每期利息

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Clever Carol went to her bank which was offering 12% interest on its savings account. She asked very nicely if instead of having 12% at the end of the year, if she could have 6% after the first 6 months and then another 6% at the end of the year. Carol and the bank talked it over and they realized that while the account would still seem like it was getting 12%, Carol would actually be earning a higher percentage. How much more will Carol earn this way?
::Clever Carol去了她的银行,银行在储蓄账户上提供了12%的利息。 她很好地问道,如果不是在年底有12%的利息,而是在年底有12%的利息,如果在头6个月之后她能有6%的利息,然后在年底再有6%的利息。 Carol和银行谈过了,他们意识到,虽然账户看起来仍然有12%的利息,但Carol实际上会得到更高的百分比。Carol这样挣多少钱?

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Compound Interest Per Period of Time
::不同时期的复合化合物利息

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Consider a bank that compounds and adds interest to accounts  $\begin{align*}k\end{align*}$ times per year. If the original percent offered is 12% then in one year that interest can be compounded:
::如果最初提供的百分比是12%,那么在一年之内,利息可以增加:

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• Once, with 12% at the end of the year $\begin{align*}(k=1)\end{align*}$
  ::一次,年底为12%(k=1)
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• Twice (semi-annually), with 6% after the first 6 months and 6% after the last six months $\begin{align*}(k=2)\end{align*}$
  ::两次(每半年两次),头6个月后6%,最后6个月后6%(k=2)
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• Four times (quarterly), with 3% at the end of each 3 months $\begin{align*}(k=4)\end{align*}$
  ::四次(季度),每3个月结束时为3%(k=4)
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• Twelve times (monthly), with 1% at the end of each month $\begin{align*}(k=12)\end{align*}$
  ::12次(每月12次),每月月底为1%(k=12)

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The intervals could even be days, hours or minutes. This is called the length of the compounding period. The number of compounding periods is how often interested is compounded. When intervals become small so does the amount of interest earned in that period, but since the intervals are small there are more of them. This effect means that there is a much greater opportunity for interest to compound.
::间隔甚至可以是天数、小时或分钟。 这叫复利期的长度。 复利期的次数是多时的复利期。 当间隔变得小时, 利息的金额也会变小, 但是由于间隔小, 利息的累积机会会更多 。 这意味着利息的复利机会要大得多 。

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Nominal interest is a number that resembles an interest rate , but it really is a sum of compound interest rates. A nominal rate of 12% compounded monthly is really 1% compounded 12 times. The formula for interest compounding  $\begin{align*}k\end{align*}$ times per year for  $\begin{align*}t\end{align*}$ years at a nominal interest rate $\begin{align*}i\end{align*}$ with present value  $\begin{align*}PV\end{align*}$ and future value  $\begin{align*}FV\end{align*}$ is:
::名义利率是一个类似于利率的数字,但实际上是复合利率的总和。 12 % 的名义复计月利率实际是1 % 复倍 12 次 。 t 年的利率以名义利率i 和现值PV 和未来价值FV 以名义利率i 的年利率K乘以K乘以年利率的公式是:

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$\begin{align*}FV=PV \left(1+\frac{i}{k}\right)^{kt}\end{align*}$
::FV=PV(1+ik)kt

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As with and compound interest, the nominal rate of interest is represented with the letter $\begin{align*}i\end{align*}$  in this formula, but the resulting rate is computed differently. A nominal rate of 12% may actually yield more than 12%.
::与复利和复利一样,名义利率在本公式中以字母i表示,但所产生的利率计算方式不同。 12%的名义利率可能实际收益超过12%。

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Let's apply the formula above to an investment of $300 at a rate of 12% compounded monthly. If you wanted to know the amount of money the person would have after 4 years, you would take the following steps:
::让我们对300美元的投资应用上面的公式,每月12 % 的复数。如果你想了解此人四年后会有多少钱,你将采取以下步骤:

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$\begin{align*}FV=?, \ PV=300, \ t=4, \ k=12, \ i=0.12\end{align*}$
::FV=? PV=300, t=4, k=12, i=0.12

