-*-紫色王家思絮絮-*-
我思無邪、我行無悔
正文
0) 開場白
前幾天推導了房屋貸款的單利模型 (見如下帖子的末尾部分)
http://blog.creaders.net/jingjibird628/user_blog_diary.php?did=131237
對 initial principal P，月利率 r，number of total payments N、基於單利 Simple Interest 的房貸而言，如果要求 equal monthly payment，那麽每月的 Mortgage Payment M 結果是：
M = P (1+N*r) / (N + N*(N-1)*r/2)
例如，如果 P = $500,000，r = 0.005 (對應年率 6%)，30-年房貸 (or N = 360)，那麽月配結果是 M = $2,049.48
Runner 同學指出了一個錯誤，就是我在推導過程中忘記了檢查某個月的本金是不是已經變成了 0，因為對單利模型而言，每月的月配都是先計入本金，直到本金變成 0 後才計入利息部分 (這應該是康門三四)，可是我疏忽了這點，因此計算結果 M 比實際值偏小。
這 裏重新修正模型。修正模型是舉手之勞，但是要具體計算出 Mortgage 月配，則要費點周折，因為此時 M 不像上麵那樣有個簡潔的顯示表達式，我隻得編個小程序計算 M。文章最後給出了 “loan amortization table”，供那些堅持現在房貸市場的貸款是基於單利模型的同學參考。
導出單利模型後，俺隨後就真是好玩朋友的一個“極端例子”做了個簡單的回答，因為這個問題比較有代表意義，寫出來供有興趣的朋友們參考。
1) 房貸的單利模型
除了將現在的房貸市場的 compound interest 換為 simple interest 外，我們 keep 所有的其它要求，例如保證 equal monthly payment，等。沿用上文的記號：
P：貸款總額，Initial Principal;
N：總月配數 (例如對 30 年貸款，N = 12 * 30 = 360)；
B_i：第 i-th month 的 Balance (i = 0, 1,2,...,N，下同，不在標注)，B_i = P_i + I_i，where；
P_i：第 i-th month B_i 中的 Principal 部分 (顯然，P_0 = P)；
I_i：第 i-th month B_i 中的 Interest 部分 (顯然，I_0 = 0)；
M：每月的 Mortgage Payment。
Define K = K (M) = [P/M]，where [x] 表示不超過 x 的最大整數，例如 [3] = 3，[3.14] = 3。這樣，我們有：
0) 最開始，Balance B_0 = P (你的 principal amount)；
1) 1st month，B_1 = (1+r) P - M，這和 Compound Interest Model 是一樣的，因為最開始賬號上沒有任何利息。現在，和 Compound Interest Model 不同，我們得嚴密跟蹤 P_1 and I_1:
B_1 = P_1 + I_1 = (P-M) + r*P
2) 2nd month，注意此時隻有 P_1 產生利息，所以
B_2 = P_1 (1+r) + I_1 - M = (P-M) (1+r) + r*P -M = (P-2M) + r*(2P-M) = P_2 + I_2
3) 3rd month， B_3 = P_2 (1+r) + I_2 - M = (P-2M) (1+r) + r*(2P-M) -M
= (P-3M) + r*(3P-3M) = P_3 + I_3
...... ......
i) i-th month ( i <= K)，B_i = (P - i*M) + (i*P - i*(i-1)*M/2) * r = P_i + I_i
...... ......
K) K-th month：B_K = (P - K*M) + (K*P - K*(K-1)*M/2) * r = P_K + I_K；
注意，因為 K = [P/M]，所以 0 <= P_K = (P - K*M) < M，意思是，這個月的本金部分已經接近 0 了，差距不足一個 M。這就是說，下個月的本金會變成 0。
K+1) (K+1)-th month：B_(K+1) = (P-K*M) (1+r) + (K*P - K*(K-1)*M/2) * r - M
= 0 + ((P - (K+1)*M) + (K*P - K*(K-1)*M/2) * r ) = P_(K+1) + I_(K+1)，
where P_(K+1) = 0, I_(K+1) = (P - (K+1)*M) + (K*P - K*(K-1)*M/2) * r;
...... ......
N) N-th month: B_N = 0 + ((P - N*M) + (K*P - K*(K-1)*M/2) * r ) = P_N + I_N，where P_N = 0。
因為第 N 月貸款必須償還完，所以 B_N = 0，所以 P - N*M + (K*P - K*(K-1)*M/2) * r = 0，代人 K 的定義，因此我們有：
P - N*M + ([P/M]*P - [P/M]*([P/M]-1)*M/2) * r = 0
這就是 Mortgage 必須滿足的方程。鑒於 [P/M] 的複雜，俺難以得出個簡潔的顯示解，隻得編個小程序來數值計算 M。對 P = $500,000，r=0.005，N=360 基於 Simple Interest 的房貸，結果如下：
複利模型 單利模型
-------------------------------------------------------------
Monthly Mortgage 2,997.75 2,186.41
Total Interest 579,190.95 287,107.6
所以如果是 Simple Interest 房貸，銀行損失大約三十萬的利息收入。
2) 回答真是好玩朋友的“極端問題”
真是好玩朋友就基於單利模型的貸款提出了個貌似荒謬的問題 (不是說真是好玩荒謬，是說問題荒謬，嗬嗬)。原問題拷貝如下：
“ 我給你舉一個極端的例子：我借100W，20年，5%年利。一年後我把100萬還掉。欠5W利息。因為利息不必再付息，我在20年後再還那5W。可不可以？（這樣合理不合理？） ”
俺 的理解是，這當然是個“合法”的 Simple Interest 貸款，因為它符合單利貸款的條件。這種貸款是不是顯得荒謬？當然，顯得荒謬，因為客戶手持了銀行5W而無須任何代價，這和市場的“公平合理”的原則是矛盾 的，對不對？這其中荒謬的根源在哪？就是因為 Simple Interest 與市場合理的要求是矛盾的。市場真正完全合理的要求，假設晝夜是不分的等，是 compound continuously，因為錢 (包括本金、利息) 在任何時刻是等價的。當然，真正做到 compound continuously 很困難，市場隻能在一個相對合理的時間區段去 compound，做到大體上的公平合理。這種荒謬的根源很好理解，換個角度看，假設 Simple Interest 是合理的，這就意味著 Principal，Interest 是兩種性質不同的錢，前者是能生蛋的金母雞，能產生利息，後者是不能生蛋的木屐，不能產生利息。所以這就意味著客戶手中最後的5萬利息是第二種錢，它不產生任何利息，它在誰手裏是一樣的，對不對？
所以，就公平、合理的角度而言，因為本金、利息部分應該等價 (從投資的角度而言，它們沒有分別)，所以無論是貸款還是儲蓄，Simple Interest 是不合理的，合理的是 Compound Interest。
3) 附錄：A Sample Amortization Table
對 P = $500,000，r=0.005，N=360 基於 Simple Interest 的房貸，版麵排列有些不整齊，懶得花力氣調整，大家湊合著看。注意第 228 到 229 月，貸款的部分本金變成了 0。
------------------ Amortization schedule ------------------
Month Balance 本金部分 利息部分 累積付款 累積付款利息部分
0 $500,000 $500,000 $0 $0 $0
1 $500,313.59 $497,813.59 $2,500 $2,186.41 -$0
2 $500,616.25 $495,627.18 $4,989.07 $4,372.82 -$0
3 $500,907.97 $493,440.77 $7,467.2 $6,559.23 -$0
4 $501,188.77 $491,254.36 $9,934.41 $8,745.64 $0
5 $501,458.63 $489,067.95 $12,390.68 $10,932.05 $0
6 $501,717.56 $486,881.54 $14,836.02 $13,118.46 $0
7 $501,965.56 $484,695.13 $17,270.43 $15,304.87 -$0
8 $502,202.62 $482,508.72 $19,693.9 $17,491.28 -$0
9 $502,428.76 $480,322.31 $22,106.45 $19,677.69 $0
10 $502,643.96 $478,135.9 $24,508.06 $21,864.1 $0
11 $502,848.23 $475,949.49 $26,898.74 $24,050.51 $0
12 $503,041.56 $473,763.08 $29,278.48 $26,236.92 $0
13 $503,223.97 $471,576.67 $31,647.3 $28,423.33 -$0
14 $503,395.44 $469,390.26 $34,005.18 $30,609.74 -$0
15 $503,555.98 $467,203.85 $36,352.13 $32,796.15 -$0
16 $503,705.59 $465,017.44 $38,688.15 $34,982.56 $0
17 $503,844.27 $462,831.03 $41,013.24 $37,168.97 $0
18 $503,972.02 $460,644.62 $43,327.4 $39,355.38 $0
19 $504,088.83 $458,458.21 $45,630.62 $41,541.79 -$0
20 $504,194.71 $456,271.8 $47,922.91 $43,728.2 -$0
21 $504,289.66 $454,085.39 $50,204.27 $45,914.61 $0
22 $504,373.68 $451,898.98 $52,474.7 $48,101.02 $0
23 $504,446.76 $449,712.57 $54,734.19 $50,287.43 $0
24 $504,508.91 $447,526.16 $56,982.75 $52,473.84 $0
25 $504,560.13 $445,339.75 $59,220.38 $54,660.25 -$0
26 $504,600.42 $443,153.34 $61,447.08 $56,846.66 -$0
27 $504,629.78 $440,966.93 $63,662.85 $59,033.07 -$0
28 $504,648.21 $438,780.52 $65,867.69 $61,219.48 $0
29 $504,655.7 $436,594.11 $68,061.59 $63,405.89 $0
30 $504,652.26 $434,407.7 $70,244.56 $65,592.3 $0
31 $504,637.89 $432,221.29 $72,416.6 $67,778.71 $0
32 $504,612.58 $430,034.88 $74,577.7 $69,965.12 -$0
33 $504,576.35 $427,848.47 $76,727.88 $72,151.53 $0
34 $504,529.18 $425,662.06 $78,867.12 $74,337.94 $0
35 $504,471.08 $423,475.65 $80,995.43 $76,524.35 $0
36 $504,402.05 $421,289.24 $83,112.81 $78,710.76 $0
37 $504,322.08 $419,102.83 $85,219.25 $80,897.17 $0
38 $504,231.19 $416,916.42 $87,314.77 $83,083.58 -$0
39 $504,129.36 $414,730.01 $89,399.35 $85,269.99 -$0
40 $504,016.6 $412,543.6 $91,473 $87,456.4 -$0
41 $503,892.91 $410,357.19 $93,535.72 $89,642.81 $0
