214 ESSAY ON PROBABILITIES. i 



expectation, an equivalent to -| or 1 -f |, or the value 

 of a perpetuity due. Consequently, since the present 

 value of 1 + 1 to be received at the death of A is | | A, 

 that of II. will be found by dividing the latter by the 

 former : or, 



The present value of 11. to be received at the end of 

 the year in which A dies, is found by subtracting the 

 present value of an annuity on the life of A from that 

 of a perpetuity, and dividing the remainder by the 

 present value of a perpetuity due, or one year's pur- 

 chase more than the present value of a perpetuity. 



EXAMPLE. (Northampton tables, 3 per cent.) What 

 is the value of 11. , to be received at the end of the 

 year in which a life of 30 shall fail ? 



Perpetuity of l at 3 per cent. 33 -3 

 Value of annuity on life of 30 l6'9 



34-3) 16-4(-478 



In the preceding rule any status may be substituted 

 for a single life, and the value of the annuity which is 

 to be paid as long as the status lasts is connected with 

 the present value of I/, to be received at the end of 

 the year in which the status fails, by the preceding 

 simple rule. 



The premium which should be paid (first down, and 

 afterwards at the end of each year), is an annuity due 

 upon the life or status, and is therefore worth -| A or 

 1 + | A year's purchase. Consequently the premium which 

 should be paid for the II. above described is the pre- 

 ceding present value divided by one year's purchase more 

 than the annuity is worth. In the example, divide '478 

 by 1 + 16*9 or 17'9, which gives -026?, so that 21. 13*. 

 6d. is the premium for insuring 100/. at the end of the 

 year in which a life of 30 fails. 



The following rule is somewhat shorter, in the case 

 in which the premium only is required, and not the 

 present value. 



QUESTION. To find the premium which should be 

 paid (first down, &c.), during the continuance of a 



