Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS R =

Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS Objectves of the Topc: Beg able to formalse ad solve practcal ad mathematcal problems, whch the subjects of loa amortsato ad maagemet of cumulatve fuds are aalysed. Assessg facal flows tme, provdg reasoed evaluatos whe comparg varous loa repaymet methods. Assessed results of the studes: Wll uderstad the methods of aalysg loa amortsato ad cumulatve fuds. Wll apply kowledge of auty whe modellg mathematcal ad real lfe cases. Wll provde mathematcally supported recommedatos. Studet achevemet Assessmet Crtera: Accurate use of terms. Approprate applcato of formulas. Accurate term ad ed aswers. Accurate aswers to questos. 3.1 Amortzato (smple autes Revew the followg terms: Perodc paymets: a smple (ordary, pad-up, deferred pad-up, deferred ordary, ordary lfe-log, pad up lfe-log, b complex (ordary, pad-up, deferred pad-up, deferred ordary, ordary lfe-log, pad-up lfe-log. All repaymets of terest-bearg debts by a seres of paymets, usually sze, made at equal tervals of tme s called a amortzato. Mortgages ad may cosumer loas are repad by ths method. We cosder a classcal problem. Suppose that a bak loas B. Ths amout plus terest s to be repad by equal paymets of R each at he ed of each perod. Further, let us assume that the bak charges terest at the omal rate of r percet compouded m tmes year (actual = r/m. Essetally, for B the bak s a auty of paymets of R each. Usg formula of a preset value of a ordary auty we obta that the mothly paymet R s R = B a. The bak ca cosder each paymet as cosstg of two parts: (1 terest o outstadg loa, ad (2 repaymet of part of the loa. The amout of the loa s the preset value of the auty. A porto of each paymet s appled agast the prcpal, ad the remader s appled agast the terest. Whe a loa s repad by a auty, t s sad to be amortzed. I aother words, a loa s amortzed whe part of each paymet s used to pay terest ad the remag part s used to reduce the outstadg prcpal. Sce each paymet reduces the outstadg prcpal, the terest porto of a paymet decreases as tmes goes o. Let us aalyze the loa descrbed above. Suppose that the prcpal s B. At the ed of the frst moth you pay R. The terest o the outstadg prcpal s I 1 = B. The balace of the paymet P 1 = R I 1 s the appled to reduce the prcpal. Hece the prcpal outstadg ow B 1 = B R 1. Further, at the ed of the secod moth, the terest s I 2 = B 1. Thus the amout of the loa repad s P 2 = R I 2, ad the outstadg prcple s B 2 = B 1 P 2. The terest due at the ed of the thrd moth s I 3 = B 2 ad so the amout of the loa repad s P 3 = R I 3. Hece the outstadg prcple s B 3 = B 2 p 3 ad etc.. The terest due at the ed of the th ad fal moth s I = B 1 ad the amout of the loa repad s B = R I. Hece the outstadg balace (prcpal s B = B 1 P. Actually, the debt should ow be pad off, ad the balace of B s due to roudg, f t s ot 0. Ofte, baks wll chage the amout of the last paymet to offset ths. A aalyss 50

of how each paymet the loa s hadled ca be gve a table called a amortzato schedule. Cosder oe example. Suppose that a bak loas 1500. Equal paymets of R at the ed of each three moth. The omal rates of 12 percet compouded mothly. Thus we have R = 1500 a 3 0.01 1500 510.03. 2.940985 The allocato of each paymet to frst cover the terval due ad the to reduce the prcpal may be show a amortzato schedule. The smplfed amortzato schedule of the loa cosdered above: No P I R B 1 1500 15 510.03 495.03 2 1004.97 10.05 510.03 499.98 3 504.99 5.05 510.03 504.98 Totals 30.10 1530.09 1499.99 Table 1 Here N0- ed of perod; B- prcpal balace; I- terest pad; R- amout pad; P- Prcpal repad at ed of perod. The total terest pad s 30.10, whch s ofte called the f ace charge. As metoed before, the total of the etres the last colum would equal the orgal prcpal were t ot for roudg errors. Whe oe s amortzg a loa, at the begg of ay perod the prcpal outstadg s preset value of the remag paymets. Usg ths fact together wth our prevous developmet, we obta the formulas lsted below that descrbe the amortzato of a terest bearg loa of B dollars, at a rate per perod, by equal paymets of R each ad such that a paymet s made at the ed of each perod. Notce below that the formula for the perodc paymet R volves a. 1. Perodc paymet: R = B ( = B. a (1 (1 + 2. Prcpal outstadg at ed of kth perod: ( 1 (1 + ( k+1 B k = Ra k = R. 3. Iterest kth paymet: 4. Prcpal cotaed kth paymet: I k = Ra k+1. P k = R(1 a k+1. 5. Total terest pad: I k = R( a or R A. k 51

