Question

Michelle takes out a $50,000 loan for 15 years which she repays with annual payments at the end of each year. The annual interest rate is 1% for the first 2 years and then increases to 5% for the remainder of the loan. Michelle's payments increase by 7% each year. Find the loan balance immediately following the tenth payment.

Answer 1

Solution:

Lets first calculate the amount to be paid by Michelle at the end of 15 years:

Amount after 2 years

Using the TVM row on BA II plus calci

PV = 50000, n=2 , I/Y =1 , PMT =0 .FV=?

Press CPT + FV we get FV = 51005.

Amunt after 15 years beginning from the theird year

PV = 51005 , N = 13, I/Y =5 , PMT =0 , FV =?

Press CPT + FV = 96177.53

Now we know Michelles payments increase by 7% each year so

Let the payment of michelle in year 1 is x

x + x(1.07) + x (1.07)^2+ ….. . . . + x (1.07)^14 = 96177.53

x { 1+ 1.07 +1.07^2 + … . . + 1.07^14} = 96177.53 ---1)

Sum to n terms of a GP is = a (r^n-1) / r-1

a =1 , r = 1.07 , n =15

We get Sn = 1(1.07^15-1)/(1.07-1) = (2.759 -1 ) / 07 = 39.4147

Now Considering 1)

x * 39.4147 = 96177.53

x = 96177.53 / 39.4147 = 2440.141

Sum till tenth payment / Sum of 10 terms of Gp = x * (a(r^10 -1)/(1-r))

2440.141 * 1(1.07^10 -1)/.07 = 2440.141 {(1.967-1)/ .07} = 2440.141 * 13.81644 = 33714.081

Loan balance left = 96177.53 - 33714.081 = 62463.4488