Joseph Louis Lagrange, ``Solution of a problem concerning annuities'' (Q2866540)

In this article, the author presents a meticulous edition of a memoir presented by Lagrange to the Berlin Academy of Sciences in 1776. Its occasion was the creation of the Königlich Preußische Allgemeine Wittwen-Verpflegungsanstalt. The memoir was never published by the academy, plausibly (as suggested by the author in the abstract) because it implied the bankruptcy of the institution, which was promoted by the powerful minister Friedrich Wilhelm von Schulenburg.NEWLINENEWLINEThe scheme presupposed an initial payment followed by yearly contributions. Lagrange at first makes a very down-to-earth direct computation, and afterwards introduces various transformations of the problem that simplify the calculations. Using the survival tables at disposition, and choosing those of Thomas Simpson (1742, using London data) and Antoine Deparcieux (1760, French-based) as presenting the highest respectively the lowest mortality rates, Lagrange shows the scheme to be in any case advantageous to the insured and disadvantageous to the bank -- most outspokenly for the youngest couples (assuming a specific interest rate, which was probably the one that was current at the epoch).