The auty formula ( 1 (1 + B = R ca be solved for to gve the umber of perods of a loa: Thus B R = 1 (1 +. 1 B = (1 + R Takg logarthm both sdes ad solvg equato by we obta ( R l R B = l(1 +. Auty due 1. Perodcal paymets: R = B ( = B. (1 + a (1 (1 + 2. Outstadg loa at the ed of k th paymet perod (balace: ( 1 (1 + ( k 1 B k = Ra 1 k = R, k = 0,..., 1. Whe k = 0, we obta balace of the loa after the frst paymet (at the begg of the frst paymet terval. 3. Amout of terest k th paymet perod: 4. Repad loa at the ed of k th perod: I k = Ra k. 5. Total amout of terest: P k = R(1 a k. I = R( (1 + a arba R B. Example A.B. amortzes a loa 30000 for a ew home obtag a 20 year mortgage at the rate of 9 percet compouded mothly. Fd (a the mothly paymet, (b the terest the frst paymet, ad (c the prcpal repad the frst paymet. We have = 0.09 12 = 0.0075, = 12 20 = 240. The the mothly paymet R = 30000 ( 0.0075 = 30000 a 240 0.0075 (1 (1.0075 240 269.92. The terest porto of the frst paymet s I 1 = 30000 0.0075 = 225. Thus the prcpal repad the frst paymet s 269.92 225 = 44.92 Example A.B. purchases a TV system for 1500 ad agrees to pay t off by mothly paymets of 75. If the store charges terest at the rate of 12 percet compouded mothly, how may moths wll t take to pay off the debt? We have that R = 75, = 0.01, B = 1500. Thus 52

( l = l(1.01 75 75 1500 0.01 = l 1.25 l 1.01 22.4. Thus we obta 22.4 moths. realty there wll be 23 paymets; however, the fal paymet wll be less tha 75. We gve a example where we cosder aother structure of the amortzato schedule. Suppose that A.B. s borrowg 7000 to by car. The loa plus terest s to be repad equal quarterly stallmets made at the ed of each quarter durg 2- years terval. Let the terest rate be 16 percet compouded quarterly. Frst we determe the quarterly paymet R. Applyg formula of preset value of ordary auty we deduce Solvg R yelds 7000 = R a 8 0.04. R = 7000 6.732745 = 1039.69. Thus, the borrowed wll make eght paymets of 1039.69 each or 8 1039.69 = 8317.52, to repay 7000 loa. Thus, the terest s 8317.52 7000 = 1317.52, I = 1317.52. I what follows we use more detal amortzato schedule, whch s gve above. Such a schedule ormally shows the paymet umber, the amout pad, the terest pad, the prcpal repad ad outstadg debt balace. No I R P M B 1 280 1039.69 759.69 759.69 6240.31 2 249.61 1039.69 790.08 1549.77 5450.23 3 218.01 1039.69 821.68 2371.45 4628.55 4 185.14 1039.69 854.55 3226 3774 5 150.96 1039.69 888.73 4114.73 2885.27 6 115.41 1039.69 924.28 5039.01 1960.99 7 78.44 1039.69 961.25 6000.26 999.74 8 39.99 1039.69 999.70 6999.96 0.04. Total 1717.56 70000 0 Table 2 Studyg the last table ote that amout prcpal reducto for a perod s the dfferece betwee the paymet R ad the terest for that perod. The equty colum s the cumulato of the prcpal reductos. The balace colum may be determed by ether of two methods: 1. As the dfferece betwee the amout of the loas ad the equty; 2. As dfferece betwee the prevous perod s balace ad the prcpal reducto for the gve perod. 53

