等額還款數(shù)學模型與計算

1、等額還款數(shù)學模型與計算在銀行貸款中，通常采用等額還款。 假定銀行貸款的年利率為 p ，貸款 K 元，分 m 年采用每月等額還款方式還清。問每月還款多少元？總共還的錢是多少？每月還款中還本金和利息各是多少元？分別考慮每月等額還款和每月等額還本金方式。并對比二者的不同。如貸款 160000 元，分 5 年還清，年利率為4.032%。給給出每種情況下每月的還款額，各自總共還款是多少？方案一：每月還總款等額解：設(shè)第 i 月還本金 xi 元，月利率 rp /12 。每月還款都為 a 元。第 i 月后所剩本金為 yi。則第 n 月后所剩本金為nynyn1xnKxii 1則第一月共還款為：ax1K .r（

2、1）第二月還款滿足：ax2y1.rx2(Kx1 )r（2）第三月還款滿足：ax3y2.rx3(Kx1x2 )r（ 3）第 n 月還款滿足： axnyn 1.rx3(Kx1x2Lxn 1 )r（ 4）第 n 1月還款滿足： a xn 1yn .rxn 1(Kx1x2L xn 1 xn )r（5）則(1)-(2) 有：x2x1 (1r )(2)-(3) 有：x3x2 (1 r )x1 (1r )2(4)-(5) 有：xn 1xn (1r )x1 (1r )n貸款分 m 年還清，共 12m個月。則有：12mxiKi112 m即x1(1 r )i1Ki 1則： x1(1r )12m1Kr所以：x1K

3、.r(1r )12m1每月還款aK .rK .rKr(1 r )12m(1r )12m1(1r )12m1其中本金每月為x1(1K .r, ,xn1 xn (1 r ) x1(1 r )nr )12m1利息每月為 L1K .r ， , Lnyn1.r總共還款 Total12ma 元如貸款160000 元，分 5 年還清，年利率為4.032%。Matlab 程序：K=160000; % 貸款金額m=5; % 還款年限p=0.04032; % 年利率r=p/12; % 月利率n=m*12;x=zeros(1,n); % 每月還本金y=zeros(1,n); % 每月所剩本金L=zeros(1,n)

4、; % 每月還利息a=K*r*(1+r)n/(1+r)n-1) ; %每月等額還款L(1)=K*r;x(1)=a-L(1);y(1)=K-x(1);for i=2:nL(i)=y(i-1)*r;x(i)=a-L(i);y(i)=y(i-1)-x(i);endfprintf(' 貸款 %6d 元 ,總共還款 %5.2f 元 nn',K,n*a);fprintf(' 月還款金額還本金還利息余本金 n');for i=1:nfprintf('%2d%5.2f%5.2f%5.2f%5.2fn',i,a,x(i),L(i),y(i);end如貸款 1600

5、0 元，分5 年還清，年利率為4.032% ，則計算結(jié)果如下（銀行的還款計劃表）：貸款 160000 元 ,總共還款176937.28 元月還款金額還本金還利息余本金12948.952411.35537.60157588.6522948.952419.46529.50155169.1932948.952427.59521.37152741.6042948.952435.74513.21150305.8652948.952443.93505.03147861.9362948.952452.14496.82145409.7972948.952460.38488.58142949.4282948.9

6、52468.64480.31140480.7792948.952476.94472.02138003.83102948.952485.26463.69135518.57112948.952493.61455.34133024.96122948.952501.99446.96130522.97132948.952510.40438.56128012.57142948.952518.83430.12125493.74152948.952527.30421.66122966.44162948.952535.79413.17120430.66172948.952544.31404.65117886.3

7、5182948.952552.86396.10115333.49192948.952561.43387.52112772.06202948.952570.04378.91110202.02212948.952578.68370.28107623.34222948.952587.34361.61105036.00232948.952596.03352.92102439.97242948.952604.76344.2099835.21252948.952613.51335.4597221.70262948.952622.29326.6694599.41272948.952631.10317.859

8、1968.31282948.952639.94309.0189328.37292948.952648.81300.1486679.56302948.952657.71291.2484021.85312948.952666.64282.3181355.21322948.952675.60273.3578679.61332948.952684.59264.3675995.01342948.952693.61255.3473301.40352948.952702.66246.2970598.74362948.952711.74237.2167887.00372948.952720.85228.106

9、5166.14382948.952730.00218.9662436.15392948.952739.17209.7959696.98402948.952748.37200.5856948.61412948.952757.61191.3554191.00422948.952766.87182.0851424.13432948.952776.17172.7948647.96442948.952785.50163.4645862.46452948.952794.86154.1043067.60462948.952804.25144.7140263.35472948.952813.67135.283

