For a random payoff with positive expectation and negative infimum, a unique price exists at which the optimal proportion of investment reaches its maximum. For a random payoff with parallel translated value, the ratio of this price to its expectation tends to converge toward a value less than or equal to 1/2 if its expectation converges to 0⁺.  In this paper, we investigate such properties by using the integral representations of a complete Bernstein function, and we establish several Abelian and Tauberian theorems.

proportion of investment, complete Bernstein function, Tauberian theorem.