Let us summarze some basc cocepts from prevous sectos. For ay facal trasacto, the value of a amout of moey chages wth tme as a result of the applcato of terest. Thus, to accumulate or brg forwards a sgle paymet R for perods at a terest rate per perod, we multply R by (1 +. To brg back a sgle paymet R for perods at a terest rate of per perod, we multply R by (1 +. To accumulate or brg forward a auty of paymets of R each, we multply R by s. To brg back a auty of paymet of R dollars each, we multply R by a. Example A.B. borrowed 15000 from SEB bak at 16% compouded quarterly. The loa agreemet requres paymet of 2500 at the ed of every three moths. Costruct a amortzato schedule. No I R P M Balace 0 0 0 0 0 15000 1 600 2500 1900 1900 13100 2 524 2500 1976 3876 11124 3 444.96 2500 2055.04 5931.04 9068.96 4 362.76 2500 2137.24 8068.28 6931.72 5 277.27 2500 2222.73 10391.01 4708.99 6 188.36 2500 2311.64 12702.65 2397.35 7 95.89 2500 2397.35 15000 0 Table 3 Tasks for the practce 1. A ew flat cost 106,000 to a perso. Whe buyg the flat, the perso kocked dow the prce by 12% ad agreed to repay the etre amout 12 years by payg equal stalmets at the ed of each quarter. The terest rate s 16% ad the terest rate s compouded every 6 moths. Calculate the followg: 1 (a The amout of the fxed stalmet. (b How much wll A.B. stll owe after 8 years? ( How much wll they pay to completely repay the loa? (d How much terest wll they pay? 2 Complete the same task f the paymets are made at the begg of the paymet perod. 3 Complete the same task f the paymets are made at the ed of the paymet perod, as provded the codtos, but by deferrg them by four years. 3.2 Amortzato of loa deferred auty case Suppose that ordary auty wth deferred l deferred perods ad paymets perods. The total deferred auty umber of perods s + l. Usg geeral preset value formula we get R = A(l(1+l a. We ote that fllg amortzato schedula the frst l paymets are empty. Thus ths case we wrte R = 0. Assume that R t t = 1,..., + l are t th paymet made at the ed of perod. 54

The R k = { 0, k l, R, k l + 1 1. k th (k 0 perod book value are B k = B k 1 P k ; or { Ra (1 + k l, l = 1,... l B k = Ra +l k, k = l + 1, + l. 2. Amout of terest k th paymet perod: { B(1 + k, k = 1,..., l I l = Ra +l k+1 k = l + 1,..., + l. 3. Prcpal cotaed k th paymet: P k = R I k, k = 1,..., + l. 4. Prcpal outstadg at ed k th perod: M k = M k 1 + P k, k = 1,..., + l or M k = B B k, k = 1,..., + l.. Example Farmer loa 100000 for e year wth 10%. Loa was deferred for 4 year. Make amortzato schedule. We have B = 100000, = 0, 1. The for the formula we obta R = 38622, 58. The R = A 5(4 1, 1 4 a 5 0,1 Nr R I P M B 0 0 0 0 0 100000 1 0 10000 10000 10000 110000 2 0 11000 11000 21000 121000 3 0 12100 12100 33100 133100 4 0 13310 13310 46410 146410 5 38622, 58 14641 23981, 58 22428, 41 122428, 42 6 38622, 58 12242, 84 26379, 74 3951, 33 96048, 68 7 38622, 58 9604, 868 29017, 712 32969, 042 67031 8 38622, 58 6703, 1 31919, 48 64888, 822 35111, 52 9 38622, 58 3511, 152 35111, 428 100000, 25 0, 092 Σ 100000 10000 0 Table 4 55

