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年金终值系数

什么是年金终值系数

年金终值系数指固定的间隔时间相等的期间(如以年为单位)分期支付(存入)1元金额,经过若干年后按复利计算的累计本利之和。

年金终值系数公式如下:

年金终值系数(Future value of an annuity factor)=F/A=(F/A,i,n)

F/A=(F/A,i,n)=\frac{(1+i)^n-1}{i}

这里F/A=(F/A,i,n)代表年金终值系数,i代表利率,n代表年数。

年金终值系数公式的推导

该公式可由复利公式推导出来,推导过程如下:

第1年存入1元到第n年的本利和 = (1 + i)n − 1

第2年存入1元到第n年的本利和 = (1 + i)n − 2

……

第(n-1)年存入1元到第n年的本利和=(1+i)

第n年存入l元 = (1 + i)0

因此,各年本利和相加得:

An | i = (1 + i)n − 1 + (1 + i)n − 2…… + (1 + i) + 1 (1)

两边乘以(1+i),则

(1 + i)An | i = (1 + i)n + (1 + i)n − 1 + …… + (1 + i) (2)

(2)-(1)得iAn | i = (1 + i)n − 1,即

A_n|i=\frac{(1+i)^n-1}{i}

年金终值系数的应用实例

例:按年利率lO%计算5年的年金终值系数如下:

A \overline{5}|10%=\frac{(1+10%)^5-1}{10%}=6.1051

为了方便年金终值系数的运用,对不同年数不同利率分别计算出年金终值系数,列示成年金终值系数表。亦称“1元普通年金终值表”。运用年金终值系数求等额支付若干年后的年金终值就很方便。

例:每年存入银行100元,按年利率l0%复利计算,第5年后的本利和(用A表示)为多少?计算如
下:

