Compound interest

Vv20

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Hello. I have this below question, lecturer says "incorrect' as i used annuity formula, not the compound interest formula (even though the loan for the annuity is compounded, dont see why i cant use what i have?)
Have a test in two days, so need to get this paper worked out a.s.a.p, so i can study.
This is my workings.

A sum of $60,000 is being borrowed at an interest rate of 6% per annum, compounded monthly, with payments made monthly.

How much will need to be repaid each month if the life of the loan is:

5 years

N = 12 x 5 = 60
I = 0.06 / 12 = 0.005
Pay = PV / ((1+i)n-1 / i
60 000 / ((1+0.005)60 -1 / 0.005)
60 000 /( 0.3488501 / 0.005)
60 000 / 69.77003
859.96809
Pay = $859.97 needs to be deposited each month




ii. 8 years

I = 0.06 / 12 = 0.005
N = 12 x 8 = 96
Pay = PV / ((1+i)n – 1) / i
60 000 / ((1+0.005)96- 1)/ 0.005
60 000 / (0.6141427 / 0.005)
60 000 / 122.82854
Pay = $122.83

I know the compound interest formula is
A = P(1+r/n)^nt

And i did that sum with my workings. But the answer i get is the total amount earned (as can be seen below). I need to find my monthly repayments

A= P(1+r/n)^nt
= 60,000 (1+(0,06/12)^60
= 60,000 (1.348850)
= $80,931.00915
So the $80,931.00915, should i divide that by 60? (total nt) = $1,348.85

I do not know. How do i rearrange the compound interest formula to get the answer im needing?

Thank you for your help.
 
Well this is my reasoning:

A sum of $60,000 is being borrowed at an interest rate of 6% per annum, compounded monthly, with payments made monthly.

How much will need to be repaid each month if the life of the loan is:

5 years

N = 12 x 5 = 60
I = 0.06 / 12 = 0.005
A = P(1+r/n)nt/nt
60,000 (1+ o.06/12)60 /60)
(60,000 (1.348850))/60
A = 80,931/60
= $1,348.85 monthly payments

ii. 8 years

I = 0.06 / 12 = 0.005
N = 12 x 8 = 96
A = P (1+r/n)nt/nt
((60,000 (1+0.06/12)96)/nt
((60,000 (1.61414))/92
A =96,848.56/92
= $1,052.70 monthly repayments
1,052.7x92 = 9,6848.56

What is the difference in total repayments between 5 and 8 year loans? Outline the reason for this difference
36,848.56 – 20,931.00
= $15,917.56 Difference between 5 year & 8year loan

Reasoning:
The 5year loan has large payments that are compounded monthly. The higher the payment, the more interest.
The 8year loan, has small payments that are compounded monthly. The smaller the payment, the less interest, but the loan goes on for longer.
 
I just glanced at this. Do not have time right now to go through your reasoning to figure out what is going on. Will answer in about 90 minutes unless someone else has done so by then. Help is on its way.
 
First off, everyone can make mistakes. I do all the time. So learn to check your work. In your first post, you said the monthly payment will be 122.83. Over 8 years that is 8*12 = 96 payments. And 96*122.83 = 11,791.86. Now that is clearly wrong. You borrowed 60,000.

So I am going through your work line by line. Go to bed. There will be a post when you wake up, but it must be very late now for you. Get some sleep. But answer me a question first. Do you know how to use excel? It will be easier to explain things if you can see why the formulas work.
 
Let's start by getting the formulas straight. I am going to assume that the formulas work with decimals rather than percents.

[MATH]p = \text {PRESENT value of a number of equal periodic payments;}\\ f = \text {FUTURE value of a stream of equal periodic payments;}\\ a = \text {AMOUNT of payment at end of each period;}\\ i = \text {interest rate PER PERIOD; and}\\ n = \text {number of payments.} [/MATH]Notice that I have changed the names of the variables a bit. I am going to work consistently with periods. It makes the formulas a little less complex, but it means that you have to convert from percents to decimals and from years to periods before you use the formulas. Make sense? Notice as well that I have tried to make the variable names appeal to memory. So n is the number of periods; no need to multiply number of periods per year times the number of years because you do that before you ever worry about formulas. And p is PRESENT VALUE, f is FUTURE VALUE, etc. I am trying to make the burden on your memory light.

I am not going to do the math to DERIVE the formulas with rigor. But I am going to sort of explain them. To take an exam, you need to memorize them and, most importantly, remember what they mean. But it is easier to remember things if they make some sense. So memorize just two formulas and write them down on your work papers along with what each means at the start of the exam. You can then use a bit of simple algebra and arithmetic to solve 99% of the problems in basic financial mathematics.

[MATH]f = a * \dfrac{(1 + i)^n - 1}{i}.[/MATH]
Does that weird looking formula make sense? Let's see

What will be the future value of a single payment of 10,000 received in one month when I can get 0.005 in interest per month.

Obviously 10,000 because I won't have anything to invest until next month. Does the formula work?

[MATH]10,000 * \dfrac{(1 + 0.005)^1 - 1}{0.005} = 10,000 * \dfrac{1.005)^1 - 1}{0.005} =\\ 10,000 * \dfrac{1.005 - 1}{0.005} = 10,000 * \dfrac{0.005}{0.005} = 10000. \ \checkmark[/MATH]However it is dangerous to check things with 1 or zero. Let's try 2 such payments. So I am going to receive two payments of 10,000 (hey I won the lottery). So I can invest the first for a month and get 50 extra. So at the end of the second month, I will have 10000 + 50 + 10000 = 20050. Let's see how well the formula works in this case.

