单利、复利和年金的计算(有附表)

单利、复利和年金的计算（有附表）一、单利的终值和现值设定I为利息；P为现值；F为终值；i为每一利息期的利率（折现率）；n为计算利息的期数。复利计算的符号标识相同。按照单利的计算法则，利息的计算公式为在计算利息时，除非特别指明，一般给出的利率均为年利率，对于不足一年的利息，以一年等于360天来折算。单利终值的计算公式如下：单利现值的计算与单利终值的计算是互逆的，由终值计算现值的过程称为折现。单利现值的计算公式为二、复利的终值和现值（一）复利终值（已知现值P，求终值F）资金时间价值通常是按复利计算的。复利不同于单利，它是“利上滚利”，既涉及本金上的利息，也涉及利上所生的利息。复利终值是指一定量的本金按复利计算若干期后的本利和。其计算公式如下：计息期为二期以上时，复利的终值大于单利的终值，时间越长，相差越大。单利是随时间的延长而按等差级数增长；复利则是按等比级数增长。在复利终值的计算公式中，表示本金为1元时，n期的复利终值，称为1元的复利终值系数，也可写成（F/P，i，n）。为了简化运算，在计算复利终值时，可通过查“复利终值系数表”求得。（二）复利现值（已知终值F，求现值P）复利现值相当于原始本金，它是指今后某一特定时间收到或付出的一笔款项，按折现率i所计算的现在时点价值。其计算公式为式中通常称作1元的复利现值系数，记作(P/F，i，n)，可以直接查阅“复利现值系数表”。上式也可写作P=F(P/F，i，n)。三、年金（A）除了上述的一次性收付款项之外，在现实经济生活中，还存在一定时期内每次等额收付的系列款项，即年金，通常用A表示。由于年金分为普通年金、即付年金、递延年金、永续年金等几种，有关终值和现值的计算方法不一样，下面分别作介绍。（一）普通年金终值的计算（已知年金A，求年金终值F）普通年金是指从第一期起，在一定时期内每期期末等额发生的系列收付款项，又称后付年金。如果年金相当于零存整取储蓄存款的零存数，那么，年金终值就是零存整取的整取数，年金终值的计算公式为式中的分式称作“年金终值系数”，记为(F/A,i,n)，可通过直接查阅“年金终值系数表”求得有关数值。上式也可写作。（二）年偿债基金的计算（已知年金终值F，求年金A）偿债基金是指为了在约定的未来某一时点清偿某笔债务或积聚一定数额的资金而必须分次等额形成的存款准备金。由于每次形成的等额准备金类似年金存款，因而同样可以获得按复利计算的利息，所以债务实际上等于年金终值，每年提取的偿债基金等于年金A。也就是说，偿债基金的计算实际上是年金终值的逆运算。其计算公式为式中的分式称作“偿债基金系数”，记为(A/F,i,n)，可通过年金终值系数的倒数推算出来。上式也可写作或（三）普通年金现值的计算（已知年金A，求年金现值P）年金现值是指一定时期内每期期末等额收付款项的复利现值之和。其现值的计算公式为式中的分式称作“年金现值系数”，记为(P/A,i,n)，可通过直接查阅“年金现值系数表”求得有关数值。上式也可以写作（四）年资本回收额的计算（已知年金现值P，求年金A）资本回收是指在给定的年限内等额回收初始投入资本或清偿所欠债务的价值指标。年资本回收额的计算是年金现值的逆运算。其计算公式为式中的分式称作“资本回收系数”，记为(A/P,i,n)，可利用年金现值系数的倒数求得。上式也可写作，或（五）即付年金的终值与现值（已知年金A，求年金现值P）即付年金是指从第一期起，在一定时期内每期期初等额收付的系列款项，也称先付年金。它与普通年金的区别仅在于付款时间的不同，前者发生在期初，后者发生在期末，两者相差（1+i）。因此，即付年金的终值可用下列公式计算：即付年金现值的计算。如前所述，n期即付年金现值与n期普通年金现值的期限相同，但由于其付款时间不同，n期即付年金现值比n期普通年金现值少折现一期。因此，在n期普通年金现值的基础上乘以(1+i)，便可求出n期即付年金的现值。其计算公式为（六）递延年金现值的计算（已知年金A，求年金现值P）递延年金是指第一次收付款发生时间与第一期无关，而是隔若干期（假设为m期，m1）后才开始发生的系列等额收付款项。它是普通年金的特殊形式，凡不是从第一期开始的年金都是递延年金。递延年金的现值可按以下公式计算：（七）永续年金现值的计算永续年金是指无限期等额收付的特种年金，可视为普通年金的特殊形式，即期限趋于无穷的普通年金。存本取息可视为永续年金的例子。此外，也可将利率较高、持续期限较长的年金视同永续年金计算。由于永续年金持续期无限，没有终止的时间，因此没有终值，只有现值。通过普通年金现值计算可推导出永续年金现值的计算公式为附录一：一元复利终值系数表 F=(1+i)n附录二：一元复利现值系数表 P=(1+i)-n附录三：一元年金终值系数表 F=(1+i)n-1/i附录四：一元年金现值系数表 P=1-(1+i)-n/i附录一：一元复利终值系数表 F=(1+i)nn1%2%3%4%5%6%7%8%9%10%11%12%13%14%15%11.01001.02001.03001.04001.05001.06001.07001.08001.09001.10001.11001.12001.13001.14001.150021.02011.04041.06091.08161.10251.12361.14491.16641.18811.21001.23211.25441.27691.29961.322531.03031.06121.09271.12491.15761.19101.22501.25971.29501.33101.36761.40491.44291.48151.520941.04061.08241.12551.16991.21551.26251.31081.36051.41161.46411.51811.57351.63051.68901.749051.05101.10411.15931.21671.27631.33821.40261.46931.53861.61051.68511.76231.84241.92542.011461.06151.12621.19411.26531.34011.41851.50071.58691.67711.77161.87041.97382.08202.19502.313171.07211.14871.22991.31591.40711.50361.60581.71381.82801.94872.07622.21072.35262.50232.660081.08291.17171.26681.36861.47