等额还款数学模型与计算

等额还款数学模型与计算
在银行贷款中，通常采用等额还款。

假定银行贷款的年利率为p ，贷款K 元，分m 年采用每月等额还款方式还清。

问每月还款多少元？总共还的钱是多少？每月还款中还本金和利息各是多少元？分别考虑每月等额还款和每月等额还本金方式。

并对比二者的不同。

如贷款160000元，分5年还清，年利率为4.032%。

给给出每种情况下每月的还款额，各自总共还款是多少？
方案一：每月还总款等额
解：设第i 月还本金i x 元，月利率/12r p =。

每月还款都为a 元。

第i 月后所剩本金为i y 。

则第n 月后所剩本金为11n n n n i i y y x K x -==-=-
∑ 则第一月共还款为：1.a x K r =+（1）
第二月还款满足：2121.()a x y r x K x r =+=+-（2）
第三月还款满足：32312.()a x y r x K x x r =+=+--（3）
第n 月还款满足：13121.()n n n a x y r x K x x x r --=+=+----（4）
第1n +月还款满足：11121.()n n n n n a x y r x K x x x x r ++-=+=+---
--（5） 则(1)-(2)有：21(1)x x r =+
(2)-(3)有：2
321(1)(1)x x r x r =+=+
(4)-(5)有：11(1)(1)n n n x x r x r +=+=+
贷款分m 年还清，共12m 个月。

则有：
即1211
1(1)m i i x r K -=+=∑ 则：121(1)1m r x K r
+-=
所以：112.(1)1
m K r x r =+- 每月还款121212.(1).(1)1(1)1
m
m m K r r a K r Kr r r +=+=+-+- 其中本金每月为112.(1)1
m K r x r =+-,…,11(1)(1)n n n x x r x r +=+=+ 利息每月为1.L K r =，…,1.n n L y r -=
总共还款12Total
m a =⨯⨯元 如贷款160000元，分5年还清，年利率为4.032%。

Matlab 程序：
K=160000;%贷款金额
m=5;%还款年限
p=0.04032;%年利率
r=p/12;%月利率
n=m*12;
x=zeros(1,n);%每月还本金
y=zeros(1,n);%每月所剩本金
L=zeros(1,n);%每月还利息
a=K*r*(1+r)^n/((1+r)^n-1);%每月等额还款
L(1)=K*r;
x(1)=a-L(1);
y(1)=K-x(1);
fori=2:n
L(i)=y(i-1)*r;
x(i)=a-L(i);
y(i)=y(i-1)-x(i);
end
fprintf('贷款%6d 元,总共还款%5.2f 元\n\n',K,n*a);
fprintf('月还款金额还本金还利息余本金\n');
fori=1:n
fprintf('%2d%5.2f%5.2f%5.2f%5.2f\n',i,a,x(i),L(i),y(i));
end
如贷款16000元，分5年还清，年利率为4.032%，则计算结果如下（银行的还款计划表）：
贷款160000元,总共还款176937.28元
月还款金额还本金还利息余本金
12948.952411.35537.60157588.65
22948.952419.46529.50155169.19
32948.952427.59521.37152741.60
42948.952435.74513.21150305.86
52948.952443.93505.03147861.93
62948.952452.14496.82145409.79
72948.952460.38488.58142949.42
82948.952468.64480.31140480.77
92948.952476.94472.02138003.83 102948.952485.26463.69135518.57 112948.952493.61455.34133024.96 122948.952501.99446.96130522.97 132948.952510.40438.56128012.57 142948.952518.83430.12125493.74 152948.952527.30421.66122966.44 162948.952535.79413.17120430.66 172948.952544.31404.65117886.35 182948.952552.86396.10115333.49 192948.952561.43387.52112772.06 202948.952570.04378.91110202.02 212948.952578.68370.28107623.34 222948.952587.34361.61105036.00 232948.952596.03352.92102439.97 242948.952604.76344.2099835.21 252948.952613.51335.4597221.70 262948.952622.29326.6694599.41 272948.952631.10317.8591968.31 282948.952639.94309.0189328.37 292948.952648.81300.1486679.56 302948.952657.71291.2484021.85 312948.952666.64282.3181355.21 322948.952675.60273.3578679.61 332948.952684.59264.3675995.01 342948.952693.61255.3473301.40 352948.952702.66246.2970598.74 362948.952711.74237.2167887.00 372948.952720.85228.1065166.14 382948.952730.00218.9662436.15 392948.952739.17209.7959696.98 402948.952748.37200.5856948.61 412948.952757.61191.3554191.00 422948.952766.87182.0851424.13 432948.952776.17172.7948647.96 442948.952785.50163.4645862.46 452948.952794.86154.1043067.60 462948.952804.25144.7140263.35 472948.952813.67135.2837449.68 482948.952823.12125.8334626.56 492948.952832.61116.3531793.95 502948.952842.13106.8328951.82 512948.952851.6897.2826100.15 522948.952861.2687.7023238.89 532948.952870.8778.0820368.02 542948.952880.5268.4417487.50 552948.952890.2058.7614597.30
562948.952899.9149.0511697.40
572948.952909.6539.308787.74
582948.952919.4329.535868.32
592948.952929.2419.722939.08
602948.952939.089.880.00
方案二：每月等额还本金方式
设月还本金相等，为b 元。

