co Eby BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 
Entering Table 35 with 2% for the values of the annuities and numbering the 
successive steps for convenience of explanation, the calculation may be outlined 7 
as follows: . i 
Agpj =34.7608867 (1) 
Qjq=13.5777093 (2) 
(3)+11= 1.9257434 (4) 
a= 3.8077287 
(4)-aq= .0057460. (5) 
Complement of (5)=1—(5)= .4942540 (6)=first factor 
(.05—.04)/.04= .25 (7)=second factor 
$=) <()=" JRA. 
The bid on one dollar is 1+£=1.1235635; consequently the bid on the whole issue is 
1.1235635 X $1, 100, 000=$1, 235,919.85. 
Example 21.—To find the price of $100,000 highway bonds, interest 5%, semi- 
annual, dated January 1, 1914, maturing $50,000 January 1, 1917, and $50,000 Janu- 
ary 1, 1919, to net the investor 4% compounded semiannually. 
In this case f=3, r=2, t=2, m=2, g=.05, j=.04, and, substituting as in the pre- 
ceding example, the required price is found to be $103,646.00. The progress of the 
loan is indicated in the following schedule. 
SCHEDULE V. 
Book value is eonems Amortizati Redempti 
Meni or principal ; Se miannual eee of prone Savane y 
ae aren eS Gane a | = 
4 $103, 646. 00 $2, 072. 92 $2, 500. 00 $427. 08 0. 00 
i LOS 218592 2, 064. 38 2, 500. 00 435. 62 0. 00 
13 102, 783. 30 2, 055. 67 2, 500. 00 444, 33 0. 00 
% 102, 338. 97 2, 046. 78 2, 500. 00 453. 22 0. 00 
24 101, 885. 75 DOS shiz 2, 500. 00 462. 28 0. 00 
3 101, 423. 47 2, 028. 47 2, 500. 00 471.53 $50, 000. 00 
34 50, 951. 94 1, 019. 04 1, 250. 00 230. 96 0. 00 i 
4 50, 720. 98 1, 014. 42 1, 250. 00 235. 58 0. 00 ; 
44 50, 485. 40 1, 009. 71 1, 250. 00 240. 29 0. 00 
5 50, 245. 11 1, 004. 89 1, 250. 00 245. 11 50, 000. 00 
Totals, 817,699.84 16,354.00 20,000.00 | 3, 646. 00 | 100, 000.00. 
Extension of term of tables.—It sometimes happens in applymeg 
formula (42) that the value of 2(f+?r) is greater.then the term given 
in the tables. In example 20 one of the required annuity values was 
adm but, if the interval between redemptions had been three years — 
instead of two, 2(f+tr)=82 would have called for the value of an 
annuity dy beyond the limits of the tables. It is easy, however, to 
extend these limits by making use of the following obvious relations: 
ymin — ymyn (43) 4 
(1 +a)m*™—= (1 +7)™(1+1), (44) 
Gnex = [1 — vm /i, (45) 
Om-n| = Am + UG, (46) | 
Saez =[(1 +2)™(1 +2)" 1s, (47) 
Sata = (1 +4) "sa + Sa: (48) 
ds 
: i 
_ th ee 
