APR和APY(名义有效率)

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In looking at an advertisement for a car you might see 2.5% APR financing on a $20,000 car. What does APR mean? What rate are they really charging you for the loan ? Different banks may offer 8.1% annually, 8% compounded monthly or 7.9% compounded continuously. How much would you really be making if you put $100 in each bank? Which bank has the best deal?
::在看汽车广告时,你可能会看到在一辆20,000美元的汽车上有2.5%的非洲复兴共和军融资。什么是非洲复兴共和军的?他们真正向您收取贷款的利率是多少?不同的银行每年可以提供8.1%,每月8 % , 每月8 % , 或7.9% 。如果你在每家银行上放100美元,你到底能赚多少?哪家银行有最好的交易?

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Nominal and Effective Rates of Interest
::名义和有效利率

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A  nominal interest rate is an interest rate in name only since a method of compounding needs to be associated with it in order to get a true effective interest rate. APR rates are nominal.  APR stands for Annual Percentage Rate . The compounding periods are usually monthly, so typically  $\begin{align*}k=12\end{align*}$ .
::名义利率是名义利率,只是名义利率是名义利率,因为为了获得真正的有效利率,需要与名义利率相结合的复利方法。 年度年利率是名义利率。 年度年利率是年百分比。 复利期通常是每月一次, 通常为k=12。

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An annual effective interest rate is the true interest that is being charged or earned. APY rates are effective rates.  APY stands for Annual Percentage Yield . It is a true rate that states exactly how much money will be earned as interest.
::年有效利率是实际收取或赚取的利息。APY利率是实际利率。APY利率是实际利率。APY利率是年收益百分率的年收益率。这个真实利率确切地说明将赚取多少钱作为利息。

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Banks, car dealerships and all companies will often advertise the interest rate that is most appealing to consumers who don’t know the difference between APR and APY. In places like loans where the interest rate is working against you, they advertise a nominal rate that is lower than the effective rate. On the other hand, banks want to advertise the highest rates possible on their savings accounts so that people believe they are earning more interest.
::银行、汽车经销商和所有公司通常都会广告宣传最能吸引消费者的利率,因为消费者不知道非洲复兴共和军和APY之间的差别。 在利率不利于你的贷款等地方,银行、汽车经销商和所有公司都刊登名义利率低于有效利率的广告。 另一方面,银行想在储蓄账户上登上尽可能高的利率广告,让人们相信自己能赚更多的利息。

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In order to calculate what you are truly being charged, or how much money an account is truly making, it is necessary to use what you have learned about compounding interest and continuous interest. Then, you can make an informed decision about what is best.
::为了计算你真正要收取的费用,或者一个账户真正要赚多少钱,必须使用你学到的利息和连续利息加在一起的知识。然后,你可以对什么是最好的作出知情的决定。

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Take a credit card that advertises 19.9% APR (annual rate compounded monthly). Say you left $1000 unpaid, how much would you owe in a year?
::使用一张信用卡,该信用卡的广告是19.9%的年利率(年利率复数每月 ) 。 假设你留下1 000美元的未付款,一年中你还欠多少?

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First recognize that 19.9% APR is a nominal rate compounded monthly.
::首先,承认19.9%的非洲复兴共和军是每月复合名义费率。

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$\begin{align*}FV =? \ PV=1000, \ i=.199, \ k=12, \ t=1\\\end{align*}$
::FV=?PV=1000,i=199,k=12,t=1

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$$\begin{align*}FV = 1000 \left(1+\frac{0.199}{12}\right)^{12} \approx \$1,218.19\end{align*}$$
::FV=1000(1+0.19912)12 $1,218.19

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Notice that $1,218.19 is an increase of about 21.82% on the original $1,000. Many consumers expect to pay only $199 in interest because they misunderstood the term APR. The effective interest on this account is about 21.82%, which is more than advertised.
::提醒1,218.19美元比原来的1,000美元增加了约21.82%。 许多消费者只希望支付199美元的利息,因为他们误解了APR这个术语。 这个账户的实际利息大约为21.82%,超过了广告公布的数额。