则/12b K m =。

月利率/12r p =。

则第一月还利息1L Kr =，总还款11x b L =+
则第二月还利息2()L K b r =-，总还款22x b L =+
第1n +月还利息1()n L K nb r +=-，总还款11n n x b L ++=+
m 年总共还利息121211[(1)]m m
i i i L L K i b r ====--∑∑=[126(121)]mK m m b r --
总还款[126(121)]Total K mK m m b r =+--
Matlab 程序为
K=160000;%贷款金额
m=5;%还款年限
p=0.04032;%年利率
r=p/12;%月利率
n=m*12;
x=zeros(1,n);%每月还总共的钱
y=zeros(1,n);%每月所剩本金
L=zeros(1,n);%每月还利息
b=K/n;%每月还本金
fori=1:n
L(i)=(K-(i-1)*b)*r;%每月还利息
x(i)=b+L(i);%每月总还钱
y(i)=K-i*b;%余本金
end
s1=sum(L);%总共利息
s2=(12*m*K-6*m*(12*m-1)*b)*r;%总共利息
Total=K+s1;%总共还款
fprintf('贷款%6d 元,总共还款%5.2f 元\n\n',K,Total);
fprintf('月还款金额还本金还利息余本金\n');
fori=1:n
fprintf('%2d%5.2f%5.2f%5.2f%5.2f\n',i,x(i),b,L(i),y(i));
end
如贷款16000元，分5年还清，年利率为4.032%，则计算结果如下：
贷款160000元,总共还款176396.80元
月还款金额还本金还利息余本金
13204.272666.67537.60157333.33
23195.312666.67528.64154666.67 33186.352666.67519.68152000.00 43177.392666.67510.72149333.33 53168.432666.67501.76146666.67 63159.472666.67492.80144000.00 73150.512666.67483.84141333.33 83141.552666.67474.88138666.67 93132.592666.67465.92136000.00 103123.632666.67456.96133333.33 113114.672666.67448.00130666.67 123105.712666.67439.04128000.00 133096.752666.67430.08125333.33 143087.792666.67421.12122666.67 153078.832666.67412.16120000.00 163069.872666.67403.20117333.33 173060.912666.67394.24114666.67 183051.952666.67385.28112000.00 193042.992666.67376.32109333.33 203034.032666.67367.36106666.67 213025.072666.67358.40104000.00 223016.112666.67349.44101333.33 233007.152666.67340.4898666.67 242998.192666.67331.5296000.00 252989.232666.67322.5693333.33 262980.272666.67313.6090666.67 272971.312666.67304.6488000.00 282962.352666.67295.6885333.33 292953.392666.67286.7282666.67 302944.432666.67277.7680000.00 312935.472666.67268.8077333.33 322926.512666.67259.8474666.67 332917.552666.67250.8872000.00 342908.592666.67241.9269333.33 352899.632666.67232.9666666.67 362890.672666.67224.0064000.00 372881.712666.67215.0461333.33 382872.752666.67206.0858666.67 392863.792666.67197.1256000.00 402854.832666.67188.1653333.33 412845.872666.67179.2050666.67 422836.912666.67170.2448000.00 432827.952666.67161.2845333.33 442818.992666.67152.3242666.67 452810.032666.67143.3640000.00 462801.072666.67134.4037333.33 472792.112666.67125.4434666.67 482783.152666.67116.4832000.00
492774.192666.67107.5229333.33
502765.232666.6798.5626666.67
512756.272666.6789.6024000.00
522747.312666.6780.6421333.33
532738.352666.6771.6818666.67
542729.392666.6762.7216000.00
552720.432666.6753.7613333.33
562711.472666.6744.8010666.67
