HIGHWAY BONDS. 95 
The discount factor.—Because of the power of money to earn 
interest the value of money depends upon the time to which it is 
referred. Then in order to compare sums of money due at different 
times, they must be referred to the same point in time. 
Formula (5) gives the principal P which will accumulate at the 
effective rate 7 in n years to the amount S. If S=1 and n=1, 
the formula gives the present value of 1 due in one year. ‘This is 
usually denoted by the symbol v, so that 
o—h/G--o) = Gd -pae. 
Similarly v?=1/(1+7)?=(1+7)~ is the present value of 1 due in 
two years, and v?=1/(1+7)"=(1+71)~ 1s the present value of 1 due 
in n years. 
The symbol v is often called the discount factor, and if it is desired 
to find the value of money n years before the point in time under con- 
sideration, it is necessary only to multiply the quantity by v’. The 
factor 1+%, which is frequently denoted by 7, is accumulative in 
character, and formula (4) shows that, to find the value of a quantity 
of money Q, n years after the point in time under consideration, it is 
necessary merely to multiply the quantity by (1 +7)”. 
Que—Q(1+1) Q Orn Oe 
t—n t ttn 
More generally, when 1 1s the effective rate per interval, the value of Q, 
at a time n intervals after the point t, is @1 +2)”=Qr”, and its value 
n intervals before point ¢ is Q(1 +2)-"=Qv". 
Annuities-certain.—An annuity is a series of payments made 
at equal wntervals during the continuance of a given status. 
The status, or condition of payment of the annuity, may take a 
variety of forms. If the status is a fixed term of years, the annuity 
is an annuity-certain. Thus payments of one hundred dollars a year 
for ten years constitute an annuity-certain. The sum of the pay- 
ments on an annuity in one year, when the payments are of the same 
amount, is the annual rent. 
Payments of twenty-five dollars are made at the end of every month for ten years. 
This is an annuity-certain with an annual rent of three hundred dollars. 
When payments are made at the end of each interval, the annuity- 
certain is said to be «wmmediate; when payments are made at the 
beginning of each interval, the annuity-certain is said to be due. 
Amount of an immediate annuity-certain.—The value of an 
annuity at the end of its term is called the amount. The amount of 
an immediate annuity-certain for n years with an annual rent 1, 
