Question

Comparing a 15 year and 30 year mortgage. Suppose you decide to purchase a house and determine you need $250000 loan. You research loan rates and see that you can get either a 15 year loan for 2.75% or a 30 year loan for 4.5%. Determine

1. The monthly payment on each loan

2. The total amount of interest paid on each loan if you pay off the loan as scheduled.

3. Compare the costs of the two loans.

4. How does the term of each loan change if you pay an extra $100 per month?

Answer 1

Solution :-

Loan Amount = $250,000

(A)

Term = 15 Years

Total Monthly payment = 15 * 12 = 180

Monthly Rate = 2.75% / 12 = 0.229%

Installment ( Monthly Payment ) = $250,000 / PVAF ( 0.229% , 180 )

= $250,000 / 147.357

= $1,696.56

In Case term = 30 Years

Total Monthly payment = 30 * 12 = 360

Monthly Rate = 4.5% / 12 = 0.375%

Installment ( Monthly Payment ) = $250,000 / PVAF ( 0.229% , 180 )

= $250,000 / 197.36

= $1,266.72

(B)

Total Interest in 15 Year Loan = ( $1,696.56 * 180 ) - $250,000 = $55,380.81

Total Interest in 30 Year Loan = ( $1,266.72 * 360 ) - $250,000 = $206,019.46

(C)

The Cost of 30 Year Loan in higher than the 15 Year Loan

(D) In Case installment of 15 Year loan = $1,696.56 + $100 = $1,796.56

$250,000 = $1,796.56 * PVAF( 0.229% , n )

Where n we need to find

$250,000 = $1,796.56 * [ 1 - ( 1 + 0.00229 )^-n ] / 0.00229

[ 1 - ( 1 + 0.00229 )^-n ] = $250,000 / $783,593.48

( 1.00229 )^-n = 0.6811

( 1.00229 )ⁿ = 1.46821

Take log Both Sides

n Log ( 1 .00229 ) = Log ( 1.46821 )

n = 0.16678687 / 0.000994134

n = 167.77

= 167.77 months

= 13.98 Years ( Approx )

Term of Loan Change = 15 - 13.98 = 1.02 Year = 12 Months Approx

In Case installment of 30 Year loan = $1,266.72 + $100 = $1,366.72

$250,000 = $1,366.72 * PVAF( 0.375% , n )

Where n we need to find

$250,000 = $1,366.72 * [ 1 - ( 1 + 0.00375 )^-n ] / 0.00375

[ 1 - ( 1 + 0.00375 )^-n ] = $250,000 / $364,458.86

( 1.00375 )^-n = 0.31405

( 1.00375 )ⁿ = 3.1842

Take log Both Sides

n Log ( 1 .00375 ) = Log ( 3.1842 )

n = 0.502999 / 0.0016255

n = 309.432

= 309.432 months

= 25.79 Years ( Approx )

Change in term of Loan = 30 - 25.79 = 4.214 Years ( 50.57 Months )

If there is any doubt please ask in comments