REVERSIONS. 



nuity), produces 208.23/. or 208/. 4r. id. nearly for the 

 value required. 



Corollary.— In like manner, the value of an annuity 

 for a given term of years, after the expiration of an- 

 other term, may be obtained by fubtracting the value for 

 the prefent term from the value during the prefent and re- 

 verfionary terms: thus, the value of an annuity of 10/. 

 during 15 years, after the expiration of 12 years, is found 

 by fubtracting 9.385 (the value for 12 years at 4/. per cent. 

 by Tab. III. Annuities) from 16.329 (the value by the 

 fame table for 27 years), and multiplying 6.944, the dif- 

 ference, into 10; which produces 69.440/. or 69/. 8j. lod. 

 nearly for the value required. 



Pkob. II. 



To find the value of the reverfion in fee of an eftate, or 

 annuity, after the extinction of a given life. 



Solution. — Deduct the value of an annuity on the given 

 life from the perpetuity ; multiply the remainder into the 

 annual produce of the eftate, or into the annuity, and the 

 product will give the value fought. 



Example. — Let the annual produce of the eftate or an- 

 nuity be 18/. the age of the pofleflor of fuch eftate or an- 

 nuity 35, and the rate of intereft 3/. per cent. By Tab. VI. 

 (Life- Annuities), the value of an annuity on a life of 35, at 

 3/. per cent, is 15.938. The perpetuity, at the fame rate, is 

 33-333 ' tne difference between thefe two values, or I7-39S> 

 multiplied into 18, produces 313.no/. or 313/. 2s. 2d. for 

 the anfwer. 



Corollary. — The reverfion in fee after two or three joint 

 lives, or after the longeft of two or three lives, is found in 

 the fame manner, by deducting the values of thofe lives from 

 the perpetuity. Thus the value of two joint lives, aged 

 30 and 35, by Tab. VIII. (L,lFE-Annuities), at 3/. percent. 

 is 12. 131, which being deducted from 33-333» and mul- 

 tiplied into 5, will give 106.01/. or 106/. or. 2d. for the value 

 of the reverfion in fee of an eftate of 5/. per annum after the 

 extinction of thofe joint lives. In like manner, the value of 

 the reverfion in fee of an eftate of 10/. per annum, after three 

 joint lives, aged 30, 35, and 40 years, and computing at 

 ^percent, may be found by Prob. IV. (LiFT.-Annuities) to 

 be 163.81/. or 163/. l6.r. 2d.; and after the longeft of 

 thofe three lives, it may be found by Prob. V. (Life An- 

 nuities) to be 59.980/. or 59/. 1 9/. jd. nearly. 



Prob. III. 

 To find the value of a given /urn, payable on the extinction 

 of a given life. 



Solution. — Deduct, as in the preceding problem, the value 

 of an annuity on the given life from the perpetuity ; multiply 

 the remainder by the given fum, and divide the product by 

 the perpetuity increafed by unity ; the quotient will be the 

 anfwer. 



Example. — Let the fum be 1000/. the age of the given 

 life 45 years, and the rate of intereft 5/. per cent. By 

 Tab. VI. (L.1FE.-Annuities), the value of an annuity, on a 

 life of 45, at 5 per cent, is 1 1.105, which being deducted 

 from 20, the perpetuity, leaves 8.895. This remainder, 

 multiplied into 1000, and divided by 21, gives 423.571/. 

 or 423/. lis. $d. for the value required. 



In the fame manner may be found the value of a given 

 fum, payable on the extinction of two or three joint lives, or 

 of the longeft of two or three lives. Thus, the value of two 

 joint lives, of 40 and 50, at4^fr cent, by Tab. IX. (Life- 

 Annuities), being 8.S34; the difference between this value 

 and 25, the perpetuity, will be 16.166 j which being mul- 

 tiplied into iooo, and the product divided by 26, will give 



621.77/. or 621/. 15J. 5</. for the prefent value of 1000/. 

 payable on the extinction of thofe joint lives. Again, by 

 the rule in the article on %as%- Annuities, and Tables VI. and 

 IX. the value of an annuity on the longeft of two lives, 

 aged 40 and 50, at 4 per cent, may be found equal to 

 15.627, which being deducted from 25, and the remainder 

 multiplied into 1000, will produce 9373, and tl- is, divided 

 by 26, will give 360.5/. or 360/. los. for the value of 1000/. 

 payable on the deceafe of the furvivor of thofe two lives. 



Remark. — It will be obferved, that the value of the re- 

 verfion of an annuity is greater than the value of the reverfion 

 of a fum, in the proportion of 1/. incrcaled by its intereft 

 for a year to 1/. ; or, which is the fame thing, in the pro- 

 portion of the perpetuity increafed by unity to the perpe- 

 tuity. In the one cafe, the payment of the annuity becomes 

 due at the end of the year, in which the life or lives become 

 extinct ; in the other cafe, the fum only becoming payable 

 at the end of that year, the annual intereft upon it cannot 

 be received till the end of the fucceeding year. See Dr. 

 Price's Treatife on Reverfionary Payments. Note E, Ap- 

 pendix. 



Prob. IV. 



To find the value of an annuity for a given term of years 

 after the extinction of any number of lives. 



Solution. — Subtract the value of an annuity on the life 

 or lives from the perpetuity ; multiply the remainder into 

 the prefent value of an annuity for the given term, and 

 divide the product by the perpetuity ; the quotient mul- 

 tiplied into the annuity will be the value fought. 



Example. — Let it be required to determine the value of 

 an annuity of 10/. for 20 years, which is not to commence 

 till the extinction of a life of 25, reckoning intereft of 

 money at 5 per cent.. By Tab. VI. [\^ife- Annuities) the 

 value of an annuity on a life of 25, is 13.567 ; this value, 

 fubtracted from 20, and multiplied into 12.4622, the value 

 of an annuity for 20 years, by Tab. III. (Annuities) 

 produces 80.1693 ; which being divided by 20, and 

 the quotient multiplied into 10, gives 40.085/. or 40/. 

 is. gd. nearly for the anfwer. By proceeding in the fame 

 manner, the value of an annuity of 15/. for 25 years, after 

 the extinction of two joint lives, aged 30 and 40, and after 

 the longeft of thofe lives, computing at 4 per cent, may be 

 refpectively found to be equal to 136/. and 70/. 8/. 2d. 

 by Tab. VI. and IX- (Life- Annuities), and Tab. III. 

 (Annuities). 



Prob. V. 



To find the value of an annuity after the deceafe of a 

 given life, or of any number of lives, during the continuance 

 of another life, or of any number of lives, to be nominated 

 at the time of fuch deceafe. 



Solution. — This differs very little from the preceding pro- 

 blem, and is anfwered much in the fame manner. Subtract 

 the value of an annuity, on the life or lives, from the per- 

 petuity ; multiply the remainder into the value of the life 

 or lives at the time of their nomination, and alfo into the 

 given annuity ; divide the product by the perpetuity, and 

 the quotient will be the anfwer. 



Example. — Required the value of an annuity of 100/. 

 after the deceafe of a perfon aged 30, during the continuance 

 of the life of a perfon to be nominated at the time of fuch 

 deceafe, whofe age may be fuppofed to be then about 15 

 years, reckoning intereft of money at 4/. per cent. 



By Tab. VI. the value of a life of 30 is 14.781, which 



being fubtracted from 25, and the remainder multiplied 



into 16.791 (the value by the fame table of an annuity on 



5 a life 