42 $503,758.28 $408,170.78 $95,587.5 $91,829.22 $0
43 $503,612.73 $405,984.37 $97,628.36 $94,015.63 $0
44 $503,456.24 $403,797.96 $99,658.28 $96,202.04 -$0
45 $503,288.82 $401,611.55 $101,677.27 $98,388.45 $0
46 $503,110.47 $399,425.14 $103,685.33 $100,574.86 $0
47 $502,921.18 $397,238.73 $105,682.45 $102,761.27 $0
48 $502,720.97 $395,052.32 $107,668.65 $104,947.68 $0
49 $502,509.82 $392,865.91 $109,643.91 $107,134.09 $0
50 $502,287.74 $390,679.5 $111,608.24 $109,320.5 -$0
51 $502,054.73 $388,493.09 $113,561.64 $111,506.91 -$0
52 $501,810.78 $386,306.68 $115,504.1 $113,693.32 -$0
53 $501,555.91 $384,120.27 $117,435.64 $115,879.73 $0
54 $501,290.1 $381,933.86 $119,356.24 $118,066.14 $0
55 $501,013.36 $379,747.45 $121,265.91 $120,252.55 $0
56 $500,725.68 $377,561.04 $123,164.64 $122,438.96 -$0
57 $500,427.08 $375,374.63 $125,052.45 $124,625.37 -$0
58 $500,117.54 $373,188.22 $126,929.32 $126,811.78 $0
59 $499,797.07 $371,001.81 $128,795.26 $128,998.19 $0
60 $499,465.67 $368,815.4 $130,650.27 $131,184.6 $0
61 $499,123.34 $366,628.99 $132,494.35 $133,371.01 $0
62 $498,770.07 $364,442.58 $134,327.49 $135,557.42 $0
63 $498,405.88 $362,256.17 $136,149.71 $137,743.83 -$0
64 $498,030.75 $360,069.76 $137,960.99 $139,930.24 -$0
65 $497,644.69 $357,883.35 $139,761.34 $142,116.65 -$0
66 $497,247.69 $355,696.94 $141,550.75 $144,303.06 $0
67 $496,839.77 $353,510.53 $143,329.24 $146,489.47 $0
68 $496,420.91 $351,324.12 $145,096.79 $148,675.88 $0
69 $495,991.12 $349,137.71 $146,853.41 $150,862.29 $0
70 $495,550.4 $346,951.3 $148,599.1 $153,048.7 $0
71 $495,098.75 $344,764.89 $150,333.86 $155,235.11 $0
72 $494,636.16 $342,578.48 $152,057.68 $157,421.52 $0
73 $494,162.64 $340,392.07 $153,770.57 $159,607.93 $0
74 $493,678.19 $338,205.66 $155,472.53 $161,794.34 $0
75 $493,182.81 $336,019.25 $157,163.56 $163,980.75 -$0
76 $492,676.5 $333,832.84 $158,843.66 $166,167.16 -$0
77 $492,159.25 $331,646.43 $160,512.82 $168,353.57 -$0
78 $491,631.07 $329,460.02 $162,171.05 $170,539.98 $0
79 $491,091.96 $327,273.61 $163,818.35 $172,726.39 $0
80 $490,541.92 $325,087.2 $165,454.72 $174,912.8 $0
81 $489,980.95 $322,900.79 $167,080.16 $177,099.21 $0
82 $489,409.04 $320,714.38 $168,694.66 $179,285.62 $0
83 $488,826.2 $318,527.97 $170,298.23 $181,472.03 $0
84 $488,232.43 $316,341.56 $171,890.87 $183,658.44 $0
85 $487,627.73 $314,155.15 $173,472.58 $185,844.85 $0
86 $487,012.1 $311,968.74 $175,043.36 $188,031.26 $0
87 $486,385.53 $309,782.33 $176,603.2 $190,217.67 -$0
88 $485,748.03 $307,595.92 $178,152.11 $192,404.08 -$0
89 $485,099.6 $305,409.51 $179,690.09 $194,590.49 -$0
90 $484,440.24 $303,223.1 $181,217.14 $196,776.9 $0
91 $483,769.95 $301,036.69 $182,733.26 $198,963.31 $0
92 $483,088.72 $298,850.28 $184,238.44 $201,149.72 $0
93 $482,396.56 $296,663.87 $185,732.69 $203,336.13 $0
94 $481,693.47 $294,477.46 $187,216.01 $205,522.54 $0
95 $480,979.45 $292,291.05 $188,688.4 $207,708.95 $0
96 $480,254.49 $290,104.64 $190,149.85 $209,895.36 $0
97 $479,518.61 $287,918.23 $191,600.38 $212,081.77 $0
98 $478,771.79 $285,731.82 $193,039.97 $214,268.18 $0
99 $478,014.04 $283,545.41 $194,468.63 $216,454.59 $0
100 $477,245.35 $281,359 $195,886.35 $218,641 -$0
101 $476,465.74 $279,172.59 $197,293.15 $220,827.41 -$0
102 $475,675.19 $276,986.18 $198,689.01 $223,013.82 -$0
103 $474,873.71 $274,799.77 $200,073.94 $225,200.23 $0
104 $474,061.3 $272,613.36 $201,447.94 $227,386.64 $0
105 $473,237.96 $270,426.95 $202,811.01 $229,573.05 $0
106 $472,403.68 $268,240.54 $204,163.14 $231,759.46 $0
107 $471,558.47 $266,054.13 $205,504.34 $233,945.87 $0
108 $470,702.34 $263,867.72 $206,834.62 $236,132.28 $0
109 $469,835.26 $261,681.31 $208,153.95 $238,318.69 $0
110 $468,957.26 $259,494.9 $209,462.36 $240,505.1 $0
111 $468,068.32 $257,308.49 $210,759.83 $242,691.51 $0
112 $467,168.46 $255,122.08 $212,046.38 $244,877.92 $0
113 $466,257.66 $252,935.67 $213,321.99 $247,064.33 $0
114 $465,335.93 $250,749.26 $214,586.67 $249,250.74 $0
115 $464,403.26 $248,562.85 $215,840.41 $251,437.15 $0
116 $463,459.67 $246,376.44 $217,083.23 $253,623.56 $0
117 $462,505.14 $244,190.03 $218,315.11 $255,809.97 $0
118 $461,539.68 $242,003.62 $219,536.06 $257,996.38 $0
119 $460,563.29 $239,817.21 $220,746.08 $260,182.79 $0
120 $459,575.96 $237,630.8 $221,945.16 $262,369.2 $0
121 $458,577.71 $235,444.39 $223,133.32 $264,555.61 $0
122 $457,568.52 $233,257.98 $224,310.54 $266,742.02 $0
123 $456,548.4 $231,071.57 $225,476.83 $268,928.43 $0
124 $455,517.35 $228,885.16 $226,632.19 $271,114.84 $0
125 $454,475.36 $226,698.75 $227,776.61 $273,301.25 $0
126 $453,422.45 $224,512.34 $228,910.11 $275,487.66 $0
127 $452,358.6 $222,325.93 $230,032.67 $277,674.07 $0
128 $451,283.82 $220,139.52 $231,144.3 $279,860.48 $0
129 $450,198.11 $217,953.11 $232,245 $282,046.89 $0
130 $449,101.46 $215,766.7 $233,334.76 $284,233.3 $0
131 $447,993.88 $213,580.29 $234,413.59 $286,419.71 $0
132 $446,875.38 $211,393.88 $235,481.5 $288,606.12 $0
133 $445,745.94 $209,207.47 $236,538.47 $290,792.53 $0
134 $444,605.56 $207,021.06 $237,584.5 $292,978.94 $0
135 $443,454.26 $204,834.65 $238,619.61 $295,165.35 $0
136 $442,292.02 $202,648.24 $239,643.78 $297,351.76 $0
137 $441,118.85 $200,461.83 $240,657.02 $299,538.17 $0
138 $439,934.75 $198,275.42 $241,659.33 $301,724.58 $0
139 $438,739.72 $196,089.01 $242,650.71 $303,910.99 $0
140 $437,533.75 $193,902.6 $243,631.15 $306,097.4 $0
141 $436,316.86 $191,716.19 $244,600.67 $308,283.81 $0
142 $435,089.03 $189,529.78 $245,559.25 $310,470.22 $0
143 $433,850.27 $187,343.37 $246,506.9 $312,656.63 $0
144 $432,600.57 $185,156.96 $247,443.61 $314,843.04 $0
145 $431,339.95 $182,970.55 $248,369.4 $317,029.45 $0
146 $430,068.39 $180,784.14 $249,284.25 $319,215.86 $0
147 $428,785.9 $178,597.73 $250,188.17 $321,402.27 $0
