Calculation of insurance premium reserves for life insurance and application of the Zillmer method against negative results in the first year

The purpose of this article is to examine the application of the Zillmer method to mitigate or eliminate the negative reserve that arises during the first year when expenses are charged to the insurance premium reserves calculated for life insurance. The net premium reserve for whole life insurance and term insurance types has been calculated using both prospective and retrospective methods. The formulas for calculating the actuarial present value of insurance payments (paid), life annuities, and premiums, which are the main components of the insurance premium reserve, as well as substitution decrements and commutation functions, are provided. Expenses affecting the reserve are classified as initial expenses (α), collection expenses (β), and administrative expenses (γ). The methodology for converting net premium reserves into gross premium reserves by charging the α, β, and γ expenses is presented. Various approaches of the Zillmer method have been analyzed to address the negative premium reserve that arises in the first year when the commission, which is part of the initial expense (α), is added.