using probability to quantify for uncertainty in inferences based on statistical data analysis. Gelman et al. (1995) [8] describes 3 steps to follow when performing Bayesian data analysis. i.e. 1. Set up a full probability model 2. Condition on the observed data 3. Evaluate the fit of the model based on the resulting posterior distribution.

Applications in Insurance Industry

The aims of this exercise are threefold:  The first aim is to show that when heterogeneity is disregarded the expected residual lifetime is underestimated.  Secondly, is that neglecting heterogeneity leads to an underestimation of the insurer's liability.  Finally, is to show the relevance of the proposed non-central Gamma frailty mixture to reflect an insurer's mortality rating.

Data Analysis and Results

Consider three hypothetical insurers i.e. insurer x, y and z. Insurer X assumes the population to be homogeneous and applies the Kenyan (KE) 2001-2003 life tables. Insurer Y assumes the population to be heterogeneous and uses frailty modeling to account for heterogeneity. Insurer Z carries out underwriting and adjusts the rates to reflect safety loadings. In this case data from Association of Kenya Insurers (AKI) 2007-2010 graduated rates is used.

Case 1: Inverse Gaussian Frailty

The inverse Gaussian frailty mixture is given by:

The frailty model: ℎ(𝑡|𝑍) = 𝑍 * ℎ 𝑜 (𝑡)

When 𝑍 = 1 the hazard corresponds to an average individual, hence ℎ(𝑡|1) = ℎ 𝑜 (𝑡). Thus the baseline hazard ℎ 𝑜 (𝑡) can be approximated parametrically i.e. ℎ 𝑜 (𝑡) ~ Weibull (𝜆, 𝑝)

Case 2: Non-central Gamma Frailty

The inverse Gaussian frailty mixture is given by:

Assumption for the baseline is:

Case 3: Gamma Frailty

The Gamma frailty mixture is given by:

Using similar assumptions for the baseline i.e.

Pension Scheme

Pension schemes are deferred annuities whose benefits are payable on retirement.

Present Value

These are annuities which commence in m (say) years' time, provided that the annuitant is then active.

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