On the Value of Contingent Assurances with Certain Limitations

SIR,—Should you consider the accompanying solutions to the two following important problems worthy of a place in your Magazine, they are at your service. You will observe that both of these problems have been previously solved by the late Mr. David Jones, in the first volume of his work on Annuities and Reversionary Payments, pages 140 and 184, but by processes very laborious, and resulting in formulae totally unfit for practical use. Mr. Peter Hardy has since solved the second problem, in the second volume of the Assurance Magazine, by a method equally if not more laborious than that above referred to; however, he possesses the claim of being the first who has put the result in a practical form. This problem was last solved in a very elegant manner, without analysis, by your very able and talented correspondent Mr. T. B. Sprague, in the last number of your Magazine, after which it would be idle for me to attempt giving a more concise or elegant solution; but having some doubt as to whether the mode of investigation he has made use of is such as would occur to the student on first reading the problem, without having previously seen the analysis, I have been induced to send you the following general and brief analytical solutions to the two problems in question, thinking they may prove interesting to your actuarial readers. I am, Sir, Your obedient Servant, WM. CURTIS OTTER, F.R.A.S. 4, Trafalgar Square, w.c.