751.59381.71821.85091.99262.14362.30452.47602.65842.85263.059091.09371.19511.30481.42331.55131.68951.83851.99902.17192.35792.55802.77313.00403.25193.5179101.10461.21901.34391.48021.62891.79081.96722.15892.36742.59372.83943.10583.39463.70724.0456111.11571.24341.38421.53951.71031.89832.10492.33162.58042.85313.15183.47863.83594.22624.6524121.12681.26821.42581.60101.79592.01222.25222.51822.81273.13843.49853.89604.33454.81795.3503131.13811.29361.46851.66511.88562.13292.40982.71963.06583.45233.88334.36354.89805.49246.1528141.14951.31951.51261.73171.97992.26092.57852.93723.34173.79754.31044.88715.53486.26137.0757151.16101.34591.55801.80092.07892.39662.75903.17223.64254.17724.78465.47366.25437.13798.1371161.17261.37281.60471.87302.18292.54042.95223.42593.97034.59505.31096.13047.06738.13729.3576171.18431.40021.65281.94792.29202.69283.15883.70004.32765.05455.89516.86607.98619.276510.7613181.19611.42821.70242.02582.40662.85433.37993.99604.71715.55996.54367.69009.024310.575212.3755191.20811.45681.75352.10682.52703.02563.61654.31575.14176.11597.26338.612810.197412.055714.2318201.22021.48591.80612.19112.65333.20713.86974.66105.60446.72758.06239.646311.523113.743516.3665211.23241.51571.86032.27882.78603.39964.14065.03386.10887.40028.949210.803813.021115.667618.8215221.24471.54601.91612.36992.92533.60354.43045.43656.65868.14039.933612.100314.713817.861021.6447231.25721.57691.97362.46473.07153.81974.74055.87157.25798.954311.026313.552316.626620.361624.8915241.26971.60842.03282.56333.22514.04895.07246.34127.91119.849712.239215.178618.788123.212228.6252251.28241.64062.09382.66583.38644.29195.42746.84858.623110.834713.585517.000121.230526.461932.9190301.34781.81142.42733.24344.32195.74357.612310.062713.267717.449422.892329.959939.115950.950266.2118（续表）n16%17%18%19%20%21%22%23%24%25%26%27%28%29%30%11.16001.17001.18001.19001.20001.21001.22001.23001.24001.25001.26001.27001.28001.29001.300021.34561.36891.39241.41611.44001.46411.48841.51291.53761.56251.58761.61291.63841.66411.690031.56091.60161.64301.68521.72801.77161.81581.86091.90661.95312.00042.04842.09722.14672.197041.81061.87391.93882.00532.07362.14362.21532.28892.36422.44142.52052.60142.68442.76922.856152.10032.19242.28782.38642.48832.59372.70272.81532.93163.05183.17583.30383.43603.57233.712962.43642.56522.69962.83982.98603.13843.29733.46283.63523.81474.00154.19594.39804.60834.826872.82623.00123.18553.37933.58323.79754.02274.25934.50774.76845.04195.32885.62955.94476.274983.27843.51153.75894.02144.29984.59504.90775.23895.58955.96056.35286.76757.20587.66868.157393.80304.10844.43554.78545.15985.55995.98746.44396.93107.45068.00458.59489.22349.892510.6045104.41144.80685.23385.69476.19176.72757.30467.92598.59449.313210.085710.915311.805912.761413.7858115.11735.62406.17596.77677.43018