148 $427,492.48 $176,411.32 $251,081.16 $323,588.68 $0
149 $426,188.13 $174,224.91 $251,963.22 $325,775.09 $0
150 $424,872.84 $172,038.5 $252,834.34 $327,961.5 $0
151 $423,546.62 $169,852.09 $253,694.53 $330,147.91 $0
152 $422,209.47 $167,665.68 $254,543.79 $332,334.32 $0
153 $420,861.39 $165,479.27 $255,382.12 $334,520.73 $0
154 $419,502.38 $163,292.86 $256,209.52 $336,707.14 $0
155 $418,132.43 $161,106.45 $257,025.98 $338,893.55 $0
156 $416,751.56 $158,920.04 $257,831.52 $341,079.96 $0
157 $415,359.75 $156,733.63 $258,626.12 $343,266.37 $0
158 $413,957 $154,547.22 $259,409.78 $345,452.78 $0
159 $412,543.33 $152,360.81 $260,182.52 $347,639.19 $0
160 $411,118.72 $150,174.4 $260,944.32 $349,825.6 $0
161 $409,683.19 $147,987.99 $261,695.2 $352,012.01 $0
162 $408,236.72 $145,801.58 $262,435.14 $354,198.42 $0
163 $406,779.31 $143,615.17 $263,164.14 $356,384.83 $0
164 $405,310.98 $141,428.76 $263,882.22 $358,571.24 $0
165 $403,831.71 $139,242.35 $264,589.36 $360,757.65 $0
166 $402,341.52 $137,055.94 $265,285.58 $362,944.06 $0
167 $400,840.38 $134,869.53 $265,970.85 $365,130.47 $0
168 $399,328.32 $132,683.12 $266,645.2 $367,316.88 $0
169 $397,805.33 $130,496.71 $267,308.62 $369,503.29 $0
170 $396,271.4 $128,310.3 $267,961.1 $371,689.7 $0
171 $394,726.54 $126,123.89 $268,602.65 $373,876.11 $0
172 $393,170.75 $123,937.48 $269,233.27 $376,062.52 $0
173 $391,604.03 $121,751.07 $269,852.96 $378,248.93 $0
174 $390,026.38 $119,564.66 $270,461.72 $380,435.34 $0
175 $388,437.79 $117,378.25 $271,059.54 $382,621.75 $0
176 $386,838.27 $115,191.84 $271,646.43 $384,808.16 $0
177 $385,227.82 $113,005.43 $272,222.39 $386,994.57 $0
178 $383,606.44 $110,819.02 $272,787.42 $389,180.98 $0
179 $381,974.12 $108,632.61 $273,341.51 $391,367.39 $0
180 $380,330.87 $106,446.2 $273,884.67 $393,553.8 $0
181 $378,676.7 $104,259.79 $274,416.91 $395,740.21 $0
182 $377,011.58 $102,073.38 $274,938.2 $397,926.62 $0
183 $375,335.54 $99,886.97 $275,448.57 $400,113.03 $0
184 $373,648.57 $97,700.56 $275,948.01 $402,299.44 $0
185 $371,950.66 $95,514.15 $276,436.51 $404,485.85 $0
186 $370,241.82 $93,327.74 $276,914.08 $406,672.26 $0
187 $368,522.05 $91,141.33 $277,380.72 $408,858.67 $0
188 $366,791.35 $88,954.92 $277,836.43 $411,045.08 $0
189 $365,049.71 $86,768.51 $278,281.2 $413,231.49 $0
190 $363,297.14 $84,582.1 $278,715.04 $415,417.9 $0
191 $361,533.64 $82,395.69 $279,137.95 $417,604.31 $0
192 $359,759.21 $80,209.28 $279,549.93 $419,790.72 $0
193 $357,973.85 $78,022.87 $279,950.98 $421,977.13 $0
194 $356,177.55 $75,836.46 $280,341.09 $424,163.54 $0
195 $354,370.32 $73,650.05 $280,720.27 $426,349.95 $0
196 $352,552.16 $71,463.64 $281,088.52 $428,536.36 $0
197 $350,723.07 $69,277.23 $281,445.84 $430,722.77 $0
198 $348,883.05 $67,090.82 $281,792.23 $432,909.18 $0
199 $347,032.09 $64,904.41 $282,127.68 $435,095.59 $0
200 $345,170.2 $62,718 $282,452.2 $437,282 $0
201 $343,297.38 $60,531.59 $282,765.79 $439,468.41 $0
202 $341,413.63 $58,345.18 $283,068.45 $441,654.82 $0
203 $339,518.95 $56,158.77 $283,360.18 $443,841.23 $0
204 $337,613.33 $53,972.36 $283,640.97 $446,027.64 $0
205 $335,696.78 $51,785.95 $283,910.83 $448,214.05 $0
206 $333,769.3 $49,599.54 $284,169.76 $450,400.46 $0
207 $331,830.89 $47,413.13 $284,417.76 $452,586.87 $0
208 $329,881.55 $45,226.72 $284,654.83 $454,773.28 $0
209 $327,921.27 $43,040.31 $284,880.96 $456,959.69 $0
210 $325,950.06 $40,853.9 $285,096.16 $459,146.1 $0
211 $323,967.92 $38,667.49 $285,300.43 $461,332.51 $0
212 $321,974.85 $36,481.08 $285,493.77 $463,518.92 $0
213 $319,970.85 $34,294.67 $285,676.18 $465,705.33 $0
214 $317,955.91 $32,108.26 $285,847.65 $467,891.74 $0
215 $315,930.04 $29,921.85 $286,008.19 $470,078.15 $0
216 $313,893.24 $27,735.44 $286,157.8 $472,264.56 $0
217 $311,845.51 $25,549.03 $286,296.48 $474,450.97 $0
218 $309,786.84 $23,362.62 $286,424.22 $476,637.38 $0
219 $307,717.24 $21,176.21 $286,541.03 $478,823.79 $0
220 $305,636.72 $18,989.8 $286,646.92 $481,010.2 $0
221 $303,545.25 $16,803.39 $286,741.86 $483,196.61 $0
222 $301,442.86 $14,616.98 $286,825.88 $485,383.02 $0
223 $299,329.54 $12,430.57 $286,898.97 $487,569.43 $0
224 $297,205.28 $10,244.16 $286,961.12 $489,755.84 $0
225 $295,070.09 $8,057.75 $287,012.34 $491,942.25 $0
226 $292,923.97 $5,871.34 $287,052.63 $494,128.66 $0
227 $290,766.92 $3,684.93 $287,081.99 $496,315.07 $0
228 $288,598.93 $1,498.52 $287,100.41 $498,501.48 $0
229 $286,420.01 $0 $286,420.01 $500,687.89 $687.89
230 $284,233.6 $0 $284,233.6 $502,874.3 $2,874.3
231 $282,047.19 $0 $282,047.19 $505,060.71 $5,060.71
232 $279,860.78 $0 $279,860.78 $507,247.12 $7,247.12
233 $277,674.37 $0 $277,674.37 $509,433.53 $9,433.53
234 $275,487.96 $0 $275,487.96 $511,619.94 $11,619.94
235 $273,301.55 $0 $273,301.55 $513,806.35 $13,806.35
236 $271,115.14 $0 $271,115.14 $515,992.76 $15,992.76
237 $268,928.73 $0 $268,928.73 $518,179.17 $18,179.17
238 $266,742.32 $0 $266,742.32 $520,365.58 $20,365.58
239 $264,555.91 $0 $264,555.91 $522,551.99 $22,551.99
240 $262,369.5 $0 $262,369.5 $524,738.4 $24,738.4
241 $260,183.09 $0 $260,183.09 $526,924.81 $26,924.81
242 $257,996.68 $0 $257,996.68 $529,111.22 $29,111.22
243 $255,810.27 $0 $255,810.27 $531,297.63 $31,297.63
244 $253,623.86 $0 $253,623.86 $533,484.04 $33,484.04
245 $251,437.45 $0 $251,437.45 $535,670.45 $35,670.45
246 $249,251.04 $0 $249,251.04 $537,856.86 $37,856.86
247 $247,064.63 $0 $247,064.63 $540,043.27 $40,043.27
248 $244,878.22 $0 $244,878.22 $542,229.68 $42,229.68
249 $242,691.81 $0 $242,691.81 $544,416.09 $44,416.09
250 $240,505.4 $0 $240,505.4 $546,602.5 $46,602.5
251 $238,318.99 $0 $238,318.99 $548,788.91 $48,788.91
252 $236,132.58 $0 $236,132.58 $550,975.32 $50,975.32
253 $233,946.17 $0 $233,946.17 $553,161.73 $53,161.73
254 $231,759.76 $0 $231,759.76 $555,348.14 $55,348.14