10、7449.68482948.952823.12125.8334626.56492948.952832.61116.3531793.95502948.952842.13106.8328951.82512948.952851.6897.2826100.15522948.952861.2687.7023238.89532948.952870.8778.0820368.02542948.952880.5268.4417487.50552948.952890.2058.7614597.30562948.952899.9149.0511697.40572948.952909.6539.308787.745

11、82948.952919.4329.535868.32592948.952929.2419.722939.08602948.952939.089.880.00方案二：每月等額還本金方式設(shè)月還本金相等，為b 元。則 bK /12m 。月利率 rp /12 。則第一月還利息L1Kr ，總還款 x1bL1則第二月還利息 L2( Kb) r ，總還款 x2 bL2第 n 1月還利息 Ln 1( Knb) r ，總還款 xn 1b Ln 1m 年總共還利息 L12m12mLi K (i1)br =12mK 6m(12m 1)bri 1i 1總還款 Total K 12mK6m(12m1)brMatlab

12、 程序為K=160000; % 貸款金額m=5; % 還款年限p=0.04032; % 年利率r=p/12; % 月利率n=m*12;x=zeros(1,n); % 每月還總共的錢y=zeros(1,n); % 每月所剩本金L=zeros(1,n); % 每月還利息b=K/n; % 每月還本金for i=1:nL(i)=(K-(i-1)*b)*r; %每月還利息x(i)=b+L(i);% 每月總還錢y(i)=K-i*b;%余本金ends1=sum(L);%總共利息s2=(12*m*K-6*m*(12*m-1)*b)*r;%總共利息Total=K+s1; %總共還款fprintf('fp

13、rintf('貸款 %6d 元 ,總共還款 %5.2f月還款金額還本金元 nn',K,Total);還利息余本金n');for i=1:nfprintf('%2d%5.2f%5.2f%5.2f%5.2fn',i,x(i),b,L(i),y(i);end如貸款 16000 元，分 5 年還清，年利率為4.032% ，則計算結(jié)果如下：貸款 160000 元 ,總共還款176396.80 元月還款金額還本金還利息余本金13204.272666.67537.60157333.3323195.312666.67528.64154666.6733186.352666

14、.67519.68152000.0043177.392666.67510.72149333.3353168.432666.67501.76146666.6763159.472666.67492.80144000.0073150.512666.67483.84141333.3383141.552666.67474.88138666.6793132.592666.67465.92136000.00103123.632666.67456.96133333.33113114.672666.67448.00130666.67123105.712666.67439.04128000.00133096.75

15、2666.67430.08125333.33143087.792666.67421.12122666.67153078.832666.67412.16120000.00163069.872666.67403.20117333.33173060.912666.67394.24114666.67183051.952666.67385.28112000.00193042.992666.67376.32109333.33203034.032666.67367.36106666.67213025.072666.67358.40104000.00223016.112666.67349.44101333.3

16、3233007.152666.67340.4898666.67242998.192666.67331.5296000.00252989.232666.67322.5693333.33262980.272666.67313.6090666.67272971.312666.67304.6488000.00282962.352666.67295.6885333.33292953.392666.67286.7282666.67302944.432666.67277.7680000.00312935.472666.67268.8077333.33322926.512666.67259.8474666.6

17、7332917.552666.67250.8872000.00342908.592666.67241.9269333.33352899.632666.67232.9666666.67362890.672666.67224.0064000.00372881.712666.67215.0461333.33382872.752666.67206.0858666.67392863.792666.67197.1256000.00402854.832666.67188.1653333.33412845.872666.67179.2050666.67422836.912666.67170.2448000.0

18、0432827.952666.67161.2845333.33442818.992666.67152.3242666.67452810.032666.67143.3640000.00462801.072666.67134.4037333.33472792.112666.67125.4434666.67482783.152666.67116.4832000.00492774.192666.67107.5229333.33502765.232666.6798.5626666.67512756.272666.6789.6024000.00522747.312666.6780.6421333.3353

19、2738.352666.6771.6818666.67542729.392666.6762.7216000.00552720.432666.6753.7613333.33562711.472666.6744.8010666.67572702.512666.6735.848000.00582693.552666.6726.885333.33592684.592666.6717.922666.67602675.632666.678.960.00如貸款 16000 元，分 5 年還清，年利率為4.032% ，兩種方案對比結(jié)果：合成 Matlab 程序%方案一：每月等額還錢K=160000; % 貸款

20、金額m=5; % 還款年限p=0.04032; % 年利率r=p/12; % 月利率n=m*12;x1=zeros(1,n); % 每月還本金y1=zeros(1,n); % 每月所剩本金L1=zeros(1,n); % 每月還利息a=K*r*(1+r)n/(1+r)n-1) ; %每月等額還款L1(1)=K*r;x1(1)=a-L1(1);y1(1)=K-x1(1);for i=2:nL1(i)=y1(i-1)*r;x1(i)=a-L1(i);y1(i)=y1(i-1)-x1(i);endTotal1=n*a; % 總共還錢%方式 2:每月等額還本金x2=zeros(1,n); % 每月還總

