186 ESSAY ON PROBABILITIES. 



plying the number of years' purchase in the perpetuity 

 by the interest (not the amount) of one pound im- 

 proved during the existence of the annuity- Thus, at 

 6 per cent., and in 50 years, l^will become (l/.-r-'054), 

 or 18*5/., so that the interest is 17'5/. This multiplied 

 by l6f, or ^%-, gives 292/. The more correct answer 

 is 29O34/., but the error is not ten shillings in a hun- 

 dred pounds. 



6. The present value of an annuity for a term of years, 

 to begin after the expiration of another term; tech- 

 nically called a deferred annuity. To find this, multiply 

 the present value of such an annuity created immediately 

 by the present value of II. due at the end of the term 

 of deferment. Thus an annuity of WL for five years, at 

 four per cent., to commence at the end of twenty years, 

 is thus found : were it to commence immediately, it 

 would (see last page) be worth 4'45 X 10/. or 44*5/., and 

 the present value of II. to be received twenty years hence 

 is '456/. Multiply 44*6 by "456, which gives 2O3/., the 

 present value required. 



' Extensive tables of the results of the preceding pro- 

 cesses are given in all works on interest or annuities ; 

 the present treatise is not meant to contain more than 

 enough to enable the student to exercise himself in first 

 principles, or the proficient to obtain approximate re- 

 sults, when no more extensive work is at hand. I now 

 pass to the consideration of annuities in general. 



Though the term annuity be generally understood as 

 meaning a sum of money paid yearly or half-yearly to 

 an individual, yet it is important that the reader should 

 consider the word as implying any sum of money paid 

 at fixed intervals, for any term, definite or indefinite 

 provided only that the payment can never be suspended 

 and resumed again. Thus II. to be paid every year 

 'for ten years provided A live as long, with an ad- 

 ditional condition that payment is to cease if C in the 

 term should die, B being alive at the time of C's death, 

 is an annuity within the meaning of the word : nothing 

 can make the payment cease at all, without making it 

 cease altogether. But II. to be paid as long as A, B, 



