节: 贷款和利息、住房贷款、自动贷款、保险 | 5.7 贷款和利息、住房贷款、自动贷款、保险

章节大纲

Loans & Interest, Home Loans, Auto Loans, Insurance
::贷款和利息、住房贷款、自动贷款、保险

Many people buy cars and houses they cannot afford. This causes major problems for both the banks and the people who have their property taken. In order to make wise choices when you purchase a car or a house, it is important to know how much you can afford to pay each period and calculate a maximum loan amount.
::许多人买不起他们买不起的汽车和房屋。这给银行和被拿走财产的人都造成了重大问题。为了在购买汽车或房屋时做出明智的选择,必须知道您能够支付多少时间和计算最高贷款额。

Universal Generalizations
::普遍化

Banks are essential to the economy, they help people save and borrow money.
::银行对经济至关重要,它们帮助人们储蓄和借钱。

Financial institutions charge interest rates to make money on the money they lend.
::金融机构收取利率,以赚取贷款。

When people borrow money they will have certain responsibilities and obligations to ensure the repayment of that loan.
::当人们借钱时,他们有一定的责任和义务确保偿还贷款。

A person’s credit score can impact how much they can borrow and at what interest rate they will have to pay to use someone else’s money.
::一个人的信用分数会影响他们能借多少贷款,以及他们为使用别人的钱而必须支付的利率。

There are economic benefits to owning your own home.
::拥有自己的家园会带来经济利益。

People who are knowledgeable about borrowing can make decisions in their best interest.
::了解借贷的人可以作出符合其最佳利益的决定。

There are different types of insurance depending on a person’s needs over time.
::保险的种类不同,视一个人在一段时间内的需求而定。

As a consumer, the rule is “buyer beware”.
::作为消费者,规则是“买方应注意”。

Guiding Questions
::问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问问

What must a bank consider before it decides to lend money?
::银行在决定借钱之前必须考虑什么?

What types of questions should a borrower ask when shopping for a loan?
::借款人在购买贷款时应问什么类型的问题?

How does a person get and maintain a good credit score?
::一个人如何获得并保持良好的信用分数?

What costs are associated with purchasing and maintaining a car?
::购买和保养汽车需要多少费用?

Which types of insurance are required by law? Which ones will you need?
::法律要求哪类保险?你需要哪类保险?

How can a person save money on auto insurance? Health insurance?
::一个人如何在汽车保险上省钱?

Buying a House
::购房

Joanna knows she can afford to pay $12,000 a year for a house loan. Interest rates are 4.2% annually and most house loans go for 30 years. What is the maximum loan she can afford? What will she end up paying after 30 years?
::Joanna知道她每年能够为房屋贷款支付12 000美元。利率是每年4.2%,大部分房屋贷款持续30年。她能支付的最高贷款是多少? 30年后她会支付什么?

Video: How to Determine the Monthly Loan Payment for a Mortgage
::视频:如何确定按抵押贷款每月贷款付款

Video: How to Determine the Payment for a Fixed Installment Loan
::视频:如何确定固定安装贷款的付款

Formulas with Examples
::带有示例的公式公式

The present value can be found from the future value using the regular compound growth formula:
::使用经常复合增长公式,可从未来价值中找到现值:

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

::PV(1+一)n=FVPV=FV(1+一)n

You also know the future value of an annuity:
::您也知道年金的未来价值 :

$\begin{aligned} F V = R \cdot \frac{(1 + i)^{n} - 1}{i} \end{aligned}$
::FV=R(1+一)n-1i

So by substitution, the formula for the present value of an annuity is:
::因此,通过替代,年金现值的公式是:

$\begin{aligned} P V = R \cdot \frac{(1 + i)^{n} - 1}{i} \cdot \frac{1}{(1 + i)^{n}} = R \cdot \frac{(1 + i)^{n} - 1}{i (1 + i)^{n}} = R \cdot \frac{1 - (1 + i)^{- n}}{i} \end{aligned}$
::PV=R__(1+一)n-1__(1+一)n=R__(1+一)n-i(1+一)n=R__(1+一)n=R__1-(1+一)-ni

The present value of a series of equal payments $\begin{aligned} R \end{aligned}$ with interest rate $\begin{aligned} i \end{aligned}$ per period for $\begin{aligned} n \end{aligned}$ periods is:
::一系列等值付款R和利率i的现值为:

$\begin{aligned} P V = R \cdot \frac{1 - (1 + i)^{- n}}{i} \end{aligned}$
::PV=R1-1-(1+1)-(i)-(ni)

Example A
::例A

What is the monthly payment of a $1,000,000 house loan over 30 years with a nominal interest rate of 6% convertible monthly?
::30年内每月1 000 000美元的房屋贷款,名义利率为每月可兑换利率6%,这是多少?