21、共的錢y2=zeros(1,n); % 每月所剩本金L2=zeros(1,n); % 每月還利息b=K/n; % 每月還本金for i=1:nL2(i)=(K-(i-1)*b)*r; %每月還利息x2(i)=b+L2(i);%每月總還錢y2(i)=K-i*b;%余本金ends1=sum(L2);%總共利息s2=(12*m*K-6*m*(12*m-1)*b)*r;%總共利息Total2=K+s1; % 總共還款fprintf(' 貸款 %6d 元 ,方案一總共還款%5.2f 元 ,方案二總共還款%5.2f 元 nn',K,Total1,Total2);fprintf('方

22、案一 :方案二 :n');fprintf(' 月還款金額還本金還利息余本金還款金額還本金還利息余本金 n');for i=1:nfprintf('%2d%5.2f%5.2f%5.2f%5.2f%5.2f%5.2f%5.2f%5.2fn',i,a,x1(i),L1(i),y1(i),x2(i),b,L2(i),y2(i);endu=1:12*m;plot(u,y1,'r',u,y2,'b');計算結(jié)果：貸款 160000元 ,方案一總共還款176937.28元 ,方案二總共還款176396.80 元方案一 :方案二 :月還款

23、金額還本金還利息余本金還款金額還本金還利息余本金12948.952411.35537.60157588.653204.272666.67537.60157333.3322948.952419.46529.50155169.193195.312666.67528.64154666.6732948.952427.59521.37152741.603186.352666.67519.68152000.0042948.952435.74513.21150305.863177.392666.67510.72149333.3352948.952443.93505.03147861.933168.43266

24、6.67501.76146666.6762948.952452.14496.82145409.793159.472666.67492.80144000.0072948.952460.38488.58142949.423150.512666.67483.84141333.3382948.952468.64480.31140480.773141.552666.67474.88138666.6792948.952476.94472.02138003.833132.592666.67465.92136000.00102948.952485.26463.69135518.573123.632666.67

25、456.96133333.33112948.952493.61455.34133024.963114.672666.67448.00130666.67122948.952501.99446.96130522.973105.712666.67439.04128000.00132948.952510.40438.56128012.573096.752666.67430.08125333.33142948.952518.83430.12125493.743087.792666.67421.12122666.67152948.952527.30421.66122966.443078.832666.67

26、412.16120000.00162948.952535.79413.17120430.663069.872666.67403.20117333.33172948.952544.31404.65117886.353060.912666.67394.24114666.67182948.952552.86396.10115333.493051.952666.67385.28112000.00192948.952561.43387.52112772.063042.992666.67376.32109333.33202948.952570.04378.91110202.023034.032666.67

27、367.36106666.67212948.952578.68370.28107623.343025.072666.67358.40104000.00222948.952587.34361.61105036.003016.112666.67349.44101333.33232948.952596.03352.92102439.973007.152666.67340.4898666.67242948.952604.76344.2099835.212998.192666.67331.5296000.00252948.952613.51335.4597221.702989.232666.67322.

28、5693333.33262948.952622.29326.6694599.412980.272666.67313.6090666.67272948.952631.10317.8591968.312971.312666.67304.6488000.00282948.952639.94309.0189328.372962.352666.67295.6885333.33292948.952648.81300.1486679.562953.392666.67286.7282666.67302948.952657.71291.2484021.852944.432666.67277.7680000.00

29、312948.952666.64282.3181355.212935.472666.67268.8077333.33322948.952675.60273.3578679.612926.512666.67259.8474666.67332948.952684.59264.3675995.012917.552666.67250.8872000.00342948.952693.61255.3473301.402908.592666.67241.9269333.33352948.952702.66246.2970598.742899.632666.67232.9666666.67362948.952

30、711.74237.2167887.002890.672666.67224.0064000.00372948.952720.85228.1065166.142881.712666.67215.0461333.33382948.952730.00218.9662436.152872.752666.67206.0858666.67392948.952739.17209.7959696.982863.792666.67197.1256000.00402948.952748.37200.5856948.612854.832666.67188.1653333.33412948.952757.61191.

31、3554191.002845.872666.67179.2050666.67422948.952766.87182.0851424.132836.912666.67170.2448000.00432948.952776.17172.7948647.962827.952666.67161.2845333.33442948.952785.50163.4645862.462818.992666.67152.3242666.67452948.952794.86154.1043067.602810.032666.67143.3640000.00462948.952804.25144.7140263.352801.072666.67134.4037333.33472948.952813.67135.2837449.682792.112666.67125.4434666.67482948.952823.12125.8334626.562783.152666.67116.4832000.