HIGHWAY BONDS. 



103 



in the first; and in the third, by adding those in the second to those 

 in the fourth as a check. 



Generalization of the annuity loan. — The preceding discussion 

 can most easily be generalized by considering the loan of a^ dollars 

 where both principal and interest at effective rate i per annum are 

 discharged by equal annual installments of 1 at the end of each year 

 for n years. The initial principal is a^ ; the interest, ia^ = l—v n ; the 

 annual payment, 1, of which 1 - (1 —v n ) =v n is applied to repayment 

 of principal. But art — v n = a^=n ; hence the outstanding principal 

 at the beginning of the second year is as=i|, as might have been 

 predicted in advance. A repetition of this process leads to the fol- 

 lowing schedule: 



Schedule II. — Showing repayment of principal and interest at effective rate i per annum 

 on a loan ofa^ by equal annual payments ofl at the end of each year for n years. 



Year. 



Principal 

 outstanding 



at begin- 

 ning of year. 



Interest due at 

 end of year. 



Annual pay- 

 ment at end 

 of year. 



Principal 



repaid at 



end of year. 



1 

 2 

 3 



h 



n 



Totals 



art 



1— vn 

 l- v n-i 



l— v n-2 



l_-j,ra-7c+ 1 



1-r 



1 

 1 



1 



1 



i 



v n-i 



v n—2 



v n-lc+ l 



V 



a n-k+H 



(n-as])/t 



n— oy 



n 



a rt 



Since this is a schedule for a loan of a^, if each item in it, apart 

 from those in the column headed "year," is divided by a ,TI and 

 multiplied by L, there results the corresponding schedule for a loan 

 of L dollars. 



For example, the items on a loan of L dollars for the Jcih year 

 would be 



iMa-j^/asi, X(l-'V»- fc+1 )/c^, i/a»|, Zv~-*+}/a$. (28) 



There are some curious properties revealed by the above schedule, 

 among which the following may be pointed out. The principal 

 repayments on an annuity loan increase in geometrical progression, 

 the factor being 1+i. The sum of these repayments is a^; the sum 

 of the annual payments is n; the total interest is n — a^; and the 

 check on the first and second columns shows that 



i(aT| + «n+ + a^)=n-a^. 



It is apparent that most of the items in the schedule can be filled 

 in directly from the art and v n tables. Having thus filled in each 