.14038.91179.748910.657111.641512.708013.862515.111616.462217.9216125.93606.58017.28768.06428.91619.849710.872211.991213.214814.551916.012017.605319.342821.236223.2981136.88587.69878.59949.596410.699311.918213.264114.749116.386318.189920.175222.358824.758827.394730.2875147.98759.007510.147211.419812.839214.421016.182218.141420.319122.737425.420728.395731.691335.339139.3738159.265510.538711.973713.589515.407017.449419.742322.314025.195628.421732.030136.062540.564845.587551.18591610.748012.330314.129016.171518.488421.113824.085627.446231.242635.527140.357945.799451.923058.807966.54171712.467714.426516.672219.244122.186125.547729.384433.758838.740844.408950.851058.165266.461475.862186.50421814.462516.879019.673322.900526.623330.912735.849041.523348.038655.511264.072273.869885.070697.8622112.45541916.776519.748423.214427.251631.948037.404343.735851.073759.567969.388980.731093.8147108.8904126.2422146.19202019.460823.105627.393032.429438.337645.259353.357662.820673.864186.7362101.7211119.1446139.3797162.8524190.04962122.574527.033632.323838.591046.005154.763765.096377.269491.5915108.4202128.1685151.3137178.4060210.0796247.06452226.186431.629338.142145.923355.206166.264179.417595.0413113.5735135.5253161.4924192.1683228.3596271.0027321.18392330.376237.006245.007654.648766.247480.179596.8894116.9008140.8312169.4066203.4804244.0538292.3003349.5935417.53912435.236443.297353.109065.032079.496897.0172118.2050143.7880174.6306211.7582256.3853309.9483374.1444450.9756542.80082540.874250.657862.668677.388195.3962117.3909144.2101176.8593216.5420264.6978323.0454393.6344478.9049581.7585705.64103085.8499111.0647143.3706184.6753237.3763304.4816389.7579497.9129634.8199807.79361025.92671300.50381645.50462078.21902619.9956附录二：一元复利现值系数表 P=(1+i)-nn1%2%3%4%5%6%7%8%9%10%11%12%13%14%15%10.99010.98040.97090.96150.95240.94340.93460.92590.91740.90910.90090.89290.88500.87720.869620.98030.96120.94260.92460.90700.89000.87340.85730.84170.82640.81160.79720.78310.76950.756130.97060.94230.91510.88900.86380.83960.81630.79380.77220.75130.73120.71180.69310.67500.657540.96100.92380.88850.85480.82270.79210.76290.73500.70840.68300.65870.63550.61330.59210.571850.95150.90570.86260.82190.78350.74730.71300.68060.64990.62090.59350.56740.54280.51940.497260.94200.88800.83750.79030.74620.70500.66630.63020.59630.56450.53460.50660.48030.45560.432370.93270.87060.81310.75990.71070.66510.62270.58350.54700.51320.48170.45230.42510.39960.375980.92350.85350.78940.73070.67680.62740.58200.54030.50190.46650.43390.40390.37620.35060.326990.91430.83680.76640.70260.64460.59190.54390.50020.46040.42410.39090.36060.33290.30750.2843100.90530.82030.74410.67560.61390.55840.50830.46320.42240.38550.35220.32200.29460.26970.2472110.89630.80430.72240.64960.58470.52680.47510.42890.38750.35