The retrospective premium reserve.

*(English)*Zbl 0721.62105Summary: For an ordinary single- or multiple-decrement life insurance policy, there is essentially only one retrospective premium reserve and its definition is given in standard textbooks on life contingencies. By contrast, for a multi-state policy, each state has its own retrospective (as well as its own prospective) reserve, there is considerable freedom of choice in its definition, and there is a corresponding variation in its properties.

This paper introduces a whole new class of sets of retrospective reserves. The new reserves automatically satisfy Thiele’s differential equation and also have other desirable properties. Each member of the class is defined by the selection of initial prospective premium reserves for states other than the conventional initial state. We note two obvious alternative choices of these parameters and show that one of them makes the corresponding set of retrospective reserves coincide with our previous suggestion in certain cases but not in general. As an application, we point out the implications for surplus accumulation.

This paper introduces a whole new class of sets of retrospective reserves. The new reserves automatically satisfy Thiele’s differential equation and also have other desirable properties. Each member of the class is defined by the selection of initial prospective premium reserves for states other than the conventional initial state. We note two obvious alternative choices of these parameters and show that one of them makes the corresponding set of retrospective reserves coincide with our previous suggestion in certain cases but not in general. As an application, we point out the implications for surplus accumulation.

##### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

##### Keywords:

Markov chain; first- and second-order valuation basis; surplus accumulation; retrospective reserves; Thiele’s differential equation; initial prospective premium reserves
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\textit{H. Wolthuis} and \textit{J. M. Hoem}, Insur. Math. Econ. 9, No. 2--3, 229--234 (1990; Zbl 0721.62105)

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##### References:

[1] | Hoem, Jan M., The versatility of the Markov chain as a tool in the mathematics of life insurance, Transactions of the 23rd international congress of actuaries, 3, 141-202, (1988), Keynote Lecture for Subject |

[2] | Ramlau-Hansen, Henrik, Hattendorff’s theorem: A Markov chain and counting process approach, Scandinavian actuarial journal, 71, 143-156, (1988) · Zbl 0659.62121 |

[3] | Ramlau-Hansen, Henrik, The emergence of profit in life insurance, Insurance: mathematics and economics, 7, 225-236, (1988) · Zbl 0683.62062 |

[4] | Wolthuis, Henk, Savings and risk processes in life contingencies, () · Zbl 0754.62083 |

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