A Php 5,000 loan is being repaid in 10 yearly payments. If interest is 8% effective, find the annual payment. 1 ( ) 10) 0.

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1 Amortzaton If a loan s repad on nstalment (whch s usually n equal amounts); then the loan s sad to be repad by the amortzaton method. Under ths method, each nstalment ncludes the repayment of prncpal and the payment for the nterest. The payment forms an annuty whose present value s equal to the orgnal loan. Example 1 A Php 5,000 loan s beng repad n 10 yearly payments. If nterest s 8% effectve, fnd the annual payment. Gven: A = 5,000; n = 10; m = 1; j = = 0.08; t = 10 If we let R be the annual payment, then we have: Solvng for R we have, 1 ( ) 10 Ra n = Ra = R ( ) = 5, ( ) R ( ) = 5,000 R = 5,000 ( ( ) 10) = Example 2 An AUV s worth Php 880,000. A down payment of 30% of the vehcles prce was made and the balance s to be pad monthly for 4 years at 24% compounded monthly. Determne the monthly payment. Gven: Cash prce: 880,000 Down : 0.3(880,000) = 264,000 Balance: 880, ,000 = 616,000 A = 616,000 n = mt = 4 12 = 48 = j m = = 0.02 From the formula: A = R ( 1 (1+) n ) R = A ( 1 (1+) n) = 616,000 ( (1.02) 48) = 20,

2 Fndng the Outstandng Prncpal Two methods of Fndng the Outstandng Prncpal 1. Prospectve method The prospectve method s used f we know the number of payments that should be made. Ths s the present value of all the payments stll to be made. 1 (1 + ) (n k) (OB p ) k = R ( ) 2. Retrospectve method Ths method s used f the number of payments s not known. We use the followng formula to get the outstandng balance. (OB r ) k = A(1 + ) k R ( (1 + )k 1 ) Example 3 A Php 20,000 debt s to be repad on nstalment every 6 months for 5 years. a) Fnd the sem-annual payment b) Fnd the outstandng balance after the 4 th payment. Assume that money s worth 12% compounded sem-annually. Soluton: a) A = 20,000; n = 10; m = 2; j = 0.12; = 0.06; 20,000 = R ( 1 (1.06) 10 ) R = 2, b) Gven that k = 4 and snce we know the total number of payments we use the prospectve method to fnd the outstandng balance after the 4 th payment (OB p ) = 2, ( 1 (1.06) 6 ) = 13,

3 Example 4 A Php 7,500 loan s amortzed annually at Php 800. The nterest s 10% effectve. Fnd the outstandng prncpal after the 5 th payment. Soluton Snce we do not know the total number of payments to be made, we cannot use the prospectve method. Gven: k = 5; A = 7,500; R = 800; = j = 10%; m = 1 (OB r ) k = A(1 + ) k R ( (1 + )k 1 ) 7,500(1.1) ( (1.1)5 1 ) = 7, An amortzaton schedule shows each payment that s broken down nto prncpal and nterest as well as the outstandng balance/prncpal after each payment. The amortzaton schedule for example 4 s shown below: R I=OB* PR=R-I OB=prevous OB -PR c (R) on Interest (I) Repayment of Prncpal at end of (PR) Outstandng Prncpal at end of (OB) 0 Php7, Php Php50.00 Php7, Php Php55.00 Php7, Php Php60.50 Php7, Php Php66.55 Php7, Php Php73.20 Php7, Php Php80.53 Php7, Php Php88.58 Php7, Php Php97.44 Php6, Php Php Php6, Php Php Php6, Php Php Php6, Php Php Php6, Php Php Php6, Php Php Php6,101.25

4 Php Php Php5, Php Php Php5, Php Php Php5, Php Php Php5, Php Php Php4, Php Php Php4, Php Php Php4, Php Php Php3, Php Php Php3, Php Php Php3, Php Php Php2, Php Php Php2, Php Php Php1, Php Php Php Php78.95 Php Php Php6.85 Php68.45 Php0.00 Take note that the last payment s an rregular payment snce the amount pad s not the regular amount that s normally pad every year. Example 5 Construct an amortzaton schedule for a Php 2,000 loan to be repad n 5 sem-annual payments f the nterest charged s 16% converted sem-annually. Soluton: Gven: A = 2,000; n = 5; = = 0.08 Computng for the regular payment we have: R = A ( 1 (1.08) 5 ) 0.08 = c on Interest Repayment of Prncpal at end of Outstandng Prncpal at end of 0 Php2, Php Php Php1, Php Php Php1, Php Php Php Php71.46 Php Php Php37.10 Php Php0.00

5 Example 6 Construct an amortzaton schedule for a Php 15,000 loan to be amortzed every 3 months for 1.5 years f the nterest charged s 12% converted quarterly. Soluton: Gven: A = 15,000; n = 6; = = Computng for the regular payment we have: R = c on Interest A ( 1 (1.03) 6 ) Repayment of Prncpal at end of = Outstandng Prncpal at end of 0 Php15, Php Php2, Php12, Php Php2, Php10, Php Php2, Php7, Php Php2, Php5, Php Php2, Php2, Php80.65 Php2, Php0.00 Example 7 A Php 7,500 loan s to be amortzed at Php 1,500 each quarter. A fnal rregular payment s made 3 months after the last regular payment. If the nterest rate s 12% converted quarterly a) fnd the number of regular payments needed b) when s the fnal rregular payment due c) how much s the fnal rregular payment d) construct an amortzaton schedule Soluton: Gven: A = 7,500; R = 1,500; = a) We frst solve for n. 1 (1 + ) n A = R ( ) 7,500 = 1,500 ( 1 (1.03) n ) 1 (7,500)() = (1.03) n 1,500

