94 BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 



Example 2. — The effective rate of interest is 6%; to find trie corresponding nominal 

 rate when interest is convertible semiannually. 

 Here i and m are given to find j; hence from formula (3) there results 



j=2[(l+.06)2-l]=2(1.06p-2=2.059126-2=.05912(i. 



It is necessary to extract the square root of 1.06. The final result shows that j— 

 5.9126%, and again the nominal rate is smaller than the corresponding effective 

 rate. 



Amount of 1 in n years at compound interest. — Let the 



effective rate of interest be i. At the end of the first year the accu- 

 mulation is 1+i. During the second year this principal 1+i will be 

 increased in the ratio of 1 to 1 + i, and will therefore amount at the 

 end of the second year to (l+i)(l+i), or (1+i) 2 . In this way at 

 the end of n years the amount is (1+i) 71 . 



Let P be the principal and S the amount of P at the end of n 

 years at compound interest at the effective rate i. Since 1 amounts 

 to (1 + i) n in n years, P would amount to Pil +i) n . There results, 

 therefore, the formula 



S=P(l+i) n . (4) 



Hence 



I>=8/(l+i) n = Sv n , (5) 



where 



«= 1/(1+0. (6) 



If in the above formulas 1+i is replaced by (1 +j/m) m , to which 

 it is equivalent according to formula (2), it follows that 



S = P(l+j/my"»; (7) 



and 

 where I* = S/(1 +j/m) '" " = Sv" in , (8) 



v=l/(l+j/m). (9) 



These formulas express the relation between P and S in terms of 

 the nominal rate j and the frequency of conversion m. The values 

 to seven places of decimals of (1 +i) n and v n for various rates of 

 interest and for 60 intervals or years are given in Tables 31 and 34. 



Example 3. — To find the amount of $12,375 at 3% compound interest in 30 years. 

 By formula (4) 



S = (1 + .03) 30 X$12,375=2.4272625X$12,375=$30,037.37. 



The value of (1.03 ) 30 was taken from Table 31. 



Example 4. — $12,375 is placed in a bank; to find the amount in 30 years if interest 

 is 3% and is compounded semiannually. 



The nominal rate of 3%, convertible twice a year, requires formula (7) with i=.03 

 and m=2. Substituting, the result is: 



S=(l+.03/2) 2X30 X$12,375=(1.015) 60 X$12,375=2.4432198X$12,375=$30,234.85. 