255 $229,573.35 $0 $229,573.35 $557,534.55 $57,534.55
256 $227,386.94 $0 $227,386.94 $559,720.96 $59,720.96
257 $225,200.53 $0 $225,200.53 $561,907.37 $61,907.37
258 $223,014.12 $0 $223,014.12 $564,093.78 $64,093.78
259 $220,827.71 $0 $220,827.71 $566,280.19 $66,280.19
260 $218,641.3 $0 $218,641.3 $568,466.6 $68,466.6
261 $216,454.89 $0 $216,454.89 $570,653.01 $70,653.01
262 $214,268.48 $0 $214,268.48 $572,839.42 $72,839.42
263 $212,082.07 $0 $212,082.07 $575,025.83 $75,025.83
264 $209,895.66 $0 $209,895.66 $577,212.24 $77,212.24
265 $207,709.25 $0 $207,709.25 $579,398.65 $79,398.65
266 $205,522.84 $0 $205,522.84 $581,585.06 $81,585.06
267 $203,336.43 $0 $203,336.43 $583,771.47 $83,771.47
268 $201,150.02 $0 $201,150.02 $585,957.88 $85,957.88
269 $198,963.61 $0 $198,963.61 $588,144.29 $88,144.29
270 $196,777.2 $0 $196,777.2 $590,330.7 $90,330.7
271 $194,590.79 $0 $194,590.79 $592,517.11 $92,517.11
272 $192,404.38 $0 $192,404.38 $594,703.52 $94,703.52
273 $190,217.97 $0 $190,217.97 $596,889.93 $96,889.93
274 $188,031.56 $0 $188,031.56 $599,076.34 $99,076.34
275 $185,845.15 $0 $185,845.15 $601,262.75 $101,262.75
276 $183,658.74 $0 $183,658.74 $603,449.16 $103,449.16
277 $181,472.33 $0 $181,472.33 $605,635.57 $105,635.57
278 $179,285.92 $0 $179,285.92 $607,821.98 $107,821.98
279 $177,099.51 $0 $177,099.51 $610,008.39 $110,008.39
280 $174,913.1 $0 $174,913.1 $612,194.8 $112,194.8
281 $172,726.69 $0 $172,726.69 $614,381.21 $114,381.21
282 $170,540.28 $0 $170,540.28 $616,567.62 $116,567.62
283 $168,353.87 $0 $168,353.87 $618,754.03 $118,754.03
284 $166,167.46 $0 $166,167.46 $620,940.44 $120,940.44
285 $163,981.05 $0 $163,981.05 $623,126.85 $123,126.85
286 $161,794.64 $0 $161,794.64 $625,313.26 $125,313.26
287 $159,608.23 $0 $159,608.23 $627,499.67 $127,499.67
288 $157,421.82 $0 $157,421.82 $629,686.08 $129,686.08
289 $155,235.41 $0 $155,235.41 $631,872.49 $131,872.49
290 $153,049 $0 $153,049 $634,058.9 $134,058.9
291 $150,862.59 $0 $150,862.59 $636,245.31 $136,245.31
292 $148,676.18 $0 $148,676.18 $638,431.72 $138,431.72
293 $146,489.77 $0 $146,489.77 $640,618.13 $140,618.13
294 $144,303.36 $0 $144,303.36 $642,804.54 $142,804.54
295 $142,116.95 $0 $142,116.95 $644,990.95 $144,990.95
296 $139,930.54 $0 $139,930.54 $647,177.36 $147,177.36
297 $137,744.13 $0 $137,744.13 $649,363.77 $149,363.77
298 $135,557.72 $0 $135,557.72 $651,550.18 $151,550.18
299 $133,371.31 $0 $133,371.31 $653,736.59 $153,736.59
300 $131,184.9 $0 $131,184.9 $655,923 $155,923
301 $128,998.49 $0 $128,998.49 $658,109.41 $158,109.41
302 $126,812.08 $0 $126,812.08 $660,295.82 $160,295.82
303 $124,625.67 $0 $124,625.67 $662,482.23 $162,482.23
304 $122,439.26 $0 $122,439.26 $664,668.64 $164,668.64
305 $120,252.85 $0 $120,252.85 $666,855.05 $166,855.05
306 $118,066.44 $0 $118,066.44 $669,041.46 $169,041.46
307 $115,880.03 $0 $115,880.03 $671,227.87 $171,227.87
308 $113,693.62 $0 $113,693.62 $673,414.28 $173,414.28
309 $111,507.21 $0 $111,507.21 $675,600.69 $175,600.69
310 $109,320.8 $0 $109,320.8 $677,787.1 $177,787.1
311 $107,134.39 $0 $107,134.39 $679,973.51 $179,973.51
312 $104,947.98 $0 $104,947.98 $682,159.92 $182,159.92
313 $102,761.57 $0 $102,761.57 $684,346.33 $184,346.33
314 $100,575.16 $0 $100,575.16 $686,532.74 $186,532.74
315 $98,388.75 $0 $98,388.75 $688,719.15 $188,719.15
316 $96,202.34 $0 $96,202.34 $690,905.56 $190,905.56
317 $94,015.93 $0 $94,015.93 $693,091.97 $193,091.97
318 $91,829.52 $0 $91,829.52 $695,278.38 $195,278.38
319 $89,643.11 $0 $89,643.11 $697,464.79 $197,464.79
320 $87,456.7 $0 $87,456.7 $699,651.2 $199,651.2
321 $85,270.29 $0 $85,270.29 $701,837.61 $201,837.61
322 $83,083.88 $0 $83,083.88 $704,024.02 $204,024.02
323 $80,897.47 $0 $80,897.47 $706,210.43 $206,210.43
324 $78,711.06 $0 $78,711.06 $708,396.84 $208,396.84
325 $76,524.65 $0 $76,524.65 $710,583.25 $210,583.25
326 $74,338.24 $0 $74,338.24 $712,769.66 $212,769.66
327 $72,151.83 $0 $72,151.83 $714,956.07 $214,956.07
328 $69,965.42 $0 $69,965.42 $717,142.48 $217,142.48
329 $67,779.01 $0 $67,779.01 $719,328.89 $219,328.89
330 $65,592.6 $0 $65,592.6 $721,515.3 $221,515.3
331 $63,406.19 $0 $63,406.19 $723,701.71 $223,701.71
332 $61,219.78 $0 $61,219.78 $725,888.12 $225,888.12
333 $59,033.37 $0 $59,033.37 $728,074.53 $228,074.53
334 $56,846.96 $0 $56,846.96 $730,260.94 $230,260.94
335 $54,660.55 $0 $54,660.55 $732,447.35 $232,447.35
336 $52,474.14 $0 $52,474.14 $734,633.76 $234,633.76
337 $50,287.73 $0 $50,287.73 $736,820.17 $236,820.17
338 $48,101.32 $0 $48,101.32 $739,006.58 $239,006.58
339 $45,914.91 $0 $45,914.91 $741,192.99 $241,192.99
340 $43,728.5 $0 $43,728.5 $743,379.4 $243,379.4
341 $41,542.09 $0 $41,542.09 $745,565.81 $245,565.81
342 $39,355.68 $0 $39,355.68 $747,752.22 $247,752.22
343 $37,169.27 $0 $37,169.27 $749,938.63 $249,938.63
344 $34,982.86 $0 $34,982.86 $752,125.04 $252,125.04
345 $32,796.45 $0 $32,796.45 $754,311.45 $254,311.45
346 $30,610.04 $0 $30,610.04 $756,497.86 $256,497.86
347 $28,423.63 $0 $28,423.63 $758,684.27 $258,684.27
348 $26,237.22 $0 $26,237.22 $760,870.68 $260,870.68
349 $24,050.81 $0 $24,050.81 $763,057.09 $263,057.09
350 $21,864.4 $0 $21,864.4 $765,243.5 $265,243.5
351 $19,677.99 $0 $19,677.99 $767,429.91 $267,429.91
352 $17,491.58 $0 $17,491.58 $769,616.32 $269,616.32
353 $15,305.17 $0 $15,305.17 $771,802.73 $271,802.73
354 $13,118.76 $0 $13,118.76 $773,989.14 $273,989.14
355 $10,932.35 $0 $10,932.35 $776,175.55 $276,175.55
356 $8,745.94 $0 $8,745.94 $778,361.96 $278,361.96
357 $6,559.53 $0 $6,559.53 $780,548.37 $280,548.37
358 $4,373.12 $0 $4,373.12 $782,734.78 $282,734.78
359 $2,186.71 $0 $2,186.71 $784,921.19 $284,921.19
360 $0.3 $0 $0.3 $787,107.6 $287,107.6
前幾天推導了房屋貸款的單利模型 (見如下帖子的末尾部分)
http://blog.creaders.net/jingjibird628/user_blog_diary.php?did=131237
