Question

A family has a $91,882, 20-year mortgage at 5.4% compounded monthly. Find the monthly payment. (Do not round until the final answer. Then, round to the nearest cent.)

Answer 1

What is your plan for this?

Answer 2

what do you mean?

Answer 3

You need a plan!

i = 0.054 -- Annual interest rate
j = 0.054/12 = 0.0045 -- Monthly Interest Rate
v = 1/(1+j) = 0.99552 -- Monthly Discount Factor

Is that a good start or am I just writing stuff?

Answer 4

I have no idea what formula to use or anything

Answer 5

This is why it is important to know how to build your own!

Here is the Basic Principles version

\(91882 = Pmt\cdot(v + v^{2} + v^{3} + ... + v^{240})\)

Do you believe this?

Answer 6

okay so I would be solving for P, m would be 0.054 and t would be 1/12?

Answer 7

what would v v2 v3 etc be

Answer 8

I cannot see your formulas or your calculator. Please do not use any terms you have not defined. What I have written is what you need. I defined I, j, v, and Pmt.

Answer 9

Each of those 'v's is the present value (when multiplied by the payment amount) of the payment made on some future date.

Answer 10

I was using terms you defined, p m and t

Answer 11

:-) That is one term. "Pmt" = the level payment amount.

Answer 12

oh okay

Answer 13

what is the next step to solving the equation?

Answer 14

First, you must believe that we have the right expression.

Pmt*v is the present value, at issue, of the first payment.
Pmt*v^2 is the present value, at issue, of the second payment.
...
Pmt*v^240 is the present value, at issue, of the last payment.

Do you see that this is so?

Answer 15

yes, I can see that

Answer 16

Then we have this expression.

\(91882=Pmt⋅(v+v^{2} +v^{3}+...+v^{240} ) \)

With a little algebra, we have written an equation that perfectly describes the original loan and all 20 years of payments. Do you agree with this?

Answer 17

yes

Answer 18

Our only remaining challenge is to add up all 240 of those v's. Do you know how to do that? Hint: You should have studied Geometric Series somewhere along the line.

Answer 19

I don't know how

Answer 20

I invite you to spend some time on that subject. Find an algebra book or something. For now, I will just tell you. You tell me if it looks familiar.

\(v + v^{2} + v^{3} + ... + v^{240} = \dfrac{v - v^{241}}{1-v} = v\cdot\dfrac{1-v^{240}}{1-v} = \dfrac{1-v^{240}}{(1+j)(1-v)} = \dfrac{1 - v^{240}}{j}\)

That last form should look familiar. It's just a bunch of algebra.

Answer 21

I haven't seen that in my lesson plans. I'm going to try to go over the lesson again. Thank you for the help.

Answer 22

You should have that last form. It is the official formula for equal, periodic payments, level interest, and 20 years of monthly payments. Are you sure you haven't seen that? It is needed to solve this problem. All the algebra simply creates it!

Answer 23

I'll double check, but maybe it is