、問(wèn)題的提出關(guān)鍵詞住房貸款；利率調(diào)整；還款負(fù)擔(dān)住房問(wèn)題是目前全國(guó)上下大家都關(guān)注的熱點(diǎn)問(wèn)題。今年春天，我們家也按揭貸款購(gòu)買了新房。在高興之余，也對(duì)按揭貸款進(jìn)行了了解，認(rèn)真研究了由中國(guó)農(nóng)業(yè)銀行印發(fā)的《個(gè)人住房貸款簡(jiǎn)介》的小冊(cè)子。小冊(cè)子中介紹了有關(guān)個(gè)人住房貸款的有關(guān)問(wèn)題。其中指明貸款額最高為擬購(gòu)買住房費(fèi)用總額的70%；貸款期限最長(zhǎng)為20年。借款人在借款期內(nèi)每月以相等的月均還款額償還銀行貸款本金和利息。個(gè)人住房貸款利率已給出，同時(shí)給出了不同貸款期限下的月均還款額、還款總額和利息負(fù)擔(dān)總和。以上這種還款方式叫做等額本息還款方式。實(shí)際上還有一種還款方式一一等額本金還款方式。這種還款方式是將本金每月等額償還，然后根據(jù)剩余本金計(jì)算利息。除了這兩種還款方式之外，還可以提前償還住房貸款。怎樣貸款比較合算，本文將探討這一方面的問(wèn)題?，F(xiàn)在需要做的內(nèi)容有：1、 根據(jù)已知利率，來(lái)求不同貸款方式下的月均還款額、還款總額和利息負(fù)擔(dān)總和；進(jìn)行比較。2、 根據(jù)規(guī)定計(jì)算提前還款的情況下月均還款額、還款總額和利息負(fù)擔(dān)總和；二、 問(wèn)題的假設(shè)假設(shè)在借款期內(nèi)利率都是不變的，都依照最初計(jì)算還款金額時(shí)的利率來(lái)計(jì)算。三、 現(xiàn)行利率時(shí)間利率短期貸款六個(gè)月以內(nèi)（含六個(gè)月）4.86六個(gè)月至一年（含一年）5.31中長(zhǎng)期一至三年（含三年）5.40三至五年（含五年）5.76五年以上 5.94四、 具體的問(wèn)題、模型的建立與求解1、 問(wèn)題：如果貸款10萬(wàn)元，10年還清。（1）：按等額本息還款方式月均還款額、還款總額和利息負(fù)擔(dān)總和各是多少？（2）：按等額本金還款方式月均還款額、還款總額和利息負(fù)擔(dān)總和各是多少？（3）：還款5年后要一次還清，第六年的還款總額是多少？2、 模型的建立（1）、等額本息還款方式等額本金還款，顧名思義就是每個(gè)月的還款額是固定的。由于還款利息是逐月減少的，因此反過(guò)來(lái)說(shuō)，每月還款中的本金還款額是逐月增加的。首先，我們先進(jìn)行一番設(shè)定：設(shè)：總貸款額=A還款次數(shù)=日 還款月利率=C月還款額=乂當(dāng)月本金還款=Yn（n=還款月數(shù)）先說(shuō)第一個(gè)月，當(dāng)月本金為全部貸款額=A,因此：第一個(gè)月的利息=AXC第一個(gè)月的本金還款額Y1=X-第一個(gè)月的利息=X-AXC第一個(gè)月剩余本金=總貸款額一第一個(gè)月本金還款額=A—(X—AXC)=AX(1+C)—X再說(shuō)第二個(gè)月，當(dāng)月利息還款額=上月剩余本金X月利率第二個(gè)月的利息=(AX(1+C)—X)XC第二個(gè)月的本金還款額Y2=X—第二個(gè)月的利息=X—(AX(1+C)—X)XC第二個(gè)月剩余本金=第一個(gè)月剩余本金—第二個(gè)月本金還款額=AX(1+C)—X—(X—(AX(1+C)—X)XC)=AX(1+C)—X—X+(AX(1+C)—X)XC=AX(1+C)X(1+C)—[X+(1+C)XX]=AX(1+C)A2—[X+(1+C)XX](1+C)A2表示(1+C)的2次方第三個(gè)月，第三個(gè)月的利息=第二個(gè)月剩余本金X月利率第三個(gè)月的利息=(AX(1+C)A2—[X+(1+C)XX])XC第三個(gè)月的本金還款額Y3=X—第三個(gè)月的利息=X—(AX(1+C)A2—[X+(1+C)XX])XC第三個(gè)月剩余本金=第二個(gè)月剩余本金—第三個(gè)月的本金還款額=AX(1+C)A2—[X+(1+C)XX](X—(AX(1+C)A2—[X+(1+C)XX])XC)=AX(1+C)A2—[X+(1+C)XX](X—(AX(1+C)A2XC+[X+(1+C)XX])XC)=AX(1+C)A2X(1+C)(X+[X+(1+C)XX]X(1+C))=AX(1+C)A3 —[X+(1+C)XX+(1+C)A2XX]上式可以分成兩個(gè)部分第一部分：AX(1+C)人3。第二部分：[X+(1+C)XX+(1+C)A2XX]=XX[1+(1+C)+(1+C)A2]通過(guò)對(duì)前三個(gè)月的剩余本金公式進(jìn)行總結(jié)，我們可以看到其中的規(guī)律：剩余本金中的第一部分=總貸款額X(1+月利率)的n次方，(其中n=還款月數(shù))剩余本金中的第二部分是一個(gè)等比數(shù)列，以(1+月利率)為比例系數(shù)，月還款額為常數(shù)系數(shù)，項(xiàng)數(shù)為還款月數(shù)n。推廣到任意月份：第n月的剩余本金=AX(1+C)An—XXSn(Sn為(1+C)的等比數(shù)列的前n項(xiàng)和)根據(jù)等比數(shù)列的前n項(xiàng)和公式：l+Z+Z2+Z3+...+Zn—1=(1—ZAn)/(l—Z)可以得出XXSn=XX(1—(1+C)An)/(1—(1+C))=XX((1+C)An—1)/C所以，第n月的剩余本金=AX(1+C)An—XX((1+C)an—1)/C由于最后一個(gè)月本金將全部還完，所以當(dāng)n等于還款次數(shù)時(shí)，剩余本金為零。