Question

You have purchased a condo and are financing a mortgage of $325,000 over 20-years with monthly payments. Your mortgage rate is quoted as APR 7.25% compounded semi-annually. After 5 years you make a lump sum payment of $50,000 towards your mortgage principal and continue with your regular payments. By approximately how many years will it reduce the amount of time taken to pay off the mortgage?

A) 4 Years

B) 2 Years

C) 6 Years

D) 3 Years

E) 5 Years

Answer 1

Solution :-

Interest rate per month = ( 1 + 0.0725 / 2 )^1/6 - 1 = 0.00595 = 0.595%

Total Monthly installments = 20 * 12 = 240

Now Value of Each Monthly Installment = $325,000 / PVAF ( 0.595% , 240 )

= $325,000 / 127.5678

= $2,547.66

Now after 5 Years Installment Remaining = ( 20 - 5 ) * 12 = 180

Now Balance outstanding after 5 Years = $2,547.66 * PVAF ( 0.595% , 180 )

= $2,547.66 * 110.274

= $280,941.19

Now after 50,000 payment , Balance Outstanding = $230,941.19

Now If installment remains same , time required to pay off the loan =

$230,941.19 = $2,547.66 * PVAF ( 0.595% , n )

90.648 = [ 1 - ( 1 + 0.00595 )^-n ] / 0.00595

0.53957 = 1 - ( 1 + 0.00595 )^-n

( 1 + 0.00595 )^-n = 0.46043

1.00595ⁿ = 2.17189

Take log both sides

n log ( 1.00595 ) = log ( 2.17189 )

n = 132

132 months approx

Now time reduced = 180 - 132 = 48 months

= 4 Years

Therefore Correct answer is (a)

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