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 (1 + 0.08) 10 Ra n = Ra 10 0.08 = R ( ) = 5,000 0.08 1 (1 + 0.08) 10 0.08 R ( ) = 5,000 R = 5,000 ( 0.08 1 (1 + 0.08) 10) = 745. 15 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: 880,000 264,000 = 616,000 A = 616,000 n = mt = 4 12 = 48 = j m = 0.24 12 = 0.02 From the formula: A = R ( 1 (1+) n ) R = A ( 1 (1+) n) = 616,000 ( 0.02 1 (1.02) 48) = 20, 082. 73
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, 717. 36 0.06 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 ) = 2,717.36 ( 1 (1.06) 6 ) = 13, 362. 14 4 0.06
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) 5 800 ( (1.1)5 1 ) = 7, 194. 75 0.1 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 Php7,500.00 1 800 Php750.00 Php50.00 Php7,450.00 2 800 Php745.00 Php55.00 Php7,395.00 3 800 Php739.50 Php60.50 Php7,334.50 4 800 Php733.45 Php66.55 Php7,267.95 5 800 Php726.80 Php73.20 Php7,194.75 6 800 Php719.47 Php80.53 Php7,114.22 7 800 Php711.42 Php88.58 Php7,025.64 8 800 Php702.56 Php97.44 Php6,928.21 9 800 Php692.82 Php107.18 Php6,821.03 10 800 Php682.10 Php117.90 Php6,703.13 11 800 Php670.31 Php129.69 Php6,573.44 12 800 Php657.34 Php142.66 Php6,430.79 13 800 Php643.08 Php156.92 Php6,273.86 14 800 Php627.39 Php172.61 Php6,101.25
15 800 Php610.13 Php189.87 Php5,911.38 16 800 Php591.14 Php208.86 Php5,702.51 17 800 Php570.25 Php229.75 Php5,472.76 18 800 Php547.28 Php252.72 Php5,220.04 19 800 Php522.00 Php278.00 Php4,942.05 20 800 Php494.20 Php305.80 Php4,636.25 21 800 Php463.63 Php336.37 Php4,299.88 22 800 Php429.99 Php370.01 Php3,929.86 23 800 Php392.99 Php407.01 Php3,522.85 24 800 Php352.28 Php447.72 Php3,075.13 25 800 Php307.51 Php492.49 Php2,582.65 26 800 Php258.26 Php541.74 Php2,040.91 27 800 Php204.09 Php595.91 Php1,445.00 28 800 Php144.50 Php655.50 Php789.50 29 800 Php78.95 Php721.05 Php68.45 30 75.3 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; = 0.16 2 = 0.08 Computng for the regular payment we have: R = A ( 1 (1.08) 5 ) 0.08 = 500.913 c on Interest Repayment of Prncpal at end of Outstandng Prncpal at end of 0 Php2,000.00 1 500.913 Php160.00 Php340.91 Php1,659.09 2 500.913 Php132.73 Php368.19 Php1,290.90 3 500.913 Php103.27 Php397.64 Php893.26 4 500.913 Php71.46 Php429.45 Php463.81 5 500.913 Php37.10 Php463.81 Php0.00
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; = 0.12 4 = Computng for the regular payment we have: R = c on Interest A ( 1 (1.03) 6 ) Repayment of Prncpal at end of = 500.913 Outstandng Prncpal at end of 0 Php15,000.00 1 2768.963 Php450.00 Php2,318.96 Php12,681.04 2 2768.96251 Php380.43 Php2,388.53 Php10,292.51 3 2768.96251 Php308.78 Php2,460.19 Php7,832.32 4 2768.96251 Php234.97 Php2,533.99 Php5,298.33 5 2768.96251 Php158.95 Php2,610.01 Php2,688.31 6 2768.96251 Php80.65 Php2,688.31 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
