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，贷款元，分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：设第月还本金元，月利率。每月还款都为元。第月后所剩本金为。则第月后所剩本金为则第一月共还款为：（1）第二月还款满足：（2）第三月还款满足：（3）第月还款满足：（4）第月还款满足：（5）则(1)-(2)有：(2)-(3)有：(4)-(5)有：贷款分年还清，共个月。则有：即则：所以：每月还款其中本金每月为,…,利息每月为，…,总共还款元如贷款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);%每月等额还款则第二月还利息，总还款第月还利息，总还款年总共还利息=总还款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:nL(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ends1=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:nfprintf('%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13204.272666.67537.60157333.3323195.312666.67528.64154666.6733186.352666.67519.68152000.0043177.392666.67510.72149333.3353168.432666.67501.76146666.6763159.472666.67492.80144000.0073150.512666.67483.84141333.3383141.552666.67474.88138666.6793132.592666.67465.92136000.00103123.632666.67456.96133333.33113114.672666.67448.00130666.67123105.712666.67439.04128000.00133096.752666.67430.08125333.33143087.792666.67421.12122666.67153078.832666.67412.16120000.00163069.872666.67403.20117333.33173060.912666.67394.24114666.67183051.952666.67385.28112000.00193042.992666.67376.32109333.33203034.032666.67367.36106666.67213025.072666.67358.40104000.00223016.112666.67349.44101333.33233007.152666.67340.4898666.67242998.192666.67331.5296000.00252989.232666.67322.5693333.33262980.272666.67313.6090666.67272971.312666.67304.6488000.00282962.352666.67295.6885333.33292953.392666.67286.7282666.67302944.432666.67277.7680000.00312935.472666.67268.8077333.33322926.512666.67259.8474666.67332917.552666.67250.8872000.00342908.592666.67241.9269333.33352899.632666.67232.9666666.67362890.672666.67224.0064000.00372881.712666.67215.0461333.33382872.752666.67206.0858666.67392863.792666.67197.1256000.00402854.832666.67188.1653333.33412845.872666.67179.2050666.67422836.912666.67170.2448000.00432827.952666.67161.2845333.33442818.992666.67152.3242666.67452810.032666.67143.3640000.00462801.072666.67134.4037333.33472792.112666.67125.4434666.67482783.152666.67116.4832000.00492774.192666.67107.5229333.33502765.232666.6798.5626666.67512756.272666.6789.6024000.00522747.312666.6780.6421333.33532738.352666.6771.6818666.67542729.392666.6762.7216000.00552720.432666.6753.7613333.33562711.472666.6744.8010666.67572702.512666.6735.848000.00582693.552666.6726.885333.33592684.592666.6717.922666.67602675.632666.678.960.00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:nL1(i)=y1(i-1)*r;x1(i)=a-L1(i);y1(i)=y1(i-1)-x1(i);endTotal1=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:nL2(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ends1=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:nfprintf('%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));endu=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12948.952411.35537.60157588.653204.272666.67537.60157333.3322948.952419.46529.50155169.193195.312666.67528.64154666.6732948.952427.59521.37152741.603186.352666.67519.68152000.0042948.952435.74513.21150305.863177.392666.67510.72149333.3352948.952443.93505.03147861.933168.432666.67501.76146666.6762948.952452.14496.82145409.793159.472666.67492.80144000.0072948.952460.38488.58142949.423150.512666.67483.84141333.3382948.952468.64480.31140480.773141.552666.67474.88138666.6792948.952476.94472.02138003.833132.592666.67465.92136000.00102948.952485.26463.69135518.573123.632666.67456.96133333.33112948.952493.61455.34133024.963114.672666.67448.00130666.67122948.952501.99446.96130522.973105.712666.67439.04128000.00132948.952510.40438.56128012.573096.752666.67430.08125333.33142948.952518.83430.12125493.743087.792666.67421.12122666.67152948.952527.30421.66122966.443078.832666.67412.16120000.00162948.952535.79413.17120430.663069.872666.67403.20117333.33172948.952544.31404.65117886.353060.912666.67394.24114666.67182948.952552.86396.10115333.493051.952666.67385.28112000.00192948.952561.43387.52112772.063042.992666.67376.32109333.33202948.952570.04378.91110202.023034.032666.67367.36106666.67212948.952578.68370.28107623.343025.072666.67358.40104000.00222948.952587.34361.61105036.003016.112666.67349.44101333.33232948.952596.03352.92102439.973007.152666.67340.4898666.67242948.952604.76344.2099835.212998.192666.67331.5296000.00252948.952613.51335.4597221.702989.232666.67322.5693333.33262948.952622.29326.6694599.412980.272666.67313.6090666.67272948.952631.10317.8591968.312971.312666.67304.6488000.00282948.952639.94309.0189328.372962.352666.67295.6885333.33292948.952648.81300.1486679.562953.392666.67286.7282666.67302948.952657.71291.2484021.852944.432666.67277.7680000.00312948.952666.64282.3181355.212935.472666.67268.8077333.33322948.952675.60273.3578679.612926.512666.67259.8474666.67332948.952684.59264.3675995.012917.552666.67250.8872000.00342948.952693.61255.3473301.402908.592666.67241.9269333.33352948.952702.66246.2970598.742899.632666.67232.9666666.67362948.952711.74237.2167887.002890.672666.67224.0064000.00372948.952720.85228.1065166.142881.712666.67215.0461333.33382948.952730.00218.9662436.152872.752666.67206.0858666.67392948.952739.17209.7959696.982863.792666.67197.1256000.00402948.952748.37200.5856948.612854.832666.67188.1653333.33412948.952757.61191.3554191.002845.872666.67179.2050666.67422948.952766.87182.0851424.132836.912666.67170.2448000.00432948.952776.17172.7948647.962827.952666.67161.2845333.33442948.952785.50163.4645862.462818.992666.67152.3242666.67452948.952794.86154.1043067.602810.032666.67143.3640000.00462948.952804.25144.7140263.352801.072666.67134.4037333.33472948.952813.67135.2837449.682792.112666.67125.4434666.67482948.952823.12125.8334626.562783.152666.67116.4832000.00492948.952832.61116.3531793.952774.192666.67107.5229333.33502948.952842.13106.8328951.822765.232666.6798.5626666.67512948.952851.6897.2826100.152756.272666.6789.6024000.00522948.952861.2687.7023238.892747.312666.6780.6421