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房贷的单利模型(正式版),兼答真玩的“极端问题”

(2013-02-07 13:38:05) 下一个
0) 开场白
前几天推导了房屋贷款的单利模型 (见如下帖子的末尾部分)
 http://blog.creaders.net/jingjibird628/user_blog_diary.php?did=131237

对 initial principal P,月利率 r,number of total payments N、基于单利 Simple Interest 的房贷而言,如果要求 equal monthly payment,那么每月的 Mortgage Payment M 结果是:
              M = P (1+N*r) / (N + N*(N-1)*r/2)

例如,如果 P = $500,000,r = 0.005 (对应年率 6%),30-年房贷 (or N = 360),那么月配结果是 M = $2,049.48

Runner 同学指出了一个错误,就是我在推导过程中忘记了检查某个月的本金是不是已经变成了 0,因为对单利模型而言,每月的月配都是先计入本金,直到本金变成 0 后才计入利息部分 (这应该是康门三四),可是我疏忽了这点,因此计算结果 M 比实际值偏小。

这 里重新修正模型。修正模型是举手之劳,但是要具体计算出 Mortgage 月配,则要费点周折,因为此时 M 不像上面那样有个简洁的显示表达式,我只得编个小程序计算 M。文章最后给出了 “loan amortization table”,供那些坚持现在房贷市场的贷款是基于单利模型的同学参考。

导出单利模型后,俺随后就真是好玩朋友的一个“极端例子”做了个简单的回答,因为这个问题比较有代表意义,写出来供有兴趣的朋友们参考。

1) 房贷的单利模型
除了将现在的房贷市场的 compound interest 换为 simple interest 外,我们 keep 所有的其它要求,例如保证 equal monthly payment,等。沿用上文的记号:

P:贷款总额,Initial Principal;
N:总月配数 (例如对 30 年贷款,N = 12 * 30 = 360);
B_i:第 i-th month 的 Balance (i = 0, 1,2,...,N,下同,不在标注),B_i = P_i + I_i,where;
P_i:第 i-th month B_i 中的 Principal 部分 (显然,P_0 = P);
I_i:第 i-th month B_i 中的 Interest 部分 (显然,I_0 = 0);
M:每月的 Mortgage Payment。

Define K = K (M) = [P/M],where [x] 表示不超过 x 的最大整数,例如 [3] = 3,[3.14] = 3。这样,我们有:

0) 最开始,Balance B_0 = P   (你的  principal amount);
1) 1st month,B_1 = (1+r) P - M,这和 Compound Interest Model 是一样的,因为最开始账号上没有任何利息。现在,和 Compound Interest Model 不同,我们得严密跟踪 P_1 and I_1:
    B_1 = P_1 + I_1 = (P-M) + r*P
2) 2nd month,注意此时只有 P_1 产生利息,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,因为 K = [P/M],所以 0 <= P_K =  (P - K*M) < M,意思是,这个月的本金部分已经接近 0 了,差距不足一个 M。这就是说,下个月的本金会变成 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。

因为第 N 月贷款必须偿还完,所以 B_N = 0,所以 P - N*M +  (K*P - K*(K-1)*M/2) * r = 0,代人 K 的定义,因此我们有:

    P - N*M +  ([P/M]*P - [P/M]*([P/M]-1)*M/2) * r = 0

这就是 Mortgage 必须满足的方程。鉴于 [P/M] 的复杂,俺难以得出个简洁的显示解,只得编个小程序来数值计算 M。对 P = $500,000,r=0.005,N=360 基于 Simple Interest 的房贷,结果如下:
                                 复利模型        单利模型
-------------------------------------------------------------
Monthly Mortgage        2,997.75           2,186.41
Total Interest            579,190.95         287,107.6

所以如果是 Simple Interest 房贷,银行损失大约三十万的利息收入。

2) 回答真是好玩朋友的“极端问题”
真是好玩朋友就基于单利模型的贷款提出了个貌似荒谬的问题 (不是说真是好玩荒谬,是说问题荒谬,呵呵)。原问题拷贝如下:

“ 我给你举一个极端的例子:我借100W,20年,5%年利。一年后我把100万还掉。欠5W利息。因为利息不必再付息,我在20年后再还那5W。可不可以?(这样合理不合理?) ”

俺 的理解是,这当然是个“合法”的 Simple Interest 贷款,因为它符合单利贷款的条件。这种贷款是不是显得荒谬?当然,显得荒谬,因为客户手持了银行5W而无须任何代价,这和市场的“公平合理”的原则是矛盾 的,对不对?这其中荒谬的根源在哪?就是因为 Simple Interest 与市场合理的要求是矛盾的。市场真正完全合理的要求,假设昼夜是不分的等,是 compound continuously,因为钱 (包括本金、利息) 在任何时刻是等价的。当然,真正做到 compound continuously 很困难,市场只能在一个相对合理的时间区段去 compound,做到大体上的公平合理。这种荒谬的根源很好理解,换个角度看,假设 Simple Interest 是合理的,这就意味着 Principal,Interest 是两种性质不同的钱,前者是能生蛋的金母鸡,能产生利息,后者是不能生蛋的木屐,不能产生利息。所以这就意味着客户手中最后的5万利息是第二种钱,它不产生任何利息,它在谁手里是一样的,对不对?

所以,就公平、合理的角度而言,因为本金、利息部分应该等价 (从投资的角度而言,它们没有分别),所以无论是贷款还是储蓄,Simple Interest 是不合理的,合理的是 Compound Interest。

3) 附录:A Sample Amortization Table
对 P = $500,000,r=0.005,N=360 基于 Simple Interest 的房贷,版面排列有些不整齐,懒得花力气调整,大家凑合着看。注意第 228 到 229 月,贷款的部分本金变成了 0。

------------------ Amortization schedule ------------------
Month   Balance     本金部分       利息部分     累积付款    累积付款利息部分
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
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紫色王家 回复 悄悄话 上次我写的那个房贷单利模型,你说了其间应该检测本金是不是0
送交者: 紫荆棘鸟 2012月12月05日08:40:18 于 [五 味 斋] 发送悄悄话
回 答:存款是月福利,贷款年单利或者多年单利,让银行怎么能活? 由 runner 于2012-12-05 08:33:56
后来我修改了,加了这个检测。

刚才一想,加了这个检测反而是错误的,最开始的那个反而对 (而且简单)。

对那个例子而言,360 months,
本金大约从 230th month 变成0,然后是负数 (意味这银行配利息给借贷方),这是理所当然的。
从大约 230 月开始,银行是应该给贷方利息,直到最后。
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