HIGHWAY BONDS. 93 
Mathematical rates.—The effective rate of wnterest is the interest 
earned by one unit of principal (one dollar) in one unit of time (one 
year) when interest is compounded at the end of each stated interval. 
The nominal rate of wterest is the total interest earned by one unit 
of principal (one dollar) in one unit of time (one year) when interest 
is not compounded at the end of each stated interval. 
It follows that the nominal and effective rates of interest coincide 
only when the stated interval is the unit of time (one year). 
Commercial rate.—In commercial transactions the rate of 
interest is usually quoted as a rate per cent, or per hundred units of 
principal, instead of a rate per unit of principal, as in the above defi- 
nitions. To find the mathematical rate as above defined, divide the 
commercial rate by 100. For example, the mathematical rate cor- 
responding to the commercial rate 6 per cent is 6/100, or .06. The 
mathematical rate is used in the following formulas. 
Relation between effective and nominal rates of interest.— 
In any transaction there is an effective rate of interest 7 and a corre- 
sponding nominal rate of interest 7. This relation can be expressed 
by an algebraic formula which involves the number of stated 
intervals, m,in one year. At the nominal rate 7, during each stated 
interval 1/mth of a year in length, one unit of principal would earn 
j/m i interest which, added to the unit, gives an amount 1+ 7/m. 
If the principal 1 accumulates in the first interval to 1 +7/m, it follows 
by proportion that the principal 1+ 7/m would accumulate in the 
second interval to (1+ 7/m)?. In like manner, at the end of the mth 
interval, the accumulation would be (1+7/m)™. The total interest 
earned in the m intervals, or one year, is the difference between the 
accumulation and the original unit of principal, which by definition 
is the effective rate of interest 2. Hence the fundamental formula: 
e— Cg) —A (1) 
or 
14+¢=(1+j/m)™. (2) 
Solving for 7, there results 
j= m|(1+4)"— 1). (3) 
The number of times, m, that interest is added, or converted into 
principal each year, is the frequency of conversion. A nominal rate 
of interest, convertible m times a year, is indicated by the sym- 
bol I(m)- 
Example 1.—The nominal rate of interest 7 on deposits is 3% and interest is added 
to the principal every six months; to find the effective rate 7. 
Here j=.03 and m=2. From formula (1) there results 
i=(1+.03/2)2—1=(1.015)2—1=.030225. 
The effective rate 3.0225% is thus slightly higher than the corresponding nominal 
rate convertible twice per annum. 
