Question

Suppose that after making 50 payments,the interest rate changes to J2=9% p.a.
a)convert the interest rate J2=9% to J12 equivalent
b)Assuming that the family seeks to accept the change in interest rates,what would be their new payment based on the new interest rate?
c)Assuming that the family seeks to continue their initial monthly payment calculated in part 1,how many full payments would be required to pay off the loan and what would be the final concluding smaller payment one period later

Answer 1

A) $(1+0.09/2)^2 =(1+i/12)^{12}$

i=0.0884=8.84%

The interest rate J12 which equivalent J2 equals 8.84%

B) $0=((S50*q_{51}-P)*q_{52}-P)....*q_{n}-P$ ) 

$0=S50*q^{n-50}-P*q^{n-51}-....P*q-P$

$0=S50*q^{n-50}-P*(q^{n-51}+q^{n-52}*...q+1)$

$S50*q^{n-50}=P*\frac{1-q^{n-50}}{1-q}$

$S50=P*\frac{1-q^{n-50}}{1-q}/q^{n-50}$

$S50=P∗ \frac{1−q^n}{1−q} /q^n$

$q_n​-\ certain\ year\ rate$

S50-sum of loan after 50 payments

P-payment

q=1.0075 (multiplier = 1+i; i=9%/12)

n- number of payments that are not made

$P'=S50*\frac{J2*(1+J2)^{n/6}}{(1+J2)^{n/6}-1}$

P'-value of new payments

C) n-number full payments

$n=12*n'$

n'-loan repayment period in years

$0=((S_{n-2}*q_{n-1}-P)*q_{n}-P$

$S_{n-2}-sum\ for\ two\ payment\ period\ to\ end$

P-payment

q=1.0075 (multiplier = 1+i; i=9%/12)

$0=S_{n-2}*q^{2n}-P*q-P$

$\frac{P*q+P}{q^{2n}}=S_{n-2}$

$PL=S_{n-2}*q^2$

PL-last payment (the final concluding smaller payment one period later)

$PL=S_{n-2}*1.015$

How many full payments would be required to pay off the loan and what would be the final concluding smaller payment one period later, depend of loan sum and payment. In question does not give this information. substitute the value in the formulas that I wrote in part c) and you will find the answer.