Consider the problem of borrowing $250,000 to buy a house.  You borrow the money at a fixed interest rate of 4.8% compounded monthly.  The term of the mortgage is for 20 years.  However, you decide against making minimum payments.  Instead you decide to pay an additional $200 every month.  How long will it take you to pay off the loan and how much have you saved on interest by paying off the loan early?

.048/12;

0.4000000000e-2

(1)

Typesetting:-RuleAssistant();

.048/12;

0.4000000000e-2

(2)

with(Finance,annuity);

[annuity]

(3)

20*12;

240

(4)

solve(annuity(payment,0.004,240)=250000,payment);

1622.393675

(5)

payment:=(5);

1622.393675

(6)

Payment:=payment+200;

1822.393675

(7)

balance:=[250000];

[250000]

(8)

Balance:=[250000];

[250000]

(9)

for n to 240 do
a:=nops(balance):
b:=1.004*balance[a]-payment:
balance:=[op(balance),b]:
end do:
a:='a':
b:='b':
nops(balance);

241

(10)

balance[241];

-0.8001e-2

(11)

for n to 240 do
a:=nops(Balance):
b:=1.004*Balance[a]-Payment:
Balance:=[op(Balance),b]:
end do:
a:='a':
b:='b':
nops(Balance);

241

(12)

balance[241];

-0.8001e-2

(13)

Balance[241];

-80335.01464

(14)

balance2:=[seq([n-1,balance[n]],n=1..241)];

[[0, 250000], [1, 249377.6063], [2, 248752.7230], [3, 248125.3402], [4, 247495.4479], [5, 246863.0360], [6, 246228.0944], [7, 245590.6131], [8, 244950.5819], [9, 244307.9905], [10, 243662.8288], [11, 243015.0864], [12, 242364.7530], [13, 241711.8183], [14, 241056.2719], [15, 240398.1033], [16, 239737.3020], [17, 239073.8575], [18, 238407.7592], [19, 237738.9965], [20, 237067.5588], [21, 236393.4353], [22, 235716.6153], [23, 235037.0881], [24, 234354.8428], [25, 233669.8685], [26, 232982.1543], [27, 232291.6892], [28, 231598.4623], [29, 230902.4624], [30, 230203.6785], [31, 229502.0995], [32, 228797.7142], [33, 228090.5114], [34, 227380.4797], [35, 226667.6079], [36, 225951.8846], [37, 225233.2984], [38, 224511.8379], [39, 223787.4916], [40, 223060.2479], [41, 222330.0952], [42, 221597.0219], [43, 220861.0163], [44, 220122.0667], [45, 219380.1613], [46, 218635.2882], [47, 217887.4357], [48, 217136.5917], [49, 216382.7444], [50, 215625.8817], [51, 214865.9915], [52, 214103.0618], [53, 213337.0803], [54, 212568.0349], [55, 211795.9133], [56, 211020.7033], [57, 210242.3924], [58, 209460.9683], [59, 208676.4185], [60, 207888.7305], [61, 207097.8917], [62, 206303.8896], [63, 205506.7115], [64, 204706.3446], [65, 203902.7763], [66, 203095.9937], [67, 202285.9840], [68, 201472.7342], [69, 200656.2314], [70, 199836.4626], [71, 199013.4148], [72, 198187.0748], [73, 197357.4294], [74, 196524.4654], [75, 195688.1696], [76, 194848.5286], [77, 194005.5290], [78, 193159.1574], [79, 192309.4003], [80, 191456.2442], [81, 190599.6755], [82, 189739.6805], [83, 188876.2455], [84, 188009.3568], [85, 187139.0005], [86, 186265.1628], [87, 185387.8298], [88, 184506.9874], [89, 183622.6216], [90, 182734.7184], [91, 181843.2636], [92, 180948.2430], [93, 