Section outline

  • Many people buy houses they cannot afford. This causes major problems for both the banks and the people who have their homes taken. In order to make wise choices when you buy 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房屋。 这给银行和搬家者都造成了重大问题。 为了在买房时做出明智的选择,重要的是要知道您能支付多少时间和计算最高贷款额。

    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?

    Annuities for Loans
    ::贷款年金

    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:

    P V ( 1 + i ) n = F V P V = F V ( 1 + i ) n

    ::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 :

    F V = R ⋅ ( 1 + i ) n − 1 i
    ::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:

    P V = R ⋅ ( 1 + i ) n − 1 i ⋅ 1 ( 1 + i ) n = R ⋅ ( 1 + i ) n − 1 i ( 1 + i ) n = R ⋅ 1 − ( 1 + i ) − n i
    ::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  R with interest rate  i per period for  n periods is:
    ::一系列等值付款R和利率i的现值为:

    P V = R ⋅ 1 − ( 1 + i ) − n i
    ::PV=R1-1-(1+1)-(i)-(ni)

    This formula can also be used to find out other information such as how much a regular payment should be and how long it will take to pay off a loan.
    ::这一公式还可以用来找出其他信息,例如定期付款应支付多少以及偿还贷款需要多长时间。

    Take a $1,000,000 house loan over 30 years with a nominal interest rate of 6% co mpounded  monthly. You are not given the monthly payments,  R . To find  R , solve for  R  in the formula given above.
    ::在30年中接受1 000 000美元的房屋贷款,名义利率为每月6%的复计利率。您没有获得每月付款,R.要找到R,在上文给出的公式中找到R。

    P V = $ 1 , 000 , 000 ,   R = ? ,   i = 0.005 ,   n = 360
    ::PV=1 000 000 000美元,R=? i=0.005,n=360

    P V = R ⋅ 1 − ( 1 + i ) − n i 1 , 000 , 000 = R ⋅ 1 − ( 1 + 0.005 ) − 360 0.005 R = 1 , 000 , 000 ⋅ 0.005 1 − ( 1 + 0.005 ) − 360 ≈ 5995.51

    ::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。

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were asked about how much Joanna can afford to take out in a loan. 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:
    ::早些时候,有人问你Joanna能承担多少贷款。Joanna知道她每年能支付12 000美元来支付房屋贷款。利率为每年4.2%,大部分房屋贷款持续30年。她能支付的最高贷款额是多少?30年后,她最终能支付多少?你可以使用当前价值公式来计算最高贷款额:

    P V = 12 , 000 ⋅ 1 − ( 1 + 0.042 ) − 30 0.042 ≈ $ 202 , 556.98
    ::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  $ 12 , 000 ⋅ 30 = $ 360 , 000 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。

    Example 2
    ::例2

    How long will it take to pay off a $20,000 car loan with a 6% annual interest rate compounded  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?
    ::以每月500美元的分期付款形式偿还20,000美元的汽车贷款,加上6%的年利率每月复利,需要多长时间? 如果你试图每月100美元的分期付款方式偿还,那又如何?

    P V = $ 20 , 000 ,   R = $ 500 ,   i = 0.06 12 = 0.005 ,   n = ?
    ::PV=20,000,R=500,i=0.0612=0.005,n=?

    P V = R ⋅ 1 − ( 1 + i ) − n i 20 , 000 = 500 ⋅ 1 − ( 1 + 0.005 ) − n 0.005 0.2 = 1 − ( 1 + 0.005 ) − n ( 1 + 0.005 ) − n = 0.8 n = − ln ⁡ 0.8 ln ⁡ 1.005 ≈ 44.74   m o n t h s

    ::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 3
    ::例3

    It saves money to pay off debt faster in order to save money on interest. As shown earlier,  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.
    ::它节省了资金以更快地偿还债务,以节省利息。 如前所述,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 factor of 13 12 ? 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? 如果月支付额增加1312倍,那么将节省多少时间和金钱? 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。

    P V = R ⋅ 1 − ( 1 + i ) − n i 300 , 000 = R ⋅ 1 − ( 1 + 0.005 ) − 360 0.005 R = $ 1 , 798.65

    ::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 13 12 ⋅ 1798.65 = 1948.54 , you would pay off the loan faster. In order to find out how much faster, you will make your unknown.
    ::如果每月付款是13121798.65=1948.54,您会更快地偿还贷款。为了知道要多快,您会说出身份不明。

    P V = R ⋅ 1 − ( 1 + i ) − n i 300 , 000 = 1948.54 ⋅ 1 − ( 1 + 0.005 ) − n 0.005 0.7698 = 1 − ( 1 + 0.005 ) − n ( 1 + 0.005 ) − n = 0.23019 n = − ln ⁡ 0.23019 ln ⁡ 1.005 ≈ 294.5   m o n t h s

    ::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。

    294.5   m o n t h s ⋅ $ 1 , 948.54 = $ 573 , 847.99
    ::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。

    Example 4
    ::例4

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

    P V = $ 160 , 000 ,   R = ? ,   n = 15 ,   i = 0.068
    ::PV=160,000美元,R=?,n=15,i=0.068

    160 , 000 = R ⋅ 1 − ( 1 + 0.068 ) − 15 0.068 R ≈ $ 17 , 345.88

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

    Example 5
    ::例5

    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.
    ::如果Francisco每月支付800美元,需要多久才能用19.9%的PAR来支付16 000美元的信用卡账单? 注意:PARA指的是每月名义可兑换汇率。

    P V = $ 16 , 000 ,   R = $ 600 ,   n = ? ,   i = 0.199 12
    ::PV=16 000美元,R=600美元,n=? i=0.19912

    16 , 000 = 600 ⋅ 1 − ( 1 + 0.199 12 ) − n 0.199 12 n = 24.50   m o n t h s

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

      Summary
    • The present value of an annuity can be calculated using the formula: P V = F V ( 1 + i ) n  
      ::可用下列公式计算年金的现值:PV=FV(1+i)n
    • This formula can be used to find out other information such as how much a regular payment should be and how long it will take to pay off a loan.
      ::这一公式可用于找出其他信息,例如定期付款应支付多少以及偿还贷款需要多长时间。

    Review
    ::回顾

    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

    		P V R

    n  (years)
    :sad n 年)

    i  (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?

    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 。