XXX11 APPENDIX THE SIXTH. 



Now, it generally happens, that the property of an 

 insurance office consists of funds invested at different 

 rates of interest, the consequence of which is, that there 

 is no absolutely rigorous method of determining the 

 profit, except by prospective calculation of the state of 

 the office for every year of the tabular duration of the 

 life of its youngest member. Supposing the insured to 

 die precisely in the manner indicated in the table, and 

 assigning the order in which the different principals are 

 to be touched, when necessary, it is then possible to 

 calculate the amount which will remain when all claims 

 are paid. The present value of this amount (the spe- 

 cies of stock in which it is to be left being known) is 

 all that can be called profit at the time of the valuation. 

 This process, however, is exceedingly laborious ; and, 

 n all probability, where yearly valuations are made, the 

 expence of making the calculation would be greater than 

 the loss prevented by taking the more simple, but less 

 accurate, method. 



If money made only simple interest, and computations 

 were performed accordingly, no difficulty would arise : 

 for =^S improving at r per pound, and =S' at r' 

 per pound, is at all times equivalent to <^P(S + S'), im- 

 proving at (Sr-t-S' r x ) -r- (S-fS') per pound : so that 

 all the different stocks might be considered as lying at 

 one average rate of interest. Such, however, is not the 

 case with compound interest. 



To introduce the question in a simple form, let us sup- 

 pose that all the stock of the office makes r per pound, 

 the rate assumed in the valuation, except only one sum, 

 H, which makes r' (less than r) per pound. If, then, 

 this sum were set down as H in the item A, the profit 

 would be overrated ; nor can we answer the question, 

 how much should it be estimated at, without some re- 

 ference to the time at which H, with its accumulations, 

 is to become necessary. If this will not be wanted for 

 n years, then H x (1 -f r') ra -s-(l+r) n is the value at 

 which it must be estimated. 