設(shè)n=B(還款次數(shù))剩余本金=AX(1+C)ab—XX((1+C)ab—1)/C=0從而得出月還款額X=AXCX(1+C)ab：((1+C)ab—1)=總貸款額X月利率X(1+月利率)A還款次數(shù)：[剩還款次數(shù)一1]將X值帶回到第n月的剩余本金公式中第n月的剩余本金=AX(1+C)An—[AXCX(1+C)ab/((1+C)ab—1)]X((1+C)An—1)/C=AX[(1+C)An—(1+C)abx((1+C)An—1)/((1+C)ab—1)]=AX[(1+C)ab—(1+C)An]/((1+C)ab—1)第n月的利息=第n—1月的剩余本金X月利率=AXCX[(1+C)ab—(1+C)A(n—1)]/((1+C)ab—1)第n月的本金還款額=X—第n月的利息=AXCX(1+C)ab/((1+C)ab—1)—AXCX(1+C)ab—(1+C)A(n—1)]/((1+C)ab—1)=AXCX(1+C)a(n—1)/((1+C)ab—1)總還款額=XXB=AXBXCX(1+C)ab：((1+C)ab—1)總利息=總還款額一總貸款額=XXB—A=AX[(BXC—1)X(1+C)ab+1]/((1+C)ab—1)(2).等額本金還款方式等額本金還款方式比較簡(jiǎn)單。顧名思義，這種方式下，每次還款的本金還款數(shù)是一樣的。因此：當(dāng)月本金還款=總貸款數(shù)：還款次數(shù)當(dāng)月利息=上月剩余本金X月利率=總貸款數(shù)X(1—(還款月數(shù)-1)：還款次數(shù))X月利率當(dāng)月月還款額=當(dāng)月本金還款+當(dāng)月利息=總貸款數(shù)X(1：還款次數(shù)+(1—(還款月數(shù)-1)：還款次數(shù))X月利率)總利息=所有利息之和=總貸款數(shù)X月利率X(還款次數(shù)一(1+2+3+???+還款次數(shù)一1)：還款次數(shù))其中1+2+3+???+還款次數(shù)一1是一個(gè)等差數(shù)列，其和為(1+還款次數(shù)一1)X(還款次數(shù)—1)/2=還款次數(shù)X(還款次數(shù)一1)/23、問(wèn)題解決：由于根據(jù)公式計(jì)算非常復(fù)雜，我們可以借助EXCEL表來(lái)計(jì)算：、等額本息還款各項(xiàng)為：累計(jì)還款總額：132,863.55元支付息款：32,863.55元

還款期次本期還款本金本期應(yīng)還利息. 本期還款后剩余本金本期還款額1612.19495.0099,387.811,107.192615.22491.9798,772.591,107.193618.27488.9298,154.321,107.194621.33485.8697,532.991,107.195624.40482.7996,908.591,107.196627.49479.7096,281.101,107.197630.60476.5995,650.501,107.198633.72473.4795,016.781,107.199636.86470.3394,379.921,107.1910640.01467.1893,739.911,107.1911643.18464.0193,096.731,107.1912646.36460.8392,450.371,107.1913649.56457.6391,800.811,107.1914652.78454.4191,148.031,107.1915656.01451.1890,492.021,107.1916659.25447.9489,832.771,107.1917662.52444.6789,170.251,107.1918665.80441.3988,504.451,107.1919669.09438.1087,835.361,107.1920672.40434.7987,162.961,107.1921675.73431.4686,487.231,107.1922679.08428.1185,808.151,107.1923682.44424.7585,125.711,107.1924685.82421.3784,439.891,107.1925689.21417.9883,750.681,107.1926692.62414.5783,058.061,107.1927696.05411.1482,362.011,107.1928699.50407.6981,662.511,107.1929702.96404.2380,959.551,107.1930706.44400.7580,253.111,107.1931709.94397.2579,543.171,107.1932713.45393.7478,829.721,107.1933716.98390.2178,112.741,107.1934720.53386.6677,392.211,107.1935724.10383.0976,668.111,107.1936727.68379.5175,940.431,107.1937731.28375.9175,209.151,107.1938734.90372.2974,474.251,107.1939738.54368.6573,735.711,107.1940742.20364.9972,993.511,107.1941745.87361.3272,247.641,107.1942749.56357.6371,498.081,107.1943753.27353.9270,744.811,107.1944757.00350.1969,987.811,107.1945760.75346.4469,227.061,107.1946764.52342.6768,462.541,107.1947768.30338.8967,694.241,107.1948772.10335.0966,922.141,107.1949775.93331.2666,146.211,107.1950