0.85 = (1.03) n log(0.85) = nlog(1.03) n = 5.4981512 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 0 1 2 3 4 5 6 1 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) = 752. 78
The tme dagram s shown below: A=7,500 Accumulate R R R R R 0 1 2 3 4 5 6 7 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 Php7,500.00 1 Php1,500.00 Php225.00 Php1,275.00 Php6,225.00 2 Php1,500.00 Php186.75 Php1,313.25 Php4,911.75 3 Php1,500.00 Php147.35 Php1,352.65 Php3,559.10 4 Php1,500.00 Php106.77 Php1,393.23 Php2,165.88 5 Php1,500.00 Php64.98 Php1,435.02 Php730.85 6 Php752.78 Php21.93 Php730.85 Php0.00
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 1,500 0.85 = (1.03) n log(0.85) = nlog(1.03) n = 5.4981512 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 ) = 730.85 5 Interest = (730.85) = 21.93 Step 4: Fnal Irregular = 730.85 + 21.93 = 752.78
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 = 15,076.19 0.04 S 4 = 15,076.19 ( (1 + 04)4 1 ) = 64,020.51 0.04 c) The followng s the snkng fund schedule Amount n Fund Interest Earned Depost Amount at end of 0 Php0.00 Php0.00 Php15,076.19 Php15,076.19 1 Php15,076.19 Php603.05 Php15,076.19 Php30,755.43 2 Php30,755.43 Php1,230.22 Php15,076.19 Php47,061.83 3 Php47,061.83 Php1,882.47 Php15,076.19 Php64,020.50 4 Php64,020.50 Php2,560.82 Php15,076.19 Php81,657.51 5 Php81,657.51 Php3,266.30 Php15,076.19 Php100,000.00
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 = 1,419.171 0.06 Interest Earned Depost Amount at end of 0 Php0.00 Php0.00 Php1,419.17 Php1,419.17 1 Php1,419.17 Php85.15 Php1,419.17 Php2,923.49 2 Php2,923.49 Php175.41 Php1,419.17 Php4,518.07 3 Php4,518.07 Php271.08 Php1,419.17 Php6,208.33 4 Php6,208.33 Php372.50 Php1,419.17 Php8,000.00 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 = 372.157 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 = 372.157 ( (1 + )9 1 ) = 3,780.81 The nterest earned n the 10 th quarter s (3,780.81)() = 113.42
c) The snkng fund schedule s shown below Amount n Fund Interest Earned Depost Amount at end of 0 Php0.00 Php0.00 Php372.16 Php372.16 1 Php372.16 Php11.16 Php372.16 Php755.48 2 Php755.48 Php22.66 Php372.16 Php1,150.30 3 Php1,150.30 Php34.51 Php372.16 Php1,556.97 4 Php1,556.97 Php46.71 Php372.16 Php1,975.83 5 Php1,975.83 Php59.27 Php372.16 Php2,407.26 6 Php2,407.26 Php72.22 Php372.16 Php2,851.64 7 Php2,851.64 Php85.55 Php372.16 Php3,309.35 8 Php3,309.35 Php99.28 Php372.16 Php3,780.78 9 Php3,780.78 Php113.42 Php372.16 Php4,266.36 10 Php4,266.36 Php127.99 Php372.16 Php4,766.51 11 Php4,766.51 Php143.00 Php372.16 Php5,281.66 12 Php5,281.66 Php158.45 Php372.16 Php5,812.27 13 Php5,812.27 Php174.37 Php372.16 Php6,358.80 14 Php6,358.80 Php190.76 Php372.16 Php6,921.72 15 Php6,921.72 Php207.65 Php372.16 Php7,501.52 16 Php7,501.52 Php225.05 Php372.16 Php8,098.73 17 Php8,098.73 Php242.96 Php372.16 Php8,713.85 18 Php8,713.85 Php261.42 Php372.16 Php9,347.42 19 Php9,347.42 Php280.42 Php372.16 Php10,000.00