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Another interesting note is that just like there are rounding conventions in this math text (4 significant digits or dollars and cents), there are legal conventions for rounding interest rate decimals. Many companies include an additional 0.0049% because it rounds down for advertising purposes, but adds additional cost when it is time to pay up. For the purposes of this concept,  ignore this addition .
::另一个有意思的注意是,正如数学文本中存在四舍五入惯例(4个重要数字或美元和美分)一样,也有四舍五入利率十进制的法律惯例。 许多公司包括额外的0.0049%,因为它是为了广告目的而四舍五入的,但是在支付时会增加额外的成本。 为了这个概念的目的,忽略这一添加。

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked about financing a car and the difference between APR and APY. A loan that offers 2.5% APR that compounds monthly is really charging lightly more than 2.5% of the initial loan per year.
::早些时候,有人问及汽车的融资问题,以及安协和APY之间的差额。 提供2.5%的安协贷款,每月的复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方复方。

$\begin{align*}\left(1+\frac{0.025}{12}\right)^{12} \approx 1.025288\end{align*}$

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They are really charging about 2.529%.
::他们真的收费 约2.529%。

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The table below shows the APY calculations for three different banks offering 8.1% annually, 8% compounded monthly and 7.9% compounded continuously.
::下表显示了三个不同银行的APY计算结果,年利率为8.1%,每月复合利率为8%,连续复合利率为7.9%。

播放段落 Bank A ::银行A	播放段落 Bank B ::B银行B银行	播放段落 Bank C ::C银行
播放段落 $$\begin{align*}FV &= PV(1+i)^t\\ FV &= 100(1+0.081)\\ FV &= \$ 108.1\end{align*}$$ ::FV=PV(1+一)tFV=100(1+0.081)FV=108.1美元 播放段落 $\begin{align*}APY=8.1 \ \%\end{align*}$ ::APY=8.1%	播放段落 $$\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ FV &= 100 \left(1+\frac{0.08}{12}\right)^{12}\\ FV &\approx 108.299\end{align*}$$ ::FV=PV(1+ik)ktFV=100(1+0.0812)12F108.299 播放段落 $\begin{align*}APY \approx 8.299 \%\end{align*}$ ::APY8.299%	播放段落 $$\begin{align*}FV &= PV \cdot e^{rt}\\ FV &= 100e^{0.079}\\ FV & \approx 108.22\end{align*}$$ ::FV=PVäertFV=100e0.079FV108.22 播放段落 $\begin{align*}APY \approx 8.22 \%\end{align*}$ ::APY8.22%

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Even though Bank B does not seem to offer the best interest rate, or the most advantageous compounding strategy, it still offers the highest yield to the consumer.
::尽管B银行似乎没有提供最佳利率或最有利的复合战略,但它仍然为消费者提供最高收益率。

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Example 2
::例2

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Three banks offer three slightly different savings accounts. Calculate the Annual Percentage Yield for each bank and choose which bank would be best to invest in.
::三家银行提供三个略微不同的储蓄账户。 计算每家银行的年收益百分比,并选择投资的最佳银行。

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Bank A offers 7.1% annual interest.
::A银行每年提供7.1%的利息。

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Bank B offers 7.0% annual interest compounded monthly.
::B银行每月提供7.0%的年利息复利。

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Bank C offers 6.98% annual interest compounded continuously.
::C银行持续提供6.98%的年利息。

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Since no initial amount is given, choose a  $\begin{align*}PV\end{align*}$ that is easy to work with like $1 or $100 and test just one year so $\begin{align*}t=1\end{align*}$ . Once you have the future value for 1 year, you can look at the percentage increase from the present value to determine the APY.
::由于没有给出初始金额, 请选择一个容易使用1美元或100美元的光电池, 并仅测试一年, 这样t=1. 一旦您拥有1年的未来价值, 您可以查看从当前值中的百分比增长, 以确定APY 。

