ASS 



ASS 



47 1. 12 s. 4(3. If the odds agalujl its failing be two to lefa 'tlian tlie gi«n tenn, and the rrprved qnotlenl beiiv 

 one, that is, if it may be expected that fome one of three divided by this laft value, incieafed by unity, will give th 

 lives, at the age of the given life, will fail in the year, the 



value of the ajfiirr.nce will be a th'ird of the firft value, 

 reckoning the fame intereil, or 31I. 148, iid. If the 

 odds be nineteen to one, or if it may be expcfted that fome 

 one out of twenty lives^ at the age of the given life, will 

 fail in a year, the value of the njpirance will be a tivcntietb 

 part of the firft value, or 4I. 153. 3d. If the odds be 

 lorty-niae to one, or only one out of lifty fuch lives aa 

 the given life can be ei:pefted to fad in the year, the value 



g- 

 give the 

 rcqun-cd value of the a(!\irance in 3. Jixed annual pzyment, 

 till either the life fails, or the term ends." 

 Example. 

 Let the term be 27 years, the hfe aged 39, the fui« 

 lool., and the interefl: 5 J>er cent. 



Solution. 

 The value of the life of a perfon whofe age is 39, for 2» 

 years, is (reckoning iiitcrell at 5 per cent, and by the 

 N/.rl/jam/i/on "^rMi of L.iTt. ylnni:i/ies)ii.igx. This value 



of the flij/i(raa« will be a /?/>ii://j part of the firft value ; that fiibtrafted from 14.643 (the value of an annuity certai* 



is, it will be l 1. 18 s. id. Now the odds of three to one for 27 years, fee Tab. III. Annuities), leaves 3.452, th 



are, according to the Northamptan Tabic of Obfervations remainder to be refcrved. The value of i 1. to be reiyived a 



(fee Mortality), the odds that a life aged 92 will not 

 drop in a year. The odds of [9 to l are tlie odds, accord- 

 ing to the fame table, that a hfe aged 65 will not drop in 

 «■ year ; and the odds of 49 to i are the odds tiiat a life 

 aged 39 will not drop in a year. It follows, therefore, 

 that i\ie value of the ajfiirance. of lOol. for a year on a life 

 aged 92 is 31!. 14s. I id. ; on a life aged 6^, 4 1. 15 3. 3d. 

 on a life aged 39, l 1. l8s. id., reckoning intereil at 5 

 per cent, if intereil be reckoned at 3 per cent, thefe values 

 willbe32l. 76.3d.; 4I. 17s.; 1 1. 18s. lod. 



The alTtirauces moll commonly praclifed are thofe on 

 fingle lives, either for a given term, or during their whole 

 continuance. When a life is afTured for a given term or 

 number of years, the value may be paid either in one Jingle 

 prefent payment, or m annual payments, to be continued till 

 the failure of tiie life, (hould that happen within the term ; 

 or if not, till the determination of the term. 



The method of finding thefe values cannot be eafily 

 . wnderftood by thofe who are unacquainted with the doc- 

 trine of life-annuities, as it has been taught by mathemati- 

 cians ; but the following obfervations niay be of ufe to 

 give fome general idea of the fubjeft. — Let us fuppofe- 

 that a perfon aged 39- years wants to affure 100 1. on 

 his life for 27 years, or till he is €,6 years of age, and 

 that he chufes to advance the proper compenfation for it in 

 a fixed annual payment, the firll to be made imm.ediately, 

 and the following payments to be continued till either the 

 term ends, or his life drops. The value of the affurance 

 for the Jirjl year, is by what has been already fhewn, 

 I 1. 18 s. id. reckoning intereil at ^percent. The value 

 of the affurance for the Iq/l year of the term, fuppoiing 

 him to have hved to the beginning of it, or to have com- 

 pleated G^, is like«nfe, by what has been already (liewn, 

 4I. 15s. 3d., reckoning all along at the fame intereil. 

