Interchangeability of the median operator with the present value operator

摘要

It is well known that the expected present value of a life annuity is smaller than the present value of an annuity certain with term equal to life expectancy. This result can be viewed as a consequence of the lack of interchangeability between the present value operator and the mathematical expectation operation. However, we prove that the median and present value operators can be interchanged; that is, the median of the present value of a life annuity equals the present value of an annuity certain whose term is the median additional years of life. At young ages and through late middle age, median additional years of life exceed life expectancy. Therefore, the median value of a life annuity exceeds an annuity certain paid to life expectancy which exceeds the expected present value of a life annuity - the fair price of a life annuity.