播放段落 Bank A ::银行A	播放段落 Bank B ::B银行B银行	播放段落 Bank C ::C银行
播放段落 $$\begin{align*}FV &= PV(1+i)^t\\ FV &= 100(1+0.071)\\ FV &= \$ 107.1\end{align*}$$ ::FV=PV(1+一)tFV=100(1+0.071)FV=1071美元 播放段落 $\begin{align*}APY=7.1 \%\end{align*}$ ::APY=7.1%	$$\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ FV &= 100 \left(1+\frac{0.07}{12}\right)^{12}\\ FV &\approx 107.229\end{align*}$$ 播放段落 $\begin{align*}APY \approx 7.2290 \%\end{align*}$ ::APY+7.2290%	播放段落 $$\begin{align*}FV &= PV \cdot e^{rt}\\ FV &= 100e^{.0698}\\ FV &\approx 107.2294\end{align*}$$ ::FV=PVäertFV=100e.0698FV107.2294 播放段落 $\begin{align*}APY \approx 7.2294 \%\end{align*}$ ::APY+7.2294%

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Bank A compounded only once per year so the APY was exactly the starting interest rate. However, for both Bank B and Bank C, the APY was higher than the original interest rates. While the APY’s are very close, Bank C offers a slightly more favorable interest rate to an investor.
::亚太银行每年只增加一次,因此亚太银行的利率正好是起始利率。 但是,对于B银行和C银行来说,亚太银行的利率都高于原利率。 虽然亚太银行的利率非常接近,但C银行却为投资者提供了略为有利的利率。

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Example 3
::例3

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The APY for two banks are the same. What nominal interest rate would a monthly compounding bank need to offer to match another bank offering 4% compounding continuously?
::两家银行的APY是相同的。 一个月复利银行需要何种名义利率来匹配另一个提供4%连续复利的银行?

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Solve for APY for the bank where all information is given, the continuously compounding bank.
::在提供所有信息的银行为APY为连续不断的复合银行解决APY问题。

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$\begin{align*}FV=PV \cdot e^{rt}=100 \cdot e^{0.04} \approx 104.08\end{align*}$
::FV=PVert=100e0.04104.08

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The APY is about 4.08%. Now you will set up an equation where you use the 104.08 you just calculated, but with the other banks interest rate.
::APY 大约为 4. 08% 。 现在您将设置一个方程式, 使用您刚刚计算的 104. 08 , 但使用其他银行利率 。

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$$\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ 104.08 &= 100 \left(1+\frac{i}{12}\right)^{12}\\ i &= 12 \left[ \left( \frac{104.08}{100}\right)^{\frac{1}{12}}-1\right] \approx 0.0400667\end{align*}$$
::FV=PV(1+ik)kt104.08=100(1+i12)12i=12[(104.08100)112-1]0.000667]

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The second bank will need to offer slightly more than 4% to match the first bank.
::第二家银行需要提供略高于4%的贷款,以与第一家银行相匹配。

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Example 4
::例4

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Which bank offers the best deal to someone wishing to deposit money?
::哪个银行向想存钱的人 提供最好的交易?

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• Bank A, offering 4.5% annually compounded
  ::A银行,每年提供4.5%
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• Bank B, offering 4.4% compounded quarterly
  ::B银行,每季提供4.4%
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• Bank C, offering 4.3% compounding continuously
  ::C银行,提供4.3%的不断复合

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The following table shows the APY calculations for the three banks.
::下表显示了三家银行的APY计算结果。

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Bank B offers the best interest rate.
::B银行提供最佳利率。

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Example 5
::例5

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What is the effective rate of a credit card interest charge of 34.99% APR compounded monthly?
::信用卡利息的每月有效利率是34.99%的非洲复兴共和军复计,实际利率是多少?