3.3 Amortzato (complex auty We cosder the stuato whe the legth of the paymet terval s dfferet from the legth of the terest coverso perod the equal debt paymets form a complex auty. The amortzato of such debs has the same prcples whch was dscussed above case smple autes. The paymets are made at the ed of the paymet tervals are obtaed by formula A c = R( 1 (1 + p R = Ra p, here p = (1 + c 1. p Cosder the example ad costruct amortzato schedule. Example A debt of 30000 wth terest at 12% compouded quarterly s to be repad by equal paymets at the ed of each year for 7 years. 1 Compute the sze of the yearly paymets; 2 Costruct a amortzato schedule. We have that A c = 30000, = 7, c = 4, = 0.03. Further p = 1.03 4 1 = 0.01255088. R = 30000 4.4851295 = 6688.77. The a amortzato schedule: No I R P M B 1 3765.26 2923.51 6688.77 2923.51 27076.49 2 3398.34 6688.77 3290.43 6213.94 23786.06 3 2985.36 6688.77 3703.41 9917.35 20082.65 4 2520.55 6688.77 4168.22 14085.57 15914.43 5 1997.40 6688.77 4691.37 18776.94 11223.06 6 1408.59 6688.77 5280.18 24057.12 5942.88 7 745.88 6688.77 5942.88 30000.00 0 Total 16821.38 46821.38 Table 5 Tasks for the Practce 1. A perso has purchased a car that cost 66,000. It has bee agreed that the debt wll be repad equal aual stalmets 6 years. The terest rate s 10%, the terest s compouded quarter. Create a loa amortzato table. 2. A ew flat cost 1,560,000. A perso has agreed to repay the etre amout 12 years by payg equal stalmets at the begg of each quarter. The terest rate s 6% ad the terest s re-calculated every quarter. Determe: (a How much wll A.B. stll owe after 8 years? (b Fll the row of the amortsato table for the paymets of year 6. (c Complete the same task whe paymets are made at the ed of the paymet perod, as provded the codtos, but by deferrg them by 4 years. 3. A.B. has borrowed 8,500 wth 18% terest, whch are re-calculated every quarter for 8 years. Equal stalmets are made every moth, at the ed of each quarter. (a Calculate the sze of the mothly stalmets. 56

(b Calculate the repad terest utl the paymet 16 clusve. (c Whch part of the loa ( per cet was repad by paymet 4. A.B borrowed 140,000 wth 12(a How may paymets wll eed to be made utl the debt s repad? (b How much terest wll be pad wth paymet 6? (c What amout of the loa wll be repad wth paymet 10? (d Create a partal loa repaymet table, whch would have the frst three ad the last three paymet rows. 3.4 Repay of the debt case of the smple terest rates Cosder the problem of the repaymet of loas whe all repad are made by equal parts of loa plus terest case smple rates. Amortzato of loa (method S1 Cosder the followg problem: repad the debt at the ed of loas perod. I ths case the debt s repad at the ed of loa perod ad the paymets cossts from the terest rb. We have I k = rb, R k = rb, k = 1,... 1 ad R = B(1 + r. The schedule of the loas s such k I R P B 0 0 0 0 B 1 rb Br 0 B 2 rb 0 rb B............... -1 rb rb 0 B rb (1 + rb B 0 totals rb B + Br ab Table 6 Amortzato of loa (method S2 Suppose that a bak loas B. As were cosdered above the debt s to be repad by equal paymets of a each at the ed of each perod plus terest from the tal amout of the debt. Assume that the bak charges terest rate of r percet year. The all paymets R are equal to B + rb Essetally, amout B s a auty of paymets whch we gve the followg table: 57

k I R P B 0 0 0 0 B 1 rb B(r + 1 B B(1 1............... -1 rb B(r + 1 B rb B(r + 1 B 0 totals rb B + Br B B(1 1 Table 7 Amortzato of loa (method S3 or lear method Suppose that a bak loas B. Ths amout plus terest s to be repad by equal paymets of B each at the ed of each perod plus terest from the remader of the loas. Further, let us assume that the bak charges terest rate of r percet year. Essetally, for B the bak s a auty of paymets whch we gve the followg table: No I R P B 0 0 0 0 B 1 rb B(r + 1 B B(1 1 2 rb(1 1 B(r(1 1 + 1 B B(1 2 3 rb(1 2 B(r(1 2 + 1 B B(1 3............... -1 rb(1 2 2 B(r(1 + 1 B r B B( r + 1 B 0 totals rb( +1 +1 B B + Br 2 2 B Table 8 here I terest, P equal parts of loa, R paymet at the ed of perod, B remader of the loa (balace. We have that B k = B ( 1 k, k = 1, 2,... s decreasg arthmetcal sequece wth dfferece ra terest of the k th perod t k t k 1 s ad frst term rb. It s clear, that a I k = rb(1 (k 1, k = 0, 1,...,. 58

