Course: 13.6 年金 | The Way To Learn

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年金

Sally knows she can earn a nominal rate of 6% convertible monthly in a retirement account, and she decides she can afford to save $1,500 from her paycheck every month. How can you use geometric series to simplify the calculation of finding the future value of all these payments? How much money will Sally have saved in 30 years?
::莎莉知道她可以在退休账户中每月赚取6%的可兑换货币,她决定每月从工资中节省1500美元。你如何使用几何序列来简化寻找所有这些付款未来价值的计算方法?萨莉在30年内能储蓄多少钱?

Annuity
::年年金

An annuity is a series of equal payments that occur periodically. The word annuity comes from annual which means yearly. You will start by working with payments that occur once at the end of each year and then delve deeper to payments that occur monthly or any period.
::年金是指一系列定期发生的同酬。年金一词来自年度,意味着每年。您首先将使用每年年底一次的付款,然后更深入地研究每月或任何时期的付款。

Assume an investor saves $\begin{aligned} R \end{aligned}$ dollars at the end of each year for $\begin{aligned} t \end{aligned}$ years in an account that earns $\begin{aligned} i \end{aligned}$ interest per period.
::假设投资者在某一账户中每年年底节省R美元,每年2年,该账户每期赚取一美元的利息。

The first payment $\begin{aligned} R \end{aligned}$ will be in the bank account for $\begin{aligned} t - 1 \end{aligned}$ years and grow to be: $\begin{aligned} R (1 + i)^{t - 1} \end{aligned}$
::第一笔付款R将在t-1年的银行账户中支付,增长为R(1+1)t-1。

The second payment $\begin{aligned} R \end{aligned}$ will be in the bank account for $\begin{aligned} t - 2 \end{aligned}$ years and grow to be: $\begin{aligned} R (1 + i)^{t - 2} \end{aligned}$
::第二笔付款R将存入银行帐户2年,增长为R(1+1)t-2。

This pattern continues until the last payment of $\begin{aligned} R \end{aligned}$ that is deposited in the account right at $\begin{aligned} t \end{aligned}$ years, so it doesn’t earn any interest at all.
::这种模式一直持续到在 t 年存入该账户右侧的最后一次RR付款,所以它根本得不到任何利息。

The account balance at this point in the future (Future Value, $\begin{aligned} F V \end{aligned}$ ) is the sum of each individual $\begin{aligned} F V \end{aligned}$ of all the payments:
::未来这一时刻的账户余额(未来价值,FV)是所有付款的每个FV的总额:

$\begin{aligned} F V = R + R (1 + i)^{1} + R (1 + i)^{2} + \dots + R (1 + i)^{t - 2} + R (1 + i)^{t - 1} \end{aligned}$
::FV=R+R(1+一)1+R(1+一)1+R(1+一)2R(1+一)2R(1+一)t-2+R(1+一)t-1

Recall that a geometric series with initial value $\begin{aligned} a \end{aligned}$ and common ratio $\begin{aligned} r \end{aligned}$ with $\begin{aligned} n \end{aligned}$ terms has sum:
::回顾具有初始值a和通用比率r与n条件n的几何序列有总和:

$\begin{aligned} a + a r + a r^{2} + \dots + a r^{n - 1} = a \cdot \frac{1 - r^{n}}{1 - r} \end{aligned}$
::a+ar+ar2ar1=a1-rn1-r

So, a geometric series with starting value $\begin{aligned} R \end{aligned}$ and common ratio $\begin{aligned} (1 + i) \end{aligned}$ has sum:
::因此,一个具有起点值R和共同比率(1+1)的几何序列总和如下:

\begin{aligned} F V & = R \cdot \frac{1 - (1 + i)^{n}}{1 - (1 + i)} \\ = R \cdot \frac{1 - (1 + i)^{n}}{- i} \\ = R \cdot \frac{(1 + i)^{n} - 1}{i} \end{aligned}

::FV=R1-(1+一)n1-(1+一)=R1-(1+一)n-1i=R(1+一)n-1i

This formula describes the relationship between $\begin{aligned} F V \end{aligned}$ (the account balance in the future), $\begin{aligned} R \end{aligned}$ (the annual payment), $\begin{aligned} n \end{aligned}$ (the number of years) and $\begin{aligned} i \end{aligned}$ (the interest per year).
::本公式描述FV(未来账户余额)、R(年度付款)、n(年数)和i(年利息)之间的关系。

