Five years ago Diane secured a bank loan of 310000 to help f

Five years ago, Diane secured a bank loan of $310,000 to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was 30 yr, and the interest rate was 10%/year compounded monthly on the unpaid balance. Because the interest rate for a conventional 30-yr home mortgage has now dropped to 7%/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 principal?


(c) If Diane decides to refinance her property by securing a 30-yr home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of 7%/year compounded monthly, what will be her monthly mortgage payment?


(d) How much less would Diane\'s monthly mortgage payment be if she refinances?

Solution

We will use the following formulae for our computations. Here L is the amount of mortgage loan, n is the number of months, c is the monthly rate of interest, P is the fixed monthly payment and B is the balance left after p months.

P = L[c(1 + c)ⁿ]/[(1 + c) ⁿ - 1]

B = L[(1 + c)ⁿ - (1 + c)^p ]/[(1 + c)ⁿ - 1]

Here L = 310000, c = 10/1200 = 1/120 0.00833333, n = 30* 12 = 360 and p = 5*12 = 60.

(a) P = 310000[ 1/120( 1 + 1/120)³⁶⁰ ] / [( 1 + 1/120)³⁶⁰ – 1 ] = 310000 [ 1/120(121/120)³⁶⁰ ] / [(121/120)³⁶⁰ – 1] = (310000/120)(19.83739935)/ (18.83739935) = (2583.3333333)(1.053085884) = $ 2720.47 ( on rounding off to the nearest cent)

(b) B = 310000 [ ( 1 + 1/120)³⁶⁰ – ( 1 + 1/120)⁶⁰ ] / [( 1 + 1/120)³⁶⁰ – 1 ] = 310000[ 19.83739935 – 1.645308934 ) / (18.83739935) = 310000( 18.19209042) / (18.83739935) = 299380. 38 = 299380 ( on rounding off to the nearest Dollar)

(c) Now, L = 299380, n = 30*12 = 360 and c = 7/1200. Then,

P = 299380[7/1200(1 + 7/1200)³⁶⁰ ] / [ (1 + 7/1200)³⁶⁰ – 1] =( 299380 * 7/ 1200) [ 8.11650/ 7.11650] = (1746.38)( 1.1405185) = $ 1991.78 on rounding off to the nearest cent )

(d) Thus Diane\'s monthly mortgage payment , if she refinances, would be lower by $ 2720.47 - $ 1991.78 = $ 728.69