Risk premiums in life insurance

Abstract

For a person insured under an insurance plan the present value of the cash flow of a premium of 1 is a random variable X , with expected value E ( X ) and variance Var ( X ), and the present value of the cash flow of a liability of 1 is a random variable Y, with expected value E (Y ) and variance Var (Y ). The authors follow the classical use of the equation PE ( X ) = LE (Y ) to derive the net premium P for the insurance of L. The net premium p for an insurance of 1 is p = P / L. To evaluate the risk of the plan, the authors derive the random variables , X . Y. from X , Y by a change due to the standard deviations ( ). x y σ σ The random variables , X . Y. are the present values, considering risk of the cash flow of a premium of1 and of a liability of 1 and the gross premium P . is derived from the equation P.E (X. ) = LE (Y. ). The gross premium p. for an insurance of 1 is p. = P. / L. There appears a risk load δ, and the gross premium p. per unit insured is p. = p +δ . For the relative risk load δ / p the following holds: δ / p = ( p. − p) / p = (P. − P) / P. More on premium and risk may be found in Gerber (1997) and Kaas et al. (2008). Assuming that n persons are insured in the insurance plan, then the estimates for the risk are reduced to / , x σ n / , y σ n where ( ) ( ) 2 / / , x σ n =Var X n ( ) ( ) 2 / / , y σ n =Var Y n respectively. Thus, the risk load δ decreases when n increases, and there appears a profit in the case of merging insurance portfolios. This profit is due to the reduced risk load and points out the advantage of enlarging n whenever possible, as in the case of using reinsurance. The standard deviation is relatively simple to calculate, and turns out to be easily used to determine the risk load on the net premium. Mualem and Zaks (2018) discussed discussed ways of splitting the emerging profit due to a reduced risk load when merging several portfolios. The authors investigate the emerging profit in the cases of the whole life insurance, the n-year endowment insurance, and the pension scheme, and suggest in the conclusion several similar cases in which the methods that are discussed may be used..