Question

Fitzgerald purchased a car for $18,980. He made a down payment of $2,240. He applied for a five-year installment loan with an interest rate of 7.4%. What is the total cost of the car after five years?

Answer 1

$20,078.40

$22,318.40

$17,978.76

Answer 2

Payments made annually?

Answer 3

It won't be less than $18,980, so you can just throw that bottom one out!

Answer 4

Doesn't say it's annually or not so I'm not sure

Answer 5

I was thinking $22,318.40

Answer 6

Well, that's just silly. The problem should say what it wants.

Let's do five annual payments at the end of the year.

18980 - 2240 = 16740 -- This is the amount financed.

I = 0.074 -- Annual interest rate

v = 1/(1+i)

Pmt we don't know. We'll have to calculate it.

16740 = Pmt(v + v^2 + v^3 + v^4 + v^5)

Do you believe this?

Answer 7

What I did was 16740(1+0.074)^5 and with that I got $23,920.86 and that answer was close to $22,318.40 so I was thinking that's the answer

Answer 8

That's no good. There is only one payment. If you make no payment at all, that is the right answer. This Fitzgerald is making payments along the way.

I should have said:

v = 1/(1+i) -- Annual Discount Factor

Okay, NOW do you believe my expression?

Answer 9

What do you mean do I believe it?

Answer 10

You need to buy into this expression.

16740 = Pmt(v + v^2 + v^3 + v^4 + v^5)

Answer 11

Okay? I've never been taught that formula though so I'm pretty confused

Answer 12

oh, the loan is almost certainly monthly payments!

Answer 13

It's not a formula. It's basic principles. You may have a formula for the present value of five payments at the end of each period. I'm just building it from scratch.

Answer 14

and indeed, the answer for monthly payments is among the choices.

Answer 15

So is it $22,318.40? Like I said?

Answer 16

No, your version still had only one payment at the end of the 5 years. It should be nowhere near that.

Answer 17

Then $20,078.40?

Answer 18

Are you happy with guessing and narrowing or would you like to prove it?

Answer 19

I don't know how to prove it that's the thing and your "basic principal" theory really confuses me... So if you want to explain that....

Answer 20

Do you have a formula for the present value of some number of payments at the end of each period?

Answer 21

You either have a formula or you have to build it. Those are the only two choices.

Answer 22

B = P(1 + i)nt

Answer 23

I think

Answer 24

or M = B i(1+i)^nt / (1+i)nt - 1

Answer 25

Not enough payments. \(B = \dfrac{1 - v^{n}}{i}\), maybe?

Answer 26

Not sure. What are M and B?

Answer 27

B is 18980, M is just what the monthly payment will equal once you plug everything in.

Answer 28

May I ask, what is this "v" you keep referring to?

Answer 29

That's why I defined it. It was the ANNUAL discount factor. It's how these formulas are built. Now that we know it's monthly, thanks to @whpalmer4's observation, we'll need a new one.

I = 0.074 -- Annual interest rate
j = I/12 -- Monthly interest rate
v = 1/(1+j) -- Monthly discount factor.

It works like this.

Right now, a payment right now is worth Pmt
Right now, a payment one month from now is worth Pmt*v
Right now, a payment two months from now is worth Pmt*v^2
Right now, a payment three months from now is worth Pmt*v^3

Making any sense?

Answer 30

No

Answer 31

If we can't build the formula, you must find the right one. That's all there is to it. I cannot help you with that because I can't see what formulas you have in front of you.