icTION st important application x # HOUSe, CAF OF othe, f ann ri tion that is amortized, Rygt'™® by magn oP though ani 8 wert NNN of periodic payments thay ueh there ancien ot — 1s widely accepted by the O° interes’ Mitenen ode as | institutions are of thig Mane othe sed. 7 ops OF REPAYING Loan p< vesically 3 methods of repeyi payment at the end of the term» ia oe sent of only interest periodicatty rm periodie payment, nderstand these methods by me ; “Rs 1,000 at Be eee and he a a Suppose of the following three ™ost common ways, an is 3 years, payment of any kind is made that, Syen ect, by the borrower : «sure loan is repaid with interest at the end ors Yearsby ee €3 year ana The amount payable = Rs, 1,000 (1.08)8 = Rs, 1, 259.71 ‘an pay only the interest re; gularly at the end of e ‘Year, The interest of Rs, 80 ‘s regularly paid at the end of every year as it falls due; and at the end of 8 years, the principal of Rs, 1,000 is repaid, >4FINANCIAL a a 236 _— en a sean in reat by aur pagers vj at tho ond of €85h YOR" f0r 3 yeu, Re. 1.000 _ pap, 388.09 6 fe) ove oe yom yearly pAYMeE IB sg SP woninina intarert ot oe Oa a principal and a every payment of Rs. 988° = wn of principal a at the amount of int Mares pala st is obvious the ,d mothod, ‘nira method, which is also 180 known gy ‘comprising both the princi ‘pal ang methods, i the thir jetail, the t ted payments Remark: 4! for the loan fs 12 We sball now dist jon of loan, = Trehe borrower adopt cous, in greater amortizati ‘py level or equal AMORTISATIO! to be amortiseds iperiods of time. Each ding principle, IN OF LOAN ifit can payment can and we dichargea by @ sequence of ext ‘A Toan is sai equal payment nade over equal ip considered as consisting of no pat Ta) Interest on the ovtstan (B) Repayment of part of the loan- ‘Thus, a loan, generally with 5 Bxe principal and interest are Paid By & sequence of equal a rate of interest, is said to be amortized, if boy 1 payments made over equal periods ot time. ‘Suppose that a loan of Rs. P is ‘amortized at a particular rate of interest i per paymer period over n periods. Let F be the periodic payment. We know that Rs. Be vrasen Perit the annuity of Ra. R at year 20r0, 18 Ab ‘the beginning of the term of loan. “ Present value of an immediate annuity of Rs. R payable at the end of very period for periods at /per payment period is given by n P=Raq: ‘The periodic payment R is given by ‘This can be further express @5 | h EE Pi R= are R= asi) ¥ 0+ i iodic payment R can also be given by aa at ae al eet is compounded continuously, the present value of an immediate . R payable at the : a at end of every year for t years at a rate of r per annum is givenFINANCIAL MATHEMAT, 8 = following tabl a Te results are show The rest Remark 1 : From the nar a | Interest @ 8% cay — [Repayment Principal | pata he end of| a2 thaend of at the end of \ pence | oustanding a the] PON very year oe tend ; Speginning of eT 9 7a 2 yeor__| 40.00 |, Bea Be oe) eed : wa 100000 | BY se | Be aaeaieg ie Re 959.30 |e para _| Re. 388.08 _| [x erates Rs. 1,164.09 | Rs. 1,000.00 | ‘Total __|_— »_ 164.0" mortization schedule, it is observed that payment to be made ay Se rrreairdiyenr to Ra. 288.04 6 He 200297 ‘Re. 28.74). The payment originally made j, toe 08, The difference of 0.01 for the third year 3 due to rounding off and can be ignored for practical purposes: oe Remark 2 : We observe that the payment of the principal goes on increasing in succenive years and the interest payment goes on decreasing: GENERAL FORMULA FOR AMORTISATION Now, we derive a general formula for amortisation of the instalment of a particular payment period for the interest portion and the capital repayment portion. Consider a loan of Rs. P to be repaid by n equal periodic payments of Rs. R. Let i be the rate of interest per period We know that P=Raq; Equal periodic payment is given by oe oi ‘Now, principal at the beginning of 1* period = R az, Interest for 1* period = R az, xi x(t] =R(1-(1+i)*) Instalment paid at the end of 1% year = R “. Principal contained in 1* payment = R—R (-( +i" =R(1+i" Principal at the end of 1% payment, period, i.e, bey of it ie, at the f 2nd payment period be, ginning = Principal at the beginning of 1* payment period =Raq, -R(1 +i" ~ Principal repaid in the 1** paymentpaaso- bw Qson sect for cond POH = Raa sme .t paid at the end of ana * ment P* Year = R spate tapal contained in the an, -_ ayment Ee *=R-Ra_¢ 3 =Ra at the besinning of 3" paym, +0000 oe! ent per Pee oni R Ueto ae Parad rads pany Bas ees ing in this manner, we can obs cotinaing acre a (g) The successive payments of Rs. R for n periods oe ‘Useful results interest portions of Fate of i per period aS RA-A+9),RO-14 ion, | we ROD (The successive principal repayments are 2, ROA sip Rd+O R(L+ ied, ROD? RO so (c) Principal outstanding at the beginning of every payment Period are Rag, RoR A=Ay wn RaQ, Ror, Now we prepare the general amortization schedule for a loan of payable for n periods at the rate of i per period. maces. Payment Principal I c Interest ati per | Principal repaid \ Instalment \ period | outstanding at the| _ period for every at the end of \ at the end of beginning of every period every period | every period ‘payment period Ray, R(-(+iy") Ra+ie R Ray RQA-GQ+iye-y | RAs? R Ray R(L-(1 + iy?) eirege R RQ-(+i? RastiP Ra-(+i) Rast nR-P P=RaqiFINANCIAL MATH, i Mi a -ORMULAS GiSATION F cs atl eo discussion, a Pi period ita joan of Rs, Ps given by Pi R=g5,71-a+i" cetanding at the beginning of period : standing at the end of (k ~ 1)*" period Principal out Present value of remaining n ~ (= 1) payment, iyn =a et i obtain the following results that descr vor an equal payments of Re. Rat the eng othe ven Principal 0 rE =R =Ra inni th period incipal repaid till the beginning of k' pe : : nee rae Loan Amount ~ Principal outstanding at the beginning of x Raj a-a (d) Interest in the k'* payment ae : = i x principal outstanding at the beginning of k* period =ixRaTqaii (e) Principal contained in the k'" payment = /*» payment — interest in the k'" payment =R-iRay gi Alterantively, principal contained in the k'” payment ‘rincipal outstanding at the beginning of k* payment. Detiog — Principal outstanding at the beginning of (k + 1) pay, (f) Total interest paid = nR - P Example 1 : Find the amortised monthly payment necessary to pay-off a personal | Rs, 60,000 at 12% per annum compounded monthly in 5 years. Solution : Loan amount P = Rs. 60,000 Rate of interest r = 12% = 0.12 Interest is compounded monthly. ee 2 Let R be the monthly payment, Periodic payment is given by Pi 1-Q4i