**5. Conclusions**

In this work, four accumulative defined contribution pension schemes with a lump sum paymen<sup>t</sup> on retirement are proposed. These schemes differ in relation to inheritance and provide various decrement factors. The availability of various schemes allows the insured to choose the most suitable scheme depending on the size of the contribution or the size of the lump sum payment. When choosing a pension scheme, the insured person may have health or family reasons, for example, elderly parents would like to ensure the future of their minor children.

The results of this work are directly related to the Russian draft law of a new system of voluntary retirement savings, called Guaranteed Retirement Plan. In particular, these results are of practical importance for the Pension Fund of the Russian Federation, as well as for private pension funds and pension actuaries in general.

Let us summarize the main results of this work. For each scheme, we use standard methods of actuarial calculations to construct the balance equation and to obtain an expression for the gross premium. A simulation model was developed to analyze the constructed schemes. The gross premium was calculated on the base of this simulation model. A comparative analysis of various schemes was also carried out. According to our calculation, we obtained


This work can be developed in two directions. It is interesting to consider schemes with a limited inheritance period. This allows one to reduce the size of the gross premium as well as the obligations of the insurer. Since some policyholders may lose jobs and/or do not have funds to pay contributions in full and on time for a certain period, it is interesting to consider schemes that take this into account.

**Author Contributions:** Conceptualization, M.S.A.-N. and S.V.A.-N.; methodology, M.S.A.-N.; software, S.V.A.-N.; validation, M.S.A.-N. and S.V.A.-N.; formal analysis, M.S.A.-N. and S.V.A.-N.; investigation, M.S.A.-N. and S.V.A.-N.; resources, S.V.A.-N.; data curation, S.V.A.-N.; writing—original draft preparation, M.S.A.-N. and S.V.A.-N.; writing—review and editing, M.S.A.-N. and S.V.A.-N.; visualization, S.V.A.-N.; supervision, M.S.A.-N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding. It was done within the framework of the Department of Mathematics and the Financial University under the Government of the Russian Federation Basic Research Programs.

**Acknowledgments:** This work is supported by the Financial University under the Government of the Russian Federation. The authors are grateful to the reviewers for their valuable comments that helped us to improve this work. We would like to thank Solovyev A.K., the head of the Department of Actuarial Calculations and Strategic Planning of the Pension Fund of the Russian Federation for useful discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.