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$\begin{align*}FV=PV \left(1+\frac{i}{k}\right)^{kt}=300 \left(1+\frac{0.12}{12}\right)^{12 \cdot 4} \approx 483.67\end{align*}$
::FV=PV(1+ik)kt=300(1+0.121212)124483.67

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Note: A very common mistake when typing the values into a calculator is using an exponent of 12 and then multiplying the whole quantity by 4 instead of using an exponent of $\begin{align*}(12 \cdot 4)=48\end{align*}$ .
::注:在将数值输入计算器时,一个非常常见的错误是使用12的指数,然后将整个数量乘以4,而不是使用12+4=48的指数。

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked about Clever Carol and the difference in amount of money she would have if her interest was compounded once a year versus twice a year. If Clever Carol earned the 12% at the end of the year she would earn $12 in interest in the first year. If she compounds it  $\begin{align*}k=2\end{align*}$ times per year then she will end up earning:
::早些时候,有人问起你Cleever Carol的情况,如果她每年的利息增加一次,而每年的利息增加两次,她会有多少钱。如果Clever Carol在年底赚到12%的利息,她第一年就能赚到12美元的利息。如果她每年加起来K=2次,她最终会挣到收入:

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$\begin{align*}FV=PV \left(1+\frac{i}{k}\right)^{kt}=100 \left(1+\frac{.12}{2}\right)^{2 \cdot 1}=\$112.36\end{align*}$
::FV=PV(1+ik)kt=100(1+122)2___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Example 2
::例2

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How many years will Matt need to invest his money at 6% compounded daily  $\begin{align*}(k=365)\end{align*}$ if he wants his $3,000 to grow to $5,000?
::如果Matt想要他的3000美元增长到5000美元,

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$\begin{align*}FV=5,000, \ PV=3,000, \ k=365, \ i=0.06, \ t=?\end{align*}$
::FV=5 000,PV=3 000,k=365,i=0.06,t=?

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F V = P V ( 1 + i k ) k t 5 , 000 = 3 , 000 ( 1 + 0.06 365 ) 365 t 5 3 = ( 1 + 0.06 365 ) 365 t ln 5 3 = ln ( 1 + 0.06 365 ) 365 t ln 5 3 = 365 t ⋅ ln ( 1 + 0.06 365 ) t = ln 5 3 365 ⋅ ln ( 1 + 0.06 365 ) = 8.514   y e a r s

$$\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ 5,000 &= 3,000 \left(1+\frac{0.06}{365}\right)^{365t}\\ \frac{5}{3} &= \left(1+\frac{0.06}{365}\right)^{365t}\\ \ln \frac{5}{3} &= \ln \left(1+\frac{0.06}{365}\right)^{365t}\\ \ln \frac{5}{3} &= 365t \cdot \ln \left(1+\frac{0.06}{365}\right)\\ t &= \frac{\ln \frac{5}{3}}{365 \cdot \ln\left (1 + \frac{0.06}{365}\right)}=8.514 \ years\end{align*}$$
::FV=PV(1+ik)kt5 000=3,000(1+0.06365365365t53)=(1+0.063653653665tln53=ln(1+0.06365365365tln53=365t}(1+0.06365Tln)=365t}(1+0.06365t=ln533665)}(1+0.06365655)=8.514

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Example 3
::例3

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What nominal interest rate compounded quarterly doubles money in 5 years?
::什么样的名义利率使季度货币在五年内翻倍?

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$\begin{align*}FV=200, \ PV=100, \ k=4, \ i=?, \ t=5\end{align*}$
::FV=200,PV=100,k=4,i=? t=5

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$$\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ 200 &= 100 \left(1+\frac{i}{4}\right)^{4 \cdot 5}\\ [2]^{\frac{1}{20}} &= \left[ \left( 1+\frac{i}{4}\right)^{20}\right]^{\frac{1}{20}}\\ 2^{\frac{1}{20}} &= 1+\frac{i}{4}\\ i &= \left(2^{\frac{1}{20}}-1\right)4 \approx 0.1411=14.11 \% \end{align*}$$
::FV=PV(1+ik)kt200=100(1+i4)4_5[2]120=[(1+i4)20]120220=1+i4i=1+i4i=(2120-1)4}0.1411=14.11%

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Example 4
::例4

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How much will Steve have in 8 years if he invests $500 in a bank that offers 8% compounded quarterly?
::如果史蒂夫投资500美元 在一个银行 提供8%的复数季度?

