Annuities include ordinary annuity, early annuity, deferred annuity and permanent annuity.
Ordinary annuity, also known as post-paid annuity, refers to the annuity received and paid at the end of each period. The present value of ordinary annuity refers to the amount that needs to be invested now in order to obtain an equal amount of funds at the end of each period.
Early annuity is an annuity paid at the beginning of each period.
Deferred annuity refers to the annuity whose first payment occurs in the second period or after the second period.
An annuity with an indefinite fixed payment is called a permanent annuity. In reality, saving principal and drawing interest can be regarded as an example of perpetual annuity.
The present value of annuity refers to the sum of the present value of the first beginning converted from the equal income or payment amount at the end of each period within the same time interval [edit this paragraph]. The present value of annuity is the inverse calculation of the final value of annuity.
Calculation formula:
P=[ 1-( 1+i) to the -n power ]/i, where p is the present value coefficient of the annuity. Let ordinary annuity 1 yuan be the present value of the annuity with the interest rate of I, n, and record it as (P/A, I, n).
Deduction process: ................. ①
Multiply formula ① by (1+i), and then:
………………………②
②-①, and then:
( 1 + i)P？ 6? 1 P = A？ 6? 1 A( 1 + i)
P( 1 + i？ 6? 1 1) = A[ 1？ 6? 1 ( 1 + i)]
∴ [Edit this paragraph] Example of calculating the present value of annuity The present value of ordinary annuity refers to the sum of the compound interest present value of income and expenditure at the end of each period in a certain period. Ordinary annuity's calculation can be represented by figure 1 [ 1]
Example 1: To get the present value of ordinary annuity at a discount of 6%, it is divided into three periods, each period is 1 yuan, which can be calculated according to Figure 2:
The present value of 1 yuan received at the end of the first period is 0.943 yuan, that of 1 yuan received at the end of the second period is 0.890 yuan, and that of 1 yuan received at the end of the third period is 0.840 yuan. The present value of the annuity is the sum of the present value of 2.673 yuan.
Example 2[ 1]: If a certain equipment is leased, the rent shall be paid at the end of each year 1200 yuan, with annual compound interest 10%. What is the present value of the total rent payable after five years?
(yuan)
4452 yuan is the present value of 1200 yuan rental annuity at the interest rate of five years 10%.
In fact, the present value of annuity and annuity are reciprocal operations. If you know the present value of an annuity, you can also get an annuity. The calculation formula is:
Among them, it is the reciprocal of the present value coefficient of annuity, also called the capital recovery coefficient, because this coefficient is used to calculate the amount of capital recovery, and capital recovery refers to the equal recovery or repayment of initial investment or debt within a given time. The calculation of capital recovery is actually to find the form of annuity from the present value of annuity.
For example, an enterprise borrows 654.38 million yuan, with an annual interest rate of 12%, and invests in a project with a project life of 10 years. How much cash should I get back at least every year?
(yuan)