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$\begin{align*}\left(1+\frac{.3499}{12}\right)^{12} \approx 1.4118\end{align*}$ or a 41.18% effective interest rate.
:1+349912)12 1.4118或41.18%的实际利率。

Summary

播放段落 APR (Annual Percentage Rate) is a nominal interest rate that usually compounds monthly (k = 12) ::APR(年百分比率)是一种名义利率,通常每月(k = 12)复利。 播放段落 APY (Annual Percentage Yield) is an effective interest rate, which accurately states how much money will be earned as interest. ::APY(年收益百分比)是有效的利率,准确说明将赚取多少作为利息。 播放段落 Companies often advertise nominal rates (APR) for loans and effective rates (APY) for savings accounts to make their offers more appealing to consumers. ::公司经常公布贷款的名义利率和储蓄账户的有效利率,使其报价对消费者更有吸引力。 播放段落 To calculate the true interest rate, it is necessary to use compounding interest and continuous interest formulas. ::为了计算真实利率,必须使用复利和连续利息公式。

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Review
::回顾

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For problems 1-4, find the APY for each of the following bank accounts.
::对于问题1-4, 找到下列各银行账户的APY。

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1. Bank A, offering 3.5% annually compounded.
::1. A银行,每年提供3.5%的复利。

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2. Bank B, offering 3.4% compounded quarterly.
::2. B银行,每季度提供3.4%的复数。

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3. Bank C, offering 3.3% compounded monthly.
::3. C银行,每月提供3.3%的复数。

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4. Bank D, offering 3.3% compounding continuously.
::4. D银行,持续提供3.3%的化合物。

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5. What is the effective rate of a credit card interest charge of 21.99% APR compounded monthly?
::5. 按21.99%的非洲复兴共和军综合月费收取的信用卡利息有效利率是多少?

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6. What is the effective rate of a credit card interest charge of 16.89% APR compounded monthly?
::6. 每月16.89%的非洲复兴共和军复交费的信用卡利息有效利率是多少?

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7. What is the effective rate of a credit card interest charge of 18.49% APR compounded monthly?
::7. 每月18.49%的非洲复兴共和军复交费的信用卡利息有效利率是多少?

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8. The APY for two banks are the same. What nominal interest rate would a monthly compounding bank need to offer to match another bank offering 3% compounding continuously?
::8. 两家银行的APY是相同的。 月复利银行需要提供何种名义利率来匹配另一家提供3%连续复利的银行?

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9. The APY for two banks are the same. What nominal interest rate would a quarterly compounding bank need to offer to match another bank offering 1.5% compounding continuously?
::9. 两家银行的APY相同,季度复利银行要提供何种名义利率,才能与另一家银行提供1.5%连续加结的利率相匹配?

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10. The APY for two banks are the same. What nominal interest rate would a daily compounding bank need to offer to match another bank offering 2% compounding monthly?
::10. 两家银行的APY是相同的:一家日间复利银行需要提供何种名义利率来与另一家银行每月提供2%复利的银行相匹配?

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11. Explain the difference between APR and APY.
::11. 解释复兴共和军与APY之间的区别。

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12. Give an example of a situation where the APY is higher than the APR. Explain why the APY is higher.
::12. 举一个例子说明APY高于APR的情况,解释为什么APY高于APR。

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13. Give an example of a situation where the APY is the same as the APR. Explain why the APY is the same.
::13. 举一个例子说明APY与APR相同的情况,解释为什么APY相同。

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14. Give an example of a situation where you would be looking for the highest possible APY.
::14. 请举一个例子,说明您将寻找尽可能高的APY的情况。

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15. Give an example of a situation where you would be looking for the lowest possible APY.
::15. 请举一个例子,说明您将寻找尽可能最低的APY的情况。

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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 。