572702.512666.6735.848000.00
582693.552666.6726.885333.33
592684.592666.6717.922666.67
602675.632666.678.960.00
如贷款16000元，分5年还清，年利率为4.032%，两种方案对比结果：
合成Matlab程序
%方案一：每月等额还钱
K=160000;%贷款金额
m=5;%还款年限
p=0.04032;%年利率
r=p/12;%月利率
n=m*12;
x1=zeros(1,n);%每月还本金
y1=zeros(1,n);%每月所剩本金
L1=zeros(1,n);%每月还利息
a=K*r*(1+r)^n/((1+r)^n-1);%每月等额还款
L1(1)=K*r;
x1(1)=a-L1(1);
y1(1)=K-x1(1);
fori=2:n
L1(i)=y1(i-1)*r;
x1(i)=a-L1(i);
y1(i)=y1(i-1)-x1(i);
end
Total1=n*a;%总共还钱
%方式2:每月等额还本金
x2=zeros(1,n);%每月还总共的钱
y2=zeros(1,n);%每月所剩本金
L2=zeros(1,n);%每月还利息
b=K/n;%每月还本金
fori=1:n
L2(i)=(K-(i-1)*b)*r;%每月还利息
x2(i)=b+L2(i);%每月总还钱
y2(i)=K-i*b;%余本金
end
s1=sum(L2);%总共利息
s2=(12*m*K-6*m*(12*m-1)*b)*r;%总共利息
Total2=K+s1;%总共还款
fprintf('贷款%6d元,方案一总共还款%5.2f元,方案二总共还款%5.2f元\n\n',K,Total1,Total2);
fprintf('方案一:方案二:\n');
fprintf('月还款金额还本金还利息余本金还款金额还本金还利息余本金\n');
fori=1:n
fprintf('%2d%5.2f%5.2f%5.2f%5.2f%5.2f%5.2f%5.2f%5.2f\n',i,a,x1(i),L1(i),y1(i),x2(i),b,L2(i),y2(i)); end
u=1:12*m;
plot(u,y1,'r',u,y2,'b');
计算结果：
贷款160000元,方案一总共还款176937.28元,方案二总共还款176396.80元
方案一:方案二:
月还款金额还本金还利息余本金还款金额还本金还利息余本金
12948.952411.35537.60157588.653204.272666.67537.60157333.33
22948.952419.46529.50155169.193195.312666.67528.64154666.67
32948.952427.59521.37152741.603186.352666.67519.68152000.00
42948.952435.74513.21150305.863177.392666.67510.72149333.33
52948.952443.93505.03147861.933168.432666.67501.76146666.67
62948.952452.14496.82145409.793159.472666.67492.80144000.00
72948.952460.38488.58142949.423150.512666.67483.84141333.33
82948.952468.64480.31140480.773141.552666.67474.88138666.67
92948.952476.94472.02138003.833132.592666.67465.92136000.00
102948.952485.26463.69135518.573123.632666.67456.96133333.33
112948.952493.61455.34133024.963114.672666.67448.00130666.67
122948.952501.99446.96130522.973105.712666.67439.04128000.00
132948.952510.40438.56128012.573096.752666.67430.08125333.33
142948.952518.83430.12125493.743087.792666.67421.12122666.67
152948.952527.30421.66122966.443078.832666.67412.16120000.00
162948.952535.79413.17120430.663069.872666.67403.20117333.33
172948.952544.31404.65117886.353060.912666.67394.24114666.67
182948.952552.86396.10115333.493051.952666.67385.28112000.00
192948.952561.43387.52112772.063042.992666.67376.32109333.33
202948.952570.04378.91110202.023034.032666.67367.36106666.67
212948.952578.68370.28107623.343025.072666.67358.40104000.00
222948.952587.34361.61105036.003016.112666.67349.44101333.33
232948.952596.03352.92102439.973007.152666.67340.4898666.67