17700 yuan is a 10-year annuity, the present value is 100000 yuan, and the interest rate is 12%, which is at least the cash amount to be recovered every year when 12% is used for borrowing and investing in10-year projects. [Edit this paragraph] Annuity present value coefficient table n1%2% 3% 4% 5% 6% 8%10%12%15%16%/ 8% 20% 22 . 9430 . 9250 . 9090 . 8920 . 87770 . 8690 . 5699 . 2020 . 8773 15453.4653.3 123. 1693.0372.9 1 32.8542.7982.6902.5882.4932.4042.36 12. 166 1.996 1.849 1.7 19 1.60454.8534.7 134.5794.45 / kloc-0/4.3294.2 123.9923.7903.6043.4333.3523.2743. 1272.9902.8632.7452.6892.4352.2 192.035 1.875 1.7366 5.7955.60 / kloc-0/5.4 175.2425.0754.9 174.6224.3554. 1 1 13.8883.7843.6843.4973.3253. 1663.0202.95 1 2.6422.3852.65438+ 067 1.983 1.82476.7286.47 16.2306.0025.7865.5825.2064.8684.5634.2884. 1604.0383.8 1 1 3.6043.4 153.2423. 1665438 +02.8022.5072.2622.057 1.88287.65 17.3257.0 1 96.7326.4636.2095.7465.3344.9674.6384.4874.3434.0773.8373.6 193.42 13.3282.9242.5 982.3302. 108 1.92 198.5668. / kloc-0/627.7867.4357. 1076.80 16.2465.7595.3284.9464.77 14.6064.3034.0303.7863.5653.4633.0654 38+092.6652.3782. 143 / kloc-0/.947 109.47 18.9828.5308. 1 107.72 17.3606.7 106. 1445.6505.2 165.0 1 84.8 334.4944. 1923.9233.68 13.5703.09 12.7 152.4 132. 168 1.965 1 1 1 0.3679.7869.2528.7608.3067.8867.6543 8+0386.4955.9375.4525.2335.0284.6564.3274.0353.7753.6563. 1472.75 12.4382. 184 1 .976 12 1 1.255 10.5759.9549.38 58.8638.3837.5366.8 136. 1945.6605.4205. 1974.7934.4394. 1 273.85 13.7253. 1902.7792.4552. 196 1.984 136543 8+02. 133 1 1.348 1 0.6349.9859.3938.8527.9037. 1036.4235.8425.5835.3424.9094.5324.2023.9 123.7803.2232.7992.4682.20465438 +0.989 14 13.003 / kloc-0/2. 106 1 1.296 10.5639.8989.2948.2447.3666.6286.0025.7245.4675.0084.6 104.2643.96 13.824 3.2482.8 / kloc-0/42.4772.209 1.993 15 13.865 12.849 1 1.937 1 1. 1 18 10.3799.7 / kloc-0/28.5597.6066 .8 106. 1425.8475.5755.09 14.6754.3 154.00 13.8593.2682.8252.4832.2 13 1.995 / kloc-0/6 14.7 17 13.57765 438+02.56 1 1 1.652 10.837 10. 1058.85 1 7.8236.9736.2655.9545.6685. 1624.7294.3564.0333.8873.2832.8332.4882.265 438+06 1.996 17 15.562 14.29 1 / kloc-0/3. 166 12. 165 1 1.274 10.4779. 12 18.02 17. 1 1 96.3726.0475.7485.2224.7744.3904.0593.9093.2942.8392.49 12.2 18 1.997 18 16.398 14.992 13.753 / kloc-0/2.659 1 1.689 10.8279.37 18.20 17.2496.4676. 1275.8 175.2734.8 124.4 1 84.0793.9273.3032.8442.4942.2 196 5438+0.998 19 17.226 15.678 14.323 13. 133 12.085 1 / kloc-0/. 1589.6038.3647.3656.5506. 1985.8775.36543 8+064.8434.44 14.0963.9423.3 102.8472.4952.220 1.99920 1 8.045 16.35 1 14.877 13.590 12.462 1 1.4 699.8 188.5 1 37.4696.6236.2595.9285.3524.8694.4604. 1 103.9533.3 152.8502.4972.220 1.9992 1 18.856 17.0654 38+0 / kloc-0/ 15.4 15 14.029 12.82 1 1 1.764 10.0 168.6487.5626.6866.3 1 25.9735.3834.89 14.4754.65 438+02 13.9633.3 192.85 12.4972.22 1 1.99922 19.660 17.658 1 5.936 14.45 1 13. 163 12.0 4 1 10.2008.77 17.6446.7426.3586.0 1 1 5.4094.9094.4884. 1293.9703.3222.8532.4982.22 1 1.9992320.455 18.292 16.443 14.856 13.488 1 2.303 10.37 18.8837.7 186.7926.3986.0445.4324.9244.4984. 1373.9763.3252.8542.4982.226543 8+0 1.999242 1 .243 18.9 13 16.935 15.246 13.798 12.550 1 0.5288.9847.7846.8356.4336.0725.4504.9374.5074.654 38+0423.98 13.3272.8552.4992.22 1 1.9992522.023 19.523 17.4 / kloc-0/3 15.622 14.093 12.783 10.6749.0777.8436 .8726.4646.0975.4664.9474.5 134. 1 473.9843.3282.8552.4992.222 1.9992622.79520. 12 1 17.876 15.982 14.375 1 3.003 10.8099. / kloc-0/607.8956.9066.4906. 1 185.4804.9564.5 194. 15 13.9873.3292.8552.4992.222 1 .9992723.55920.706654 38+08.327 16.329 14.643 13.2 10 10.9359.2377.9426.9356.5 136. 1365.49 1 4.9634.5244. 1543.9903.3302.8562.49 92.222 1.9992824.3 162 1.28 1 18.764 16.663 14.898 / kloc-0/3.406 1 1.05 19.3067.9846.9606.5336. 15 25.50 14.9694.5284. 1563.9923.33 12.8562.4992.222 / kloc-0/.9992925.0652 1.844 19. 188 16.983 15. 14 165 438+03.590 1 1. 1 589.3698.02 16.9836.5506. 1655.5094.9744.53 14. 1583.9933.33 12.8562.4992.222 1.9993025.8 0722.396 1 9.600 17.292 15.372 13.764 1 1.2579.4268.0557.0026.5656. 1775.5 164.9784.5334. 1 603.9953.3322.85 62.4992.222 1.9994032.83427.35523. 1 14 19.792 17. 159 15.046 1 1 .9249.7798.2437. 1056.64 16.233 5.5484.9964.5434. 1653.9993.3332.8572.4992.222 1.9995039. 1963 1 .42325.7292 1.482 18.255 15.76 1 12.2339.9 148.3047. 1326.6606.2465.5544.9994.5454. 1663.9993.3332.8572.4992.222 1.999