Question

Aya and Harumi would like to buy a house and their dream house costs $500,000. They have $50,000 saved up for a down payment but would still need to take out a mortgage loan for the remaining $450,000 and they’re not sure whether they could afford the monthly loan payments. The bank has offered them an interest rate of 4.25%, compounded monthly. How much would they have to be able to afford to pay each month in order to pay off their mortgage in 25 years? What if Aya and Harumi could only afford a monthly payment of $2,000? What would be the maximum mortgage amount they could afford to borrow from the bank, if all the other conditions were the same? What is the total amount that would be paid to the lender over 25 years?

Answer 1

Solution :-

Cost of House = $500,000

Down Payment = $50,000

Amount Mortgaged = $500,000 - $50,000 = $450,000

Interest Rate Per Month ( r ) = 4.25 % / 12 = 0.35417%

Now If the Mortgage is paid in 25 Years  

Total Monthly Installments Paid ( n ) = 25 * 12 = 300 Payments

Now The amount they need to pay each Month = Amount Mortgage / PVAF ( r , n )

= $450,000 / PVAF ( 0.35417% , 300 )

= $450,000 / 184.5912

= $2,437.82

(b) Now if they paid monthly $2,000 , time to pay loan =

$450,000 = $2,000 * PVAF ( n , 300 )

$450,000 = $2,000 * [ 1 - ( 1 + 0.0035417 )^-n ] / 0.0035417

= 0.796875 = [ 1 - ( 1 + 0.0035417 )^-n ]

( 1 + 0.0035417 )^-n = 0.203125

( 1.0035417 )ⁿ = 4.9231

Take Log Both Sides

n log ( 1.0035417 ) = Log ( 4.9231 )

n = 450.85 = 451 Months ( Approx )

= 37 Years 6 Months 25 Days ( Approx )

(C)

If Monthly Payment = $2,000

Rate of Interest Per month = 0.35417%

Number of Months of Payments = 300

Now Amount Mortgaged = $2000 * PVAF ( 0.35417% , 300 )

= $2,000 * 184.5912

= $369,182.39

The maximum mortgage amount they could afford to borrow from the bank = $369,182.39

(D) The total amount that would be paid to the lender over 25 years = $2000 * 300 = $600,000

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