Question

A young couple buying their first home borrow $50,000 for 30 years at 7.5%, compounded monthly, and make payments of $349.61. After 5 years, they are able to make a one-time payment of $2000 along with their 60th payment.

(a) Find the unpaid balance immediately after they pay the extra $2000 and their 60th payment. (Round your answer to the nearest cent.)
$

(b) How many regular payments of $349.61 will amortize the unpaid balance from part (a)? (Round your answer to the nearest whole number.)
payments

(c) How much will the couple save over the life of the loan by paying the extra $2000? (Use your answer from part (b). Round your answer to the nearest cent.)
$

Answer 1

A young couple buying their first home borrow $50,000 for 30 years at 7.5%, compounded monthly, and make payments of $349.61. After 5 years, they are able to make a one-time payment of $2000 along with their 60th payment.

We can use PV of an Annuity formula to calculate the monthly payment of loan

PV = PMT * [1-(1+i) ^-n)]/i

Where PV = $50,000

PMT = Monthly payment =?

n = N = number of payments = 30 years *12 months = 360 month

i = I/Y = interest rate per year = 7.5%, therefore monthly interest rate is 7.5%/12 = 0.625% per month

Therefore,

$50,000 = PMT* [1- (1+0.00625)^-360]/0.00625

= $349.61

Monthly payment is $349.61

(a) Find the unpaid balance immediately after they pay the extra $2000 and their 60th payment.

The balance on this loan be at the end of the 60th payment

PV = PMT * [1-(1+i) ^-n)]/i

Where,

The balance on this loan at the end of 60th payment, Present value (PV) =?

PMT = Monthly payment = $349.61

n = N = number of remaining payments = 360 -60 = 300 payments

i = I/Y = interest rate per year = 7.5%, therefore monthly interest rate is 7.5%/12 = 0.625% per month

Therefore,

PV = $349.61* [1- (1+0.00625)^-300]/0.00625

= $47,308.72

The unpaid balance immediately after they pay the extra $2000

= $47,308.72 - $2,000 = $45,308.72

(b) How many regular payments of $349.61 will amortize the unpaid balance from part (a)?

Now calculate time period in following manner

PV = PMT * [1-(1+i) ^-n)]/i

Where PV = $45,308.72

PMT = Monthly payment =$349.61

n = N = number of payments =?

i = I/Y = interest rate per year = 7.5%, therefore monthly interest rate is 7.5%/12 = 0.625% per month

Therefore,

$45,308.72 = $349.61* [1- (1+0.00625)^-n]/0.00625

Or n = 266.54 months or 267 months

(c) How much will the couple save over the life of the loan by paying the extra $2000?

If not paid the extra $2000; the remaining loan payments was for 300 months

After paying extra $2000; the remaining loan payments was for 266.54 months

Therefore, savings over the life of the loan by paying the extra $2000 = 300 * $349.61 - 266.54 * $349.61

= $104,882.18 - $93,184.57

= $11,697.60

After reducing $2,000 extra from this amount net savings over the life of the loan = $11,697.60 - $2,000

=$9,697.60