Mortality graduation using a rating up approach

Modeling the risk profile of a heterogeneous life from unreported risks is complex. Insurance firms routinely disregard unreported risk factors perhaps because of difficulties in modeling. The scientific interest of the study is to account for unreported heterogeneity factors using a compound process. For comparison reasons the gamma (mostly applied in literature) and non-central gamma compound process are suggested with the generalized exponential and generalized Weibull baselines to account for unreported heterogeneity. In this study, maximum likelihood estimation is used to calibrate the base force of mortality distributions using a large Kenyan insurer term insurance data. Subsequently, the performance of the candidate models described above is compared following the criteria information values. The findings show that the non-central gamma generalized Weibull is significant to model the insurers’ claims events.

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Introduction

Rating up theory was first proposed by Beard (1959) [2] to account for age-pattern of mortality, where the term "longevity factor" was used instead of rating up. Homogeneity is presumed with respect of reported risk factors in standard models. The implication is that study subjects are pooled in the same risk profile at a given age. However, statistical evidence suggest a different model as indicated by several researchers, such as Su & Sherris (2012) [12] ; Gatzert et al., (2012) [7] ; Onchere (2013) [10] ; Fong (2015) [6] ; Olivieri & Pitacco (2016) [9] and Pitacco (2018) [11] with references therein. To adequately price and allocate reserves that represent the insurance contracts all relevant factors affecting mortality needs to be considered. Neglecting unreported heterogeneity could result in biased insurance products pricing and reserves allocation. This research outlines a heterogeneous model that will improve the underwriting process to ensure fair pricing and reserving of life products consistent with the insured risk. The mortality model selected in valuation determines how term insurance and annuity products are priced (Batty et al., (2010) [1] ; Gildas et al., (2018) [8] . Life insurance-based rating up models measure population-level heterogeneity caused by unreported risks. On the other hand, heterogeneity caused by reported risk factors is determined during underwriting before issuing a policy to guarantee an optimal assignment of premium equivalent to insured risk for each contract. Excluding relevant factors or relying solely on age and sex may contribute to incorrectly priced assurance products. Term insurance contract is a policy in which a certain payment, say, KES A per annum, is provided if the insured passes on within a stated period, say m years. This benefit is determined from the EPV as 𝐴 ∫ 𝑣 𝑡 𝑆 𝑥 (𝑡)ℎ(𝑥 + 𝑡)𝑑𝑡 𝑚 0 (1) Where: 𝑥 is the insured's age, 𝑣 𝑡 represents the present value interest factor, 𝑆 𝑥 (𝑡)represents the survivor function and ℎ(𝑥 + 𝑡) the intensity rate. The type of mortality model applied to ℎ(𝑥 + 𝑡) influences how the policy is priced.

Methodology

For this study, we considered 732 term insurance contracts between calendar years 2010-2015 from a large Kenyan insurance company. Demographic information of policyholders includes; age, date of signing the contract and mortality date. This data-set will be used to compute the times-to-death and crude intensity rates experienced by the insured in the 22-64 age group. Our aims of study are firstly to show that when the gamma is applied as a rating up distribution the intensity rates are overestimated at all ages compared to the non-central gamma (NCG). Secondly, is to show the relevance of the NCG rating up mixture to graduate the insurance firm's crude intensity rates. Here, force mortality is presumed to be a pair-wise constant, assuming a fixed value across all ages consistent with Brouhns et al., (2002) [3] and Dodd et al., (2018) [5] assumption. We further assumed our rating up model has no observed covariates as only survival data was obtained for this analysis also in life insurance due to the underwriting procedures groups should be homogeneous with respect to observed covariates. Finally, and as given in the actual dataset we presumed that policyholders purchase term assurance policy at the age of 22-64 years.

The proposed model

The generalized Weibull (GW) density function is represented as

We can express the survivor, baseline hazard and cumulative hazard rates as

The generalized exponential (GE) function is derived from the GW distribution when 𝜌 = 1. Putting 𝜌 = 1 in Equations (2, 3) leads to

Firstly, using GE distribution as the baseline intensity for gamma rating up, we can generate an intensity rate for gamma-GE rating up mixture as follows

The NCG is further adapted as a rating up distribution resulting in the NCG-GE intensity rate given by

Where, 𝑏 < 1, 𝑏 > 1, 𝑎𝑛𝑑 𝑏 = 1 represent decreasing, increasing, and constant GE intensity rate, respectively.

