94 BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 
Example 2.—The effective rate of interest is 6%; to find the corresponding nominal 
rate when interest is convertible semiannually. 
Here 7 and m are given to find 7; hence from formula (3) there results 
j=2[(1-+.06)2 —1]=2(1.06)? -2=2.059126 —2—=.059126. 
It is necessary to extract the square root of 1.06. The final result shows that 7= 
5.9126%, and again the nominal rate is smaller than the corresponding effective 
rate. 
Amount of 1 inn years at compound interest.—Let the 
effective rate of interest be 7. At the end of the first year the accu- 
mulation is 1+7. During the second year this principal 1 +7 will be 
increased in the ratio of 1 to 1+7, and will therefore amount at the 
end of the second year to (1+7)(1+2), or (1+72)?. In this way at 
the end of n years the amount is (1 +7)”. 
Let P be the principal and S the amount of P at the end of n 
years at compound interest at the effective rate 7. Since 1 amounts 
to (1+72)" in n years, P would amount to P(1+7)". There results, 
therefore, the formula 
pee Oba Ce (4) 
Hence 
P=S/1+6)"=Se, (5) 
where 
v=1/(1+%). (6) 
If in the above formulas 1+7 is replaced by (1+7/m)™, to which 
it is equivalent according to formula (2), it follows that 
S=P( +j/m)", (7) 
and . 
where P=S8/1+)/m)r = Som, (8) 
v=1/(1+j/m). (9) 
These formulas express the relation between P and S in terms of 
the nominal rate 7 and the frequency of conversion m. The values 
to seven places of decimals of (1+7)" and v” 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)° x $12,375=2.4272625 X $12 ,375=$30,037.37. 
The value of (1.03)°° 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 7=.05 
and m=2. Substituting, the result is: 
S=(1+.03/2)?°°x $12 375=(1.015)° x $12,375=2 4432198 x $12,375=$30,234.85, 
