56 FOREST VALUATION 



value to $7.1067 if placed at compound interest at 4 per cent for 

 fifty years. 



The value of all other formulae are derived from the expression 



i.op n . The values for discount, , are obtained by reducing 



the fraction to a decimal. In the above case, gives a 



7.1067 fe 



discount factor of 0.1407, which means that $1.00 payable in 

 fifty years has a present value, at 4 per cent, of $0.1407. 



These values may be expressed in tables, which make such 

 calculations unnecessary. (Table VI, appendix.) 



79. Future Value of Temporary Annual Rentals. The 

 future value, or cost, of a rental continuing for a temporary 

 period represented by n annual payments, equals the sum of 

 all these items plus interest upon each from the date of pay- 

 ment to the end of the year n. 



The payments are regarded as occurring in each instance at 

 the end of the year, although they might be made during the 

 year. The final payment thus incurs no interest charge. 



Let r = annual rental. 



Then, final rental for year n = r. 



Rental for year n i, in year n = r X i.op. 



Rental for n 2, in year n = r X i.op 2 . 



Rental for first year, in year n = r X i.op n ~ l . 



These terms form an increasing series showing a geometrical 

 progression. The ratio of increase or multiple is the factor i.op. 

 The sum of these terms gives the total value or expense of the 

 rental at the end of the period. The calculation of these separate 

 values is unnecessary. The sum may be obtained by formula. 



Let F r = total value of rental. 



(1) F r =r + r(i.o/0+r(i.o/> 2 )+r(i.o/> 3 )+. +r(i.o/>"- 1 ). 

 Multiply the equation by the ratio i.op. 



(2) V r (i.op) = r(i.op)+r(i.of)+r(i.of)-\ \-r(i.op n }. 



Subtract (i) from (2). 



V r (i.op)- V r =r(i.op n )-r, 

 V r (i.op- i) =r(i.op n - i), 



rr _ ; .(i.o/> n -i), 

 i.op i 



