先付年金——（年金二）

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遇事要有韧性，要懂得随机应变。You gotta be flexible, prepared to roll with anything.

先付年金

Annuity Due

与普通年金不同，先付年金是指每期期初支付年金Unlike anordinary annuity, a prepaid annuity is an annuity paid at the beginning of eachperiod. 例题（Example）计算为期3年，每年年初支付200元的年金终值与现值，利率为10%。The calculation period is three years. The final and present value of an annuity of 200 yuan is paid at the beginning of each year. The interest rate is 10%.

终值

Final Value

先付年金终值即将每一期收到的现金流按照复利形式折现到第3期末。比如第0期收到的200元，将按照10%的利率再投资3年，在第3年年末的价值是200×（1 10%）³；第1期末收到的200元再投资两年，在第3期末的价值是200×（1 10%）²；第二期末收到200元再投资一年，在第3年年末的价值是200×（1 10%）

The final value of the prepaid annuity is to discount the cash flow received in each period to the end of the third period in the form of compound interest.For example, 200 yuan received in the period 0 will be reinvested for three years at 10% interest rate, and the value at the end of the third year is 200 x (1 10%) ³.At the end of the first phase, 200 yuan was reinvested for two years, and the value at the end of the third phase was 200 x(1 10%) ².At the end of the second period, 200 yuan was received for reinvestment for one year, and the value at the end of the thirdyear was 200 x (1 10%).先画出现金流图（Draw a cash flow chart）

年金终值计算如下

The final value calculated as follows

FV=200×(1 10%) 200×(1 10%)² 200×(1 10%)³=728.2

现值

Present Value

先付年金现值即将每一期收到的现金流安复利形式折算到第0期，比如第0期收到200元；

第一期末收到200元，将按照10%的利率折现到第0期价值200×(1 10%)^-1；第二期末收到200元，折现到第0期的价值是200×(1 10%)^-2The present value of the prepaid annuity is to discount the cash flow received in each period to the zero period in the form of compound interest, such as 200 yuan received in the zero period.At the end of the first period, 200 yuan will be received and discounted at 10% interest rate to the zero period value of 200x (1 10%) ^{- 1}.At the end of the second period, 200 yuan was received, and the value discounted to the zero period was 200 x (1 10%)^-2.画出现金流图(Draw a golden flow chart)

年金现值计算如下:The present value of an annuity is calculated as follows:PV=200 200×(1 10%)^-1 200×(1 10%)^-2=547.11