6 0.85 = (1.03) n log(0.85) = nlog(1.03) n = Ths tells us that we need at least 5 regular payments and an rregular payment. b) The fnal rregular payment s due on the 6 th payment specfcally on the 1.5 year 3 months R R R R R year c) If we let x be the amount of the rregular payment. We obtan the followng equaton of values x = A(1 + ) 6 R ( (1 + )n 1 ) (1 + ) n x = 7,500(1.03) 6 1,500 ( (1.03)5 1 ) (1.03) =

7 The tme dagram s shown below: A=7,500 Accumulate R R R R R Ordnary annuty of 5 payments R ( (1 + )n 1 ) = S Accumulate S for 1 perod/ 3months d) The amortzaton schedule s shown below c on Interest Repayment of Prncpal at end of Outstandng Prncpal at end of 0 Php7, Php1, Php Php1, Php6, Php1, Php Php1, Php4, Php1, Php Php1, Php3, Php1, Php Php1, Php2, Php1, Php64.98 Php1, Php Php Php21.93 Php Php0.00

8 Another method of computng the fnal rregular payment makes use of the retrospectve method. Step 1: Wth the gven values for R, A and I; obtans n and determne the number of regular payments that should be made Step 2: Determne the outstandng balance after the last regular payment. Step 3: Fnd the nterest pad on the outstandng balance on step 2 Step 4: Fnd the rregular payment by addng the nterest pad n Step 3 and the outstandng balance n step 2. Alternatve soluton for Example 7 Step 1: We frst solve for n. 1 (1 + ) n A = R ( ) 7,500 = 1,500 ( 1 (1.03) n ) 1 (7,500)() = (1.03) n 1, = (1.03) n log(0.85) = nlog(1.03) n = Ths tells us that we need at least 5 regular payments and an rregular payment. Step 2: Step 3: (OB r ) k = A(1 + ) k R ( (1 + )k 1 ) (OB r ) 5 = 7,500(1.03) 5 1,500 ( (1.03)5 1 ) = Interest = (730.85) = Step 4: Fnal Irregular = =

9 Snkng Fund If a person sees the need to have a certan sum at some future date, he mght accumulate a fund by makng perodc deposts. Such a fund s called a snkng fund. The amount n the fund after any k th depost s gven by: Example 8 S k = D ( (1 + )k 1 ) A man expects to buy a condomnum worth Php 100,000 n 3 years. He decdes to put hs savngs every 6 months n a fund that earns 8% converted sem-annually. a) How much should he put n the fund sem-annually b) How much s n the fund after the 4 th depost c) Construct a snkng fund schedule Soluton: a) Let D be the sem-annual depost. Snce the sem-annual depost forms an ordnary annuty, whose amount s Php 100,000, we have the followng: b) The amount of the fund after the 4 th depost s 100,000 = D ( (1.04)6 1 ) D = 15, S 4 = 15, ( (1 + 04)4 1 ) = 64, c) The followng s the snkng fund schedule Amount n Fund Interest Earned Depost Amount at end of 0 Php0.00 Php0.00 Php15, Php15, Php15, Php Php15, Php30, Php30, Php1, Php15, Php47, Php47, Php1, Php15, Php64, Php64, Php2, Php15, Php81, Php81, Php3, Php15, Php100,000.00

10 Example 9 In order to have Php 8,000 n 5 years, a man deposts each year n a snkng fund earnng 6% effectve. Fnd the annual depost and construct a snkng fund schedule. Gven: S k = D ( (1 + )k 1 Amount at the Begnnng of the S k = 8,000; k = 5; = 0.06 ) 8,000 = D ( (1.06)5 1 ) D = 1, Interest Earned Depost Amount at end of 0 Php0.00 Php0.00 Php1, Php1, Php1, Php85.15 Php1, Php2, Php2, Php Php1, Php4, Php4, Php Php1, Php6, Php6, Php Php1, Php8, Example 10 A father wshes to have Php 10,000 n 5 years. He sets up a snkng fund, depostng a sum every 3 months n a bank that pays 12% nterest converted quarterly. a) Fnd the quarterly deposts b) Fnd the nterest earned n the 10 th quarter c) Construct a snkng fund schedule Gven: S k = 10,000; j = 0.12; m = 4; = a) He wll be makng 20 deposts n 5 years. And after the 2th depost he should have Php 10,000 n the fund, thus we have S 20 = D ( (1 + )20 1 ) 10,000 = D ( (1.03)20 1 ) D = b) To get the nterest earned on the 10 th quarter we frst compute for the amount n the fund on the 9 th quarter and multply the amount by the nterest rate for one perod. From ths we have the followng: S 9 = ( (1 + )9 1 ) = 3, The nterest earned n the 10 th quarter s (3,780.81)() =

11 c) The snkng fund schedule s shown below Amount n Fund Interest Earned Depost Amount at end of 0 Php0.00 Php0.00 Php Php Php Php11.16 Php Php Php Php22.66 Php Php1, Php1, Php34.51 Php Php1, Php1, Php46.71 Php Php1, Php1, Php59.27 Php Php2, Php2, Php72.22 Php Php2, Php2, Php85.55 Php Php3, Php3, Php99.28 Php Php3, Php3, Php Php Php4, Php4, Php Php Php4, Php4, Php Php Php5, Php5, Php Php Php5, Php5, Php Php Php6, Php6, Php Php Php6, Php6, Php Php Php7, Php7, Php Php Php8, Php8, Php Php Php8, Php8, Php Php Php9, Php9, Php Php Php10,000.00