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 Calculating loan repayments

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A couple wishes to borrow $400,000 to buy a house. The bank has off ered them a 30-year loan repayable in 360 equal monthly instalments (at the end of each month) at a nominal interest rate of 5% per annum, compounded monthly.


After 100 monthly payments of the loan, interest rates rise to 6% per annum, compounded monthly. What should the remaining 260 monthy payments be?

 Aug 12, 2015

Best Answer 

 #1
avatar+118608 
+8

Mmm

A couple wishes to borrow $400,000 to buy a house. The bank has off ered them a 30-year loan repayable in 360 equal monthly instalments (at the end of each month) at a nominal interest rate of 5% per annum, compounded monthly.

First I think that you need to work out the monthly payments.

This is a present value of an ordinary annuity problem

PV=400000,       i=0.05/12=   1/240        n=360

$$\\400000=C*\frac{1-(1+1/240)^{-360}}{1/240}\\\\
C=400000\div\frac{1-(1+1/240)^{-360}}{1/240}\\\\$$

 

$${\frac{{\mathtt{400\,000}}}{\left({\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{240}}}}\right)}^{\left(-{\mathtt{360}}\right)}\right)}{\left({\frac{{\mathtt{1}}}{{\mathtt{240}}}}\right)}}\right)}} = {\mathtt{2\,147.286\: \!492\: \!048\: \!555\: \!939\: \!3}}$$

 

ok so the monthly payment is  $2147.29

 


After 100 monthly payments of the loan, interest rates rise to 6% per annum, compounded monthly. What should the remaining 260 monthy payments be?
 
Now you need to work out how much is still owing after 100 months worth of payments.
 
There will still be 260 payments to go at $2147.29 each  at 5%pa =  1/240  per month
 
To do this you need to use the present value of an ordinary annuity formula again.
 
 
$$\\PV=2147.29*\frac{1-(1+1/240)^{-260}}{1/240}\\\\$$
 
 
$${\frac{{\mathtt{2\,147.29}}{\mathtt{\,\times\,}}\left({\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{240}}}}\right)}^{-{\mathtt{260}}}\right)}{\left({\frac{{\mathtt{1}}}{{\mathtt{240}}}}\right)}} = {\mathtt{340\,528.552\: \!212\: \!129\: \!782\: \!205\: \!6}}$$
 
 Ok so after 100 months you owe   $340528.55
 
 
Now you need to use the present value formula again only this time
the interest rate is 6%pa = 0.005 per month
 
 
$$\\340528.55=C*\frac{1-(1.005)^{-260}}{0.005}\\\\
C= 340528.55\div \frac{1-(1.005)^{-260}}{0.005}\\\\$$
 
 
 
$${\frac{{\mathtt{340\,528.55}}}{\left({\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{1.005}}\right)}^{-{\mathtt{260}}}\right)}{{\mathtt{0.005}}}}\right)}} = {\mathtt{2\,343.352\: \!972\: \!831\: \!645\: \!026\: \!7}}$$
 
 
 So the remaining payments will be  $2343.35 per month 
 
 Aug 12, 2015

3+0 Answers

 #1
avatar+118608 
+8
Best Answer

Mmm

A couple wishes to borrow $400,000 to buy a house. The bank has off ered them a 30-year loan repayable in 360 equal monthly instalments (at the end of each month) at a nominal interest rate of 5% per annum, compounded monthly.

First I think that you need to work out the monthly payments.

This is a present value of an ordinary annuity problem

PV=400000,       i=0.05/12=   1/240        n=360

$$\\400000=C*\frac{1-(1+1/240)^{-360}}{1/240}\\\\
C=400000\div\frac{1-(1+1/240)^{-360}}{1/240}\\\\$$

 

$${\frac{{\mathtt{400\,000}}}{\left({\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{240}}}}\right)}^{\left(-{\mathtt{360}}\right)}\right)}{\left({\frac{{\mathtt{1}}}{{\mathtt{240}}}}\right)}}\right)}} = {\mathtt{2\,147.286\: \!492\: \!048\: \!555\: \!939\: \!3}}$$

 

ok so the monthly payment is  $2147.29

 


After 100 monthly payments of the loan, interest rates rise to 6% per annum, compounded monthly. What should the remaining 260 monthy payments be?
 
Now you need to work out how much is still owing after 100 months worth of payments.
 
There will still be 260 payments to go at $2147.29 each  at 5%pa =  1/240  per month
 
To do this you need to use the present value of an ordinary annuity formula again.
 
 
$$\\PV=2147.29*\frac{1-(1+1/240)^{-260}}{1/240}\\\\$$
 
 
$${\frac{{\mathtt{2\,147.29}}{\mathtt{\,\times\,}}\left({\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{240}}}}\right)}^{-{\mathtt{260}}}\right)}{\left({\frac{{\mathtt{1}}}{{\mathtt{240}}}}\right)}} = {\mathtt{340\,528.552\: \!212\: \!129\: \!782\: \!205\: \!6}}$$
 
 Ok so after 100 months you owe   $340528.55
 
 
Now you need to use the present value formula again only this time
the interest rate is 6%pa = 0.005 per month
 
 
$$\\340528.55=C*\frac{1-(1.005)^{-260}}{0.005}\\\\
C= 340528.55\div \frac{1-(1.005)^{-260}}{0.005}\\\\$$
 
 
 
$${\frac{{\mathtt{340\,528.55}}}{\left({\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{1.005}}\right)}^{-{\mathtt{260}}}\right)}{{\mathtt{0.005}}}}\right)}} = {\mathtt{2\,343.352\: \!972\: \!831\: \!645\: \!026\: \!7}}$$
 
 
 So the remaining payments will be  $2343.35 per month 
 
Melody Aug 12, 2015
 #2
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+5

Brilliant answer Melody, the working matches up perfectly with my notes and I can follow it clearly, thanks alot =)

 Aug 12, 2015
 #3
avatar+118608 
0

You are very welcome.

Thanks for the feedback :))

 Aug 12, 2015

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