Question

When Gustavo and Serrana bought their home, they had a 5.1% loan with monthly payments of $870.60 for 30 years. After making 78 monthly payments, they plan to refinance for an amount that includes an additional $35,000 to remodel their kitchen. They can refinance at 4.5% compounded monthly for 25 years with refinancing costs of $625 included with the amount refinanced.

A) Find the amount refinanced. (Round your answer to the nearest cent.)

(b) Find their new monthly payment. (Round your answer to the nearest cent.)

(c) How long will it take to pay off this new loan if they pay $1200 each month? (Round your answer up to the next whole number.)
payments

Answer 1

The formula used to calculate the fixed monthly payment (P) required to fully amortize a loan of L dollars over a term of n months at a monthly interest rate of r is P = L [r(1 +r)ⁿ]/[(1 + r)ⁿ - 1].

Here, P =$, r=5.1/1200=17/4000, and n=30*12=360.Therefore, 870.60 = L* (17/ 4000)*[(1+17/4000)³⁶⁰] / [(1+17/4000)³⁶⁰-1] = L*(17/4000)*4.603228626/3.603228626 so that L = 870.60* (4000/17)* 3.603228626 /4.603228626 = $ 160346 (On rounding off to the nearest dollar).

The formula used to calculate the remaining loan balance (B) of a fixed payment loan of $ L, after p months is B = L [(1+ r) ⁿ - (1+ r) ^p]/ [(1 + r) ⁿ - 1]. Here, p= 78 so that B= 160346[(1+17/4000)³⁶⁰-(1+17/4000)⁷⁸] / (1+17/4000)³⁶⁰-1] = 160346*[ 4.603228626 -1.39207794]/ (3.603228626) = $ 142898.28 (On rounding off to the nearest cent). Thus, the amount refinanced is $ 142898.28 +$ 625 = $ 143523.28.

(b). The new rate of interest is 4.5/1200 = 3/800 so that the new monthly payment is 143523.28*(3/800)[(1+(3/800)^25*12]/ [(1+(3/800)^25*12 -1] =538.2123*3.073742528/2.073742528 = $ 797.75(On rounding off to the nearest cent).

(c). If instead of $ 797.75, a sum of $ 1200 is paid each month, let the loan of $ 143523.28 get repaid in n months. Then 1200=143523.28*(3/800)[(1+3/800)ⁿ]/ [(1+3/800)ⁿ -1] or,[(1+3/800)ⁿ]/ [(1+3/800)ⁿ -1] = 1200*800/3*143523.28 = 2.229603448. Now, let (1+3/800)ⁿ = x. Then we have x/(x-1) = 2.229603448 or, 2.229603448x -x =2.229603448 or, 1.229603448 x = 2.229603448 so that x = 2.229603448/1.229603448 or, x = 1.813270329. Thus, (1+3/800)ⁿ = 1.813270329 or, (803/800)ⁿ=1.813270329. Now, on taking log of both the sides, we get n(log 803-log 800)=log 1.813270329 so that n=log 1.813270329/(log 803-log 800) = 0.258462555/(2.904715545-2.903089987) =0.258462555/0.00162555801 = 158.9992811 =159 ( on rounding off to the nearest whole no.). Thus, the new loan would get repaid in 159 months if $ 1200 is repaid each month.