(1)用公式计算

A=100 \times A \overline{5}|10%

=100 \times \frac{(1-10%)^5-1}{10%}=610.51

(2)查表计算

A=100 \times 6.1051=610.51

年金终值系数表

期数1%2%3%4%5%6%7%8%9%10%11%12%13%14%15%16%17%18%19%20%21%22%23%24%25%26%27%28%29%30%
11.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
22.01002.02002.03002.04002.05002.06002.07002.08002.09002.10002.11002.12002.13002.14002.15002.16002.17002.18002.19002.20002.21002.22002.23002.24002.25002.26002.27002.28002.29002.3000
33.03013.06043.09093.12163.15253.18363.21493.24643.27813.31003.34213.37443.40693.43963.47253.50563.53893.57243.60613.64003.67413.70843.74293.77763.81253.84763.88293.91843.95413.9900
44.06044.12164.18364.24654.31014.37464.43994.50614.57314.64104.70974.77934.84984.92114.99345.06655.14055.21545.29135.36805.44575.52425.60385.68425.76565.84805.93136.01566.10086.1870
55.10105.20405.30915.41635.52565.63715.75075.86665.98476.10516.22786.35286.48036.61016.74246.87717.01447.15427.29667.44167.58927.73967.89268.04848.20708.36848.53278.69998.87009.0431
66.15206.30816.46846.63306.80196.97537.15337.33597.52337.71567.91298.11528.32278.53558.75378.97759.20689.44209.68309.929910.183010.442310.707910.980111.258811.544211.836612.135912.442312.7560
77.21357.43437.66257.89838.14208.39388.65408.92289.20049.48729.783310.089010.404710.730511.066811.413911.772012.141512.522712.915913.321413.739614.170814.615315.073515.545816.032416.533917.050617.5828
88.28578.58308.89239.21429.54919.897510.259810.636611.028511.435911.859412.299712.757313.232813.726814.240114.773315.327015.902016.499117.118917.762318.430019.122919.841920.587621.361222.163422.995323.8577
99.36859.754610.159110.582811.026611.491311.978012.487613.021013.579514.164014.775715.415716.085316.785817.518518.284719.085919.923420.798921.713922.670023.669024.712525.802326.940428.128729.369230.663932.0150
1010.462210.949711.463912.006112.577913.180813.816414.486615.192915.937416.722017.548718.419719.337320.303721.321522.393123.521324.708925.958727.273828.657430.112831.643433.252934.944936.723538.592640.556442.6195
1111.566812.168712.807813.486414.206814.971615.783616.645517.560318.531219.561420.654621.814323.044524.349325.732927.199928.755130.403532.150434.001335.962038.038840.237942.566145.030647.638850.398553.317856.4053
1212.682513.412114.192015.025815.917116.869917.888518.977120.140721.384322.713224.133125.650227.270729.001730.850232.823934.931137.180239.580542.141644.873747.787750.895054.207757.738661.501365.510069.780074.3270
1313.809314.680315.617816.626817.713018.882120.140621.495322.953424.522726.211628.029129.984732.088734.351936.786239.404042.218745.244548.496651.991355.745959.778864.109768.759673.750679.106684.852991.016197.6250
1414.947415.973917.086318.291919.598621.015122.550524.214926.019227.975030.094932.392634.882737.581140.504743.672047.102750.818054.840959.195963.909569.010074.528080.496186.949593.9258101.4654109.6117118.4108127.9125
1516.096917.293418.598920.023621.578623.276025.129027.152129.360931.772534.405437.279740.417543.842447.580451.659556.110160.965366.260772.035178.330585.192292.6694100.8151109.6868119.3465129.8611141.3029153.7500167.2863
1617.257918.639320.156921.824523.657525.672527.888130.324333.003435.949739.189942.753346.671750.980455.717560.925066.648872.939079.850287.442195.7799104.9345114.9834126.0108138.1085151.3766165.9236181.8677199.3374218.4722
1718.430420.012121.761623.697525.840428.212930.840233.750236.973740.544744.500848.883753.739159.117665.075171.673078.979287.068096.0218105.9306116.8937129.0201142.4295157.2534173.6357191.7345211.7230233.7907258.1453285.0139
1819.614721.412323.414425.645428.132430.905733.999037.450241.301345.599250.395955.749761.725168.394175.836484.140793.4056103.7403115.2659128.1167142.4413158.4045176.1883195.9942218.0446242.5855269.8882300.2521334.0074371.5180
1920.810922.840625.116927.671230.539033.760037.379041.446346.018551.159156.939563.439770.749478.969288.211898.6032110.2846123.4135138.1664154.7400173.3540194.2535217.7116244.0328273.5558306.6577343.7580385.3227431.8696483.9734
2022.019024.297426.870429.778133.066036.785640.995545.762051.160157.275064.202872.052480.946891.0249102.4436115.3797130.0329146.6280165.4180186.6880210.7584237.9893268.7853303.6006342.9447387.3887437.5726494.2131558.1118630.1655
2123.239225.783328.676531.969235.719339.992744.865250.422956.764564.002572.265181.698792.4699104.7684118.8101134.8405153.1385174.0210197.8474225.0256256.0176291.3469331.6059377.4648429.6809489.1098556.7173633.5927720.9642820.2151
2224.471627.299030.536834.248038.505243.392349.005755.456862.873371.402781.214392.5026105.4910120.4360137.6316157.4150180.1721206.3448236.4385271.0307310.7813356.4432408.8753469.0563538.1011617.2783708.0309811.9987931.04381067.2796
2325.716328.845032.452936.617941.430546.995853.436160.893369.531979.543091.1479104.6029120.2048138.2970159.2764183.6014211.8013244.4868282.3618326.2369377.0454435.8607503.9166582.6298673.6264778.7707900.19931040.35831202.04651388.4635
2426.973530.421934.426539.082644.502050.815658.176766.764876.789888.4973102.1742118.1552136.8315158.6586184.1678213.9776248.8076289.4945337.0105392.4842457.2249532.7501620.8174723.4610843.0329982.25111144.25311332.65861551.64001806.0026
2528.243232.030336.459341.645947.727154.864563.249073.105984.700998.3471114.4133133.3339155.6196181.8708212.7930249.2140292.1049342.6035402.0425471.9811554.2422650.9551764.6054898.09161054.79121238.63631454.20141706.80312002.61562348.8033
2629.525633.670938.553044.311751.113559.156468.676579.954493.3240109.1818127.9988150.3339176.8501208.3327245.7120290.0883342.7627405.2721479.4306567.3773671.6330795.1653941.46471114.63361319.48901561.68181847.83582185.70792584.37413054.4443
2730.820935.344340.709647.084254.669163.705874.483887.3508102.7231121.0999143.0786169.3740200.8406238.4993283.5688337.5024402.0323479.2211571.5224681.8528813.6759971.10161159.00161383.14571650.36121968.71912347.75152798.70613334.84263971.7776
2832.129137.051242.930949.967658.402668.528180.697795.3388112.9682134.2099159.8173190.6989227.9499272.8892327.1041392.5028471.3778566.4809681.1116819.2233985.54791185.74401426.57191716.10072063.95152481.58602982.64443583.34384302.94705164.3109
2933.450438.792245.218952.966362.322773.639887.3465103.9659124.1354148.6309178.3972214.5828258.5834312.0937377.1697456.3032552.5121669.4475811.5228984.06801193.51291447.60771755.68352128.96482580.93943127.79843788.95834587.68015551.80166714.6042
3034.784940.568147.575456.084966.438879.058294.4608113.2832136.3075164.4940199.0209241.3327293.1992356.7868434.7451530.3117647.4391790.9480966.71221181.88161445.15071767.08132160.49072640.91643227.17433942.02604812.97715873.23067162.82418729.9855
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