# Refinance Car Loan

Refinance Question Car Loan:

Two years ago, you purchased a \$20,000 car, putting \$4,000 down and borrowing the rest. Your loan was a 36-month fixed rate loan at a stated rate of 6.0% per year. You paid a non-refundable application fee of \$100 at that time in cash. Interest rates have fallen during the last two years and a new bank now offers to refinance your car by lending you the balance due at a stated rate of 5.0% per year. You will use the proceeds of this loan to pay off the old loan. Suppose the new loan over the residual loan life requires a \$200 non-refundable application fee. Given all this information, should you refinance? How much do you gain/lose if you do?

1. Yes, gain \$30.32

2. No, lose \$30.32

3. No, lose \$169.68

4. Yes, gain \$169.68

Can anyone explain how to do this question? I am not getting any of these answers when completing the questions.

Thanks!

• For this and your other question, you could work out the full amortization schedule to see the difference in the two scenarios. There's not a formula that you can apply easily to get the right answer. Jul 18, 2017 at 17:27
• "I am not getting any of these answers when completing the questions." What answer are you getting, and how are you getting there? Jul 18, 2017 at 17:40

Step 1: Figure out where you are now. You are 2 years into a three year loan, what is the balance?

Step 2: Figure out how much interest you will pay if you stick with the existing loan.

Step 3: Figure out how much interest you will pay if you refi. A one year loan at 5% for the balance in step 1. The add \$200.

Step 4: Compare.

The most difficult part is step 1.

Given

``````PV = principal
i  = periodic rate
m  = number of periods
d  = periodic payment

d  = PV i (1 + 1/((1 + i)^m - 1))           Formula 1
``````

the balance `b` remaining in month `x` is

``````b  = (d + (1 + i)^x (i PV - d))/i           Formula 2
``````

\$4000 is paid down, plus \$100 in cash, requiring a loan of \$16000

``````  PV =  16000
i  =  0.06/12
m  =  36

∴ d  =  PV i (1 + 1/((1 + i)^m - 1))  =  486.75
``````

Balance after two years

``````  x  =  24
∴ b  =  (d + (1 + i)^x (i PV - d))/i  =  5655.53
``````

Refinancing the balance

``````  PV =  b
i  =  0.05/12
m  =  12

∴ d  =  PV i (1 + 1/((1 + i)^m - 1))  =  484.16
``````

Original remaining payments less new ones

``````(486.75 * 12) - (484.16 * 12 + 200)  =  -168.92
``````

You would lose \$168.92 by refinancing. Close enough to option 3.

Formulae

1. Formula for periodic payment - loan payment formula 1. Formula for loan balance - inhomogeneous difference equation (Arne Jensen, Aalborg Uni.) 