對 initial principal P，月利率 r，number of total payments N、基於單利 Simple Interest 的房貸而言，如果要求 equal monthly payment，那麽每月的 Mortgage Payment M 結果是：
M = P (1+N*r) / (N + N*(N-1)*r/2)
例如，如果 P = $500,000，r = 0.005 (對應年率 6%)，30-年房貸 (or N = 360)，那麽月配結果是 M = $2,049.48
Runner 同學指出了一個錯誤，就是我在推導過程中忘記了檢查某個月的本金是不是已經變成了 0，因為對單利模型而言，每月的月配都是先計入本金，直到本金變成 0 後才計入利息部分 (這應該是康門三四)，可是我疏忽了這點，因此計算結果 M 比實際值偏小。
這 裏重新修正模型。修正模型是舉手之勞，但是要具體計算出 Mortgage 月配，則要費點周折，因為此時 M 不像上麵那樣有個簡潔的顯示表達式，我隻得編個小程序計算 M。文章最後給出了 “loan amortization table”，供那些堅持現在房貸市場的貸款是基於單利模型的同學參考。
導出單利模型後，俺隨後就真是好玩朋友的一個“極端例子”做了個簡單的回答，因為這個問題比較有代表意義，寫出來供有興趣的朋友們參考。
1) 房貸的單利模型
除了將現在的房貸市場的 compound interest 換為 simple interest 外，我們 keep 所有的其它要求，例如保證 equal monthly payment，等。沿用上文的記號：
P：貸款總額，Initial Principal;
N：總月配數 (例如對 30 年貸款，N = 12 * 30 = 360)；
B_i：第 i-th month 的 Balance (i = 0, 1,2,...,N，下同，不在標注)，B_i = P_i + I_i，where；
P_i：第 i-th month B_i 中的 Principal 部分 (顯然，P_0 = P)；
I_i：第 i-th month B_i 中的 Interest 部分 (顯然，I_0 = 0)；
M：每月的 Mortgage Payment。
Define K = K (M) = [P/M]，where [x] 表示不超過 x 的最大整數，例如 [3] = 3，[3.14] = 3。這樣，我們有：
0) 最開始，Balance B_0 = P (你的 principal amount)；
1) 1st month，B_1 = (1+r) P - M，這和 Compound Interest Model 是一樣的，因為最開始賬號上沒有任何利息。現在，和 Compound Interest Model 不同，我們得嚴密跟蹤 P_1 and I_1:
B_1 = P_1 + I_1 = (P-M) + r*P
2) 2nd month，注意此時隻有 P_1 產生利息，所以
B_2 = P_1 (1+r) + I_1 - M = (P-M) (1+r) + r*P -M = (P-2M) + r*(2P-M) = P_2 + I_2
3) 3rd month， B_3 = P_2 (1+r) + I_2 - M = (P-2M) (1+r) + r*(2P-M) -M
= (P-3M) + r*(3P-3M) = P_3 + I_3
...... ......
i) i-th month ( i <= K)，B_i = (P - i*M) + (i*P - i*(i-1)*M/2) * r = P_i + I_i
...... ......
K) K-th month：B_K = (P - K*M) + (K*P - K*(K-1)*M/2) * r = P_K + I_K；
注意，因為 K = [P/M]，所以 0 <= P_K = (P - K*M) < M，意思是，這個月的本金部分已經接近 0 了，差距不足一個 M。這就是說，下個月的本金會變成 0。
K+1) (K+1)-th month：B_(K+1) = (P-K*M) (1+r) + (K*P - K*(K-1)*M/2) * r - M
= 0 + ((P - (K+1)*M) + (K*P - K*(K-1)*M/2) * r ) = P_(K+1) + I_(K+1)，
where P_(K+1) = 0, I_(K+1) = (P - (K+1)*M) + (K*P - K*(K-1)*M/2) * r;
...... ......
N) N-th month: B_N = 0 + ((P - N*M) + (K*P - K*(K-1)*M/2) * r ) = P_N + I_N，where P_N = 0。
因為第 N 月貸款必須償還完，所以 B_N = 0，所以 P - N*M + (K*P - K*(K-1)*M/2) * r = 0，代人 K 的定義，因此我們有：
P - N*M + ([P/M]*P - [P/M]*([P/M]-1)*M/2) * r = 0
這就是 Mortgage 必須滿足的方程。鑒於 [P/M] 的複雜，俺難以得出個簡潔的顯示解，隻得編個小程序來數值計算 M。對 P = $500,000，r=0.005，N=360 基於 Simple Interest 的房貸，結果如下：
複利模型 單利模型
-------------------------------------------------------------
Monthly Mortgage 2,997.75 2,186.41
Total Interest 579,190.95 287,107.6
所以如果是 Simple Interest 房貸，銀行損失大約三十萬的利息收入。
2) 回答真是好玩朋友的“極端問題”
真是好玩朋友就基於單利模型的貸款提出了個貌似荒謬的問題 (不是說真是好玩荒謬，是說問題荒謬，嗬嗬)。原問題拷貝如下：
“ 我給你舉一個極端的例子：我借100W，20年，5%年利。一年後我把100萬還掉。欠5W利息。因為利息不必再付息，我在20年後再還那5W。可不可以？（這樣合理不合理？） ”
俺 的理解是，這當然是個“合法”的 Simple Interest 貸款，因為它符合單利貸款的條件。這種貸款是不是顯得荒謬？當然，顯得荒謬，因為客戶手持了銀行5W而無須任何代價，這和市場的“公平合理”的原則是矛盾 的，對不對？這其中荒謬的根源在哪？就是因為 Simple Interest 與市場合理的要求是矛盾的。市場真正完全合理的要求，假設晝夜是不分的等，是 compound continuously，因為錢 (包括本金、利息) 在任何時刻是等價的。當然，真正做到 compound continuously 很困難，市場隻能在一個相對合理的時間區段去 compound，做到大體上的公平合理。這種荒謬的根源很好理解，換個角度看，假設 Simple Interest 是合理的，這就意味著 Principal，Interest 是兩種性質不同的錢，前者是能生蛋的金母雞，能產生利息，後者是不能生蛋的木屐，不能產生利息。所以這就意味著客戶手中最後的5萬利息是第二種錢，它不產生任何利息，它在誰手裏是一樣的，對不對？
所以，就公平、合理的角度而言，因為本金、利息部分應該等價 (從投資的角度而言，它們沒有分別)，所以無論是貸款還是儲蓄，Simple Interest 是不合理的，合理的是 Compound Interest。
3) 附錄：A Sample Amortization Table
對 P = $500,000，r=0.005，N=360 基於 Simple Interest 的房貸，版麵排列有些不整齊，懶得花力氣調整，大家湊合著看。注意第 228 到 229 月，貸款的部分本金變成了 0。
------------------ Amortization schedule ------------------
Month Balance 本金部分 利息部分 累積付款 累積付款利息部分
0 $500,000 $500,000 $0 $0 $0
1 $500,313.59 $497,813.59 $2,500 $2,186.41 -$0
2 $500,616.25 $495,627.18 $4,989.07 $4,372.82 -$0
3 $500,907.97 $493,440.77 $7,467.2 $6,559.23 -$0
4 $501,188.77 $491,254.36 $9,934.41 $8,745.64 $0
5 $501,458.63 $489,067.95 $12,390.68 $10,932.05 $0
6 $501,717.56 $486,881.54 $14,836.02 $13,118.46 $0
7 $501,965.56 $484,695.13 $17,270.43 $15,304.87 -$0
8 $502,202.62 $482,508.72 $19,693.9 $17,491.28 -$0
9 $502,428.76 $480,322.31 $22,106.45 $19,677.69 $0
10 $502,643.96 $478,135.9 $24,508.06 $21,864.1 $0
11 $502,848.23 $475,949.49 $26,898.74 $24,050.51 $0
12 $503,041.56 $473,763.08 $29,278.48 $26,236.92 $0
13 $503,223.97 $471,576.67 $31,647.3 $28,423.33 -$0
14 $503,395.44 $469,390.26 $34,005.18 $30,609.74 -$0
15 $503,555.98 $467,203.85 $36,352.13 $32,796.15 -$0
16 $503,705.59 $465,017.44 $38,688.15 $34,982.56 $0
17 $503,844.27 $462,831.03 $41,013.24 $37,168.97 $0
18 $503,972.02 $460,644.62 $43,327.4 $39,355.38 $0
19 $504,088.83 $458,458.21 $45,630.62 $41,541.79 -$0
20 $504,194.71 $456,271.8 $47,922.91 $43,728.2 -$0
21 $504,289.66 $454,085.39 $50,204.27 $45,914.61 $0
22 $504,373.68 $451,898.98 $52,474.7 $48,101.02 $0
23 $504,446.76 $449,712.57 $54,734.19 $50,287.43 $0
24 $504,508.91 $447,526.16 $56,982.75 $52,473.84 $0
25 $504,560.13 $445,339.75 $59,220.38 $54,660.25 -$0
26 $504,600.42 $443,153.34 $61,447.08 $56,846.66 -$0
27 $504,629.78 $440,966.93 $63,662.85 $59,033.07 -$0
28 $504,648.21 $438,780.52 $65,867.69 $61,219.48 $0
29 $504,655.7 $436,594.11 $68,061.59 $63,405.89 $0
30 $504,652.26 $434,407.7 $70,244.56 $65,592.3 $0
31 $504,637.89 $432,221.29 $72,416.6 $67,778.71 $0
32 $504,612.58 $430,034.88 $74,577.7 $69,965.12 -$0
33 $504,576.35 $427,848.47 $76,727.88 $72,151.53 $0
34 $504,529.18 $425,662.06 $78,867.12 $74,337.94 $0
35 $504,471.08 $423,475.65 $80,995.43 $76,524.35 $0
36 $504,402.05 $421,289.24 $83,112.81 $78,710.76 $0
37 $504,322.08 $419,102.83 $85,219.25 $80,897.17 $0
38 $504,231.19 $416,916.42 $87,314.77 $83,083.58 -$0
39 $504,129.36 $414,730.01 $89,399.35 $85,269.99 -$0
40 $504,016.6 $412,543.6 $91,473 $87,456.4 -$0
41 $503,892.91 $410,357.19 $93,535.72 $89,642.81 $0
42 $503,758.28 $408,170.78 $95,587.5 $91,829.22 $0
43 $503,612.73 $405,984.37 $97,628.36 $94,015.63 $0
44 $503,456.24 $403,797.96 $99,658.28 $96,202.04 -$0
