Question

Chelly purchased a pool for $6,120 using a six-month deferred payment plan with an interest rate of 22.65%. She did not make any payments during the deferment period. What will Chelly’s monthly payment be if she must pay off the pool within five years after the deferment period?

Answer 1

You will need a compounding period with that. Monthly? Semi-annually?

Answer 2

monthly, I'm pretty sure

Answer 3

Okay, without any payments, finding the Future Value of the original cost, six months later, should be no problem. Let's see what you get.

Answer 4

8317.08?

Answer 5

$6,846.63 -- Not very close. How did you get that?

Answer 6

I don't even know xD i just multiplied stuff

Answer 7

Here's a plan. Never do that.

Annual Interest Rate: i = 0.2265
Monthly Interest Rate: j = You Tell Me.

Answer 8

22.65/12? = 1.8? or?

Answer 9

0.018?

Answer 10

You have the right idea, but...

Please use WAY more decimal places. AT LEAST as many as will calculate to the penny to numbers you will encounter. \(\log_{10}(6120) = 3.787\), almost 4, so use at least 6 decimal places. Please try again.

Answer 11

0.018875? So lost....

Answer 12

Lost with what? We just converted an annual interest rate to a monthly interest rate. Are you lost on that?

Answer 13

Did I do it right?

Answer 14

Perfect. Ready to move on?

Answer 15

Yes! :)

Answer 16

Check it out:

Monthly Interest Rate
j = 0.018875

6120*0.018875 = 115.515
6120*0.01888 = 115.5456
6120*0.0189 = 115.668
6120*0.019 = 116.28
6120*0.02 = 122.4

It makes quite a difference how we wound. Be sensitive to that.

Next, we have to get used to the idea of Present Value and Future Value.

Now that we have j, the monthly interest rate, let's define two more things we will need to fully understand what it going on.

r = 1+j = 1.018875

v = 1/r = 1/1.018875 = 0.981475

Let's just be okay with the arithmetic for now. Do you believe?

Answer 17

Okay:)

Answer 18

Now, what do those do?

r, I like to call an "Accumulation Factor" It's purpose is to advance a value one month in the interest world.

$1000 today, is the same as $1000*r = 1000*(1.018875) = $1018.88 in one month.

v, I like to call a "Discount Factor" It's purpose is to reverse a value one month in the interest world.

$1000 one month from now, is the same as $1000*v = $1000*(0.981475) = $981.48 today.

You must be feeling pretty good about how this works. r moves forward one month in value and v moves backwards one month in value.

Answer 19

got it!

Answer 20

You might be asking, if I use one and then the other, do I get back where I started? Except for some rounding considerations, yes. Go ahead and try it.

5000 * r * v = ??

Answer 21

Hmm...I'm not that sure....show me? :)

Answer 22

Do you have a calculator? You will need one.

5000 * r * v = 5000 * 1.018875 * 0.981475 = ??

What do you get?

Answer 23

500.000170?

Answer 24

You missed a decimal place, there. s/b 5000.00170
Notice how this is off by about 1/5 cent. This is good. This is also why we picked 6 decimal places. Look what happens when we use only five decimal places.

5000 * 1.01888 * 0.98148 = 5000.051712 -- And we're off a whole nickle. This is no coincidence. We picked six decimal places so that it would do that and the error would be small.

Answer 25

Okay:)

Answer 26

Now what?

Answer 27

Now, we need to tackle the periodic payments.

From the problem statement.
"What will Chelly’s monthly payment be if she must pay off the pool within five years after the deferment period?"

First, "within" is an utterly silly word. We have to pick a time frame. How about EXACTLY 5 years? "within" is just meaningless in this context.

Answer 28

okay

Answer 29

How many monthly payments in 5 years?

Answer 30

60?

Answer 31

Let's just use "P" to represent the level, monthly payment for 5 years. I'll write a little chart so portray exactly when these payments will be made.

P - One month from now
P - two months from now
P - three months from now
.
.
.
P - 30 months from now
P - 31 months from now
.
.
.
P - 58 months from now
P - 59 months from now
P - 60 months from now

Do you agree with this very long listing?

Answer 32

yes

Answer 33

Our challenge, for the day, is to get this series of payments to have the same Present Value as what is owed. We need to find a level monthly payment such that these 60 spaced out payments have the Present Value of $6,846.63.

Answer 34

okaaay....

Answer 35

Series of similar questions.

Using "v", that we talked about before, what is the Present Value of the first Payment?

The 1st Payment has the value "P" one month from now. What is it's value today?

Answer 36

I really don't know, I'm about to give up. v is 0.98148? whats P?

Answer 37

Don't give up. We're almost there.

We defined "P" earlier.

Let's just use "P" to represent the level, monthly payment for 5 years.

v = 0.948175 -- Not sure why it turned out rounded one decimal place shorter..

Anyway, this is of no consequence. We just have to know what the symbols represent.

The first payment has a value of P one month from now.
We need to discount this payment to today. Use 'v' to do this.

What do you get?

Answer 38

Not sure still...

Answer 39

Review.

v, I like to call a "Discount Factor" It's purpose is to reverse a value one month in the interest world.

$1000 one month from now, is the same as $1000*v = $1000*(0.981475) = $981.48 today.

Give it another go.

It's worth "P" in one month. What is it worth today?

Answer 40

Honestly, I don't know. I give up.

Answer 41

All you had to say was Pv.

Come back when you're ready.

Answer 42

Okay. Well. Keep going then. I can't give up.