Suppose that the portion of surplus allotted to a particular policy at the division of profits, is applied so as to relieve the assured from the annual payments after arriving at an age to he determined by the amount of that portion or cash bonus, the other conditions of the policy remaining the same; or again, suppose that the cash bonus is so applied as to convert the assurance into an endowment assurance, payable at an age to be determined:–it is proposed to investigate formulæ to apply to these two cases.

In the first case, the cash bonus must be equal to the present value at the time of division of profits of a deferred annuity, of which the annual payments are equal to the premium for the assurance, the number of years for which the annuity is deferred being unknown; and the problem, in fact, is to find the number of years.