Analyzing the effect of low interest rates on the surplus participation of life insurance policies with different annual interest rate guarantees

Abstract

We analyze the effects of a prevailing low interest rates regime on investment decisions of insurance companies and on the risk/return profile of participating life insurance policies with different contractually guaranteed minimum annual return. Our analysis is based on German legislation and a stylized insurance company with two cohorts of insured persons having different minimal return guarantees. Our findings shed some light on the non-trivial interrelation between profit distribution, minimum guarantees, and resulting profitability for the different cohorts. Moreover, we investigate the complex role of the risk reserve that allows insurance companies to redistribute profits in time and, less obviously, also between the cohorts.

Appendix: Optimization

At each time point $t_{j-1}$, we choose an optimal investment strategy $\varvec{\pi }_{t_{j-1}} = (\pi ^{(1)}_{t_{j-1}},\pi ^{(2)}_{t_{j-1}},\pi ^{(3)}_{t_{j-1}})$ such that the probability of the portfolio return $m(t_{j}) := \pi ^{(1)}_{t_{j-1}} r_{t_{j}} + \pi ^{(2)}_{t_{j-1}} b_{t_{j}} + \pi ^{(3)}_{t_{j-1}} \mu _{t_{j}}$ being less than the portfolio guarantee $g(t_{j-1})$ is minimized, i.e.,

Abbreviating

$$\begin{aligned} \varvec{d} := \left( \begin{array}{lll} r_{t_{j-1}}\,e^{-\kappa _1\triangle t} + \theta _1\big (1-e^{-\kappa _1\triangle t}\big ) \\ b_{t_{j-1}}\,e^{-\kappa _2\triangle t} + \theta _2\big (1-e^{-\kappa _2\triangle t}\big ) \\ \theta _3\triangle t \end{array}\right) ,\quad \varvec{\sigma } := \left( \begin{array}{l}\sigma _1\,\sqrt{\frac{1-e^{-2\kappa _1\triangle t}}{2\kappa _1}} \\ \sigma _2\,\sqrt{\frac{1-e^{-2\kappa _2\triangle t}}{2\kappa _2}} \\ \sigma _3 \end{array}\right) , \end{aligned}$$

then $m(t_{j})$ is normally distributed with mean $\varvec{\pi }'_{t_{j-1}}\,\varvec{d}$ and variance $\varvec{\pi }_{t_{j-1}}'\,(\varvec{\sigma }\varvec{\sigma }'\, .\, \Sigma )\,\varvec{\pi }_{t_{j-1}}$ (where . denotes point-wise matrix-multiplication and $'$ transpose). The portfolio guarantee $g(t_j)$ is a constant. In the following, we write the covariance matrix as $V:= \varvec{\sigma }\varvec{\sigma }'\, .\, \Sigma$.

If returns are normally distributed, the objective function can easily be evaluated if one recalls basic properties of the truncated normal distribution (see e.g., [13]), i.e.,

where $\phi (\,\cdot \,)$, respectively $\Phi (\,\cdot \,)$, denote the density, respectively distribution function, of the standard normal distribution.