Secondly, the GW distribution is proposed as the base force of mortality with the gamma rating up giving the gamma-GW (G-GW) intensity rate expressed as

Similarly, the NCG-GW intensity rate is described as

where (𝜌 ≤ 1) and (𝑏𝜌 ≤ 1) represent monotonically decreasing function and (𝜌 ≥ 1) and (𝑏𝜌 ≥ 1) represents monotonically increasing function; unimodal if (𝜌 < 1) and (𝑏𝜌 > 1) and bath-tub shaped if (𝜌 > 1) and (𝑏𝜌 < 1).

Data analysis, results and discussions 3.1 Parameter Estimation

Butt & Haberman (2004) [4] apply the rating up-based survival model to insurance. The authors first consider different choices of models and then apply them to two large life insurance mortality datasets. The results indicate a potential range 𝜎 2 ≈ (2.916, 14.444) in an insured population with 𝜎 2 = 14% for the heterogeneous case. From the findings of the investigation by Butt & Haberman (2004) [4] , in this study we consider 𝜎 2 = 14% for the heterogeneous case.

The MLE approach is concerned with obtaining parameter values say, (𝑏, 𝜆, 𝜌) that maximizes the probability of observing the data D given those parameters, 𝑝(𝐷| 𝑏, 𝜆, 𝜌). The likelihood function gives the probability of the observed sample generated by the model. Generally, maximization of the likelihood function to find the ML estimates is done algebraically, but can be computational intensive. In this study, the MLE algorithm is implemented using stats 4 package in R.

For the Generalized Weibull with PDF 𝑓 (𝑡) = 𝑏(1 -𝑒𝑥𝑝 (-𝜆𝑡 𝜌 )) 𝑏-1 • 𝜆𝜌𝑡 𝜌-1 • 𝑒𝑥𝑝 (-𝜆𝑡 𝜌 ); 𝑡 > 0, 𝜌, 𝑏, 𝜆 > 0.

The log-likelihood function is derived as

For the Generalized Exponential with PDF

The log-likelihood function is derived as

The parameter estimates and criteria values is shown in the table below;

Discussion

Stats 4 provides the AIC and BIC measures that penalizes both excessive use of parameters and poor data fitting. From the above results, the least AIC and BIC suggests that the GW gives a better fit. The G-GW and NCG-GW rating up models given in Equations ( 6) and ( 7) respectively are as shown in Figure 1 where t is the time-to-death, σ 2 = 0.14, λ = 0.031367, b = 80.3076, ρ = 1.3477; h0(t)~GW (0.031367, 80.3076, 1.3477).

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Figure 1Close ×

Construction of Crude Intensity Rates from Real Term Insurance Dataset

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In Figure 1 graduation is done using the G-GW and NCG-GW rating up model both calibrated on the real term assurance timesto-death data. This is compared with the real term assurance intensity rates. As shown the G-GW overestimates the intensity rate at all ages compared to the NCG-GW model. The NCG-GW is observed to fit well to the actual claims experience hazards. The chi-square test Table 2 and Kolmogorov-Smirnov (KS) hypothesis test Table 3 for overall goodness of fit is significant for NCG-GW. The chi-squared goodness-of-fit test has p-value greater than 0.01, indicating that the model fits well. Similarly, for KS goodness of fit test p-value > 0.01, indicating goodness of fit for the distribution.

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Table 3Close ×

Goodness of fit using Kolmogorov-Smirnov test

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Name	Value
Chi-squared statistic	1722
Degree of freedom	1681
Chi-squared p-value	0.2379

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Chi-squared Goodness-of-fit of NCG-GW to the Crude Intensity Rates

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Discussion

Our goal in this paper is to model unreported heterogeneity by applying the gamma and NCG rating up distributions. We applied our model to real term insurance times-to-death data. Based on maximum likelihood, the GW baseline fits better as the AIC & BIC is lower than the GE baseline. As shown in Figure 1 the G-GW model overestimates the intensity rates at all ages compared to the NCG-GW model. According to chi-squared goodness of fit test Table 2 and KS hypothesis test Table 3, the NCG-GW shows better fitness to insurer's claims experience.

Conclusion

Life Insurance-Based rating up models measure population-level heterogeneity caused by unreported risks. On the other hand, heterogeneity caused by reported risk factors is determined during underwriting before issuing a policy to guarantee an optimal

Figures and tables from the paper

Baseline model	Parameter estimates	AIC	BIC
1. Generalized Exponential	λ = 0.14537, b = 286.57	4995.128	5004.32
2. Generalized Weibull	λ = 0.031367, b = 80.3076, ρ = 1.3477	4960.588	4974.376

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Base Force of Mortality Parameter Estimates

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