Applyg the formula for the sum of the terms of arthmetcal sequece we the deduce, that total amout of the terest s rb+ rb = +1 B. The sequece of the paymets R 2 2 k = B +I k s decreasg arthmetc sequece wth dfferece Br. The the total amout of the all paymets s B + + 1 rb. 2 Tasks for the practce 1. A loa of 20,000, take for 5 years, must be repad by stalmets every 6 moths. The ordary terest rate of the loa s 12%. Create a loa repaymet table: 1 Usg method Pl; 2 Usg method P2; 3 Usg method P3. 2. Whe mplemetg a vestmet project, a amout of 100,000 was borrowed for 20 years wth the ordary terest of 6%. A part of the loa or the terest wll be repad at the ed of each quarter. 1 Determe what the costs of facg ths project would be f the followg methods were appled: a Pl; b P2; c P3. 2 What s the balace value of the debt at the ed of year 10? a Applyg method Pl; b applyg method P2; c applyg method P3. 3 How much terest was pad o the loa utl the ed of year 12 clusvely: a applyg method Pl; b applyg method P2; c applyg method P3. 3. Whe expadg a busess, a etrepreeur has take a loa of 500,000 for 10 years wth the terest of 6%. The loa s repad every quarter. 1 Determe what fxed stalmets would eed to be pad at the ed of each quarter f: a The method P3 (lear was used to repay the loa; b The smple ordary auty method was used to repay the loa; c Compare the costs of both facg methods. 2 Create a amortsato table of the last two years f method P3 s appled for repayg the loa. 4. A three-year bll, whose terest rate s 8% s repad by quarterly paymets, whch cover the terest, equal stalmets. The loa s repad wth the last paymet, whch s coducted at maturty. Create the amortsato table for coverg the bll. 3.5 Skg fuds Defto The terest bearg fud whch paymets are made at perodc tme tervals to provde a desred sum of moey at a specfed future pot tme s called skg fud. Skg fuds usually volve large sums of moey used by both the prvate sector ad the publc sector to repay loas, face future captal acqustos, provde for the replacemet of deprecable plat ad equpmet ad recover vestmets depletable atural resources. The ma problem dealg wth skg fuds s that of determg sze of the perodc paymets, whch wll accumulate to kow a future amout. These paymets form a auty whch the accumulated value s kow. Depedg o whether the perodc paymets are made 1 at the ed or 2 at the begg of each paymet perod, the auty formed s a ordary auty or a auty due. Depedg o whether or ot the paymet terval s equal the legth a to the terest coverso perod, the auty formed s a smple auty or a complex auty. We troduce formulas defg amouts of skg fuds. I what follows R sze of perodc paymets; rate of the paymet perod; umber of coverse perods. 59

1 For the skg fuds case smple auty wth paymets at the ed of each paymet tervals, the amout of the skg fud fd by ( (1 + 1 S = R =: R s ; 2 For the skg fuds case smple auty wth paymets at the begg of each paymet tervals, the amout of the skg fud fd by ( (1 + S 1 = (1 + R =: R(1 + s 3 For the skg fuds case complex auty wth paymets at the ed of each paymet tervals, the amout of the skg fud fd by ( (1 + p S c 1 = R =: R s p, p 4 For the skg fuds case complex auty wth paymets at the begg of each paymet tervals, the amout of the skg fud fd by ( (1 + p S c 1 = (1 + p R =: R(1 + p s p. p here, both cases 3 4, p = (1 + c 1, c s the umber of terest coverso perods per paymet terval. Suppose, that a perso decded to accumulate a sum of moey by makg perodc deposts to a fud. At the ed of a specfed tme perod the deposts plus the terest eared equal the desre accumulated amout. Such a fud s called a skg fud. Geeral formulas Ordary auty 1. Perodcal paymets: R = 2. Balace at the ed of k t perod: 3. Iterest k th perod: 4. Total amout: S ( = S s (1 + 1. ( (1 + k 1 S k = Rs k = R, k = 1, 2,.... I k = Rs k 1, k = 1, 2,.... R(s arba S R. Auty due Nr dcate umber of perod ad R s doe at the begg of ths perod ad balace S k - at the ed of ths perod. 1. Perodcal paymets: R = 2. k th perod balace: S ( = A (1 + s (1 + ((1 + 1 ( (1 + k 1 S k = R(1 + s k = R(1 + k, k = 0, 1, 2,.... 60.

