Marcus C. Christiansen

Who is Changing Health Insurance Coverage? Empirical Evidence on Policyholder Dynamics

Long-term health insurance contracts provide policyholders with the option of lapsing coverage or switching to another tariff within the same insurance company. We empirically analyze policyholder behavior regarding contract commitment in a large dataset of German private health insurance contracts. We show that short-term as well as long-term premium development, along with premium adjustment frequency, affect lapse and tariff switch rates. Moreover, the sales channel has a strong impact on switching behavior, indicating that policyholder choice is not fully independent of sales representatives. Our results are important for risk assessment and risk management of portfolios of health insurance contracts and provide better understanding of the dynamics of policyholder behavior in health insurance.

Deterministic mean-variance-optimal consumption and investment

In dynamic optimal consumption–investment problems one typically aims to find an optimal control from the set of adapted processes. This is also the natural starting point in case of a mean-variance objective. In contrast, we solve the optimization problem with the special feature that the consumption rate and the investment proportion are constrained to be deterministic processes. As a result we get rid of a series of unwanted features of the stochastic solution including diffusive consumption, satisfaction points and consistency problems. Deterministic strategies typically appear in unit-linked life insurance contracts, where the life-cycle investment strategy is age dependent but wealth independent. We explain how optimal deterministic strategies can be found numerically and present an example from life insurance where we compare the optimal solution with suboptimal deterministic strategies derived from the stochastic solution.

First-order mortality rates and safe-side actuarial calculations in life insurance

In this paper, we discuss how to define conservative biometric bases in life insurance. The first approach is based on cumulative hazard (or survival probabilities), the second one on the hazard itself, and the third one on the rate of increase of the hazard. The second case has been studied in the literature and the sum-at-risk plays a central role in defining safe-side requirements. The two other cases appear to be new and concepts related to sum-at-risk are defined.

Stress Scenario Generation for Solvency and Risk Management

We derive worst-case scenarios in a life insurance model in the case where the interest rate and the various transition intensities are mutually dependent. Examples of this dependence are that (a) surrender intensities and interest rates are high at the same time, (b) mortality intensities of a policyholder as active and disabled, respectively, are low at the same time, and (c) mortality intensities of the policyholders in a portfolio are low at the same time. The set from which the worst-case scenario is taken reflects the dependence structure and allows us to relate the worst-case scenario-based reserve, qualitatively, to a Value-at-Risk-based calculation of solvency capital requirements. This brings out perspectives for our results in relation to qualifying the standard formula of Solvency II or using a scenario-based approach in internal models. Our results are powerful for various applications and the techniques are non-standard in control theory, exactly because our worst-case scenario is deterministic and not adapted to the stochastic development of the portfolio. The formalistic results are exemplified in a series of numerical studies.

Safety margins for unsystematic biometric risk in life and health insurance

In multistate life and health insurances, the pattern of states of the policyholder is random, thus exposing the insurer to an unsystematic biometric risk. For this reason safety margins are added on premiums and reserves. But in contrast to non-life insurance, traditionally the safety margins are not chosen explicitly but implicitly in form of a valuation basis of first order. If we define the implicit margins bottom-up, we are not able to control the level of safety that we finally reach for premiums and reserves. If we use a top-down approach, that means that we directly calculate explicit margins for premiums and reserves and then choose implicit safety margins that correspond to the explicit margins, we are able to control the total portfolio risk, but we have the problem that it is unclear how to allocate the total margin to partial margins for different transitions at different ages. Although the allocation of the total margin to the partial (implicit) margins is not relevant for the total portfolio risk, we have to pay attention since it can have a great effect on the calculation of surplus. In this paper we calculate asymptotic probability distributions for premiums and reserves of second order by using the functional delta method. As a result, we can not only determine the actual level of safety that is induced by given implicit safety margins, but we can also linearly decompose the total randomness of a portfolio to contributions that the different transition rates at different ages make to the total uncertainty. As a result we do not only get new insight into the sources of unsystematic biometric risk, but we also obtain a useful tool that allows to construct reasonable principles for the allocation of the total safety margin to implicit margins with respect to transitions and ages.

On the forward rate concept in multi-state life insurance

Similarly to the notion of modeling credit risk by using forward credit default spread rates, mortality risk in life insurance contracts is nowadays often modeled by using forward mortality (spread) rates. More recently, this concept has also been discussed for more complex life insurances that include multiple lives or intermediate states that correspond to the health status of the insured. For consistency purposes and for technical reasons, most authors assume that the underlying financial and demographic events are stochastically independent.In the present paper, we study sufficient and necessary conditions under which general transition forward rates are indeed consistent with respect to the relevant insurance claims. This shows the theoretical limitations of the forward rate concept in life insurance. Our study is based on a model where the underlying financial and demographical developments are diffusion processes driven by a multivariate Brownian motion. This allows us to investigate independence properties by analyzing the asymptotic behavior of mixed (conditional) moments. In particular, we obtain that for joint life and disability insurance policies, some specific demographic events need to be dependent in order to ensure consistency.

Modeling and forecasting duration-dependent mortality rates

Mortality data of disabled individuals are studied and parametric modeling approaches for the force of mortality are discussed. Empirical observations show that the duration since disablement has a strong effect on mortality rates. In order to incorporate duration effects, different generalizations of the Lee-Carter model are proposed. For each proposed model, uniqueness properties and fitting techniques are developed, and parameters are calibrated to mortality observations of the German Pension Insurance. Difficulties with coarse tabulation of the empirical data are solved by an age-period-duration Lexis diagram. Forecasting is demonstrated for an exemplary model, leading to the conclusion that duration dependence should not be neglected. While the data shows a clear longevity trend with respect to age, significant fluctuations but no systematic trend is observed for the duration effects.

Integral and differential equations for the moments of multistate models in health insurance

The moments of the random future liabilities of health insurance policies are key quantities for studying distributional properties of the future liabilities. Assuming that the randomness of the future health status of individual policyholders can be described by a semi-Markovian multistate model, integral and differential equations are derived for moments of any order and for the moment generating function. Different representations are derived and discussed with a view to numerical solution methods.

Eine Fortschreibung des Rehabilitationsbedarfs in Deutschland bis 2040 anhand demografischer Faktoren

The demographic changes in Germany leads to a significant shift in the composition of the population and the workforce, this affecting the future need for medical rehabilitation. This paper estimates the future change in rehabilitation demand based on a forecast for demographic changes till 2040. First, the sensitivity of the rehabilitation demand with respect to demographic factors is estimated. Second, the demographic factors are projected by stochastic methods, resulting in forecasts for the future need for medical rehabilitation. The projections show that the short-term demand is likely to rise. Theoretically, yearly wage increases of about 2.2% are needed for covering the increasing medical rehabilitation costs from 2010 till 2017. For the mid-term demand, the model predicts a slight decline in rehabilitation cases. Considering all these facts, the budget for rehabilitation will probably not cover the future costs for rehabilitation. However, the long-term forecast is subject to considerable uncertainty.

Dynamics of solvency risk in life insurance liabilities

We describe the time dynamics of the solvency level of life insurance contracts by representing the solvency level and the underlying risk sources as the solution of a forward–backward system. This leads to an additive decomposition of the total solvency level with respect to time and different risk sources. The decomposition turns out to be an intuitive tool to study risk sensitivities. We study the forward–backward system and discuss two methods to obtain explicit representations: via linear partial differential equations and via a Monte Carlo method based on Malliavin calculus.