13.4 连续利息

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  连续连续利息

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  Clever Carol realized that she makes more money when she convinces the bank to give her 12% in two chunks of 6% than only one time at 12%. Carol knew she could convince them to give her 1% at the end of each month for a total of 12% which would be even more than the two chunks of 6%. As Carol makes the intervals smaller and smaller, does she earn more and more money from the bank? Does this extra amount ever stop or does it keep growing forever?
  ::聪明的卡罗尔意识到,当她说服银行在两块6 % 中给她12 % , 而不是在12 % 。 卡罗尔知道,她可以说服银行在每个月底给她1 % , 总共12 % , 比6 % 两块6 % 还要多。 随着卡罗尔的间隔越来越小,她从银行挣的钱是否越来越多?这一额外数额是停止还是永远增长?

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  Continuous Interest
  ::连续连续利息

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  Calculus deals with adding up an infinite number of infinitely small amounts. Using calculus, we can derive the value $\begin{array}{r}e\end{array}$  to help us understand what happens as $\begin{array}{r}k\end{array}$ , the number of compounding periods , approaches infinity. The number $\begin{array}{r}e\end{array}$  is used frequently in finance and other fields to represent this type of continuous growth.
  ::微积分处理的是将无限数量的小数量相加。使用微积分,我们可以得出e值来帮助我们理解 k 、 复合期的数量、 无穷的方法。 e 数字经常用于金融和其他领域来代表这种持续增长。

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  $\begin{array}{r}e\approx {\left(1+\frac{1}{k}\right)}^k\approx 2.71828\dots \end{array}$  as  $\begin{array}{r}k\end{array}$  approaches infinity
  ::e( 1+1k) k2. 71828... k 接近无限

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  This means that even when there are an infinite number of infinitely small compounding periods, there will be a limit on the interest earned in a year. The term for infinitely small compounding periods is continuous compounding.  A   c ontinuously compounding interest rate is the rate of growth proportional to the amount of money in the account at every instantaneous moment in time. It is equivalent to infinitely many but infinitely small compounding periods.
  ::这意味着,即使有无限数目的无限小的复利期,一年的利息也将有一定的限度。无限小的复利期是连续的复利期。一个连续的复利率是每个瞬间与账户金额成正比的增长率。它相当于无限多但无限小的复利期。

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  The formula for finding the future value of a present value invested at a continuously compounding interest rate  $\begin{array}{r}r\end{array}$  for  $\begin{array}{r}t\end{array}$  years is:
  ::以连续复利利率r/ t年投资的现值的未来价值的计算公式是:

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  $\begin{array}{r}FV=PV\cdot e^{rt}\end{array}$
  ::FV = PV-iert

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  Applying this formula, you can determine what the future value of $360 invested for 6 years at a continuously compounding rate of 5% is.
  ::应用此公式, 您可以确定未来360美元投资6年的价值, 以5%的连续复利率计算 。

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  $\begin{array}{r}FV=?,PV=360,r=0.05,t=6\end{array}$
  ::FV=?PV=360,r=0.05,t=6

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  $\begin{array}{r}FV=PV\cdot e^{rt}=360e^{0.05\cdot 6}=360e^{0.30}\approx 485.95\end{array}$
  ::FV=PV-ert=360e0.05-6=360e0.30485.95

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to compare the amount of money Clever Carol would make using different rates of compounding. Clever Carol could calculate the returns on each of the possible compounding periods for one year.
  ::早些时候,有人要求你比较 Clever Carol 使用不同的复合率可以赚到多少钱。 Clever Carol 可以计算一年中每个可能的复合期的回报率。

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  For once per year,  $\begin{array}{r}k=1\end{array}$ :
  ::每年一次,k=1:

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  $\begin{array}{r}FV=PV(1+i{)}^t=100(1+0.12{)}^1=112\end{array}$
  ::FV=PV(1+一)t=100(1+0.12)1=112

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  For twice per year,  $\begin{array}{r}k=2\end{array}$ :
  ::每年两次,k=2:

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  $\begin{array}{r}FV=PV(1+i{)}^t=100{\left(1+\frac{0.12}{2}\right)}^2=112.36\end{array}$
  ::FV=PV(1+一)t=100(1+0.1222)2=112.36

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  For twelve times per year,  $\begin{array}{r}k=12\end{array}$ :
  ::每年12次,k=12:

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  $\begin{array}{r}FV=PV(1+i{)}^t=100{\left(1+\frac{0.12}{12}\right)}^{12}\approx 112.68\end{array}$
  ::FV=PV(1+一)t=100(1+0.1212121212)12112.68

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  At this point Carol might notice that while she more than doubled the number of compounding periods, she did not more than double the extra pennies. The growth slows down and approaches the continuously compounded growth result.
  ::此时此刻,卡罗尔或许会注意到,虽然她把复合期增加了一倍多,但她没有把额外零用金增加一倍多。 增长放缓,并接近持续复合增长结果。

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  For continuously compounding interest:
  ::为持续增加利息:

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  $\begin{array}{r}FV=PV\cdot e^{rt}=100\cdot e^{0.12\cdot 1}\approx 112.75\end{array}$
  ::FV=PVert=100e0.121112.75

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  No matter how many times Clever Carol might convince her bank to compound the 12% over the course of each year, the most she can earn from the original $100 is around $12.75 in interest.
  ::无论有多少次Clever Carol能说服她的银行 将每年的12%加起来, 她最多能从原来的100美元赚到的利息是12.75美元左右。