3. Iterest k th perod: 4. Total amout of terest: I k = S k 1, k = 0, 1,... 1. R((1 + s arba S R. Example Cosder a cotractor foreseeg the eed for a ew truck 4 years from ow. The prce of the truck s forecast to be 20000. The cotractor wshes to accumulate ths amout by settg asde semaual paymets of R each for 4 years. Each paymet of ths skg fud ears terest at 10 percet compouded semaually. The cotractor must determe the semaual paymet R. Sce the semaual paymet costtute a auty wth a amout of 20000, the S = R s. Solvg for R yelds R = 20000 s 8 0.05 2094.44. Thus, the semaual paymets R = 2094.44 plus terest wll accumulate to S = 20000. Note that the cotractor wll make eght paymets of 2094.44 each, or 8 2094.44 = 16755.52. Therefore, the terest eared s 20000 16755.52 = 3244.48. Ths result are summarzed the table, whch s called a schedule of the skg fud, whch more detals shows all the process of the paymets. I the skg fud schedule 1 paymet umber, 2 perodc paymet, 3 terest eared by the fud, 4 the crease the fud ad 5 accumulated balace wll be show. Costruct a skg fud schedule for the last example above. We have that R = 2094.44 ; = 8, = 0.05. Set No paymet umber; P - perodc paymet; I terest eared by the fud; B Balace accumulated balace creasg of the balace ths tme momet. No R I B 1 2094.44 0 0 2094.44 2 2094.44 104.72 2199.16 4293.60 3 2094.44 214.68 2309.12 6602.72 4 2094.44 330.14 2424.58 9027.30 5 2094.44 451.37 2545.81 11573.11 6 2094.44 578.66 2673.10 14246.21 7 2094.44 712.31 2806.75 17052.96 8 2094.44 852.65 2947.09 20000.05 Totals 16755.52 3244.53 20000.05 Table 8 61

Example A.B. compay wats to provde for replacemet of equpmet estmated 60000 seve years from ow. To do so the compay set up a skg fud to whch the compay wll pay equals sums of moey at the begg of each of the ext seve years. Iterest pad by the fud s 11.5% compouded aually. 1 Fd the sze of aual paymet to the fud; 2 What s a total paymet to the fud by A.B.?; 3 How much of the fud wll be terest? We have that S = 60000, = 7, = 0.115. The 1 60000 = 1.115 R s 7 0.115. From the last equalty we deduce R = 5416.42. 2 The total pad to the fud by A.B. wll be 7 5416.42 = 37914.94. 3 The terest eared by the fud wll be 60000 37914.94 = 22085.06. No. P I B 1 5416.42 622.89 6039.31 6039.31 2 5416.42 1317.41 6733.83 12773.14 3 5416.42 2091.8 7508.22 20281.36 4 5416.42 2955.24 8371.66 28653.02 5 5416.42 3917.99 9334.41 37987.43 6 5416.42 4991.44 10407.86 48395.29 7 5416.42 6188.35 11604.77 60000.06 Totals 37914.94 22085.12 60000.06 Table 9 We deal wth the skg fuds whch accumulate future value usg lear method (smple terest case. We assume, that terest s calculated from the balace value, ad accumulated terest ot captalzed. The balace s creased begg of paymet perod. I ths case we have the followg schedule of the skg fuds: Nr. R I B 1 R 0 R R 2 R rr R(1 + r 2R............... R rr( 1 R(1 + r( 1 R Σ R r ( 1 2 R R S Table 10 62