The formula is extraordinarily flexible and will work even when payments occur monthly instead of yearly by rethinking what, $\begin{aligned} R, i \end{aligned}$ and $\begin{aligned} n \end{aligned}$ mean. The resulting Future Value will still be correct. If $\begin{aligned} R \end{aligned}$ is monthly payments, then $\begin{aligned} i \end{aligned}$ is the interest rate per month and $\begin{aligned} n \end{aligned}$ is the number of months.
::公式非常灵活,即使每月支付而不是每年支付,也会起作用,重新思考R、i和n的意思。由此产生的未来价值仍然正确。如果R是每月支付,那么i是每月利率,n是月数。

Take an IRA (special type of savings account). If Lenny saves $5,000 a year at the end of each year for 35 years at an interest rate of 4%, he can determine what his Future Value will be using the formula.
::采用IRA(特殊类型的储蓄账户 ) 。如果Lenny每年年底以4%的利率在35年内每年节省5 000美元,那么他可以确定他的未来价值将使用公式。

$\begin{aligned} R = 5, 000, i = 0.04, n = 35, F V = ? \end{aligned}$
::R=5,000, i=0.04, n=35, FV=?

$\begin{aligned} F V & = R \cdot \frac{(1 + i)^{n} - 1}{i} \\ F V & = 5, 000 \cdot \frac{(1 + 0.04)^{35} - 1}{0.04} \\ F V & = $ 368, 281.12 \end{aligned}$

::FV=R__(1+1)n-1iFV=5,000__(1+0.04)35-10.04FV=368,281.12美元。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem where Sally wanted to know how much she will have if she can earn a nominal 6% interest rate compounded monthly in a retirement account where she decides to save $1500 from her paycheck every month for thirty years.
::早些时候,你遇到一个问题莎莉想知道如果她能每月在退休账户里赚到6%的名义利率她能赚到多少钱她决定30年来每月从她的薪水中节省1500美元

$\begin{aligned} F V = ?, i = \frac{0.06}{12} = 0.005, n = 30 \cdot 12 = 360, R = 1, 500 \end{aligned}$
::FV=? ,i=0.0612=0.005,n=30=12=360,R=1 500

\begin{aligned} F V & = R \cdot \frac{(1 + i)^{n} - 1}{i} \\ F V & = 1, 500 \cdot \frac{(1 + 0.005)^{360} - 1}{0.005} \\ F V & \approx 1, 506, 772.56 \end{aligned}

::FV=R__(1+1)n-1-iFV=1,500(1+0.005)360-10.005FV=1,506,772.56。

Example 2
::例2

How long does Mariah need to save if she wants to retire with a million dollars and saves $10,000 a year at 5% interest?
::如果她想用一百万美元退休,每年以5%的利息节省一万美元,那么Mariah需要储蓄多久?

$\begin{aligned} F V = 1, 000, 000, R = 10, 000, i = 0.05, n = ? \end{aligned}$
::FV=1 000 000,R=10 000,i=0.05,n=?

$\begin{aligned} F V & = R \cdot \frac{(1 + i)^{n} - 1}{i} \\ 1, 000, 000 & = 10, 000 \cdot \frac{(1 + 0.05)^{n} - 1}{0.05} \\ 100 & = \frac{(1 + 0.05)^{n} - 1}{0.05} \\ 5 & = (1 + 0.05)^{n} - 1 \\ 6 & = (1 + 0.05)^{n} \\ n & = \frac{\ln 6}{\ln 1.05} \approx 36.7 y e a r s \end{aligned}$

::FV=R(1+1)n-1i1 000 000 000=10 000(1+0.05 000 000=10 000(1+0.05 000)-1005100=(1+0.05 000)-10055=(1+0.05 000)n-16=(1+0.05)n=(1+0.05)nn=一6ln 1.0536.7岁

Example 3
::例3

How much will Peter need to save each month if he wants to buy an $8,000 car with cash in 5 years? He can earn a nominal interest rate of 12% compounded monthly.
::如果彼得想在5年内用现金买一辆8000美元的汽车,每月需要储蓄多少? 他每月可以赚取12%的每月名义利率。

