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等额本金与等额本息

日期:2015-01-30 22:55:30
  
最后更新日期:2017-03-10 09:46:18
花了点时间了解一下常见的收益计算的方法。一般房贷的月供计算分为两种, 等额本息(等额)与等额本金(递减法)。等额本息,就是你每个月的月供是一样多的,月供包括本息,利息是未偿还的本金乘以月息。直观理解就是,假设你月供1000,第一个月偿还的本金是200,利息是800,第二个月偿还的本金是250,利息是750。等额本金,就是你每个月偿还的本金是一样多的,而利息是未偿还的本金乘以月息。
1.等额本息偿还方法,月供数学计算为:
等额本息还款公式推导 设贷款总额为A,银行月利率为β,总期数为m(个月),月还款额设为X:
则各个月所欠银行贷款为:
第一个月后还欠银行A(1+β)-X
第二个月后还欠银行[A(1+β)-X](1+β)-X = A(1+β)^2-X[1+(1+β)]
第三个月后还欠银行{[A(1+β)-X](1+β)-X}(1+β)-X = A(1+β)^3-X[1+(1+β)+(1+β)^2]

由此可得第n个月后所欠银行贷款为:
A(1+β)^n-X[1+(1+β)+(1+β)^2+…+(1+β)^(n-1)] = A(1+β)^n-X[(1+β)^n-1]/β
由于还款总期数为m,也即第m月刚好还完银行所有贷款,因此有:
A(1+β)^m-X[(1+β)^m-1]/β = 0
由此求得:
X = Aβ(1+β)^m/[(1+β)^m-1]


2.等额本金偿还方法,月供数学计算为:
等额本金还款公式推导 设贷款总额为A,银行月利率为β,总期数为m(个月):
第一个月需还款(A/m) + Aβ
第二个月需还款[(A-第一个月还款)/(m-1)] + (A-第一个月还款)β
第三个月需还款[(A-第一个月还款-第二个月还款)/(m-2)] + (A-第一个月还款-第二个月还款)β
...
可以看出等额本金的方法,月供是递减的。
关于此种方法的递推,比较麻烦,因为后一个月的剩余的本金是总本金减去前面总共偿还的本金的形式,结论肯定比较复杂了。呵呵,不搞递推了,不如直接一个python脚本搞定。
[code lang="cpp"]
# cat month.py
# encoding: utf-8
#!/usr/bin/python
import sys

'''
total 总本金, months 总月供月数,p 月息
'''
def month_pay(total,months,p):
lefttotal=total
monthpay=0.0
for i in range(months):
lefttotal=lefttotal-monthpay
monthpay = (lefttotal/months)+lefttotal*p
print "%d month pay %d" % (i,monthpay)
months=months-1

