Having thus, as I believe, demonstrated that life insurance calculations have nothing to do with probabilities, I come back to the idea of risk. This, as I pointed out at starting, must be taken from the theory of probabilities, or more precisely, from that part of it which has been cultivated since the beginning of this century, by Lagrange, Gauss, Laplace, and others, viz., the method of least squares. In that method is defined the idea of the “mean error,” which is considered as the measure of the danger to which we are exposed in a single case. This “mean error” is the square root of the sum of all the squares of the errors divided by their number; and the squares of the errors themselves are formed from the deviations of all the single cases from the average or most probable value. In insurances depending upon life and death, the value is also calculated according to the average, so that when all the assured are dead, if the mortality has followed the mean numbers given by the table of mortality, and the additions to the premiums for the expenses of management are disregarded, there will be neither surplus nor deficiency. This average value is the so-called net premium, which may be either a single premium or may be payable for a term of years agreed on beforehand. But we can calculate beforehand from the mortality table all the deviations, or the gains and losses, which can arise from the earlier or later death of the lives assured. Squaring all these deviations, and dividing the sum of the squares by their number, and taking the square root of this sum, we get the value of the mean danger or the risk attaching to a single insurance. For further elucidation some applications of this process will now be given.
Valuation of Guaranteed Unitized Participating Life Insurance under MEGB2 Distribution