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 7 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 ee 
progress of the loan. 
SCHEDULE J.—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- 
annually. 
| 
ese Principal outstand- : ee ; lp 1 . 
Year. ing Bt hese of Interest for interval. |Semiannual payment. eee 
4 $100, 000. 00 $2, 500. 00 $18, 155. 00 $15, 655. 00 
it 84, 345. 00 2, 108. 63 18, 155. 00 16, 046. 37 
13 68, 298. 63 1, 707. 47 18, 155. 00 16, 447. 53 
2 51, 851. 10 I, 296.28 18, 155. 00 16, 858. 72 
Ds 34, 992. 38 874. 81 18, 155. 00 17, 280. 19 
3 Lprmb2 aug 442.81 18, 155. 00 Lge U2 SUS 
Totals 397, 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 isretired. When preparing such 
aschedule, 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, 
—— ae = $84,345. 00, 
where the annuities are taken at 24 per cent. Similarly, by succes- 
sively employing aq, az, dy, and az, all at 24 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 
