## 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.

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## Notes

- 1.
- 2.
In Germany this is regulated via the minimum funding ordinance (Mindestzuführungsverordnung), see http://www.gesetze-im-internet.de/mindzv/index.html.

- 3.
Grosen and Jørgensen [10] state that “many insurers now face claims from distinct groups of liability holders distinguished by different guaranteed interest rates in their policies. This raises the problem of how to avoid inequitable treatment of different classes of policyholders within the same fund. Some companies [...] have tremendous concerns over the definition of a correct and fair distribution policy [...]”.

- 4.
In practice, insurance companies earn additional money by the difference between true and best-estimate actuarial assumptions, i.e., in their assumptions on administrative costs or mortality. This income is, however, rather stable over time and is thus—for simplicity—neglected in this analysis. Instead, we concentrate on the financial risks.

- 5.
In Germany, at least \(90\,\%\) of returns have to be distributed to the policyholders (compare the minimum funding ordinance “Mindestzuführungsverordnung”).

- 6.
In Germany, the regulator tries to (at least) mitigate the unequal treatment by rules (3) and (4): If a high guarantee contract benefits by rules (3) or (4), it might not benefit fully from a high surplus in later years as part of a compensation for the other cotracts.

- 7.
Historic daily data was obtained from Reuters (ticker names: DE10YT=RR, GREXP, and GDAXI).

- 8.
Mortality tables are, for example, available from the German government agency

*Statistisches Bundesamt*, see https://www.destatis.de. - 9.
Note that these also include hidden risk reserves that can be accumulated, for example, by exploiting accounting rules.

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## Acknowledgments

We want to thank Alexander Kling, Jochen Ruß, Ralf Werner, and the participants of the 2014 research seminar TU München/University Duisburg-Essen for fruitful discussions on the topic.

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## Additional information

P. Hieber acknowledges funding by the *German Association of Insurance Science* (DVfVW).

## Appendix: Optimization

### 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

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.

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Hieber, P., Korn, R. & Scherer, M. Analyzing the effect of low interest rates on the surplus participation of life insurance policies with different annual interest rate guarantees.
*Eur. Actuar. J. * **5, **11–28 (2015). https://doi.org/10.1007/s13385-014-0102-3

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### Keywords

- Life insurance policy
- Minimal return guarantees
- Surplus distribution