6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1776. 



paid yearly. But it is well known, that generally they are paid half-yearly, and 

 sometimes quarterly ; and that this is a circumstance which always adds to their 

 value. The difference between the values of annuities, according as they are 

 paid in these different ways, being no where stateil with accuracy, Dr. P. has 

 thought that an attempt to do this may be of some use. But, for particulars, 

 see the last edition of Dr. P.'s Treatise on Reversionary Payments. 



From the rules and examples there given it may be gatliered, that the differ- 

 ence between the values of annuities on lives payable yearly, half-yearly, quarterlv, 

 and momently, increases continually with the ages; but, if not secured by land, 

 this difference can never be so great as a quarter of a year's purchase in the case 

 of annuities payable yearly and half-yearly ; f of a year's purchase in the case of 

 annuities payable yearly and quarterly ; and half a year's purchase in the case of 

 annuities payable yearly and momently. , ■ 



Mr. Simpson, in his Treatise on the Doctrine of life-annuities, p. 78, and in 

 his Select Exercises, p. 283, has given a quarter of a year's purchase as the addi- 

 tion always to be made to the value of a life-annuity payable yearly, in order to 

 obtain its value payable half yearly ; and -f- of a year's purchase, if its value 

 payable quarterly is required. But it appears that these are too large additions ; 

 and, whatever be the rate of interest, or the number of lives, a 5th of a year's 

 purchase will be generally niore than a sufficient addition, if the value of the 

 annuity is desired payable half-yearly ; and y\ of a year's purchase, if the value 

 of the annuity is desired payable quarterly. Mr. De Moivre's rules, in p. 85 

 of his book, on life-annuities, for finding tlie values of life-annuities payable 

 half-yearly and quarterly from their values payable yearly, are still less correct ; 

 for they suppose the difterence between these values the same, whether the an- 

 nuities are life-annuities, or annuities certain. 



Mr. Dodson, in the 1st question in the 3d volume of his Mathematical Re- 

 pository, has given a rule for finding the value of an annuity secured by land, 

 and payable yearly, which coincides with that here given ; and Mr. De Moivre, 

 in p. 338 of his Treatise on the Doctrine of Chances, has given a theorem for 

 this purpose, which also brings out nearly the same answers. But Mr. Simp- 

 son, in prob. 1, p. 323, of his Select Exercises, makes the excess of the value 

 of such an annuity above the value of an annuity payable yearly, but not secu- 

 red by land, double to the same excess derived from Mr. Dodson's and Mr. 

 De Moivre's rules. The truth is, that Mr. Dodson's rule gives the exact value ; 

 and that Mr. Simpson's problem gives the value, not of an annuity secured by 

 land and payable yearly, but of an annuity secured by land and payable 

 momently; and also, that his method of solution implies a rate of interest some- 

 what less when the annuity is payable momently, than when it is payable yearly. 



