ASS 



that is, in tlio prcfcnt iiidance, by 12.979, wliich will mak« 

 the rtquirtil vahu nf thcafTurance 2.941 i. iiiilcadof 3.l8al. 

 oral. 18 s. icJ. iiidead of 3I. 3 s. 9d. 



The roafoii of addinpr unity to the values of lives taken 

 from the tables is, tliat in all the tables the values of annui- 

 ties on lives are given on the fuppofition that the lirft pay- 

 ment is not to be made till the end of a year. If therefore 

 the fiill yearly payment is to be made immediately, the 

 value mull exceed that in the tables by one year's pur- 

 cliafe. 



Remark II. 

 The perpetuity meins the value of the fee-fimple of an 

 eflatc, which is found by dividing tool, by its intereil lor 

 a year. I'or example, if the rate of intcreit be 5 1. /«r 

 ant. tool, divided by 5 gives 20 for the perpetuity ; if 

 the rate of inttieft be 4, 3!, or 3 pir ant. lool. divided 

 by 4, 3.5 or 3, will give' 25, 23.571, or 33.333 for the 

 perpetuity. 



Remark III. 



If inftead of a grofs/um, an ejlate or a peip:liml annuity is 

 to be aflured during the whole duration of a life, the value 

 in TifmgL- payment will be " the value of the life fubtracU-d 

 from the perpetuity, and the remainder miikiplicd by the 

 aiinnity, or by the rent of the cilate." — And the value in 

 annual payments ieginning immcili,ii:-/y will be " the fiiigle pay- 

 ment divided by the value of the life incrcafed by unity." 

 — Univerfally, it ought to be remembered that the tifur- 

 ance of an ejlale or annuity after any given life or lives, is 

 worth as much more than the njjurance of a correfponding 

 fum, as tool, increafed 

 than tool. — Thus the prefent values, in fit 



ASS 



the alTurancc for n years is S X 



+ -,+ 

 ar 





I 

 »~ 



I 



r' 



a — J-\-ii" 



+ 



+ 



-\ , &C. 15 = [- 



«/•' /■ ar 



t I It I in r 



a +a -ft; _f 



- - + ■ 

 r 



a — I 



! — ii'-\-a" 



— + - 



c — a'-^ii'-\-a"' 

 ar' 



(fuppofing / to be the number of pcr- 



fons who have died in the n- i''' year). The feiics 



+ ■ 



-„'+.<" 



kc, - - -\ — IS known to exprcls tlie va- 



lue of an annuity on tlie life of A for n years, and the fe- 



ne3 1 -1- — -| to exprcls the value or an 



■ r r- r' r" 



annuity ccrlaii) for n ycai's. Call the firft of thcfe ferics 



A, and the fecond N, then will the whole of the above 



feries be = N — A — 



N 



A 



X N-A + 



perpetuity {or p), 



Now fincc — 



iir' 

 r— I 



■ IS cqua 



will be 



and 



I 



1 to tl 

 /' 



d by its fntereil for a year is greater ^^^^^^ ^,^^ ^^,,^^,^ ^^,,_; ^^^ ^,^^ affurf.ite'of S for « years will 

 e prelent values, in Jingle ana annual •' 



payments of the ajfurana- of an ellate of 5 1. per aim. for 

 ever, and of a icol. in money on the whole duration, or 

 on any part of an iifiigned life, are to one another (intereil 

 being at ^ per cent.) as 105 1. to tool. The rtafon of the 

 difference is, that the algebraical calculations, by which 

 thefe values are determined, fuppofe that xhegrofs fum and 



be — -^ X N— A -\ — agreeable to the rule 



p-\- 1 ar" 



above. 



given 



If the aCTurance be for the whole continuance of life, the 



fraction 



mp 



vaniflies, N becomes equal to the perpetuity, 



the iirfl yearly payment of the Pv.nuUy are to be received at ^^.j p^ j^ ^i,g ygj^g ^f ^„ ari„uity for the whole life of A, 

 the fame time after the extinction of the lives. It is eafy g 



to fee, that this is a circumftance which mull make the lat- fo that in this cafe the exprcffion becomes fimply = --—- 



ter of more value. . , . , . , , . . , r ,- ,■ 1 ' 



This fpecimen is fufficicnt to explain the general nature X /-A, winch is the rule given in words for finding the 



and principles of affurances on fmgle lives, and to teach value of an affurance on the whole poffible duration of the 



in all cafes the method of tindi'.>g the values of fuch af- hie 01 A. 



furances. To tliofe who with to be further informed on If the affnrance be that of an £/7<!/<.' or a />,r/c/!W,j;:n,v//y, 



this fubjtft, it may not be improper to add the following t'le value of each payment of fuch annuity depending on the 



mathematical dcmouRrations of the rules which have been failure of the hfe of A in one, two , three, &c. y ears to n 

 Let a be the number of perfons living at the 



given above. 



age of any given life A ; let a', a", a'", &c. be the number 

 of pcrfons who have ditd in the ill, 2d, 3d, &c. year after 

 the age of A ; let r be 1 1. increaftd by its intereil for a 

 year, and S the fum to be affured. The probability that A 



dies in the ill year is — , the valUe therefore of the af- 

 a 



furance in that year is . The probabiLty that A dies 



in the 2d, after having furvived the ill year, is 



• „ , 1 a — a' 

 years will be 



in one, 

 I 



« - «' + a" 



a-a'-\-a"+a"' 



and the value of the 



fee-fiinple after n years, depending on the contingency of 



A having died in the mean tiu'.c, will be ; the whole 



ar" 



value, therefore, of the affurance will be N — A + — 



and 



and 



multiplied into the annuity ; or fimply ^ — A multiplied into 

 fuch annuity, if the affurance is to be continued during 

 the whole duration of A's life. For the more ample dif- 

 cufTion of tliis fubjeft, the reader is referred to Mr. Simp- 

 fon's " Seled Exercifes," Dr. Price's " Treatife on Rever- 

 fionary Payments," and iVIr. Morgan's " Doflrine of An- 

 nuities and Affurances dated and explained." 



AJfurauccs may be made on any number oi joint lives, or on 



refpcaively. The whole value, therefore, of '"^^ ^""S'J^ °^ *")' '"^*- ^"**^* ^""^ finding the values of fuch 

 «r* ' affurance* 



o.a 

 «onfequently the value of the affurance in the 2d year is — '■ — 



ar' 



In like manner, the value of the affurance in the 3d, 4th, 5th, 



..... n"^ year, fuppofing m to denote the number of pcrfons 



who have died in the nth or laft year, is - — -, -^ 



S.a' 



s. 



