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Five years ago, Diane secured a bank loan of $300,000 to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was 35 yr, and the interest rate was 6%/year compounded monthly on the unpaid balance. Because the interest rate for a conventional 35-yr home mortgage has now dropped to 4%/year compounded monthly, Diane is thinking of refinancing her property. How much less would Diane's monthly mortgage payment be if she refinances?

Guest Jun 20, 2014
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10+0 Answers

 #1
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Do you have an answer already and want to know how to get to it?

If so, can you post the answer in please? 

Melody  Jun 20, 2014
 #2
avatar+91510 
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I am going to answer this in a series of posts because it is long and I am going to have to do some homework along the way

FIRST

first you will need to find the initial repayments.

This is the present value of an ordinary anuity so you will use this formula

$$A=R\times \frac{1-(1+i)^{-n}}{i}$$

A=300000

i=6%/12=0.5%=0.005

n=35*12=420

$$\begin{array}{rll}
300000&=& R\times \frac{1-(1.005)^{-420}}{0.005}\\\\
300000&=& R\times 175.3802262\\\\
R&=&\$ 1710.57 \mbox{ to the nearest cent}
\end{array}$$

Okay now I have to work out how to find the amount that will still be owing after 5 years.

Melody  Jun 20, 2014
 #3
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NOW you will need to work out the present (5years down the track) value of the loan

This is the present value of an ordinary anuity so you will use this formula (same formula)

$$A=R\times \frac{1-(1+i)^{-n}}{i}$$

A=??

R=$1710.57

i=6%/12=0.5%=0.005

n=30*12=360 months

$$\begin{array}{rll}
A&=& \$ 1710.57\times \frac{1-(1.005)^{-360}}{0.005}\\\\
A&=& \$ 1710.57\times 166.79161444\\\\
A&=&\$ 285\;308.73 \mbox{ to the nearest cent}
\end{array}$$

------------------------------------------------------------

So after 5 years Dianne owes the bank $285 308.73

Melody  Jun 20, 2014
 #4
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now I need to work out what Diane's new repayments will be if her interest rate drops to 4%

30 years is remaining on the loan.

This is the present value of an ordinary anuity so you will use this formula (same formula)

$$A=R\times \frac{1-(1+i)^{-n}}{i}$$

A=285308.73

R=$??

i=4%/12=0.3repeater%=0.003333333....

n=30*12=360 months

$$\begin{array}{rll}
285\;308.73&=& R\times \frac{1-(1.00\dot3)^{-360}}{0.00\dot3}\\\\
285\;308.73&=& R\times 209.4612296\\\\
R&=&\$ 1362.11 \mbox{ to the nearest cent}
\end{array}$$

------------------------------------------------------------

So her payments dropped from $1710.57 to $1362.11

$${\mathtt{1\,710.57}}{\mathtt{\,-\,}}{\mathtt{1\,362.11}} = {\mathtt{348.46}}$$

This would save Diane $348.46

------------------------------------------------------------

Now all of this is assuming that I have made no careless mistakes.

At least the numbers pass the sensibleness test (I think)

Now, please tell me if you have the answer in the back of a book or somewhere.

Melody  Jun 20, 2014
 #5
avatar+2353 
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I thought I could let wolframalpha check it for you.

Maybe I overestimated a little;

reinout-g  Jun 20, 2014
 #6
avatar+2353 
0

I checked your answer,

seems correct to me, but I'm no genius in this field of expertise.

I also googled it to be sure and the problem has been used a million times only with different numbers.

Here's an example; https://answers.yahoo.com/question/index?qid=20110309151602AAcBisG 

 

This also agrees with your method of operation, so it should be correct 

reinout-g  Jun 20, 2014
 #7
avatar+91510 
0

Thanks reinout. Is that wolfram|Alpha output suposed to show something?

Melody  Jun 20, 2014
 #8
avatar+2353 
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Well yeah,

I found it pretty ironic to see that he interprets the problem

a 300000 35-year mortgage with 6% interest computed monthly

as the problem

300000*35

 

It's like wolframalpha is 4 years old. 

reinout-g  Jun 20, 2014
 #9
avatar+91510 
0

Yes, it was a strange interpretation - I'll give you that!

Melody  Jun 20, 2014
 #10
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Sorry I lost the answer for the other one cause I accidentally deleted the page. This is another similar question with different values!



Five years ago, Diane secured a bank loan of $330,000 to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was 30 yr, and the interest rate was 12%/year compounded monthly on the unpaid balance. Because the interest rate for a conventional 30-yr home mortgage has now dropped to 10%/year compounded monthly, Diane is thinking of refinancing her property. How much less would Diane's monthly mortgage payment be if she refinances? Please round the answer to the nearest cent.

Select the correct answer.

 $572.09  
 $567.16  
 $549.69  
 $544.24  
 $566.11

 

Guest Jun 20, 2014

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