050.31730.28750.26070.23660.2149120.88740.78850.70140.62460.55680.49700.44400.39710.35550.31860.28580.25670.23070.20760.1869130.87870.77300.68100.60060.53030.46880.41500.36770.32620.28970.25750.22920.20420.18210.1625140.87000.75790.66110.57750.50510.44230.38780.34050.29920.26330.23200.20460.18070.15970.1413150.86130.74300.64190.55530.48100.41730.36240.31520.27450.23940.20900.18270.15990.14010.1229160.85280.72840.62320.53390.45810.39360.33870.29190.25190.21760.18830.16310.14150.12290.1069170.84440.71420.60500.51340.43630.37140.31660.27030.23110.19780.16960.14560.12520.10780.0929180.83600.70020.58740.49360.41550.35030.29590.25020.21200.17990.15280.13000.11080.09460.0808190.82770.68640.57030.47460.39570.33050.27650.23170.19450.16350.13770.11610.09810.08290.0703200.81950.67300.55370.45640.37690.31180.25840.21450.17840.14860.12400.10370.08680.07280.0611210.81140.65980.53750.43880.35890.29420.24150.19870.16370.13510.11170.09260.07680.06380.0531220.80340.64680.52190.42200.34180.27750.22570.18390.15020.12280.10070.08260.06800.05600.0462230.79540.63420.50670.40570.32560.26180.21090.17030.13780.11170.09070.07380.06010.04910.0402240.78760.62170.49190.39010.31010.24700.19710.15770.12640.10150.08170.06590.05320.04310.0349250.77980.60950.47760.37510.29530.23300.18420.14600.11600.09230.07360.05880.04710.03780.0304300.74190.55210.41200.30830.23140.17410.13140.09940.07540.05730.04370.03340.02560.01960.0151（续表）n16%17%18%19%20%21%22%23%24%25%26%27%28%29%30%10.86210.85470.84750.84030.83330.82640.81970.81300.80650.80000.79370.78740.78130.77520.769220.74320.73050.71820.70620.69440.68300.67190.66100.65040.64000.62990.62000.61040.60090.591730.64070.62440.60860.59340.57870.56450.55070.53740.52450.51200.49990.48820.47680.46580.455240.55230.53370.51580.49870.48230.46650.45140.43690.42300.40960.39680.38440.37250.36110.350150.47610.45610.43710.41900.40190.38550.37000.35520.34110.32770.31490.30270.29100.27990.269360.41040.38980.37040.35210.33490.31860.30330.28880.27510.26210.24990.23830.22740.21700.207270.35380.33320.31390.29590.27910.26330.24860.23480.22180.20970.19830.18770.17760.16820.159480.30500.28480.26600.24870.23260.21760.20380.19090.17890.16780.15740.14780.13880.13040.122690.26300.24340.22550.20900.19380.17990.16700.15520.14430.13420.12490.11640.10840.10110.0943100.22670.20800.19110.17560.16150.14860.13690.12620.11640.10740.09920.09160.08470.07840.0725110.19540.17780.16190.14760.13460.12280.11220.10260.09380.08590.07870.07210.06620.06070.0558120.16850.15200.13720.12400.11220.10150.09200.08340.07570.06870.06250.05680.05170.04710.0429130.14520.12990.11630.10420.09350.08390.07540.06780.06100.05500.04960.04470.04040.03650.0330140.12520.11100.09850.08760.07790.06930.06180.05510.04920.04400.03930.03520.03160.02830.0254150.10790.09490