45 $503,288.82 $401,611.55 $101,677.27 $98,388.45 $0
46 $503,110.47 $399,425.14 $103,685.33 $100,574.86 $0
47 $502,921.18 $397,238.73 $105,682.45 $102,761.27 $0
48 $502,720.97 $395,052.32 $107,668.65 $104,947.68 $0
49 $502,509.82 $392,865.91 $109,643.91 $107,134.09 $0
50 $502,287.74 $390,679.5 $111,608.24 $109,320.5 -$0
51 $502,054.73 $388,493.09 $113,561.64 $111,506.91 -$0
52 $501,810.78 $386,306.68 $115,504.1 $113,693.32 -$0
53 $501,555.91 $384,120.27 $117,435.64 $115,879.73 $0
54 $501,290.1 $381,933.86 $119,356.24 $118,066.14 $0
55 $501,013.36 $379,747.45 $121,265.91 $120,252.55 $0
56 $500,725.68 $377,561.04 $123,164.64 $122,438.96 -$0
57 $500,427.08 $375,374.63 $125,052.45 $124,625.37 -$0
58 $500,117.54 $373,188.22 $126,929.32 $126,811.78 $0
59 $499,797.07 $371,001.81 $128,795.26 $128,998.19 $0
60 $499,465.67 $368,815.4 $130,650.27 $131,184.6 $0
61 $499,123.34 $366,628.99 $132,494.35 $133,371.01 $0
62 $498,770.07 $364,442.58 $134,327.49 $135,557.42 $0
63 $498,405.88 $362,256.17 $136,149.71 $137,743.83 -$0
64 $498,030.75 $360,069.76 $137,960.99 $139,930.24 -$0
65 $497,644.69 $357,883.35 $139,761.34 $142,116.65 -$0
66 $497,247.69 $355,696.94 $141,550.75 $144,303.06 $0
67 $496,839.77 $353,510.53 $143,329.24 $146,489.47 $0
68 $496,420.91 $351,324.12 $145,096.79 $148,675.88 $0
69 $495,991.12 $349,137.71 $146,853.41 $150,862.29 $0
70 $495,550.4 $346,951.3 $148,599.1 $153,048.7 $0
71 $495,098.75 $344,764.89 $150,333.86 $155,235.11 $0
72 $494,636.16 $342,578.48 $152,057.68 $157,421.52 $0
73 $494,162.64 $340,392.07 $153,770.57 $159,607.93 $0
74 $493,678.19 $338,205.66 $155,472.53 $161,794.34 $0
75 $493,182.81 $336,019.25 $157,163.56 $163,980.75 -$0
76 $492,676.5 $333,832.84 $158,843.66 $166,167.16 -$0
77 $492,159.25 $331,646.43 $160,512.82 $168,353.57 -$0
78 $491,631.07 $329,460.02 $162,171.05 $170,539.98 $0
79 $491,091.96 $327,273.61 $163,818.35 $172,726.39 $0
80 $490,541.92 $325,087.2 $165,454.72 $174,912.8 $0
81 $489,980.95 $322,900.79 $167,080.16 $177,099.21 $0
82 $489,409.04 $320,714.38 $168,694.66 $179,285.62 $0
83 $488,826.2 $318,527.97 $170,298.23 $181,472.03 $0
84 $488,232.43 $316,341.56 $171,890.87 $183,658.44 $0
85 $487,627.73 $314,155.15 $173,472.58 $185,844.85 $0
86 $487,012.1 $311,968.74 $175,043.36 $188,031.26 $0
87 $486,385.53 $309,782.33 $176,603.2 $190,217.67 -$0
88 $485,748.03 $307,595.92 $178,152.11 $192,404.08 -$0
89 $485,099.6 $305,409.51 $179,690.09 $194,590.49 -$0
90 $484,440.24 $303,223.1 $181,217.14 $196,776.9 $0
91 $483,769.95 $301,036.69 $182,733.26 $198,963.31 $0
92 $483,088.72 $298,850.28 $184,238.44 $201,149.72 $0
93 $482,396.56 $296,663.87 $185,732.69 $203,336.13 $0
94 $481,693.47 $294,477.46 $187,216.01 $205,522.54 $0
95 $480,979.45 $292,291.05 $188,688.4 $207,708.95 $0
96 $480,254.49 $290,104.64 $190,149.85 $209,895.36 $0
97 $479,518.61 $287,918.23 $191,600.38 $212,081.77 $0
98 $478,771.79 $285,731.82 $193,039.97 $214,268.18 $0
99 $478,014.04 $283,545.41 $194,468.63 $216,454.59 $0
100 $477,245.35 $281,359 $195,886.35 $218,641 -$0
101 $476,465.74 $279,172.59 $197,293.15 $220,827.41 -$0
102 $475,675.19 $276,986.18 $198,689.01 $223,013.82 -$0
103 $474,873.71 $274,799.77 $200,073.94 $225,200.23 $0
104 $474,061.3 $272,613.36 $201,447.94 $227,386.64 $0
105 $473,237.96 $270,426.95 $202,811.01 $229,573.05 $0
106 $472,403.68 $268,240.54 $204,163.14 $231,759.46 $0
107 $471,558.47 $266,054.13 $205,504.34 $233,945.87 $0
108 $470,702.34 $263,867.72 $206,834.62 $236,132.28 $0
109 $469,835.26 $261,681.31 $208,153.95 $238,318.69 $0
110 $468,957.26 $259,494.9 $209,462.36 $240,505.1 $0
111 $468,068.32 $257,308.49 $210,759.83 $242,691.51 $0
112 $467,168.46 $255,122.08 $212,046.38 $244,877.92 $0
113 $466,257.66 $252,935.67 $213,321.99 $247,064.33 $0
114 $465,335.93 $250,749.26 $214,586.67 $249,250.74 $0
115 $464,403.26 $248,562.85 $215,840.41 $251,437.15 $0
116 $463,459.67 $246,376.44 $217,083.23 $253,623.56 $0
117 $462,505.14 $244,190.03 $218,315.11 $255,809.97 $0
118 $461,539.68 $242,003.62 $219,536.06 $257,996.38 $0
119 $460,563.29 $239,817.21 $220,746.08 $260,182.79 $0
120 $459,575.96 $237,630.8 $221,945.16 $262,369.2 $0
121 $458,577.71 $235,444.39 $223,133.32 $264,555.61 $0
122 $457,568.52 $233,257.98 $224,310.54 $266,742.02 $0
123 $456,548.4 $231,071.57 $225,476.83 $268,928.43 $0
124 $455,517.35 $228,885.16 $226,632.19 $271,114.84 $0
125 $454,475.36 $226,698.75 $227,776.61 $273,301.25 $0
126 $453,422.45 $224,512.34 $228,910.11 $275,487.66 $0
127 $452,358.6 $222,325.93 $230,032.67 $277,674.07 $0
128 $451,283.82 $220,139.52 $231,144.3 $279,860.48 $0
129 $450,198.11 $217,953.11 $232,245 $282,046.89 $0
130 $449,101.46 $215,766.7 $233,334.76 $284,233.3 $0
131 $447,993.88 $213,580.29 $234,413.59 $286,419.71 $0
132 $446,875.38 $211,393.88 $235,481.5 $288,606.12 $0
133 $445,745.94 $209,207.47 $236,538.47 $290,792.53 $0
134 $444,605.56 $207,021.06 $237,584.5 $292,978.94 $0
135 $443,454.26 $204,834.65 $238,619.61 $295,165.35 $0
136 $442,292.02 $202,648.24 $239,643.78 $297,351.76 $0
137 $441,118.85 $200,461.83 $240,657.02 $299,538.17 $0
138 $439,934.75 $198,275.42 $241,659.33 $301,724.58 $0
139 $438,739.72 $196,089.01 $242,650.71 $303,910.99 $0
140 $437,533.75 $193,902.6 $243,631.15 $306,097.4 $0
141 $436,316.86 $191,716.19 $244,600.67 $308,283.81 $0
142 $435,089.03 $189,529.78 $245,559.25 $310,470.22 $0
143 $433,850.27 $187,343.37 $246,506.9 $312,656.63 $0
144 $432,600.57 $185,156.96 $247,443.61 $314,843.04 $0
145 $431,339.95 $182,970.55 $248,369.4 $317,029.45 $0
146 $430,068.39 $180,784.14 $249,284.25 $319,215.86 $0
147 $428,785.9 $178,597.73 $250,188.17 $321,402.27 $0
148 $427,492.48 $176,411.32 $251,081.16 $323,588.68 $0
149 $426,188.13 $174,224.91 $251,963.22 $325,775.09 $0
150 $424,872.84 $172,038.5 $252,834.34 $327,961.5 $0
