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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!

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    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. – D Stanley Jul 18 '17 at 17:27
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    "I am not getting any of these answers when completing the questions." What answer are you getting, and how are you getting there? – Grade 'Eh' Bacon Jul 18 '17 at 17:40
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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.

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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

Applying your figures

$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

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

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