260 Theorems in the Doctrine of Annuities. 



deduce the values of annuities, payable at intervals shorter than a 

 year. An annuity of 1, payable half yearly, is equal to two annui- 

 ties of J, the one beginning as usual at the end of the year, the 

 other anticipated by half a year ; and the value of this portion is 

 greater than the other by half a year's payment, that is, by J : so 

 that " We may always find the value of a Ufe annuity payable half 

 yearly by adding a quarter of a year to the tabular value of the 

 same annuity ^ 



In the same manner, an annuity payable quarterly may be di- 

 vided into four parts, the value of the three, which become pay- 

 able before the year's end, being greater than the fourtli by 



— - X — , — X — and — x — > respectively ; and the sum 

 4 4 4 2 4 4 



of the whole being greater than the simple value of the annuity by 



16 "^ 16 16 16 8 ' ^" 



" For quarterly payments y we must add^ of a year's value to 



8 



t^e computation made on the supposition of an?iual payments." 



It is easy to show, by continuing the operation, that the limit of 

 this anticipation of the payment, by the continual bisection of the 

 interval, would at last afford us the addition of half a yearly pay- 

 ment for the value of a daily or hourly payment of a proportional 

 part of the given annuity. 



It may also be observed, that when we reckon at three per cent. 

 per annum, an annuity payable half yearly is nearly the same that 

 would be granted on a life a year older, if payable annually : a 

 year and a half, if we suppose the payment to be quarterly ; and 

 two years, if daily or hourly. 



A. B. C. D. 



r, Waterloo Place ^ "iO June, 1826. 



