Question

Question 15 Jenny takes out a loan of $27,000 from Bendigo Bank for her small business at 11.00% p.a. compounded monthly and promises to pay it back over six years with equal monthly payments. Twenty-two months after taking out the loan (just after the twenty-second payment is made), she decides to refinance her loan at a lower rate of 8.00% p.a. compounded monthly offered by Qudos Bank for the remaining term of the loan. Assuming she can do so immediately and there are no refinancing costs or charges, what will her new monthly payments be? a. $490.87 b. $473.40 c. $636.77 d. $484.39 e. $513.92

Answer 1

Here, first we need to find the amount of payment per month.

For that we can use the Present Value of Annuity(PVA) Formula:

1+ 1/ana -1 PVA A i/a(1/a)a

Where,
PVA = Present Value of Annuity
A = Annuity
i = rate of interest
n = number of years
a = number of payments per year
na = number of payments

$27000 = A\left [ \frac{(1+0.11/12)^{6*12}-1}{0.11/12(1+0.11/12)^{6*12}} \right ]$

$27000 = A\left [ \frac{0.928983847}{0.017682352} \right ]$

$A = \frac{27000}{52.5373463}$

Therefore, Monthly Payment = $513.92

Now, we have to find out the balance of loan after 22 payments:

For that we can use the formula:

$Balance~Of~Loan = PV(1+i/a)^{na}-A\left [ \frac{(1+i/a)^{na}-1}{i/a} \right ]$
Where,
PV = Present value / original balance
A = Annuity / Payment
i = rate of interest
a = number of payments per year
n = number of years
na = total number of payments

$Balance~Of~Loan = 27000(1+0.11/12)^{22}-513.92\left [ \frac{(1+0.11/12)^{22}-1}{0.11/12} \right ]$

$Balance~Of~Loan = 33002.55-513.92\left [ \frac{(1+0.11/12)^{22}-1}{0.11/12} \right ]$

Therefore, Balance of loan after 22 payments is $20,538.58

Now we have to find the monthly payment on this amount for the rest of the period at the rate of 8% p.a interest

Total period = 72

Therefore, Number of payments left = 72 - 22 = 50

Now let us use the first formula again and find the amount of payment per month:

1+ 1/ana -1 PVA A i/a(1/a)a

$20538.58 = A\left [ \frac{(1+0.08/12)^{50}-1}{0.08/12(1+0.08/12)^{50}} \right ]$

$20538.58 = A\left [ \frac{0.394069456}{0.009293796} \right ]$

$A = \frac{20538.58}{42.401344}$

Therefore, her new monthly payments will be $484.39