779.77327.4265,366.441,107.1951783.63323.5664,582.811,107.1952787.51319.6863,795.301,107.1953791.40315.7963,003.901,107.1954795.32311.8762,208.581,107.1955799.26307.9361,409.321,107.1956803.21303.9860,606.111,107.1957807.19300.0059,798.921,107.1958811.19296.0058,987.731,107.1959815.20291.9958,172.531,107.1960819.24287.9557,353.291,107.1961823.29283.9056,530.001,107.1962827.37279.8255,702.631,107.1963831.46275.7354,871.171,107.1964835.58271.6154,035.591,107.1965839.71267.4853,195.881,107.1966843.87263.3252,352.011,107.1967848.05259.1451,503.961,107.1968852.25254.9450,651.711,107.1969856.46250.7349,795.251,107.1970860.70246.4948,934.551,107.1971864.96242.2348,069.591,107.1972869.25237.9447,200.341,107.1973873.55233.6446,326.791,107.1974877.87229.3245,448.921,107.1975882.22224.9744,566.701,107.1976886.58220.6143,680.121,107.1977890.97216.2242,789.151,107.1978895.38211.8141,893.771,107.1979899.82207.3740,993.951,107.1980904.27202.9240,089.681,107.1981908.75198.4439,180.931,107.1982913.24193.9538,267.691,107.1983917.76189.4337,349.931,107.1984922.31184.8836,427.621,107.1985926.87180.3235,500.751,107.1986931.46175.7334,569.291,107.1987936.07171.1233,633.221,107.1988940.71166.4832,692.511,107.1989945.36161.8331,747.151,107.1990950.04157.1530,797.111,107.1991954.74152.4529,842.371,107.1992959.47147.7228,882.901,107.1993964.22142.9727,918.681,107.1994968.99138.2026,949.691,107.1995973.79133.4025,975.901,107.1996978.61128.5824,997.291,107.1997983.45123.7424,013.841,107.1998988.32118.8723,025.521,107.1999993.21113.9822,032.311,107.19100998.13109.0621,034.181,107.191011,003.07104.1220,031.111,107.191021,008.0499.1519,023.071,107.191031,013.0394.1618,010.041,107.191041,018.0489.1516,992.001,107.191051,023.0884.1115,968.921,107.191061,028.1479.0514,940.781,107.191071,033.2373.9613,907.551,107.191081,038.3568.8412,869.201,107.191091,043.4963.7011,825.711,107.191101,048.6558.5410,777.061,107.191111,053.8453.359,723.221,107.191121,059.0648.138,664.161,107.191131,064.3042.897,599.861,107.191141,069.5737.626,530.291,107.191151,074.8732.325,455.421,107.191161,080.1927.004,375.231,107.191171,085.5321.663,289.701,107.191181,090.9116.282,198.791,107.191191,096.3110.881,102.481,107.191201,102.485.460.00 1,107.94、等額本金還款各項(xiàng)為:累計(jì)還款總額：129,947.80元支付息款：29,947.80元

