Abraham Zaks
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3 publications
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Joining insured groups: how to split the emerging profit
Elinor Mualem , Abraham Zaks doi: http://dx.doi.org/10.21511/ins.08(1).2017.03Insurance Markets and Companies Volume 8, 2017 Issue #1 pp. 29-33
Views: 835 Downloads: 135 TO CITE АНОТАЦІЯIn the process of evaluating the premium of an insurance plan, one considers the risk arising from various uncertainties. The authors suppose for a plan whose net premium is p and the standard deviation is σ the premium including the risk factor will be p + 3σ for a given member, and 3σ reflects the risk. For a group of n members with the same premium p and with standard deviation σ, the premium including the risk factor will be p + 3σ/√n where 3σ/√n reflects the risk for each member of the group. The authors study the emerging profit in case of n insured groups each with its own premium and its own risk when all the n insured groups merge into a single group uniting all insured members. They prove that there emerge a profit due to joining the n groups into a single one due to a reduced total risk of the n separate insured groups when merging into a single group. The emerging profit between the various groups may be divided using the Shapley values method or using utility functions for each group. The auhors discuss various reasonable ways to split the emerging profit between the n groups and show that the split of the profit depends on the chosen method. The main tools are techniques of game theory, in particular those of cooperative games.
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Fair split of profit generated by n parties
Elinor Mualem , Abraham Zaks doi: http://dx.doi.org/10.21511/ins.09(1).2018.01Insurance Markets and Companies Volume 9, 2018 Issue #1 pp. 1-5
Views: 875 Downloads: 118 TO CITE АНОТАЦІЯThe authors studied the process of merging insured groups, and the splitting of the profit that arises in the process due to the fact that the risk for the merged group is essentially reduced. There emerges a profit and there are various ways of splitting this profit between the combined groups. Techniques from game theory, in particular cooperative game theory turn out to be useful in splitting of the profit. The authors proceed in this paper to apply techniques of utility theory to study the possibility of a fair split of that profit. In this research, the authors consider a group of n parties 1,...,n such that each of them has a corresponding utility function u1(x),...,un(x) . Given a positive amount of money C, a fair split of C is a vector (c1,...,cn) in Rn, such that c1 +...cn = C and u1(c1) = u2(c2) = ... = un(cn). The authors presume the utility functions to be normalized, that is ui(c) = 1 for each party i, i = 1, ... ,n. The authors show that a fair split exists and is unique for any given set of utility functions u1(x), ..., un(x), and for any given amount of money C. The existence theorem follows from observing simplexes. The uniqueness follows from the utility functions being strictly increasing. An example is given of normalizing some utility functions, and evaluating the fair split in special cases. In this article, the authors study the case of merging two groups (or more) of insured members, they provide an evaluation of the emerging benefit in the process, and the splitting of the benefit between the groups.
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Risk premiums in life insurance
Elinor Mualem , Abraham Zaks doi: http://dx.doi.org/10.21511/ins.10(1).2019.01Insurance Markets and Companies Volume 10, 2019 Issue #1 pp. 1-8
Views: 2567 Downloads: 274 TO CITE АНОТАЦІЯ
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