102 



BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 



Installment annuity loan. — The preceding example shows how 

 the function 1/a^ may be employed to determine the periodical fixed 

 payment which in n years will discharge both principal and interest 

 on a loan. It is to be noted particularly that the lender receives 

 interest throughout the term of the loan on all outstanding principal. 

 The following schedule, based on the above example, illustrates the 

 progress of the loan. 



Schedule I. — Showing repayment of principal and interest on a loan of $100,000 by six 

 equal semiannual payments, each of $18,155; interest 5 per cent, compounded semi- 



a n anally. 



Year. 



Principal outstand- 

 ing at beginning of 



interval. 



Interest for interval. 



Semiannual payment. 



Principal repayment 

 for interval. 



1 



I 2 



u 



2" 

 2 A 



3 



Totals 



$100, 000. 00 



84, 345. 00 

 68, 298. 63 

 51,851.10 

 34, 992. 38 

 17.712.19 



$2, 500. 00 



2, 108. 63 



1, 707. 47 



1, 296. 28 



874. 81 



442. 81 



$18, 155. 00 

 18, 155. 00 

 18, 155. 00 

 18, 155. 00 

 18, 155. 00 

 18, 155. 00 



$15, 655. 00 

 16,046.37 

 16, 447. 53 

 16, 858. 72 

 17, 280. 19 

 17,712.19 



357, 199. 30 8, 930. 00 



108, 930. 00 



100, 000. 00 



The initial invested principal of $100,000 earns $2,500 interest dur- 

 ing the first half year; the first payment of $18,155.00 takes care of 

 this and there remains a balance of $15,655.00 which goes to reduce 

 the outstanding principal to $84,345.00, beginning with the second 

 half year. This process is repeated until the end of the third year, 

 when the last outstanding principal is retired. When preparing such 

 a schedule, the work can be checked by adding the columns. It is evi- 

 dent from the nature of the calculations that, for example, if the first 

 row were omitted from this schedule, the remaining five would repre- 

 sent the schedule for a loan of $84,345.00 on the same terms as the 

 original loan, except that it would be discharged in two and one-half 

 years by five equal semiannual payments. It must therefore be the 

 present value of the five payments, that is, 



8100,000 



where the annuities are taken at 2§ per cent. Similarly, by succes- 

 sively employing a^\, ajj, aj\, and a x -\, all at 2\ per cent, as multi- 

 pliers, the figures in the first column of principal outstanding at the 

 beginning of the interval could be obtained. When these are known, 

 the figures in the second column are obtained by multiplying the cor- 

 responding figures in the first column by the interest rate for the 

 interval, .025; in the fourth, by successive subtractions of the figures 