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$\begin{align*}PV = 500, \ t=8, \ i=8 \%, \ FV=?, \ k=4 \end{align*}$
::PV=500, t=8, i=8%, FV=?, k=4

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$\begin{align*}FV = PV \left(1+\frac{i}{k}\right)^{kt}=500 \left(1+\frac{0.08}{4}\right)^{4 \cdot 8}=\$942.27\end{align*}$
::FV=PV(1+ik)kt=500(1+0.084)4_8=942.27美元。

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Example 5
::例5

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How many years will Mark need to invest his money at 3% compounded weekly  $\begin{align*}(k=52)\end{align*}$ if he wants his $100 to grow to $400?
::如果马克想把100美元增加到400美元,

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$\begin{align*}FV = 400, \ PV=100, \ k=52, \ i=0.03, \ t=?\end{align*}$

$$\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ 400 &= 100 \left(1+\frac{0.03}{52}\right)^{52 \cdot t}\\ \ln 4 &= \ln \left(1+ \frac{0.03}{52}\right)^{52t}\\ t &= \frac{\ln 4}{52 \cdot \ln \left(1+\frac{0.03}{52}\right)}=46.22 \ years\end{align*}$$
::FV=400, PV=100, k=52, i=0.03, t=?FV=PV(1+ik)kt400=100(1+0.0352)52tln4=ln(1+0.0352)52tn4=ln(1+0.0352)52tt=ln4520(1+0.0352)=46.22年

Summary

播放段落 Compound interest can be compounded at different intervals, such as annually, semi-annually, quarterly, or monthly, which affects the total interest earned. ::复利可每隔不同时期,例如每年、半年、每季度或每月,使复利复利复利,这影响到所得利息总额。 播放段落 Nominal interest is a sum of compound interest rates ::名义利息是复合利率的合计 播放段落 The formula for interest compounding $\begin{align*}k\end{align*}$  times per year for $\begin{align*}t\end{align*}$  years at a nominal interest rate $\begin{align*}i\end{align*}$  with present value $\begin{align*}PV\end{align*}$  and future value $\begin{align*}FV\end{align*}$  is: $\begin{align*}FV = PV(1 + i/k)^{kt}\end{align*}$   ::以名义利率i和现值PV及未来价值FV的利率公式是:FV=PV(1+i/k)kt

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Review
::回顾

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1. What is the length of a compounding period if $\begin{align*}k=12\end{align*}$ ?
::1. 如果k=12,复合期的长度是多少?

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2. What is the length of a compounding period if $\begin{align*}k=365\end{align*}$ ?
::2. 如果k=365,复合期的长度是多少?

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3. What would the value of  $\begin{align*}k\end{align*}$ be if interest was compounded every hour?
::3. 如果每小时利息加在一起,k的价值会如何?

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4. What would the value of  $\begin{align*}k\end{align*}$ be if interest was compounded every minute?
::4. 如果每分钟利息都加在一起,k的价值会如何?

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5. What would the value of  $\begin{align*}k\end{align*}$ be if interest was compounded every second?
::5. 如果利息每秒加在一起,k的价值会是什么?

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For problems 6-15, find the missing value in each row using the compound interest formula.
::对于6-15号问题,使用复利公式在每行中找到缺失值。

播放段落 Problem Number ::问题编号	$\begin{align*}PV\end{align*}$	$\begin{align*}FV\end{align*}$	$\begin{align*}t\end{align*}$	$\begin{align*}i\end{align*}$	$\begin{align*}k\end{align*}$
6.	$1,000		7	1.5%	12
7.	$1,575	$2,250	5		2
8.	$4,000	$5,375.67		3%	1
9.		$10,000	12	2%	365
10.	$10,000		50	7%	52
11.	$1,670	$3,490	10		4
12.	$17,000	$40,000	25		12
13.	$12,000		3	5%	365
14.		$50,000	30	8%	4
15.		$1,000,000	40	6%	2

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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 。