In this situation you will do all calculations in months instead of years. An adjustment in the interest rate and the time is required and the answer needs to be clearly interpreted at the end.
::在这种情况下,您将用几个月而不是几年来做所有计算。利率和时间需要调整,答案需要在结尾处清楚解释。

$\begin{aligned} F V = 8, 000, R = ?, i = \frac{0.12}{12} = 0.01, n = 5 \cdot 12 = 60 \end{aligned}$
::FV=8,000,R=? i=0.1212=0.01,n=512=60

\begin{aligned} F V & = R \cdot \frac{(1 + i)^{n} - 1}{i} \\ 8, 000 & = R \cdot \frac{(1 + 0.01)^{60} - 1}{0.01} \\ R & = \frac{8, 000 \cdot 0.01}{(1 + 0.01)^{60} - 1} \approx 97.96 \end{aligned}

::FV=R__(1+1)n-18,000=R__(1+0.01)60-10.01R=8000-0.01(1+0.01)60-1\97.96。

Peter will need to save about $97.96 every month.
::Peter每月需要节省97.96美元

Example 4
::例4

At the end of each quarter, Fermin makes a $200 deposit into a mutual fund. If his investment earns 8.1% interest compounded quarterly, what will his annuity be worth in 15 years?
::在每个季度结束时,费尔明向一个共同基金存款200美元。如果他的投资每季赚取8.1%的利息,那么他15年的年金值是多少?

Quarterly means 4 times per year.
::季度是指每年4次。

$\begin{aligned} F V = ?, R = 200, i = \frac{0.081}{4}, n = 60 \end{aligned}$
::FV=? R=200, i=0.0814, n=60

$\begin{aligned} F V = 200 \cdot \frac{{(1 + \frac{0.081}{4})}^{60} - 1}{\frac{0.081}{4}} \approx $ 23, 008.71 \end{aligned}$
::FV=200(1+0.0814)60-10.0814_23 008.71美元

Example 5
::例5

What interest rate compounded semi-annually is required to grow $25 semi-annual payments to $500 in 8 years?
::在8年内将25美元半年付款增加到500美元,每半年需要何种利率加起来每半年一次的利率?

$\begin{aligned} F V = 500, R = 25, i = ?, n = 8 \cdot 2 = 16 \end{aligned}$ . Note that the calculation will be done in months. At the end you will convert your answer to years.
::FV=500, R=25, i=?, n=82=16。请注意, 计算将在几个月内完成。结尾处您将把答案转换为年数。

\begin{aligned} F V & = R \cdot \frac{(1 + i)^{n} - 1}{i} \\ 500 & = 25 \cdot \frac{(1 + i)^{16} - 1}{i} \\ 20 i & = (1 + i)^{16} - 1 \\ 0 & = (1 + i)^{16} - 20 i - 1 \end{aligned}

::FV=R(1+一)n-1i500=25(1+一)16-1i20i=(1+一)16-10=(1+一)16-20i-1

Using a graphing calculator, we find this equation has roots at $\begin{aligned} i = 0 \end{aligned}$ and $\begin{aligned} i = 0.0290 \end{aligned}$ . Since $\begin{aligned} i \neq 0 \end{aligned}$ , the semi-annual interest rate is $\begin{aligned} i = 0.0290 = 2.90 % \end{aligned}$ for a nominal annual interest rate of $\begin{aligned} 5.80 % \end{aligned}$ .
::使用图形计算器, 我们发现这个方程式的根值是 i=0 和 i= 0.0290。自 i0 开始, 半年利率为 i= 0.0290 = 2. 90%, 名义年利率为 5. 80% 。

Summary

An annuity is a series of equal payments that occur periodically
::年金是指定期发生的一系列同酬

The future value (FV) of an annuity can be calculated using a geometric series formula: $\begin{aligned} F V = R * \frac{((1 + i)^{n} - 1)}{i}, \end{aligned}$ where $\begin{aligned} R \end{aligned}$ is the payment, $\begin{aligned} i \end{aligned}$ is the interest rate per period, and n is the number of periods.
::年金的未来值(FV)可以用几何序列公式计算:FV=R*((1+1)n-1)i,R为付款,i为每个时期的利率,n为周期数。

The formula can be adapted for different payment frequencies, such as monthly or yearly, by adjusting the interest rate and number of periods accordingly.
::公式可相应调整利率和期间数,以适应不同的支付频率,例如月度或年度。

Review
::回顾

1. At the end of each month, Rose makes a $400 deposit into a mutual fund. If her investment earns 6.1% interest compounded monthly, what will her annuity be worth in 30 years?
::1. 每月月底,Rose向共同基金存款400美元,如果她的投资每月赚取6.1%的利息,她30年的年金价值是多少?