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  Example 2
  ::例2

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  What is the continuously compounding rate that will grow $100 into $250 in just 2 years?
  ::两年内就会增长100美元至250美元,

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  $\begin{array}{r}PV=100,FV=250,r=?,t=2\end{array}$
  ::PV=100,FV=250,r=? t=2

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  $$\begin{array}{rl}FV& =PV\cdot e^{rt}\\ 250& =100\cdot e^{2r}\\ 2.5& =e^{2r}\\ \ln 2.5& =2r\\ r& =\frac{\ln 2.5}{2}\approx 0.4581=45.81\mathrm{\%}\end{array}$$

  
  ::FV=PVert250=100e2r2.5=e2rln=2.5=2rr=ln2.520.4581=45.81%

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  Example 3
  ::例3

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  What amount invested at 7% continuously compounding yields $1,500 after 8 years?
  ::投资额是7%,连续增加1500美元,在8年后达到1500美元?

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  $\begin{array}{r}PV=?FV=1,500,t=8,r=0.07\end{array}$
  ::PV=? FV=1500, t=8, r=0.07

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  $$\begin{array}{rl}FV& =PV\cdot e^{rt}\\ 1,500& =PV\cdot e^{0.07\cdot 8}\\ PV& =\frac{1,500}{e^{0.07\cdot 8}}\approx \mathrm{\$}856.81\end{array}$$

  
  ::FV=PV@ert1500=PV0.07_8PV=1,500e0.07_88_856.81美元。

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  Example 4
  ::例4

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  What is the future value of $500 invested for 8 years at a continuously compounding rate of 9%?
  ::未来500美元投资8年的价值是多少?

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  $\begin{array}{r}FV=500e^{8\cdot 0.09}\approx 1027.22\end{array}$
  ::FV=500e80.091027.22

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  Example 5
  ::例5

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  What is the continuously compounding rate which grows $27 into $99 in just 4 years?
  ::在短短四年内, 27美元将增至99美元,

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  $\begin{array}{r}99=27e^{4r}\end{array}$
  ::99=27e4r

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  Solving for  $\begin{array}{r}r\end{array}$  yields:  $\begin{array}{r}r=0.3248=32.48\mathrm{\%}\end{array}$
  ::r 产量的溶剂:r=0.3248=32.48%

  Summary

  播放段落 Continuous interest deals with adding up an infinite number of infinitely small amounts, using calculus to understand the growth as the number of compounding periods approaches infinity. ::持续利息涉及将无限数量、无限小的数量相加,利用微积分来理解增量,因为增量期的数量接近无限。 播放段落 The number  $\begin{array}{r}e\end{array}$  is used frequently in finance and other fields to represent continuous growth, and is approximately 2.71828. ::e 数字经常用于金融和其他领域,以代表持续增长,约为2.71828。 播放段落 The formula for continuously compounding interest is  $\begin{array}{r}FV=PV\cdot e^{rt}\end{array}$   ::连续增加利息的公式是 FV=PVert

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  Review
  ::回顾

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  For problems 1-10, find the missing value in each row using the continuously compounding interest formula.
  ::对于问题1-10, 使用连续复利公式在每行查找缺失值 。

  播放段落 Problem Number ::问题编号	$\begin{array}{r}PV\end{array}$	$\begin{array}{r}FV\end{array}$	$\begin{array}{r}t\end{array}$	$\begin{array}{r}r\end{array}$
  1.	$1,000		7	1.5%
  2.	$1,575	$2,250	5
  3.	$4,500	$5,500		3%
  4.		$10,000	12	2%
  5.	$1,670	$3,490	10
  6.	$17,000	$40,000	25
  7.	$10,000	$18,000		5%
  8.		$50,000	30	8%
  9.		$1,000,000	40	6%
  10.	$10,000		50	7%

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  11. How long will it take money to double at 4% continuously compounding interest?
  ::11. 将4%的利息翻一番需要多久才能持续增加利息?

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  12. How long will it take money to double at 3% continuously compounding interest?
  ::12. 将3%的利息翻一番需要多久才能持续增加利息?

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  13. Suppose you have $6,000 to invest for 12 years. How much money would you have in 12 years if you earned 3% simple interest? How much money would you have in 12 years if you earned 3% continuously compounding interest?
  ::13. 假设你有6000美元投资12年。12年中,如果你赚取3%的简单利息,你将有多少钱?12年中,如果你连续赚取3%的利息,你将有多少钱?

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  14. Suppose you invest $2,000 which earns 5% continuously compounding interest for the first 12 years and then 8% continuously compounding interest for the next 8 years. How much money will you have after 20 years?
  ::14. 假设你投资2 000美元,在头12年中,赚取5%的利息,连续增加利息5%,然后连续增加8%的利息,在今后8年中,持续增加利息。20年后,你有多少钱?

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  15. Suppose you invest $7,000 which earns 1.5% continuously compounding interest for the first 8 years and then 6% continuously compounding interest for the next 7 years. How much money will you have after 15 years?
  ::15. 假设你投资7 000美元,前8年赚取1.5%,连续增加利息1.5倍,后6个百分点,后7年持续增加利息6倍。 15年后,你有多少钱?

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  Review (Answers)
  ::回顾(答复)

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  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 。