APPENDIX D. 
THEORY OF INTEREST APPLIED TO BOND CALCULATIONS. 
Introduction.—This appendix presents briefly the application of 
the theory of compound interest to highway bonds. There are six 
important quantities in terms of which the solution of most problems 
can be expressed. If 7is the rate of interest and n the term of years, 
these quantities are: 
The accumulation of 1 at the end of 7 years, 1”; 
The accumulation of an annuity of 1 per annum at the end of n 
years, 8a} 
The annual sinking fund which will accumulate to 1 at the end of 
nm years, 1/87; 
The present value of 1 due in 7 years, v"; 
The present value of an annuity of 1 per annum for 7 years, ay; 
and 
The annuity for n years which 1 will purchase, or the annuity 
necessary to discharge a debt of 1 in n years with interest, 1/az. 
The first three are accumulative functions and the last three are 
discount or present value functions. 
The mathematical formulas for these six quantities are: 
(pga y\n ats ibs is It bs | aes 
r =(1+4) S7| = 4 Sz (1+4)"-1 
ra a —1-44)-" i i 
a =(1+4) Se an = 7 al = Ct ‘a +4)” 
The values of most of these functions are given more or less com- 
pletely in published tables of interest.+ 
Definitions.—J/nterest may be defined as the consideration for the 
use of capital. The capital is called the principal. 3 
The rate at which a given principal is earning interest requires 
the adoption of some interval as the unit of time, and this is usually 
the year. 
It is clear that interest when received may be added to the principal 
and so in turn earn interest. This process, called compounding, takes 
place at the end of stated intervals, as every three months, six months, 
or year. 
1 At the end of this appendix, pages 116 to 127, are short tables to seven places for 60 intervais and 
14 interest rates. 
92 
