Question

You would like to have $260,000 in 18 years by making regular deposits at the end of each month in an annuity that pays an annual interest rate of 4.5% compounded monthly. How much of the $260,000 comes from interest? In your calculations, do not round until the final answer. Then, round the monthly payment to the nearest dollar.

Answer 1

Do you have an accumulation formula for payments at the END of the period, or shall we build it?

Answer 2

Build it please

Answer 3

Okay, bt it will lake a little patience and you will need to answer questions along the way.

Define:

P = the Monthly Payment
i = 0.045 the annual interest rate to be compounded montly.
j = i/12 = 0.00375 the monthly interest rate.
r = 1+j = 1.00375 the monthly accumulation factor.

Let's stop there and see if you are seeing where all that came from.

Answer 4

ok I am pretty certain how you get that information. I have a couple of formulas but I am not sure of which to use. with one i got 188,288 for the part that is interest but i do not think that is correct

Answer 5

Well, let's keep building and see which one we're supposed to use.

Question: What is the value at the end of the VERY LAST payment made? Just the last payment. It is made on the day that we need $260,000. We're going to pay "P", because we defined that. How much interest has this LAST payment earned and what is it worth?

Answer 6

I am not quite sure of that

Answer 7

It's the very last payment.
It's paid in the very last day.
It has earned NO interest.
When it is time to have $260,000, the value of the last peyment is "P".

Make sense?

Answer 8

332?

Answer 9

Where did that come from? If you don't know which formula to use, why are you using formulas just to guess? Shall we find the right formula and be SURE?!

Answer 10

I used the one the instructor posted but I am happy to learn the correct way

Answer 11

she did not

Answer 12

That's the problem with formulas. You need to know when they are applicable and when they are not. Did the instructor write the formula for accumulation with payments at the END of each compounding period? You say she did not. Then it's just not clear, is it? Shall we keep building?

Answer 13

yes please

Answer 14

We have the result that the very last payment of P has the value P, right?
What is the value of the second to last payment - the one paid just one month earlier?

Don't worry. Just a couple more questions and we'll be there.

Answer 15

I dont mean to be difficult, but I really do not know

Answer 16

It is the accumulated value of P, with interest credited for one month. Right?

That makes it P*r or P*(1+j) or P*(1 + i/12). This needs to make sense before we can move on.

Answer 17

Perhaps I will do some more research as I am very confused. My class is online so I don't have any help other than videos. Ty very much for your help and patience :)

Answer 18

We are SO CLOSE.

Answer 19

I do not understand how to get P

Answer 20

We're building the machinery to do that. All we need to know right now is that there IS a value. Since we don't know it, right now, we're just calling it "P".

Pay "P" every month
Value of Last Payment: P*r^0 = P -- Paid on the date that we care
Value of 2nd to Last Payment: P*r^1 -- One month's interest
Value of 3rd to Last Payment: P*r^2 -- Two month's interest
Value of 4nd to Last Payment: P*r^3 -- Three month's interest

We can worry about P later. Right now, we need the right formula to help us find it.

Answer 21

That would be great. The formula are what's confusing

Answer 22

Okay, now stare at that list I just wrote.

0 month's interest uses r^0
1 month's interest uses r^1
2 month's interest uses r^2
3 month's interest uses r^3

Pretty obviuos patern, isn't it?

Answer 23

Yep

Answer 24

Perfect. Now we're going to have to take a giant leap. We have 18 years of monthly payments. 18*12 = 216. If we write the value of ALL the payments (each and every one!), they will look like this:

P
Pr
Pr^2
Pr^3
Pr^4
.
.
.
Pr^214
Pr^215

Did I get them all?

Answer 25

That's a lot but I'm following

Answer 26

Excellent. The remaining task is to add them up. Have you EVER added up a list like this? You may have seen it like this: "What is the fractional form of 0.813813813813...".

It's also important to notice that I started with "0" and ended with "215", so I did get all 216 payments.

Answer 27

I have never done that

Answer 28

Well, rather than fry your brain, today, we can save that. Perhaps the formula that it would produce will be sufficient for now:

\(P + Pr + Pr^{2} + ... + Pr^{215} = P\cdot\left(1 + r + r^{2} + ... + r^{215}\right)\)

It's just the Distributive Property.

Answer 29

So you have to add each one?

Answer 30

No, no... please don't do that. This is the magic part. I can show you, but to save you from an early demise, I'll just tell you.

\(1 + r + r^{2} + ... + r^{215} = \dfrac{1 - r^{216}}{1-r}\)

You may have something floating around in your head concerning "Finite Geometric Series". That's where this comes from.

Answer 31

So you use the rate from the problem in that formula?

Answer 32

We MADE IT!!!! The original problem statement can no be restated:

\(260000 = P\cdot\dfrac{1-r^{216}}{1-r}\)

And we can solve for P! You thought we never would, right?

\(P = 260000\cdot\dfrac{1-r}{1-r^{216}}\)

Anyone remember what 'r' is?

Answer 33

Rate=.045

Answer 34

Remember those defintions a LONG time ago?

i = 0.045 annual interest rate to be compounded monthly
j = i/12 = 0.00375 monthly interest rate
r = 1+j = 1.00375 monthly accumulation factor

There it is. r = 1.00375

Can you finish?

Answer 35

I must be doing the math wrong or entering it in the calculator incorrectly

Answer 36

Just a piece at a time...

\(r = 1.00375\)

\(1-r = 1 - 1.00375 = -0.00375\)

\(r^{216} = 2.244505\)

\(1-r^{216} = 1 - 2.244505 = -1.244505\)

\dfrac{-0.00375}{-1.1244505} = 0.00301324

\(260000*0.00301324 = 783.45\)

Answer 37

I was doing it all at once in the calculator. Makes much more sense that way. Much easier than the formula the teacher posted. I will now try a few more problems with this information. Thank you so much for not giving up in me. I really appreciate it

Answer 38

Hopefully, it is the SAME formula as the tacher posted. Make sure, when you see a formula to be SURE you know.

1) Is it for Present Value or for Accumulation?
2) Is it for Payments at the Beginning of the Period or at the End of the period?