ANNUITIES. 



r=rr=^ 



EXAMPLE. What is the present value 

 of 50/. per annum for 6 years, at 5 per 

 cent, compound interest ? 



i= 2531. 15 S . 8J. 



But such questions are much more readi- 

 ly answered by the following table. 



TABLE II. 



Shewing the present value of an annuity 

 of 11. for any number of years not ex- 

 ceeding 100, at 5 per cent, per annum, 

 compound interest. 



EXAMPLE 1. What is the present va- 

 ue of an annuity of 63/. to continue for 21 

 years ? 



The value in the table against 21 years 



is 12,821153, which multiplied by 63 

 gives the answer 807/. 14s. 7d. 



EXAMPLE 2 --What present sum is 

 equivalent to a nett rent of 201. per an- 

 num for 69 years ? 



The value in the table against 69 years 

 is 19,309,:iO, which multiplied by 20 

 gives the answer 36Z. 3s. lid. 



If any of the annuities in the above ta- 

 ble, instead of being for an absolute term 

 of years, had been subject to cease, if a 

 given life should fail during the term, it 

 is evident that the value would have been 

 lessened in proportion to the probability 

 of the life failing ; and that if. instead of 

 being for a certain number of years, the 

 annuity depended wholly on the uncer- 

 tain continuance of a given life or lives, 

 its value must be ascertained by the pro- 

 bable duration of such life or lives. In 

 order to compute the value of LIFE AN- 

 NUITIES, therefore, it is necessary to 

 have recourse to tables that exhibit the 

 number of persons, which, out of a cer- 

 tain number born, are found to be living 

 at the end of every subsequent year of 

 human life, which thus shew what are 

 termed the probabilities of life. 



Various tables of this kind have been 

 formed by the different writers on this 

 subject, as Dr. Halley, Mr. Thomas Simp- 

 son, M. Kersseboom, M. de Parcieux, 

 Dr. Price, Dr Haygarth, Mr. W'argentin, 

 M. Susmilch, and others; and the true 

 method of computing the value of life 

 annuities,according to the probabilities of 

 any table of mortality, is laid down by 

 Mr. Wilh'am Morgan as follows : 



" Was it certain that a person of a given 

 age would live to the end of a year, the 

 value of an annuity of 11. on such a life 

 would be the present sum that would in- 

 crease in a year to the value of a life one 

 year older, together with the value of the 

 single payment of 11. to be made at the 

 end of a year ; that is, it would be 11. to- 

 gether with the value of a life aged one 

 year older than the given life, multiplied 

 by the value of 11. payable at the end of 

 a year. Call the value of a life of one 

 year older than the given life N, and the 

 value of 11. payable at the end of a year 



; then will the value of an annuity on 

 r 



the given life, on the supposition of a cer- 

 tainty, be i -f -X N = - X 1+N. But 



the fact is, that it is uncertain whether the 

 given life will exist to the end of the year 

 or not ; this last value, therefore, must be 

 diminished in the proportion of this un- 