Here accumulated total amout s S = r ( 1 R + R. 2 Task for the practse 1. A perso has decded to save. The bak has offered the lear method for 10 years by payg stalmets every quarter. The terest rate s 10%. 1 Determe what stalmets the perso wll have to pay every quarter f durg the perod they hope to accumulate 200000. 2 Formulate the 5th ad 6th rows of paymets of the amortsato table. 3 What percetage of terest wll be cluded ths amout? Self-cotrol exercses 1. Create the geeral formulas, based o whch t would be possble formulate the row of ay paymet perod the amortsato table, whe: a The auty s ordary pad-up; b The auty s ordary pad-up ad deferred by paymet k. 2. A.B. has purchased a car that cost 36000. They kocked the prce dow by 4,000 ad agreed that the debt wll be repad equal stalmets wth 15 years, by payg the stalmets at the ed of each quarter. The terest rate s 14 (a Determe the sze of the fxed stalmet; (b How much wll they stll owe after 10 years? (c How much they wll pay total after 15 years? (d How much terest wll they pay? As: (a 1282.84 (b 18232.24 ( 80970.40 (d 44970.40. 3. A debt of 1000000 wth the terest rate of 15%, whch are re-calculated every year, s repad at the ed of each year wth a perod of 7 years. Create a loa amortsato table. Determe the sze of the aual stalmets, the total amouts pad ad the costs of the loa. As: Istalmets 240360; total amout 1682525; costs 682525. 4. A.B. has borrowed 920000 wth the terest rate of 13 As: Pad total 1494316; terest pad 574316. 5. A.B. has borrowed 8500000 wth the terest rate of 18%, whch s re-calculated every 8 years. Equal stalmets are also made every quarter, at the ed of each quarter. (a Calculate the sze of the quarterly paymets. (b Calculate the terest pad utl paymet 16, clusve; (c What part of the loa was repad wth paymet 20? As: (a 506287 (b 266724 (c 285683. 6. A.B has borrowed 2400000 wth the terest of 17%, whch are re-calculated every 6 moths. The debt s repad by stalmets of 250000 at the ed of each half of the year. (a How may paymets wll eed to be made utl the debt s repad? (b How much terest wll be pad wth paymet 6? (c What amout of the loa wll be pad wth paymet 10? (d Create a partal loa repaymet table, whch would clude the frst three ad the last three paymet rows ad the last balace row. As: (a = 20.750427 (b 180832 (c 95857 (d Total 5189503; 2789503; 2400000. 7. A debt of 2500000 s repad by stalmets of 350,000, whch are made at the ed of each half of the year. The terest of 21% s re-calculated every 6 moths: (a How may paymets wll eed to be doe utl the loa s repad? (b What wll be the amout of the last paymet? 63

