Data

In this case study, we use mortality experience data for a Universal Life (UL) portfolio covering insured lives aged 18 to 99 years, observed over calendar years 2008 to 2017, inclusive.

The data contains multiple risk factors. In this report, we investigate the effects of gender, smoker category, and the Cost of Insurance (COI) type on mortality. Note that there are two COI types in this portfolio: Annual Renewable Term (ART) insurance where the COI increases annually, and Level (LVL) insurance where the COI remains constant over the entire premium payment period.

Selecting Mortality Law

We fit mortality laws for the force of mortality \(\mu_x\) to data at the level of the individual using Maximum Likelihood Estimation. We then compare the model fit to the crude hazard rates by age, i.e., the ratio of deaths to central exposure by age-group. Comparing the fit to the crude rates visually and in terms of the deviance residuals, we can tell which model is most suitable. The AIC score also provides a quantitative measure for model choice.

Adding Risk Factors

Gender and smoker category are the most important risk factors to model. We do not split our data and create models for MNS, MSM, FNS, and FSM, separately - we create one single model that results in different \(\mu_x\)-rates based on risk factor combination.

Including risk factors in the model is done by estimating an adjustment to any of the standard parameters \(\theta = (\alpha, \beta, \epsilon, \rho)\), depending on whether a given life \(i\) is in risk group \(j\).

\[\theta_i = \theta_{baseline} + \sum_{j=1}^{m} z_{ij}\theta_j\]

The indicator parameter is unity, \(z_{i,j} = 1\), if person \(i\) is in risk group \(j\), otherwise \(z_{i,j}=0\).

We refer to adjustments to the INTERCEPT parameter \(\alpha\) as Main Effects, adjustments to the age parameter \(\beta\) as AGE interactions, and to the Makeham parameter as MAKEHAM interactions. Note that having different values for the BEARD parameter by risk group is not very common, however, because mortality differentials generally disappear at advanced ages. This phenomenon is referred to as mortality convergence.

We successively add main effects and interaction parameters to improve the model. The three main criteria for a better model are (1) that each new parameter has a statistically significant estimate, (2) that adding the parameter reduces the AIC score by at least four units, an (3) that the corresponding residual chart shows greater randomness than before inclusion of the new parameter.

This bottom-up method of model building is illustrated below for the risk factors GENDER and SMOKER category.

Selection effect

Just as we model risk factors by age, we can also model selection effects by policy duration. The chart of residuals for a model without selection plotted against policy duration below shows that a selection effect is present in the data.

To model the selection effect, we split the experience data for each life into phases. In this example, policy durations up to five years fall within \(Phase\ 1 := [0,5)\) and \(Phase\ 2 := [5, \infty)\) contains the exposure for policies five years or older. Using the DURATION parameter \(\gamma\) as a coefficient for policy duration \(t\), we can model simple log-linear select effects by phase according to the equation below.

\[\mu_{x,t} = e^{\epsilon} +e^{\alpha + \beta x + \gamma t}\]

The table below summarises the experience data by policy year phase. Note that we only observe a fraction of the total deaths during the first five policy years.

Mortality Trend

Within the survival model framework we include mortality trends by treating calendar year \(y\) as another independent variable with a \(TIME\) coefficient \(\delta\).

\[\mu_{x,t,y} = e^{\epsilon} +e^{\alpha + \beta x + \gamma t + \delta (y \ -2000)}\]

Discussion and Outlook

The final model shown in the table above incorporates all our findings: It demonstrates that we have modelled the age-dependence of mortality, included the selection effect in early policy years, as well as the mortality trend by calendar year and anti-selective mortality deterioration, all in one model. Besides the “usual suspects” of gender and smoking status, we were able to incorporate COI Type as a risk factor. We demonstrate that lives covered by policies with annually increasing COI (ART) experience greater ongoing deterioration of mortality with policy duration than Level COI business, as the estimate for the DURATION:COI-TYPE interaction is statistically significant. It’s value implies that ART business experiences ongoing deterioration at a rate of around \(2.8\%\) per policy year, while the mortality of Level COI business worsens at a rate of around \(2.2\%\) per policy year. Both effects are masked by an ongoing mortality improvement trend of around \(3.4\%\) per year.

Furthermore, our modelling demonstrated that it would be inaccurate to simply use a mortality table with greater slope for ART business, as an AGE:COI-TYPE interaction was not significant in the presence of the duration parameter \(\gamma\).

A model for mortality rates which are to be used in an actuarial application should also properly account for differences in mortality by policy size, as shown in the exploratory analysis. Differentiating mortality by face amount band can be implemented in a similar way as shown for the other risk factors.

The Appendices show mortality rates \(q_x\) against attained age, as well as a review of crude hazard rates by each risk factor compared to the corresponding final model expected deaths.

Appendix: Mortality rates

The models developed in this note can be used to derive actuarial \(q_x\) type rates for any given valuation date. Note that we do not recommend applying the TIME trend \(\delta\) beyond the observation period of the mortality study.

Appendix: Risk Factor Analyses