S Siipptemerit to the Value of Uwreasing Life Annuities. 



Now let the theorems in the paper referred to, as they occur 

 in order, be severally denoted by P. Q.R. S. (— ."') the ex- 

 pectation of the given life recelvhig Jzl a year hence; c. c the 

 complements of the same, and of a life one year older. Then 

 •after effecting the neces;»ary substitutions, and by reduction, the 

 resulting equations may be represented as follows : 



S.r — 2 /■ . f' + c — '■\ 

 ■ B-D + c-l. + c^.^A-AA}- ,,._, \B — T^ — .x J 



cA 

 P= _ 



€ix—\) 



B being the value of an annuity 1. 2. 3, 4. for the term certain 

 equal to c, and D that of ^'1 for like term. 



Q= c(A-AA)+c'.(.V-AA').^a:. 



Now if P' denote the value as found by the first formula for 

 a life 1 veav older than A, ue shall obtain 



R = 



-I 

 oA+:pv/iSx 



And, b= 



' a 



Any given sum being proposed to be advanced in lieu of an 

 equivalent increasing life annuity, one general instance of the 

 utilitv of these formula; then would be, to assign what the ratio 

 of the series should be, because allied in principle to that which 

 determines the \alue of the annuity sought ; and I conceive there 

 is a numerous class of individuals in tiie speculative order of so- 

 ciety, and of others having no prospective views as to the future 

 disposition of tlieir property, but who, prudently contemplating 

 a time of adversity iiappcning, would be induced to stipulate for 

 the gram of an annuitv beginniiig with some small annual payment, 

 and increabing upwards, so that it is rendered an object the move 

 inviting as the duration of life is prolonged. These persons, it may 

 i)e imagined, would infer a peculiar policy in this mode of assur- 

 ance, independent of the pleasing reflections it must inspire at a 

 season (as we jjerceive but too commonly evidenced) when human 

 nature hath to recount the sublunary ills of an eventfid and che- 

 quered life. I aim not at objects merely speculative and curious, 

 but practically advantageous. 



Yet I am sensible of an existing cause that tends to counter- 

 act a spirit of stimulus for life annuity business in a quarter where 

 most naturally it would be expected, arising from the tables at 

 this time so indiscriminately resorted to being very defective and 

 ill-suited thereto; anj indeed its objects, at first confined to the 



chartered 



