112 



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 

 as follows : 



a m =34.7608867 



(1) 



ajg ( =13.5777093 



(2) 



ay -«^=21.1831774 



(3) 



(3)-h11= 1.9257434 



(4) 



a Tj = 3.8077287 





(4)-n*_ ( = .5057460 



(5) 



Complement of (5)=l-(5)= .4942540 



(6)=first factor 



(.05-.04)/.04= .25 



(7)= second factor 



/j=(6)X(7)= .1235635. 





The bid on one dollar is 1+^=1.1235635; consequently the bid on the whole issue is 

 1.1235635X$1, 100,000=41, 235, 919. 85. 



Example 21. — To find the price of $100,000 highway bonds, interest b%, 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/=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. 



Year. 



Book value 

 or principal 

 at beginning 

 of half-year. 



Se miannual 

 interest of 2%. 



Semiannual 



dividend of 



2i% on bonds. 



Amortization 



of premium 



at end of 



half-year. 



Redemption 

 payment 

 at end of 

 half-year. 



1 



H 



2 



2i 



3 



H 



4 



4i 



5 



Totals 



$103, 646. 00 



103, 218. 92 



102, 783. 30 



102, 338. 97 



101, 885. 75 



101, 423. 47 



50, 951. 94 



50, 720. 98 



50, 485. 40 



50, 245. 11 



$2, 072. 92 

 2, 064. 38 

 2, 055. 67 

 2, 046. 78 

 2, 037. 72 

 2, 028. 47 

 1, 019. 04 

 1, 014. 42 

 1, 009. 71 

 1. 004. 89 



$2, 500. 00 

 2, 500. 00 

 2, 500. 00 

 2, 500. 00 

 2, 500. 00 

 2, 500. 00 

 1, 250. 00 

 1, 250. 00 

 1, 250. 00 

 1, 250. 00 



$427. 08 

 435. 62 

 444. 33 

 453. 22 

 462. 28 

 471.53 

 230. 96 

 235. 58 

 240. 29 

 245. 11 



0.00 



0.00 



0.00 



0.00 



' 0.00 



$50, 000. 00 



0.00 



0.00 



0.00 



50, 000. 00 



817, 699. 84 



16,354.00 , 20,000.00 1" 3,646.00 



100, 000. 00 



Extension of term of tables. — It sometimes happens in applying 

 formula (42) that the value of 2(f+tr) is greater then the term given 

 in the tables. In example 20 one of the required annuity values was 

 agoi but, if the interval between redemptions had been three years 

 instead of two, 2(f+tr) = S2 would have called for the value of an 

 annuity a^i beyond the limits of the tables. It is easy, however, to 

 extend these limits by making use of the following obvious relations : 



v m+n = v m v n^ (43) 



(l+i) m+n =(l+i) m {l +i) n , (44) 



a S qr^ = [l-<y' B v»]/t, (45) 



oteri = a>m\ + v m a^, (46) 



*ttH = [(l+i)™(l+i)»-iyi, (47) 



s^i\ = (l+i) n s^ + s^. (48) 