還款期次本期還款本金本期應(yīng)還利息本期還款后剩余本金本期還款額1833.33495.0099,166.671,328.332833.33490.8898,333.341,324.213833.33486.7597,500.011,320.084833.33482.6396,666.681,315.965833.33478.5095,833.351,311.836833.33474.3895,000.021,307.717833.33470.2594,166.691,303.588833.33466.1393,333.361,299.469833.33462.0092,500.031,295.3310833.33457.8891,666.701,291.2111833.33453.7590,833.371,287.0812833.33449.6390,000.041,282.9613833.33445.5089,166.711,278.8314833.33441.3888,333.381,274.7115833.33437.2587,500.051,270.5816833.33433.1386,666.721,266.4617833.33429.0085,833.391,262.3318833.33424.8885,000.061,258.2119833.33420.7584,166.731,254.0820833.33416.6383,333.401,249.9621833.33412.5082,500.071,245.8322833.33408.3881,666.741,241.7123833.33404.2580,833.411,237.5824833.33400.1380,000.081,233.4625833.33396.0079,166.751,229.3326833.33391.8878,333.421,225.2127833.33387.7577,500.091,221.0828833.33383.6376,666.761,216.9629833.33379.5075,833.431,212.8330833.33375.3875,000.101,208.7131833.33371.2574,166.771,204.5832833.33367.1373,333.441,200.4633833.33363.0072,500.111,196.3334833.33358.8871,666.781,192.2135833.33354.7570,833.451,188.0836833.33350.6370,000.121,183.9637833.33346.5069,166.791,179.8338833.33342.3868,333.461,175.7139833.33338.2567,500.131,171.5840833.33334.1366,666.801,167.4641833.33330.0065,833.471,163.3342833.33325.8865,000.141,159.2143833.33321.7564,166.811,155.0844833.33317.6363,333.481,150.9645833.33313.5062,500.151,146.8346833.33309.3861,666.821,142.7147833.33305.2560,833.491,138.5848833.33301.1360,000.161,134.4649833.33297.0059,166.831,130.3350833.33292.8858,333.501,126.2151833.33288.7557,500.171,122.0852833.33284.6356,666.841,117.9653833.33280.5055,833.511,113.8354833.33276.3855,000.181,109.7155833.33272.2554,166.851,105.5856833.33268.1353,333.521,101.4657833.33264.0052,500.191,097.3358833.33259.8851,666.861,093.2159833.33255.7550,833.531,089.0860833.33251.6350,000.201,084.9661833.33247.5049,166.871,080.8362833.33243.3848,333.541,076.7163833.33239.2547,500.211,072.5864833.33235.1346,666.881,068.4665833.33231.0045,833.551,064.3366833.33226.8845,000.221,060.2167833.33222.7544,166.891,056.0868833.33218.6343,333.561,051.9669833.33214.5042,500.231,047.8370833.33210.3841,666.901,043.7171833.33206.2540,833.571,039.5872833.33202.1340,000.241,035.4673833.33198.0039,166.911,031.3374833.33193.8838,333.581,027.2175833.33189.7537,500.251,023.0876833.33185.6336,666.921,018.9677833.33181.5035,833.591,014.8378833.33177.3835,000.261,010.7179833.33173.2534,166.931,006.5880833.33169.1333,333.601,002.4681833.33165.0032,500.27998.3382833.33160.8831,666.94994.2183833.33156.7530,833.61990.0884833.33152.6330,000.28985.9685833.33148.5029,166.95981.8386833.33144.3828,333.62977.7187833.33140.2527,500.29973.5888833.33136.1326,666.96969.4689833.33132.0025,833.63965.3390833.33127.8825,000.30961.2191833.33123.7524,166.97957.0892833.33119.6323,333.64952.9693833.33115.5022,500.3194