2. What interest rate compounded quarterly is required to grow a $40 quarterly payment to $1000 in 5 years?
::2. 5年内将季度付款40美元增至1 000美元,每季需要何种利率,再加上何种利率?

3. How many years will it take to save $10,000 if Sal saves $50 every month at a 2% monthly interest rate?
::3. 如果Sal每月以2%的月利率节省50美元,节省10 000美元需要多少年?

4. How much will Bob need to save each month if he wants to buy a $33,000 car with cash in 5 years? He can earn a nominal interest rate of 12% compounded monthly.
::4. Bob如果想在5年内用现金购买一辆33 000美元的汽车,每月需要储蓄多少?他每月可赚取12%的每月名义利率。

5. What will the future value of his IRA be if Cal saves $5,000 a year at the end of each year for 35 years at an interest rate of 8%?
::5. 如果Cal每年年底以8%的利率在35年内每年节省5 000美元,他的爱尔兰共和军的未来价值会如何?

6. How long does Kathy need to save if she wants to retire with four million dollars and saves $10,000 a year at 8% interest?
::6. 如果凯西想用400万美元退休,每年以8%的利息节省10 000美元,那么她需要储蓄多久?

7. What interest rate compounded monthly is required to grow a $416 monthly payment to $80,000 in 10 years?
::7. 将10年中的416美元每月付款增加到80 000美元,每月需要何种利率(每月加息)?

8. Every six months, Shanice makes a $1000 deposit into a mutual fund. If her investment earns 5% interest compounded semi-annually, what will her annuity be worth in 25 years?
::8. 每六个月,Shanice向共同基金存款1 000美元,如果她的投资每半年赚取5%的利息,每半年再加5%的利息,她25年的年金值是多少?

9. How much will Jen need to save each month if she wants to put $60,000 down on a house in 5 years? She can earn a nominal interest rate of 8% compounded monthly.
::9. Jen如果希望在五年内把6万美元花在一所房子里,每月需要多少钱? 她可以每月赚取8%的名义利率。

10. How long does Adrian need to save if she wants to retire with three million dollars and saves $5,000 a year at 10% interest?
::10. 如果Adrian想用300万美元退休,每年以10%的利息节省5 000美元,那么她需要节省多长时间?

11. What will the future value of her IRA be if Vanessa saves $3,000 a year at the end of each year for 40 years at an interest rate of 6.7%?
::11. 如果Vanessa以6.7%的利率在40年内每年年底每年节省3 000美元,她的爱尔兰共和军的未来价值会如何?

12. At the end of each quarter, Justin makes a $1,500 deposit into a mutual fund. If his investment earns 4.5% interest compounded quarterly, what will her annuity be worth in 35 years?
::12. 每个季度结束时,Justin向共同基金存款1 500美元,如果他的投资每季赚取4.5%的利息,再加季度,她35年的年金值是多少?

13. What will the future value of his IRA be if Ted saves $3,500 a year at the end of each year for 25 years at an interest rate of 5.8%?
::13. 如果Ted在25年内每年年底以5.8%的利率每年节省3 500美元,他的爱尔兰共和军的未来价值会如何?

14. What interest rate compounded monthly is required to grow a $300 monthly payment to $1,000,000 in 35 years?
::14. 在35年内,将每月支付300美元增加到1 000 000美元,每月需要何种利率(每月加息)?

15. How much will Katie need to save each month if she wants to put $55,000 down in cash on a house in 2 years? She can earn a nominal interest rate of 6% compounded monthly.
::15. 如果Katie想在两年内将55,000美元现金存入一所房屋,每月需要储蓄多少? 她可以每月赚取6%的名义利率。

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

年金

Annuity ::年年金

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)