Solution: $\begin{aligned} P V = $ 1, 000, 000, R = ?, i = 0.005, n = 360 \end{aligned}$
::解决方案:PV=100万美元,R=? i=0.005,n=360

$\begin{aligned} P V & = R \cdot \frac{1 - (1 + i)^{- n}}{i} \\ 1, 000, 000 & = R \cdot \frac{1 - (1 + 0.005)^{- 360}}{0.005} \\ R & = \frac{1, 000, 000 \cdot 0.005}{1 - (1 + 0.005)^{- 360}} \approx 5995.51 \end{aligned}$

::PV=R1-1-(1+一)-(一+一)-(一+一)-(一+一)-(一)-(一+0.005)-(三)-(三)0.005R=1 000 000-(一+0.005)-(三)-(三)-(五)-(五)-(五)-(五)-(五)-(五)-51)。

It is remarkable that in order to pay off a $1,000,000 loan you will have to pay $5,995.51 a month, every month, for thirty years. After 30 years, you will have made 360 payments of $5995.51, and therefore will have paid the bank more than $2.1 million, more than twice the original loan amount. It is no wonder that people can get into trouble taking on more debt than they can afford.
::令人瞩目的是,为了偿还1 000 000美元的贷款,你必须每月支付5 995.51美元,每月支付5 995.51美元,持续30年。 30年后,你将支付360美元5995.51美元,因此将支付银行超过210万美元,比原始贷款数额高出一倍多。难怪人们会陷入麻烦,承担超过他们承受得起的债务。

Example B
::例例BB

How long will it take to pay off a $20,000 car loan with a 6% annual interest rate convertible monthly if you pay it off in monthly installments of $500? What about if you tried to pay it off in monthly installments of $100?
::如果每月分期支付500美元,那么用每月6%的可兑换年利率每月偿还20 000美元的汽车贷款还要花多久才能兑现? 如果每月分期支付100美元,那么,如果每月分期支付100美元,还要花多久才能偿还这笔20 000美元的汽车贷款?

Solution: $\begin{aligned} P V = $ 20, 000, R = $ 500, i = \frac{0.06}{12} = 0.005, n = ? \end{aligned}$
::解决办法:PV=20,000美元,R=500,i=0.0612=0.005,n=?

$\begin{aligned} P V & = R \cdot \frac{1 - (1 + i)^{- n}}{i} \\ 20, 000 & = 500 \cdot \frac{1 - (1 + 0.005)^{- n}}{0.005} \\ 0.2 & = 1 - (1 + 0.005)^{- n} \\ (1 + 0.005)^{- n} & = 0.8 \\ n & = - \frac{\ln 0.8}{\ln 1.005} \approx 44.74 m o n t h s \end{aligned}$

::PV=R1-1-(1+1)-ni200=5001-1-(1+0.005)-n0.0050.2=1-1-1+0.0005-n(1+0.005)-n=0.8n=0.8nln_0.8n1.00544.74个月

For the $100 case, if you try to set up an equation and solve, there will be an error. This is because the interest on $20,000 is exactly $100 and so every month the payment will go to only paying off the interest. If someone tries to pay off less than $100, then the debt will grow.
::对于100美元的案例,如果你试图设置一个方程和解决,就会有一个错误。这是因为20,000美元的利息恰好是100美元,因此每个月的支付将只用来偿还利息。如果有人试图偿还不到100美元,那么债务就会增加。

Example C
::例例C

It saves money to pay off debt faster in order to save money on interest. As shown in Example A, interest can more than double the cost of a 30 year mortgage. This example shows how much money can be saved by paying off more than the minimum.
::它节省资金以更快地偿还债务,以节省利息。如例A所示,利息可以比30年抵押贷款成本高出一倍以上。这个例子显示了通过支付超过最低限额的贷款可以节省多少钱。

Suppose a $300,000 loan has 6% interest convertible monthly with monthly payments over 30 years. What are the monthly payments? How much time and money would be saved if the monthly payments were larger by a fraction of $\begin{aligned} \frac{13}{12} \end{aligned}$ ? This is like making 13 payments a year instead of just 12. First you will calculate the monthly payments if 12 payments a year are made.
::假设30万美元的贷款每月有6%的可兑换利息,每月支付超过30年。月支付是多少? 如果月支付额比1312年多一点,那么会节省多少时间和多少钱? 这就像每年支付13笔,而不是仅仅12笔。首先,如果每年支付12笔,你将计算月支付额。

\begin{aligned} P V & = R \cdot \frac{1 - (1 + i)^{- n}}{i} \\ 300, 000 & = R \cdot \frac{1 - (1 + 0.005)^{- 360}}{0.005} \\ R & = $ 1, 798.65 \end{aligned}

