HIGHWAY BONDS. 99 
In most cases in the illustrative tables in the body of this bulletin, 
for simplicity of presentation, annual payments and annual interest 
are assumed, whereas in practice usually annual payments and semi- 
annual interest are employed. 
Present value of an immediate annuity-certain.—The present 
value of an immediate annuity-certain for n years, with annual rent 
1 payable at the end of each year, is designated by the symbol aj. 
It is equal to the sum of the present values of 1, due at the suc- 
ceeding yearly intervals. By formula (5) the present value of 1, 
due at the end of one year at the effective rate of interest 7, is 
v—1/(1 +2); at the end of two years, v?=1/(1+72)?, ete. ....... 
at the end of n years, v™=1/(1+7)". Hence 
Gy] SOS Ge eas se + yu 
OE Le a eee an) 
147° (141)? (1+7)” 
The sum of this geometric series 1s 
leo? LaGeore 
Pe Sipe a 
ani (19) 
and its values are given in Table 35. 
Example 12.—To find the present value at 3% of an annual payment of $56,325 
at the end of each year for thirty years. 
Referring to Table 35, it is seen that agqy at 3% is 19.6004413, and therefore the 
equired present value is 
19.6004413 X $56 ,325=$1, 103,994.86. 
While the above demonstration relates to an annuity of 1 per 
annum, payable at the end of each year, the same principles apply 
to finding the present value of an annuity of 1 per annum, payable 
in p installments during each year. The present value of such an 
annuity is designated by the symbol a”, and its value is repre- 
sented by the following formula: 
2 aan i vad Ui Onis, 
peas DICE CO) eat) ple) | 
In formulas (19) and (20) the values of the annuities are expressed 
in terms of the effective rate 7. If (1+7) 1s replaced by (1 +7/m)™ 
in accordance with formula (2), there result the present values of 
the same annuities expressed as follows in terms of the nominal rate 
of interest 7, with frequency of conversion m: 
ye i @ +)/m) —mn 1 
an = (1+ j/m)™"—1 ; (2 ) 
(20) 
and 
Ihe QUE LO) ee (22) 
a® ete See ai eee ee 
a pers) ee 
