Theorems in the Doctrine of A^inuities. 259 



time, they may all be superseded by a very simple mode of confii- 

 dering the subject, derived from the progressive fluctuation to 

 which every common annuity must necessarily be liable. 



It is perfectly well known, tliat stocks of every kind, besides 

 their accidental variations, are subject to gradual elevations in their 

 true values, during the intervals of the payment of the dividends , 

 and to a sudden depression, the moment that each dividend has 

 been paid, equal in magnitude to the amount of the dividend : 

 and it is equally obvious, that a life annuity becomes continually 

 more valuable, as the period of the payment approaches, and loses 

 at once the value of one payment the moment that payment has 

 been managed. 



An annuity, for example, of which a payment is due on a given 

 day, is more valuable than an annuity purchased on that day, and 

 payable a year after, by the amount of a year's payment ; and " The 

 value of an annuity beginning to he -payable at any intermediate 

 time between the day of purchase and its Jirst anniversary^ will be 

 greater than the simple tabular value of the annuity by a sum pro- 

 portional to the anticipation of the payments /' the increase of the 

 value being very nearly uniform, when we suppose the anticipation 

 to be gradually increased : this increase of the value depending 

 obviously on the greater probability as well as on the greater 

 proximity of each payment, and proceeding from day to day by 

 very nearly equal increments. Thus if we wished to purchase an 

 annuity of 100/. a year, and its value were 1000/., upon the ordi- 

 nary supposition of the payments commencing after the end of a 

 year ; supposing that we desired to have the first payment made 

 at the end of nine months, and the subsequent payments at intervals 

 of a year as usual : we should have to add 25/. to the purchase 

 money, making it 1025/. at whatever rate of interest the value 

 might have been computed. If we began at six months, 50/., and 

 if at three months, 75/. must be added to the purchase : it being 

 obvious that an additional 100/. would be equivalent to an an- 

 ticipation of twelve months, or to an immediate payment of a 

 year's annuity. 



From this simple and incontestible principle, it is very easy to 



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