180049.6423], [94, 179147.4472], [95, 178241.6433], [96, 177332.2162], [97, 176419.1514], [98, 175502.4343], [99, 174582.0503], [100, 173657.9848], [101, 172730.2230], [102, 171798.7502], [103, 170863.5515], [104, 169924.6120], [105, 168981.9167], [106, 168035.4507], [107, 167085.1988], [108, 166131.1459], [109, 165173.2768], [110, 164211.5762], [111, 163246.0288], [112, 162276.6192], [113, 161303.3320], [114, 160326.1516], [115, 159345.0625], [116, 158360.0491], [117, 157371.0956], [118, 156378.1863], [119, 155381.3053], [120, 154380.4368], [121, 153375.5648], [122, 152366.6734], [123, 151353.7464], [124, 150336.7677], [125, 149315.7211], [126, 148290.5903], [127, 147261.3590], [128, 146228.0107], [129, 145190.5290], [130, 144148.8974], [131, 143103.0993], [132, 142053.1180], [133, 140998.9368], [134, 139940.5388], [135, 138877.9073], [136, 137811.0252], [137, 136739.8756], [138, 135664.4414], [139, 134584.7055], [140, 133500.6506], [141, 132412.2595], [142, 131319.5148], [143, 130222.3992], [144, 129120.8951], [145, 128014.9850], [146, 126904.6512], [147, 125789.8761], [148, 124670.6419], [149, 123546.9308], [150, 122418.7248], [151, 121286.0060], [152, 120148.7563], [153, 119006.9576], [154, 117860.5917], [155, 116709.6404], [156, 115554.0853], [157, 114393.9079], [158, 113229.0898], [159, 112059.6125], [160, 110885.4573], [161, 109706.6054], [162, 108523.0381], [163, 107334.7366], [164, 106141.6818], [165, 104943.8548], [166, 103741.2365], [167, 102533.8077], [168, 101321.5492], [169, 100104.4417], [170, 98882.46582], [171, 97655.60200], [172, 96423.83074], [173, 95187.13238], [174, 93945.48724], [175, 92698.87552], [176, 91447.27734], [177, 90190.67278], [178, 88929.04180], [179, 87662.36430], [180, 86390.62008], [181, 85113.78888], [182, 83831.85036], [183, 82544.78408], [184, 81252.56954], [185, 79955.18614], [186, 78652.61320], [187, 77344.82998], [188, 76031.81562], [189, 74713.54920], [190, 73390.00972], [191, 72061.17608], [192, 70727.02710], [193, 69387.54154], [194, 68042.69804], [195, 66692.47516], [196, 65336.85138], [197, 63975.80512], [198, 62609.31466], [199, 61237.35824], [200, 59859.91400], [201, 58476.95998], [202, 57088.47414], [203, 55694.43436], [204, 54294.81842], [205, 52889.60402], [206, 51478.76876], [207, 50062.29016], [208, 48640.14564], [209, 47212.31254], [210, 45778.76812], [211, 44339.48952], [212, 42894.45380], [213, 41443.63794], [214, 39987.01882], [215, 38524.57322], [216, 37056.27784], [217, 35582.10928], [218, 34102.04404], [219, 32616.05854], [220, 31124.12910], [221, 29626.23194], [222, 28122.34320], [223, 26612.43890], [224, 25096.49498], [225, 23574.48728], [226, 22046.39156], [227, 20512.18346], [228, 18971.83852], [229, 17425.33220], [230, 15872.63986], [231, 14313.73674], [232, 12748.59802], [233, 11177.19874], [234, 9599.513855], [235, 8015.518235], [236, 6425.186633], [237, 4828.493705], [238, 3225.414005], [239, 1615.921986], [240, -0.8001e-2]]

