In this note, we consider a problem and its implication. The problem is the control of the flow of new business in a portfolio in a certain type of policy. Supposing we know in advance what the investment implications of a certain trend of reserves will be so that we can decide a programme of development of the fund purely from investment consideration. If the flow of new business is ‘controllable’, e.g. by the amount of advertising, commission, and reward to the selling staff allowed from time to time, what volume of new business is required in order to build up to a predetermined reserve pattern? In solving this problem, we will use a mathematical integral employed in electricity, physics, chemistry and other fields. This is so fundamental in other fields that perhaps it will solve many more of our problems.

In this note, emphasis is given to the amount of insurance in force because that is customarily done in the United States.