151 $423,546.62 $169,852.09 $253,694.53 $330,147.91 $0
152 $422,209.47 $167,665.68 $254,543.79 $332,334.32 $0
153 $420,861.39 $165,479.27 $255,382.12 $334,520.73 $0
154 $419,502.38 $163,292.86 $256,209.52 $336,707.14 $0
155 $418,132.43 $161,106.45 $257,025.98 $338,893.55 $0
156 $416,751.56 $158,920.04 $257,831.52 $341,079.96 $0
157 $415,359.75 $156,733.63 $258,626.12 $343,266.37 $0
158 $413,957 $154,547.22 $259,409.78 $345,452.78 $0
159 $412,543.33 $152,360.81 $260,182.52 $347,639.19 $0
160 $411,118.72 $150,174.4 $260,944.32 $349,825.6 $0
161 $409,683.19 $147,987.99 $261,695.2 $352,012.01 $0
162 $408,236.72 $145,801.58 $262,435.14 $354,198.42 $0
163 $406,779.31 $143,615.17 $263,164.14 $356,384.83 $0
164 $405,310.98 $141,428.76 $263,882.22 $358,571.24 $0
165 $403,831.71 $139,242.35 $264,589.36 $360,757.65 $0
166 $402,341.52 $137,055.94 $265,285.58 $362,944.06 $0
167 $400,840.38 $134,869.53 $265,970.85 $365,130.47 $0
168 $399,328.32 $132,683.12 $266,645.2 $367,316.88 $0
169 $397,805.33 $130,496.71 $267,308.62 $369,503.29 $0
170 $396,271.4 $128,310.3 $267,961.1 $371,689.7 $0
171 $394,726.54 $126,123.89 $268,602.65 $373,876.11 $0
172 $393,170.75 $123,937.48 $269,233.27 $376,062.52 $0
173 $391,604.03 $121,751.07 $269,852.96 $378,248.93 $0
174 $390,026.38 $119,564.66 $270,461.72 $380,435.34 $0
175 $388,437.79 $117,378.25 $271,059.54 $382,621.75 $0
176 $386,838.27 $115,191.84 $271,646.43 $384,808.16 $0
177 $385,227.82 $113,005.43 $272,222.39 $386,994.57 $0
178 $383,606.44 $110,819.02 $272,787.42 $389,180.98 $0
179 $381,974.12 $108,632.61 $273,341.51 $391,367.39 $0
180 $380,330.87 $106,446.2 $273,884.67 $393,553.8 $0
181 $378,676.7 $104,259.79 $274,416.91 $395,740.21 $0
182 $377,011.58 $102,073.38 $274,938.2 $397,926.62 $0
183 $375,335.54 $99,886.97 $275,448.57 $400,113.03 $0
184 $373,648.57 $97,700.56 $275,948.01 $402,299.44 $0
185 $371,950.66 $95,514.15 $276,436.51 $404,485.85 $0
186 $370,241.82 $93,327.74 $276,914.08 $406,672.26 $0
187 $368,522.05 $91,141.33 $277,380.72 $408,858.67 $0
188 $366,791.35 $88,954.92 $277,836.43 $411,045.08 $0
189 $365,049.71 $86,768.51 $278,281.2 $413,231.49 $0
190 $363,297.14 $84,582.1 $278,715.04 $415,417.9 $0
191 $361,533.64 $82,395.69 $279,137.95 $417,604.31 $0
192 $359,759.21 $80,209.28 $279,549.93 $419,790.72 $0
193 $357,973.85 $78,022.87 $279,950.98 $421,977.13 $0
194 $356,177.55 $75,836.46 $280,341.09 $424,163.54 $0
195 $354,370.32 $73,650.05 $280,720.27 $426,349.95 $0
196 $352,552.16 $71,463.64 $281,088.52 $428,536.36 $0
197 $350,723.07 $69,277.23 $281,445.84 $430,722.77 $0
198 $348,883.05 $67,090.82 $281,792.23 $432,909.18 $0
199 $347,032.09 $64,904.41 $282,127.68 $435,095.59 $0
200 $345,170.2 $62,718 $282,452.2 $437,282 $0
201 $343,297.38 $60,531.59 $282,765.79 $439,468.41 $0
202 $341,413.63 $58,345.18 $283,068.45 $441,654.82 $0
203 $339,518.95 $56,158.77 $283,360.18 $443,841.23 $0
204 $337,613.33 $53,972.36 $283,640.97 $446,027.64 $0
205 $335,696.78 $51,785.95 $283,910.83 $448,214.05 $0
206 $333,769.3 $49,599.54 $284,169.76 $450,400.46 $0
207 $331,830.89 $47,413.13 $284,417.76 $452,586.87 $0
208 $329,881.55 $45,226.72 $284,654.83 $454,773.28 $0
209 $327,921.27 $43,040.31 $284,880.96 $456,959.69 $0
210 $325,950.06 $40,853.9 $285,096.16 $459,146.1 $0
211 $323,967.92 $38,667.49 $285,300.43 $461,332.51 $0
212 $321,974.85 $36,481.08 $285,493.77 $463,518.92 $0
213 $319,970.85 $34,294.67 $285,676.18 $465,705.33 $0
214 $317,955.91 $32,108.26 $285,847.65 $467,891.74 $0
215 $315,930.04 $29,921.85 $286,008.19 $470,078.15 $0
216 $313,893.24 $27,735.44 $286,157.8 $472,264.56 $0
217 $311,845.51 $25,549.03 $286,296.48 $474,450.97 $0
218 $309,786.84 $23,362.62 $286,424.22 $476,637.38 $0
219 $307,717.24 $21,176.21 $286,541.03 $478,823.79 $0
220 $305,636.72 $18,989.8 $286,646.92 $481,010.2 $0
221 $303,545.25 $16,803.39 $286,741.86 $483,196.61 $0
222 $301,442.86 $14,616.98 $286,825.88 $485,383.02 $0
223 $299,329.54 $12,430.57 $286,898.97 $487,569.43 $0
224 $297,205.28 $10,244.16 $286,961.12 $489,755.84 $0
225 $295,070.09 $8,057.75 $287,012.34 $491,942.25 $0
226 $292,923.97 $5,871.34 $287,052.63 $494,128.66 $0
227 $290,766.92 $3,684.93 $287,081.99 $496,315.07 $0
228 $288,598.93 $1,498.52 $287,100.41 $498,501.48 $0
229 $286,420.01 $0 $286,420.01 $500,687.89 $687.89
230 $284,233.6 $0 $284,233.6 $502,874.3 $2,874.3
231 $282,047.19 $0 $282,047.19 $505,060.71 $5,060.71
232 $279,860.78 $0 $279,860.78 $507,247.12 $7,247.12
233 $277,674.37 $0 $277,674.37 $509,433.53 $9,433.53
234 $275,487.96 $0 $275,487.96 $511,619.94 $11,619.94
235 $273,301.55 $0 $273,301.55 $513,806.35 $13,806.35
236 $271,115.14 $0 $271,115.14 $515,992.76 $15,992.76
237 $268,928.73 $0 $268,928.73 $518,179.17 $18,179.17
238 $266,742.32 $0 $266,742.32 $520,365.58 $20,365.58
239 $264,555.91 $0 $264,555.91 $522,551.99 $22,551.99
240 $262,369.5 $0 $262,369.5 $524,738.4 $24,738.4
241 $260,183.09 $0 $260,183.09 $526,924.81 $26,924.81
242 $257,996.68 $0 $257,996.68 $529,111.22 $29,111.22
243 $255,810.27 $0 $255,810.27 $531,297.63 $31,297.63
244 $253,623.86 $0 $253,623.86 $533,484.04 $33,484.04
245 $251,437.45 $0 $251,437.45 $535,670.45 $35,670.45
246 $249,251.04 $0 $249,251.04 $537,856.86 $37,856.86
247 $247,064.63 $0 $247,064.63 $540,043.27 $40,043.27
248 $244,878.22 $0 $244,878.22 $542,229.68 $42,229.68
249 $242,691.81 $0 $242,691.81 $544,416.09 $44,416.09
250 $240,505.4 $0 $240,505.4 $546,602.5 $46,602.5
251 $238,318.99 $0 $238,318.99 $548,788.91 $48,788.91
252 $236,132.58 $0 $236,132.58 $550,975.32 $50,975.32
253 $233,946.17 $0 $233,946.17 $553,161.73 $53,161.73
254 $231,759.76 $0 $231,759.76 $555,348.14 $55,348.14
255 $229,573.35 $0 $229,573.35 $557,534.55 $57,534.55
256 $227,386.94 $0 $227,386.94 $559,720.96 $59,720.96
257 $225,200.53 $0 $225,200.53 $561,907.37 $61,907.37
