HIGHWAY BONDS. 93 



Mathematical rates. — The effective rate of interest 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 interest 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 i and a corre- 

 sponding nominal rate of interest j. This relation can be expressed 

 by an algebraic formula which involves the number of stated 

 intervals, m, in one year. At the nominal rate j, during each stated 

 interval 1/mth of a year in length, one unit of principal would earn 

 j/m in interest which, added to the unit, gives an amount 1 +j/m. 

 If the principal 1 accumulates in the first interval to 1 +j/m, it follows 

 by proportion that the principal 1+j/m would accumulate in the 

 second interval to (1 +j/m) 2 . In like manner, at the end of the mth 

 interval, the accumulation would be (1 +j/m) 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 i. Hence the fundamental formula: 



i=(l+j/m)™-l (1) 



or 



l+i=(l+j/m) m . (2) 



Solving for j, there results 



j = m[(l+iyi™-l]. (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 j {m) . 



Example 1. — The nominal rate of interest j on deposits is 3% and interest is added 

 to the principal every six months; to find the effective rate i. 

 Here j=. 03 and m=2. From formula (1) there results 



;=(l+.03/2) 2 -l==(1.015) 2 -l=.030225. 



The effective rate 3.0225 % is thus slightly higher than the corresponding nominal 

 rate convertible twice per annum. 