::PV=R1-1-(1+一)-NI300 000=R1-1-(1+0.005)-300005R=1,798.65美元。

After 30 years, you will have paid $647,514.57, more than twice the original loan amount.
::30年后,您将支付647,514.57美元,超过原始贷款金额的两倍。

If instead, the monthly payment was $\begin{aligned} \frac{13}{12} \cdot 1798.65 = 1948.54 \end{aligned}$ , you would pay off the loan faster. In order to find out how much faster, you will make your unknown.
::如果换句话说,每月的付款是13121798.65=1948.54,你会更快地偿还贷款。为了了解速度有多快,您会说出身份不明。

\begin{aligned} P V & = R \cdot \frac{1 - (1 + i)^{- n}}{i} \\ 300, 000 & = 1948.54 \cdot \frac{1 - (1 + 0.005)^{- n}}{0.005} \\ 0.7698 & = 1 - (1 + 0.005)^{- n} \\ (1 + 0.005)^{- n} & = 0.23019 \\ n & = - \frac{\ln 0.23019}{\ln 1.005} \approx 294.5 m o n t h s \end{aligned}

::PV=R1-1-(1+1)-(1+一)-(ni300 000=1948.541-(1+0.0005)-(1+0.0005)-(n0.0050.7698)=1-(1+0.0005)-n(1+0.005)-n=0.23019n)-n=0.23019n00.2301900=1.005294.5个月

294.5 months is about 24.5 years. Paying fractionally more each month saved more than 5 years of payments.
::294.5个月大约为24.5年,每月支付额略高,节省了5年以上。

$\begin{aligned} 294.5 m o n t h s \cdot $ 1, 948.54 = $ 573, 847.99 \end{aligned}$
::294.5个月 1,948.54美元=573,847.99美元。

The loan ends up costing $573,847.99, which saves you more than $73,000 over the total cost if you had paid over 30 years.
::贷款最终耗资573 847.99美元,如果30年以上偿还了贷款,比总费用节省了73 000多美元。

Concept Problem Revisited
::重新审视问题的概念

Joanna knows she can afford to pay $12,000 a year to pay for a house loan. Interest rates are 4.2% annually and most house loans go for 30 years. What is the maximum loan she can afford? What does she end up paying after 30 years? You can use the present value formula to calculate the maximum loan:
::Joanna知道她每年能够支付12 000美元来支付房屋贷款。利率是每年4.2%,大部分房屋贷款持续30年。她能支付的最大贷款是多少?她30年后会支付多少?你可以使用当前价值公式来计算最高贷款额:

$\begin{aligned} P V = 12, 000 \cdot \frac{1 - (1 + 0.042)^{- 30}}{0.042} \approx $ 202, 556.98 \end{aligned}$
::PV=12 000%1-(1+0.042)-30.042 $202 556.98

For 30 years she will pay $12,000 a year. At the end of the 30 years she will have paid $\begin{aligned} $ 12, 000 \cdot 30 = $ 360, 000 \end{aligned}$ total
::在30年中,她每年将支付12 000美元,在30年结束时,她将支付12 000美元30=360 000美元。

Guided Practice
::实践指南实践指南实践指南实践指南实践指南实践指南实践指南

1. Mackenzie obtains a 15 year student loan for $160,000 with 6.8% interest. What will her yearly payments be?
::1. Mackenzie获得15年学生贷款160,000美元,利息6.8%。她的年付款额是多少?

2. How long will it take Francisco to pay off a $16,000 credit card bill with 19.9% APR if he pays $800 per month? Note: APR in this case means nominal rate convertible monthly .
::2. Francisco每月支付800美元,需要多久才能用19.9%的PAR支付16 000美元的信用卡帐单? 注意:PARP指的是每月可兑换名义汇率。

3. What will the monthly payments be on a credit card debt of $8,000 with 34.99% APR if it is paid off over 3 years?
::3. 如果在3年中付清8 000美元和34.99%的卢旺达爱国军信用卡债务,则每月支付这笔8 000美元和34.99%的卢旺达爱国军债务将如何?