As: (a = 13.884418 (b 311309. 8. A debt of 1750000 s repad by equal stalmets of 285000 at the ed of each year. The terest of 14(a Determe how may paymets wll eed to be made utl the debt s repad. (b Determe the costs of the loa of the frst three years. (c Whch part of the loa wll be repad year 7? (d Create a debt amortsato table by dcatg the frst three ad the last three loa repaymet years ad the balace row. As: (a = 16.294188 (b 746405 (c 70775 (d balace row 4647884; 2897884; 1750000. 9. A loa has bee take for 5 years, t has bee agreed that t wll be repad by stalmets every 6 moths, whe the smple terest rate s 12%. Create a loa repaymet table: 1 Usg method P1. 2 Usg method P2. 3 Usg method P3. 10. Whe mplemetg a vestmet project, a amout of 100,000 was borrowed for 20 years wth the smple terest of 6%. A part of the loa or the terest wll be pad at the ed of each quarter. 1 Determe what the costs of facg ths project would be f the followg methods were appled: a P1; b P2; c P3. 2 What s the debt balace value at the ed of year 10? a applyg method P1; b applyg method P2; c applyg method P3. 3 How much terest wll be pad o the loa utl the ed of year 12 clusve? a applyg method P1; b applyg method P2; c applyg method P3. 11. A etrepreeur has borrowed a loa of 500000 for 10 years wth the terest rate of 6%; the terest rate s re-calculated every quarter. 1 Determe what fxed stalmets would eed to be pad at the ed of each quarter f: a Method P3 (lear was appled to repay the loa; b The ordary auty method was appled to repay the loa; 2 Compare the facg costs of both methods; 3 Create the amortsato table of the last two years f method P3 s appled to repay the loa. 12. A.B. at the begg of each moth trasfers 900 to a accout whch at the ed of the stalmets they expect to accumulate 72,500. The terest rate s 12%, the terest s re-calculated ever moth. (a How may paymets wll eed to be doe utl the desred result s acheved? (b What wll be the amout of the last stalmet? As: (a = 160.53831 (b 485.59. 13. After purchasg a boat that cost 330000 va leasg, the future every quarter 43000 wll have to be pad. Paymets are deferred by three years. The value of moey durg ths etre perod s 20%. The terest s recalculated every quarter. (a How may paymets wll eed to be made utl the debt s repad? (b What s the amout of the last leasg fee? As: : (a = 21.888956 (b 38.328. 14. A trasport compay, order to reew the vehcle park, seeks to accumulate 11000000 5 years. At the ed of each half of the year, from the proft they trasfer a fxed amout to the cumulatve accout. The terest rate of the accout s 17.5%. The terest rate s calculated every 6 moths. (a Determe the sze of the fxed amout. (b What wll the balace of the accout be after paymet 3? (c Determe the amout of terest that wll accumulate after makg paymet 6. As: (a 732.706 (b 2396063 (c 381784. It s ecessary to kow the followg: Loa repaymet methods, comparg methods by determg ther effectveess, creatg loa amortsato tables, calculatg the values of ay depedetly chose rows ( the amortsato tables, creatg cumulatve fud tables ad calculatg the values of ay perod, aalysed the cumulatve fud tables, coductg calculatos the case of smple terest ad auty. Homework exercses 1. Make geeral formulas for the amortzato of loa ad skg fuds case auty due. 64

2. A.B. borrowed 140000 for replacemet of equpmet. The debt s repad stalmets of 12000 made at the ed of every three moths. (a If terest s 10% compouded quarterly, how may paymets are eeded? (b How much wll Comfort Swm owe after two years? (c How much of the 12th paymet s terest? (d How much of the prcpal wll be repad by the 20th paymet? (e Costruct a partal amortzato schedule showg detals of the frst three paymets, the last three paymets ad totals. 3. A debt of 100000 s repad equal mothly stalmets over four years. Iterest s 15% compouded quarterly. (a What s the sze of the mothly paymets? (b What wll be the total cost of borrowg? (c What s the outstadg balace after oe year? (d How much of the 30th paymet s terest? (e Costruct a partal amortzato schedule showg detals of the frst three paymets, the last three paymets ad totals. 4. A cotract worth 80000 provdes beefts of 20000 at the ed of each year. The beefts are deferred for te years ad terest s 9% compouded quarterly. (a How may paymets are to be made uder the cotract? (b What s the sze of the last beeft paymet? 5. Mr.A.B borrowed 800000 from hs Credt Uo. He agreed to repay the loas by makg equal quarterly paymets for fve years (at the ed of quarter. Iterest rate s 15%. (a What s the sze of the quarterly paymet? (b How much wll the loa cost hm? (c How much wll Mr.A.B owe after three years? (d How much terest wll he pay hs 16th paymet? (e How much of the prcpal wll he repay by hs 14th paymet? (f Prepare a partal amortzato schedule showg detals of the frst three paymets, Paymets 8, 9, 10, the last three paymets ad totals. Use method S3 (Lear 6. A skg fud of 100000 s to be created by equal aual paymets at the begg of each sx moth for seve years. Iterest eared by the fud s 17.5% compouded aually. (a Compute the aual depost to the fud. (b Costruct a skg fud schedule showg totals. 7. A skg fud of 100000 s to be created by equal aual paymets at the ed of each sx moth for seve years. Iterest eared by the fud s 17.5% compouded aually. (a Compute the aual depost to the fud. (b Costruct a skg fud schedule showg totals. 8. Fd balace of the skg fud after 10 years f at the begg of each year are deposte 5000. Iterest eared by the fud s 22%. Make amortzato schedula f case lear method a S1; b S2; c S3. 65