242948.952604.76344.2099835.212998.192666.67331.5296000.00
252948.952613.51335.4597221.702989.232666.67322.5693333.33
262948.952622.29326.6694599.412980.272666.67313.6090666.67
272948.952631.10317.8591968.312971.312666.67304.6488000.00
282948.952639.94309.0189328.372962.352666.67295.6885333.33
292948.952648.81300.1486679.562953.392666.67286.7282666.67
302948.952657.71291.2484021.852944.432666.67277.7680000.00
312948.952666.64282.3181355.212935.472666.67268.8077333.33
322948.952675.60273.3578679.612926.512666.67259.8474666.67
332948.952684.59264.3675995.012917.552666.67250.8872000.00
342948.952693.61255.3473301.402908.592666.67241.9269333.33
352948.952702.66246.2970598.742899.632666.67232.9666666.67
362948.952711.74237.2167887.002890.672666.67224.0064000.00
372948.952720.85228.1065166.142881.712666.67215.0461333.33
382948.952730.00218.9662436.152872.752666.67206.0858666.67
392948.952739.17209.7959696.982863.792666.67197.1256000.00
402948.952748.37200.5856948.612854.832666.67188.1653333.33
412948.952757.61191.3554191.002845.872666.67179.2050666.67
422948.952766.87182.0851424.132836.912666.67170.2448000.00
432948.952776.17172.7948647.962827.952666.67161.2845333.33
442948.952785.50163.4645862.462818.992666.67152.3242666.67
452948.952794.86154.1043067.602810.032666.67143.3640000.00
462948.952804.25144.7140263.352801.072666.67134.4037333.33
472948.952813.67135.2837449.682792.112666.67125.4434666.67
482948.952823.12125.8334626.562783.152666.67116.4832000.00
492948.952832.61116.3531793.952774.192666.67107.5229333.33
502948.952842.13106.8328951.822765.232666.6798.5626666.67
512948.952851.6897.2826100.152756.272666.6789.6024000.00
522948.952861.2687.7023238.892747.312666.6780.6421333.33
532948.952870.8778.0820368.022738.352666.6771.6818666.67
542948.952880.5268.4417487.502729.392666.6762.7216000.00
552948.952890.2058.7614597.302720.432666.6753.7613333.33
562948.952899.9149.0511697.402711.472666.6744.8010666.67
572948.952909.6539.308787.742702.512666.6735.848000.00
582948.952919.4329.535868.322693.552666.6726.885333.33
592948.952929.2419.722939.082684.592666.6717.922666.67
602948.952939.089.880.002675.632666.678.960.00
分析：
从结果来看，等额还贷款与等额还本金两者对比来看，贷款160000元,分5年还清，方案一总共还款176937.28元,方案二总共还款176396.80元。

因此等额还款对银行更有利，因此银行采用的此方式。

如果采用提前还贷，如半年后一次还清，则方案一要还145409.79元，而方案二只需要还144000元。

如一年后一次还清，方案一要还130522.97元，而方案二只需要还128000.00元。

如两年后一次还清，方案一要还99835.21元，而方案二只需要还96000元。

但采用方案二对贷款人来说，前29个月都比方案一每个月还款多。

从图一也可以看出，方案一每月余下的本金比方案二多，这这说明方案二是一种更积极的还款方式。

图一两种方案每月所余本金对比图(红线代表方案一，蓝线代表方案二)。