Five years ago, Diane secured a bank loan of $370,000 to help finance the purcha
ID: 2799829 • Letter: F
Question
Five years ago, Diane secured a bank loan of $370,000 to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was 30 years, and the interest rate was 10% per year compounded monthly on the unpaid balance. Because the interest rate for a conventional 30-year home mortgage has now dropped to 7% per year compounded monthly, Diane is thinking of refinancing her property. (Round your answers to the nearest cent.) (a) What is Diane's current monthly mortgage payment? (b) What is Diane's current outstanding balance? (c) If Diane decides to refinance her property by securing a 30-year home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of 7% per year compounded monthly, what will be her monthly mortgage payment? Use the rounded outstanding balance. (d) How much less would Diane's monthly mortgage payment be if she refinances? Use the rounded values from parts (a)Explanation / Answer
a) The monthly mortgage payments can be computed by the dividing the mortgage amount with the present value interest factor annuity (PVIFA) at the given rate -
PMT = Mortgage Amount / PVIFA(Rate, Period)
10% is annual rate. Since it is compounded monthly, we need monthly rate - 10% / 12 = 0.83333333333%.
No. of periods = 30 years x 12 months per year = 360
PMT = $370000 / (PVIFA 0.83333333333%, 360) = $370000 / 113.95081998 = $3247.01480923 or $3247.01
b) Again, we use the above formula and this time use the monthly payments computed above. The only difference being the PVIFA which will be computed for 25 x 12 = 300 periods this time (as 5 years have passed).
PMT = Loan Balance / PVIFA (0.83333333333%, 300)
Or, $3247.01 = Loan Balance / 110.04723006
Or, Loan Balance = $3247.01 x 110.04723006 = $357324.456477 or $357324.46 (The answer would be more accurate if you use $3247.01480923)
c) Now, we use the annual rate of 7% or monthly rate of 7%/ 12 = 0.58333333333%. Same formula, same period, above loan balance and the new rate -
PMT = Loan Balance / PVIFA (0.58333333333%, 300)
or, PMT = $357324.46 / 141.48690344 = $2525.49494908 or $2525.50 (Again, more accuracy if you no rounded off figures)
d) Savings in monthly payment = $3247.01 - $2525.50 = $721.51
NOTE : Present value factor (PVF) is computed as 1 / (1 + rate)n where, rate in our case is 0.00833333333 in the first case and 0.00583333333 in the second case, n being the year for which it is calculated. Like for year 1, it would 1 / (1.00833333333)1 = 0.99173553719, for year 4 it would be 1 / (1.00833333333)4 = 0.9673497036
To compute Present value Interest factor annuity (PVIFA), you need to compute PVF for 360 or 300 periods as the case maybe and add them all. Fairly easy but a bit lengthly if you do it on a calculator.
In case you have a PVFA table available with you, you can use that as well.
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