.08350.07360.06490.05730.05070.04480.03970.03520.03120.02770.02470.02190.0195160.09300.08110.07080.06180.05410.04740.04150.03640.03200.02810.02480.02180.01930.01700.0150170.08020.06930.06000.05200.04510.03910.03400.02960.02580.02250.01970.01720.01500.01320.0116180.06910.05920.05080.04370.03760.03230.02790.02410.02080.01800.01560.01350.01180.01020.0089190.05960.05060.04310.03670.03130.02670.02290.01960.01680.01440.01240.01070.00920.00790.0068200.05140.04330.03650.03080.02610.02210.01870.01590.01350.01150.00980.00840.00720.00610.0053210.04430.03700.03090.02590.02170.01830.01540.01290.01090.00920.00780.00660.00560.00480.0040220.03820.03160.02620.02180.01810.01510.01260.01050.00880.00740.00620.00520.00440.00370.0031230.03290.02700.02220.01830.01510.01250.01030.00860.00710.00590.00490.00410.00340.00290.0024240.02840.02310.01880.01540.01260.01030.00850.00700.00570.00470.00390.00320.00270.00220.0018250.02450.01970.01600.01290.01050.00850.00690.00570.00460.00380.00310.00250.00210.00170.0014300.01160.00900.00700.00540.00420.00330.00260.00200.00160.00120.00100.00080.00060.00050.0004附录三：一元年金终值系数表 F=(1+i)n-1/in1%2%3%4%5%6%7%8%9%10%11%12%13%14%15%11.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.000022.01002.02002.03002.04002.05002.06002.07002.08002.09002.10002.11002.12002.13002.14002.150033.03013.06043.09093.12163.15253.18363.21493.24643.27813.31003.34213.37443.40693.43963.472544.06044.12164.18364.24654.31014.37464.43994.50614.57314.64104.70974.77934.84984.92114.993455.10105.20405.30915.41635.52565.63715.75075.86665.98476.10516.22786.35286.48036.61016.742466.15206.30816.46846.63306.80196.97537.15337.33597.52337.71567.91298.11528.32278.53558.753777.21357.43437.66257.89838.14208.39388.65408.92289.20049.48729.783310.089010.404710.730511.066888.28578.58308.89239.21429.54919.897510.259810.636611.028511.435911.859412.299712.757313.232813.726899.36859.754610.159110.582811.026611.491311.978012.487613.021013.579514.164014.775715.415716.085316.78581010.462210.949711.463912.006112.577913.180813.816414.486615.192915.937416.722017.548718.419719.337320.30371111.566812.168712.807813.486414.206814.971615.783616.645517.560318.531219.561420.654621.814323.044524.34931212.682513.412114.192015.025815.917116.869917.888518.977120.140721.384322.713224.133125.650227.270729.00171313.809314.680315.617816.626817.713018.882120.140621.495322.953424.522726.211628.029129.984732.088734.35191414.947415.973917.086318.291919.598621.015122.550524.214926.019227.975030.094932.392634.882737.581140.50471516.096917.293418.598920.023621.578623.276025.129027.152129.360931.772534.405437.279740.417543.842447.58041617.257918.639320.156921.824523.657525.672527.888130.324333.003435.949739.189942.753346.671750.980455.71751718.430420.012121.761623.697525.840428.2