After attempting to solve the Guided Practice Questions, check below for the correct solutions.
::在试图解决《实践指南》中的问题之后,请在下面查询正确的解决办法。

Answers:
::答复:

1. $\begin{aligned} P V = $ 160, 000, R = ?, n = 15, i = 0.068 \end{aligned}$
::1.PV=160,000美元,R=?,n=15,i=0.068

\begin{aligned} 160, 000 & = R \cdot \frac{1 - (1 + 0.068)^{- 15}}{0.068} \\ R & \approx $ 17, 345.88 \end{aligned}

::160,000=R1-(1+0.068)-150.068R 17,345.88美元

2. $\begin{aligned} P V = $ 16, 000, R = $ 600, n = ?, i = \frac{0.199}{12} \end{aligned}$
::2.PV=16 000美元,R=600美元,n=? i=0.19912

\begin{aligned} 16, 000 & = 600 \cdot \frac{1 - {(1 + \frac{0.199}{12})}^{- n}}{\frac{0.199}{12}} \\ n & = 24.50 m o n t h s \end{aligned}

::16,000=6001-1-(1+0.19912)-n0.19912n=24.50个月

3. $\begin{aligned} P V = $ 8, 000, R = ?, n = 36, i = \frac{0.3499}{12} \end{aligned}$
::3. PV=8,000美元,R=?,n=36,i=0.349912

\begin{aligned} 8, 000 & = R \cdot \frac{1 - {(1 + \frac{0.3499}{12})}^{- 36}}{\frac{0.3499}{12}} \\ R & = $ 361.84 \end{aligned}

::8 000=R1-(1+0.349912)-360.349912R=361.84美元

Activity
::活动活动活动活动

Complete the Independent Practice below as directed by your teacher.
::完成以下由老师指导的独立实践。

Independent Practice
::独立实践

For problems 1-10, find the missing value in each row using the present value for annuities formula.
::对于问题1-10, 使用年金公式的现值在每行中查找缺失值。

Problem Number ::问题编号	$\begin{aligned} P V \end{aligned}$	$\begin{aligned} R \end{aligned}$	$\begin{aligned} n \end{aligned}$ (years) : n 年)	$\begin{aligned} i \end{aligned}$ (annual) ::i (每年)	Periods per year ::年年期期年年期年年年年期年年年年年期年年年年年年年年年年年年年年年年年年年年年年年年年年年
1.		$4,000	7	1.5%	1
2.	$15,575		5	5%	4
3.	$4,500	$300		3%	12
4.		$1,000	12	2%	1
5.	$16,670		10	10%	4
6.		$400	4	2%	12
7.	$315,000	$1,800		5%	12
8.		$500	30	8%	12
9.		$1,000	40	6%	4
10.	$10,000		6	7%	12

11. Charese obtains a 15 year student loan for $200,000 with 6.8% interest. What will her yearly payments be?
::11. Charese以20万美元和6.8%的利息获得15年学生贷款,为期15年。

12. How long will it take Tyler to pay off a $5,000 credit card bill with 21.9% APR if he pays $300 per month? Note: APR in this case means nominal rate convertible monthly .
::12. 如果泰勒每月支付300美元,他需要多久才能用21.9%的PAR支付5 000美元的信用卡帐单? 注意:PTR指的是每月名义可兑换汇率。

13. What will the monthly payments be on a credit card debt of $5,000 with 24.99% APR if it is paid off over 3 years?
::13. 如果3年以上付清5 000美元的信用卡债务和24.99%的复兴共和军债务,每月偿还额会是多少?

14. What is the monthly payment of a $300,000 house loan over 30 years with a nominal interest rate of 2% convertible monthly?
::14. 30年来每月以每月2%可兑换名义利率支付300 000加元住房贷款是多少?

15. What is the monthly payment of a $270,000 house loan over 30 years with a nominal interest rate of 3% convertible monthly?
::15. 30年来每月以每月3%可兑换名义利率偿还270 000美元住房贷款是多少?

Learning Extension
::学习推广

Complete the Learning Extension activity on the EconEdLink website by following your teacher's instructions.
::根据教师的指示完成EconEdLink网站的学习推广活动。

Visit the lesson titled on the EconEdLink website. Follow your teachers's instructions for completing this activity.
::访问EconEdLink网站上的题为“EconEdLink”的课目,按照教师的指示完成这项活动。

章节大纲

Loans & Interest, Home Loans, Auto Loans, Insurance ::贷款和利息、住房贷款、自动贷款、保险

Buying a House ::购房

Video: How to Determine the Monthly Loan Payment for a Mortgage ::视频:如何确定按抵押贷款每月贷款付款

Guided Practice ::实践指南 实践指南 实践指南 实践指南 实践指南 实践指南 实践指南

Activity ::活动活动活动活动