 If, therefore, the value of the affurance for the whole 27 

 years is to be one conilant fum payable at the beginning 

 of every, year, that fum, it is obvious, ought to hi greater 

 than the Jirjl, and lefs than the LJl ; or a fum which is 

 fome mean between 1 1. iSs. id. and 4I. 15 s. 3d. The 

 rule for finding this mean in all cafes is the following, 



le 

 cfxveA at 

 the end of 27 years is .26785, by Tab. II. under tlie article 

 Annuities. The probability that the life of a perfos 

 aged 39 fliall fail in 27 years, is, by the Northampton Table, 

 (fee Mortalitv) xfrl' ^'"■'^ the perpetuity is 20. Thefe 

 numbers multiplied by one another, and 3.452 added t» 

 the produsS, make 6.568, which multiplied into 100 1. the 

 given fum, and divided by 21, the perpetuity increafed bf 

 unity, gives 31.276 for the quotient to be referved. 



The value of an annuity on a life of 39 for 26 years, if 

 II. 019. Dividing therefore 31.276 (the refervird quotient) 

 by 12.019, "f ^'^ value of the above annuity, with unity- 

 added, we have 2.60 1., or 2 1. 12 s., which is the required 

 value, in Jlxed amival payments, of the affurance of 1 00 1. 

 on the given life for 27 years, reckoning intereil at 5 per 

 cent. 



The value of the fame affurance in one prefent payment it 

 the quotient rcferved above, or 3 1 1. 5 s. 6 d. ; in other' 

 words, it is the value of an annuity of 2 1. 12 s. for 26 year* 

 on a life of 39 ; the firll payment of which is to be made im- 

 mediately, and the remaining ones at the beginning of each 

 year ; or, it is the fura ariting in the foregoing operation 

 before the divilion by the value of the life for the term of 

 26 years. 



If the ajfurance is to be made for the whole pofTible 

 duration of the life, the method of finding 'the value will 

 be more fimple, and the rule for this purpofe is as follows. 

 " From the perpetuity fubtraft the value of the given life, 

 and multiply the remainder by the given fum, and this lall 

 produft divided by the perpetuity, increafed by unity, will 

 give the value in ^fmgle prefent payment. And this payment, 

 divided by the value of the life, will give the value of the 

 affurance in annual payments during the continuance of the 

 life." 



Example. 

 Let the age of the life be, as in the lafl example, 39 ; 

 the fum to be afTured for its whole duration 100 1. ; and 

 the rate of intereil 5 per cent. The value of the life, ac- 

 cording to the Northampton Table (fee Life Annuities), it 

 11.979. The value of the life fubtratled from 20 {the per- 

 petuity J is 8.021, which multiplied by 100, the given fum. 



" From tl'.e value of an annuity certain for the gnven and divided by 21, the perpetuity increafed by unitv, gives 



term, found by Tab. III. under the article Annuities, fub- 

 ftraft the v;due of the life for the given term, found by 

 , the method explained under the article l^wt-Annuities, and 

 refene the remainder. Multiply the value of 1 1. due at 

 the end of the given term (found by Tab. I. under the 

 article ylnnuities), by the perpetuity (fee Remark II.), and 

 alfo by the probability (fee Mortality), that the givm 

 life fhall fail in the given term. This produft being added 

 to the referved remainder, let the total be multiplied by the 

 fum to be affured, and afterwards divided by the perpetuity 



8.195 1. or 38 1.4s. for the value in Zl ftngle pr.smtnt of 

 the ajfurance of lOol. for the whole duration of a life aged 

 39, reckoning intereil at 5 psr cent. And this payment 

 divi^led by 11.97915 3.188 1. or 3 1. 3 s. 9 d. the value of 

 the fame affurance in annual payments during the continuance 

 of the life. 



Remark I. 

 If the value of the ajfurance is dcfircd in annual payment^ 

 the firll of which, inflead of being made at the end of the 

 year as the preceding rule f'uppofes, is to he made immedi- 



incrcafed by unity, then let this quotient be refri^ed. Find ctlely, the value in a fmgle payment (found as directed above) 

 sext the value of an annuity on the given life for one year mult be divided by the value of the life incrtaj'ed bj unity y 



ii» a that 