(15)

Balance2:=[seq([n-1,Balance[n]],n=1..241)];

[[0, 250000], [1, 249177.6063], [2, 248351.9230], [3, 247522.9370], [4, 246690.6350], [5, 245855.0038], [6, 245016.0301], [7, 244173.7005], [8, 243328.0016], [9, 242478.9199], [10, 241626.4419], [11, 240770.5540], [12, 239911.2425], [13, 239048.4938], [14, 238182.2941], [15, 237312.6296], [16, 236439.4864], [17, 235562.8506], [18, 234682.7083], [19, 233799.0454], [20, 232911.8479], [21, 232021.1016], [22, 231126.7923], [23, 230228.9058], [24, 229327.4277], [25, 228422.3437], [26, 227513.6394], [27, 226601.3003], [28, 225685.3118], [29, 224765.6593], [30, 223842.3282], [31, 222915.3038], [32, 221984.5713], [33, 221050.1159], [34, 220111.9227], [35, 219169.9767], [36, 218224.2629], [37, 217274.7663], [38, 216321.4717], [39, 215364.3639], [40, 214403.4277], [41, 213438.6477], [42, 212470.0086], [43, 211497.4949], [44, 210521.0912], [45, 209540.7819], [46, 208556.5513], [47, 207568.3838], [48, 206576.2636], [49, 205580.1750], [50, 204580.1020], [51, 203576.0287], [52, 202567.9391], [53, 201555.8172], [54, 200539.6468], [55, 199519.4117], [56, 198495.0956], [57, 197466.6823], [58, 196434.1553], [59, 195397.4982], [60, 194356.6945], [61, 193311.7276], [62, 192262.5808], [63, 191209.2374], [64, 190151.6806], [65, 189089.8936], [66, 188023.8595], [67, 186953.5612], [68, 185878.9817], [69, 184800.1039], [70, 183716.9106], [71, 182629.3845], [72, 181537.5083], [73, 180441.2646], [74, 179340.6360], [75, 178235.6048], [76, 177126.1535], [77, 176012.2644], [78, 174893.9198], [79, 173771.1018], [80, 172643.7925], [81, 171511.9740], [82, 170375.6282], [83, 169234.7370], [84, 168089.2822], [85, 166939.2456], [86, 165784.6089], [87, 164625.3536], [88, 163461.4613], [89, 162292.9134], [90, 161119.6914], [91, 159941.7765], [92, 158759.1499], [93, 157571.7928], [94, 156379.6863], [95, 155182.8113], [96, 153981.1488], [97, 152774.6797], [98, 151563.3847], [99, 150347.2445], [100, 149126.2398], [101, 147900.3511], [102, 146669.5588], [103, 145433.8433], [104, 144193.1850], [105, 142947.5640], [106, 141696.9606], [107, 140441.3547], [108, 139180.7264], [109, 137915.0556], [110, 136644.3221], [111, 135368.5057], [112, 134087.5860], [113, 132801.5426], [114, 131510.3551], [115, 130214.0028], [116, 128912.4651], [117, 127605.7213], [118, 126293.7505], [119, 124976.5318], [120, 123654.0442], [121, 122326.2667], [122, 120993.1781], [123, 119654.7571], [124, 118310.9824], [125, 116961.8326], [126, 115607.2862], [127, 114247.3216], [128, 112881.9172], [129, 111511.0512], [130, 110134.7017], [131, 108752.8468], [132, 107365.4645], [133, 105972.5327], [134, 104574.0291], [135, 103169.9315], [136, 101760.2175], [137, 100344.8647], [138, 98923.85052], [139, 97497.15224], [140, 96064.74718], [141, 94626.61250], [142, 93182.72528], [143, 91733.06250], [144, 90277.60108], [145, 88816.31780], [146, 87349.18940], [147, 85876.19248], [148, 84397.30358], [149, 82912.49912], [150, 81421.75544], [151, 79925.04878], [152, 78422.35530], [153, 76913.65104], [154, 75398.91196], [155, 73878.11394], [156, 72351.23272], [157, 70818.24398], [158, 69279.12328], [159, 67733.84610], [160, 66182.38780], [161, 64624.72368], [162, 63060.82890], [163, 61490.67854], [164, 59914.24758], [165, 58331.51090], [166, 56742.44326], [167, 55147.01936], [168, 53545.21376], [169, 51937.00094], [170, 50322.35526], [171, 48701.25100], [172, 47073.66232], [173, 45439.56330], [174, 43798.92788], [175, 42151.72992], [176, 40497.94316], [177, 38837.54126], [178, 37170.49776], [179, 35496.78608], [180, 33816.37954], [181, 32129.25138], [182, 30435.37472], [183, 28734.72254], [184, 27027.26776], [185, 25312.98316], [186, 23591.84142], [187, 21863.81512], [188, 20128.87670], [189, 18386.99854], [190, 16638.15286], [191, 14882.31180], [192, 13119.44738], [193, 11349.53150], [194, 9572.535955], [195, 7788.432424], [196, 5997.192479], [197, 4198.787574], [198, 2393.189049], [199, 580.368130], [200, -1239.704072], [201, -3067.056563], [202, -4901.718464], [203, -6743.719013], [204, -8593.087564], [205, -10449.85359], [206, -12314.04668], [207, -14185.69654], [208, -16064.83300], [209, -17951.48600], [210, -19845.68562], [211, -21747.46204], [212, -23656.84556], [213, -25573.86662], [214, -27498.55576], [215, -29430.94366], [216, -31371.06110], [217, -33318.93902], [218, -35274.60846], [219, -37238.10056], [220, -39209.44664], [221, -41188.67810], [222, -43175.82648], [223, -45170.92346], [224, -47174.00082], [225, -49185.09050], [226, -51204.22454], [227, -53231.43512], [228, -55266.75454], [229, -57310.21524], [230, -59361.84978], [231, -61421.69086], [232, -63489.77130], [233, -65566.12406], [234, -67650.78224], [235, -69743.77904], [236, -71845.14784], [237, -73954.92210], [238, -76073.13546], [239, -78199.82168], [240, -80335.01464]]

(16)

with(plots):

plot1:=listplot(balance2,color=black,thickness=3,gridlines=true,tickmarks=[[seq(40*n,n=0..6)],[seq(25000*n,n=-3..10)]],font=[Helvetica,Bold,9]):

plot2:=listplot(Balance2,color=red,thickness=3,gridlines=true,tickmarks=[[seq(40*n,n=0..6)],[seq(25000*n,n=-3..10)]],font=[Helvetica,Bold,9]):

display(plot1,plot2);

 

payment;

1622.393675

(17)

Payment;

1822.393675

(18)

balance2[241];

[240, -0.8001e-2]

(19)

Balance2[201];

[200, -1239.704072]

(20)

240*payment-250000;

139374.4820

(21)

200*Payment-250000;

114478.7350

(22)

(21)-(22);

24895.7470

(23)

By paying an additional $200 every month, you have reduced the cost of the loan by almost $25,000.  Not bad!  That's actually a pretty big savings, especially in today's dubious economy.

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