if __name__=="__main__":
if len(sys.argv)<4:
print "%s <total> <months> <p>" % sys.argv[0]
sys.exit(0)
total = float(sys.argv[1])
months = int(sys.argv[2])
p = float(sys.argv[3])
month_pay(total,months,p)
[/code]
若一套房子60w,不算首付,20年还清,年利息按4.25%计算,那么结果如下:
# python ./month.py 600000 240 0.00354166666667
0 month pay 4625
1 month pay 4599
2 month pay 4574
3 month pay 4549
4 month pay 4524
5 month pay 4499
6 month pay 4475
7 month pay 4450
8 month pay 4426
9 month pay 4401
10 month pay 4377
11 month pay 4353
12 month pay 4329
13 month pay 4305
14 month pay 4282
15 month pay 4258
16 month pay 4234
17 month pay 4211
18 month pay 4188
19 month pay 4165
20 month pay 4141
21 month pay 4118
22 month pay 4096
23 month pay 4073
24 month pay 4050
25 month pay 4028
26 month pay 4005
27 month pay 3983
28 month pay 3961
29 month pay 3939
30 month pay 3917
31 month pay 3895
32 month pay 3873
33 month pay 3852
34 month pay 3830
35 month pay 3809
36 month pay 3787
37 month pay 3766
38 month pay 3745
39 month pay 3724
40 month pay 3703
41 month pay 3682
42 month pay 3661
43 month pay 3641
44 month pay 3620
45 month pay 3600
46 month pay 3579
47 month pay 3559
48 month pay 3539
49 month pay 3519
50 month pay 3499
51 month pay 3479
52 month pay 3459
53 month pay 3440
54 month pay 3420
55 month pay 3401
56 month pay 3381
57 month pay 3362
58 month pay 3343
59 month pay 3324
60 month pay 3305
61 month pay 3286
62 month pay 3267
63 month pay 3248
64 month pay 3230
65 month pay 3211
66 month pay 3193
67 month pay 3175
68 month pay 3156
69 month pay 3138
70 month pay 3120
71 month pay 3102
72 month pay 3084
73 month pay 3066
74 month pay 3049
75 month pay 3031
76 month pay 3013
77 month pay 2996
78 month pay 2979
79 month pay 2961
80 month pay 2944
81 month pay 2927
82 month pay 2910
83 month pay 2893
84 month pay 2876
85 month pay 2859
86 month pay 2843
87 month pay 2826
88 month pay 2809
89 month pay 2793
90 month pay 2777
91 month pay 2760
92 month pay 2744
93 month pay 2728
94 month pay 2712
95 month pay 2696
96 month pay 2680
97 month pay 2664
98 month pay 2648
99 month pay 2633
100 month pay 2617
101 month pay 2602
102 month pay 2586
103 month pay 2571
104 month pay 2556
105 month pay 2540
106 month pay 2525
107 month pay 2510
108 month pay 2495
109 month pay 2480
110 month pay 2465
111 month pay 2451
112 month pay 2436
113 month pay 2421
114 month pay 2407
115 month pay 2392
116 month pay 2378
117 month pay 2364
118 month pay 2349
119 month pay 2335
120 month pay 2321
121 month pay 2307
122 month pay 2293
123 month pay 2279
124 month pay 2265
125 month pay 2252
126 month pay 2238
127 month pay 2224
128 month pay 2211
129 month pay 2197
130 month pay 2184
131 month pay 2170
132 month pay 2157
133 month pay 2144
134 month pay 2131
135 month pay 2118
136 month pay 2105
137 month pay 2092
138 month pay 2079
139 month pay 2066
140 month pay 2053
141 month pay 2040
142 month pay 2028
143 month pay 2015
144 month pay 2003
145 month pay 1990
146 month pay 1978
147 month pay 1966
148 month pay 1953
149 month pay 1941
150 month pay 1929
151 month pay 1917
152 month pay 1905
153 month pay 1893
154 month pay 1881
155 month pay 1869
156 month pay 1858
157 month pay 1846
158 month pay 1834
159 month pay 1823
160 month pay 1811
161 month pay 1800
162 month pay 1788
163 month pay 1777
164 month pay 1765
165 month pay 1754
166 month pay 1743
167 month pay 1732
168 month pay 1721
169 month pay 1710
170 month pay 1699
171 month pay 1688
172 month pay 1677
173 month pay 1666
174 month pay 1655
175 month pay 1645
176 month pay 1634
177 month pay 1624
178 month pay 1613
179 month pay 1603
180 month pay 1592
181 month pay 1582
182 month pay 1571
183 month pay 1561
184 month pay 1551
185 month pay 1541
186 month pay 1531
187 month pay 1521
188 month pay 1511
189 month pay 1501
190 month pay 1491
191 month pay 1481
192 month pay 1471
193 month pay 1461
194 month pay 1452
195 month pay 1442
196 month pay 1432
197 month pay 1423
198 month pay 1413
199 month pay 1404
200 month pay 1394
201 month pay 1385
202 month pay 1376
203 month pay 1366
204 month pay 1357
205 month pay 1348
206 month pay 1339
207 month pay 1330
208 month pay 1321
209 month pay 1312
210 month pay 1303
211 month pay 1294
212 month pay 1285
213 month pay 1276
214 month pay 1267
215 month pay 1258
216 month pay 1250
217 month pay 1241
218 month pay 1232
219 month pay 1224
220 month pay 1215
221 month pay 1207
222 month pay 1198
223 month pay 1190
224 month pay 1181
225 month pay 1173
226 month pay 1164
227 month pay 1156
228 month pay 1148
229 month pay 1139
230 month pay 1131
231 month pay 1123
232 month pay 1114
233 month pay 1106
234 month pay 1098
235 month pay 1089
236 month pay 1081
237 month pay 1072
238 month pay 1062
239 month pay 1051
其实不论是等额本息还是等额本金偿还方式,利息都是一样。因为利息本来就是本金与利率和年限的乘积,不过等额本金每次偿还的本金多,所以总利息要少一些。如果你拿这笔钱,以高于4.25%的年化率去投资,那么你可以选择等额本息。看了数据后,囧了,一句话,屌丝的春天离不开奋斗啊。