258 $223,014.12 $0 $223,014.12 $564,093.78 $64,093.78
259 $220,827.71 $0 $220,827.71 $566,280.19 $66,280.19
260 $218,641.3 $0 $218,641.3 $568,466.6 $68,466.6
261 $216,454.89 $0 $216,454.89 $570,653.01 $70,653.01
262 $214,268.48 $0 $214,268.48 $572,839.42 $72,839.42
263 $212,082.07 $0 $212,082.07 $575,025.83 $75,025.83
264 $209,895.66 $0 $209,895.66 $577,212.24 $77,212.24
265 $207,709.25 $0 $207,709.25 $579,398.65 $79,398.65
266 $205,522.84 $0 $205,522.84 $581,585.06 $81,585.06
267 $203,336.43 $0 $203,336.43 $583,771.47 $83,771.47
268 $201,150.02 $0 $201,150.02 $585,957.88 $85,957.88
269 $198,963.61 $0 $198,963.61 $588,144.29 $88,144.29
270 $196,777.2 $0 $196,777.2 $590,330.7 $90,330.7
271 $194,590.79 $0 $194,590.79 $592,517.11 $92,517.11
272 $192,404.38 $0 $192,404.38 $594,703.52 $94,703.52
273 $190,217.97 $0 $190,217.97 $596,889.93 $96,889.93
274 $188,031.56 $0 $188,031.56 $599,076.34 $99,076.34
275 $185,845.15 $0 $185,845.15 $601,262.75 $101,262.75
276 $183,658.74 $0 $183,658.74 $603,449.16 $103,449.16
277 $181,472.33 $0 $181,472.33 $605,635.57 $105,635.57
278 $179,285.92 $0 $179,285.92 $607,821.98 $107,821.98
279 $177,099.51 $0 $177,099.51 $610,008.39 $110,008.39
280 $174,913.1 $0 $174,913.1 $612,194.8 $112,194.8
281 $172,726.69 $0 $172,726.69 $614,381.21 $114,381.21
282 $170,540.28 $0 $170,540.28 $616,567.62 $116,567.62
283 $168,353.87 $0 $168,353.87 $618,754.03 $118,754.03
284 $166,167.46 $0 $166,167.46 $620,940.44 $120,940.44
285 $163,981.05 $0 $163,981.05 $623,126.85 $123,126.85
286 $161,794.64 $0 $161,794.64 $625,313.26 $125,313.26
287 $159,608.23 $0 $159,608.23 $627,499.67 $127,499.67
288 $157,421.82 $0 $157,421.82 $629,686.08 $129,686.08
289 $155,235.41 $0 $155,235.41 $631,872.49 $131,872.49
290 $153,049 $0 $153,049 $634,058.9 $134,058.9
291 $150,862.59 $0 $150,862.59 $636,245.31 $136,245.31
292 $148,676.18 $0 $148,676.18 $638,431.72 $138,431.72
293 $146,489.77 $0 $146,489.77 $640,618.13 $140,618.13
294 $144,303.36 $0 $144,303.36 $642,804.54 $142,804.54
295 $142,116.95 $0 $142,116.95 $644,990.95 $144,990.95
296 $139,930.54 $0 $139,930.54 $647,177.36 $147,177.36
297 $137,744.13 $0 $137,744.13 $649,363.77 $149,363.77
298 $135,557.72 $0 $135,557.72 $651,550.18 $151,550.18
299 $133,371.31 $0 $133,371.31 $653,736.59 $153,736.59
300 $131,184.9 $0 $131,184.9 $655,923 $155,923
301 $128,998.49 $0 $128,998.49 $658,109.41 $158,109.41
302 $126,812.08 $0 $126,812.08 $660,295.82 $160,295.82
303 $124,625.67 $0 $124,625.67 $662,482.23 $162,482.23
304 $122,439.26 $0 $122,439.26 $664,668.64 $164,668.64
305 $120,252.85 $0 $120,252.85 $666,855.05 $166,855.05
306 $118,066.44 $0 $118,066.44 $669,041.46 $169,041.46
307 $115,880.03 $0 $115,880.03 $671,227.87 $171,227.87
308 $113,693.62 $0 $113,693.62 $673,414.28 $173,414.28
309 $111,507.21 $0 $111,507.21 $675,600.69 $175,600.69
310 $109,320.8 $0 $109,320.8 $677,787.1 $177,787.1
311 $107,134.39 $0 $107,134.39 $679,973.51 $179,973.51
312 $104,947.98 $0 $104,947.98 $682,159.92 $182,159.92
313 $102,761.57 $0 $102,761.57 $684,346.33 $184,346.33
314 $100,575.16 $0 $100,575.16 $686,532.74 $186,532.74
315 $98,388.75 $0 $98,388.75 $688,719.15 $188,719.15
316 $96,202.34 $0 $96,202.34 $690,905.56 $190,905.56
317 $94,015.93 $0 $94,015.93 $693,091.97 $193,091.97
318 $91,829.52 $0 $91,829.52 $695,278.38 $195,278.38
319 $89,643.11 $0 $89,643.11 $697,464.79 $197,464.79
320 $87,456.7 $0 $87,456.7 $699,651.2 $199,651.2
321 $85,270.29 $0 $85,270.29 $701,837.61 $201,837.61
322 $83,083.88 $0 $83,083.88 $704,024.02 $204,024.02
323 $80,897.47 $0 $80,897.47 $706,210.43 $206,210.43
324 $78,711.06 $0 $78,711.06 $708,396.84 $208,396.84
325 $76,524.65 $0 $76,524.65 $710,583.25 $210,583.25
326 $74,338.24 $0 $74,338.24 $712,769.66 $212,769.66
327 $72,151.83 $0 $72,151.83 $714,956.07 $214,956.07
328 $69,965.42 $0 $69,965.42 $717,142.48 $217,142.48
329 $67,779.01 $0 $67,779.01 $719,328.89 $219,328.89
330 $65,592.6 $0 $65,592.6 $721,515.3 $221,515.3
331 $63,406.19 $0 $63,406.19 $723,701.71 $223,701.71
332 $61,219.78 $0 $61,219.78 $725,888.12 $225,888.12
333 $59,033.37 $0 $59,033.37 $728,074.53 $228,074.53
334 $56,846.96 $0 $56,846.96 $730,260.94 $230,260.94
335 $54,660.55 $0 $54,660.55 $732,447.35 $232,447.35
336 $52,474.14 $0 $52,474.14 $734,633.76 $234,633.76
337 $50,287.73 $0 $50,287.73 $736,820.17 $236,820.17
338 $48,101.32 $0 $48,101.32 $739,006.58 $239,006.58
339 $45,914.91 $0 $45,914.91 $741,192.99 $241,192.99
340 $43,728.5 $0 $43,728.5 $743,379.4 $243,379.4
341 $41,542.09 $0 $41,542.09 $745,565.81 $245,565.81
342 $39,355.68 $0 $39,355.68 $747,752.22 $247,752.22
343 $37,169.27 $0 $37,169.27 $749,938.63 $249,938.63
344 $34,982.86 $0 $34,982.86 $752,125.04 $252,125.04
345 $32,796.45 $0 $32,796.45 $754,311.45 $254,311.45
346 $30,610.04 $0 $30,610.04 $756,497.86 $256,497.86
347 $28,423.63 $0 $28,423.63 $758,684.27 $258,684.27
348 $26,237.22 $0 $26,237.22 $760,870.68 $260,870.68
349 $24,050.81 $0 $24,050.81 $763,057.09 $263,057.09
350 $21,864.4 $0 $21,864.4 $765,243.5 $265,243.5
351 $19,677.99 $0 $19,677.99 $767,429.91 $267,429.91
352 $17,491.58 $0 $17,491.58 $769,616.32 $269,616.32
353 $15,305.17 $0 $15,305.17 $771,802.73 $271,802.73
354 $13,118.76 $0 $13,118.76 $773,989.14 $273,989.14
355 $10,932.35 $0 $10,932.35 $776,175.55 $276,175.55
356 $8,745.94 $0 $8,745.94 $778,361.96 $278,361.96
357 $6,559.53 $0 $6,559.53 $780,548.37 $280,548.37
358 $4,373.12 $0 $4,373.12 $782,734.78 $282,734.78
359 $2,186.71 $0 $2,186.71 $784,921.19 $284,921.19
360 $0.3 $0 $0.3 $787,107.6 $287,107.6
送交者: 紫荊棘鳥 2012月12月05日08:40:18 於 [五 味 齋] 發送悄悄話
回 答:存款是月福利，貸款年單利或者多年單利，讓銀行怎麽能活？ 由 runner 於2012-12-05 08:33:56
後來我修改了，加了這個檢測。
剛才一想，加了這個檢測反而是錯誤的，最開始的那個反而對 (而且簡單)。
對那個例子而言，360 months，
本金大約從 230th month 變成0，然後是負數 (意味這銀行配利息給借貸方)，這是理所當然的。
從大約 230 月開始，銀行是應該給貸方利息，直到最後。