[MATH]10,000 * \dfrac{(1 + 0.005)^2 - 1}{0.005} = 10,000 * \dfrac{(1.005)^2 - 1}{0.005} = 10,000 \dfrac{1.010025 - 1}{0.005} = \\ 10,000 * \dfrac{0.010025}{0.005} = 10,000 * \dfrac{10.025}{5} = 10,000 * 2.005 = 20050. \ \checkmark[/MATH]That is not a proof that the formula works, but it is a demonstration that you can easily validate on your own if you are uncertain of your memory. Here is the other formula you must remember.

[MATH]f = p(1 + i)^n.[/MATH]
This formula applies across the board, not just to payment streams.

Now let's put together these two formulas algebraically to determine what is the annuity payment that generates a specific present value.

[MATH]f = p(a + i)^n \text { and } f = a * \dfrac{(1 + i)^n - 1}{i} \implies\\ p(1 + i)^n = a * \dfrac{(1 + i)^n - 1}{i} \implies \\ a = p * \dfrac{i(1 + i)^n}{(1 + i)^n - 1}.[/MATH]But you did not use that formula. And hence all your problems.

So let's solve the problem correctly.

[MATH]a = 60,000 * \dfrac{0.005 * 1.005^{60}}{1.005^{60} - 1} \approx 1,159.97.[/MATH]
You will not get the right answer if you use the wrong formula. That is why it is so important to know what a formula means. Now I suggest you do the other computation yourself. Send me the answer to make sure you are now on track. And only AFTER we see what the numbers come out to be does it make sense to explain the result.
 
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Hi JeffM
Thank you for taking the time to post a reply.
Thank you for your formula and walkthrough for an annuity.. You have the ability to lay everything out clearly. (Hopefully your a teacher and pity your not mine).
I left High School in 1992, so this is difficult to say the least.
I rang my local bank yesterday, and asked, if they knew the formula needed. Of course the person didnt, saying "I left school 40 years ago, and i don't know". (Rang bank as they do loans and repayments).arraagghhhhh, more grey hairs.
Thank you again.
 
If you are still awake, we can continue right away. (By the way, I am not surprised that an employee at the bank did not know the math; it is done by computer and probably only a very small number of people know the formulas with possibly none knowing the math behind the formulas.)

You did not tell me whether you have excel available and know how to use it, and I have been assuming that your algebra is not too rusty.

Nor have I ever been employed as a teacher. I was an oficer of a bank for 32 years and then a director of that bank for another 10.
 
Hi JeffM. Sorry i haven't replied to you reply as yet. 'Just not enough hours in the day', as the saying goes.
May i ask, what country you are from? Im Australia. Just good to know time differences.
Wow. I'm impressed at your banking career, very impressive. You were correct at stating the staff at bank didn't know formula, it was only computer based.
Yes, i have Excel available. Only basic skills. Putting the FV, PMT, sort of formulas, very new as i haven"t done.
I studied your formula, much like what is meant to be used here. I was first confused by the i(1+i) = 0.005+1.005. But i am figuring, anything times 1 is equal to itself, where the i * i. Figured one 'i' is one, and the other is the original rate. Sorry if i'm confusing you.
Sorry for not replying earlier, but need at least days break from study. Overload is worse than no study. I will post the answer to my original question by Monday, fingers crossed correct.
Thank you once again JeffM, for bothering to help me.(you actually help, more than my lecturer). Your the reason, when i close this laptop, i will be smiling.
 
Always happy to give someone a smile or a beer.

I'm in the eastern time zone of the US (but well inland of the coast, west of the Allegheny Mountains). Not sure quite how that relates to Oztime, but I used to play bridge online with a woman from Singapore: the time difference was 12 hours exactly. Right now I am trying to figure out whether your Monday is my Sunday or Tuesday or maybe Friday. Perhaps when it is Monday morning for you, it is Tuesday evening for me, and when it is Monday evening for you , it is Sunday morning for me. It makes no difference to me, I'm retired now.

Here is the way I'd like to go. You try using the formula on the remaining half of the problem by yourself. Let's make sure there are no glitches there. The whole point of a formula is that all you MUST know is which is the corect formula to use and what numbers to plug into it. But that creates no understanding at all.

So for the next step after confirming your grasp of the formula, use the private message box here to send me your email address, and I'll send you an excel spreadsheet that shows all the numbers that get collapsed into that weird looking formula. The whole thing will make at least some intuitive sense if you see all those numbers organized in a sensible way.

As for i and (1 + i), i represents interest on one dollar of principal for one period at the agreed rate, 1 represents one dollar of principal, so 1 + i represents a payment of one dollar of principal plus interest on that dollar at the end of a single period. Buried in that fraction with exponents is the simple interest formula you probably learned in elementary school applied to one dollar (or rupee or ruble or whatever) for just one period.

To really explain why the formula is what is takes a bit of algebra beyond the basics. You may not care to delve into that. I can explain it of course; it is not really that hard, but it may require learning some math tools that go beyond anything that you learned in school.
 
Hi. To Jeffm. I have been trying to get the answer your formula created. I must be missing something, it is not working out, (and this is from your completed formula.

This is the formula im following
a=p * i(1+i)^n
(1+i)^n-1
60,000 * (0.005+1.005)^60
(1.005)^60-1
=$1,159.97

This is my workings
60,000 * 1.81669
0.348850
60,000 * 5.20765
= 312,459.22 so not that answer
Try
60,000/ 5.20765
= 11,521.511 not that answer.
I have tried cancelling top and bottom, in the fraction.
60,000 * 0.005
-1
= - 300 obviously not that answer
I have tried lots of different ways (been trying for hours), but none are working,
 
Hope you are able to understand that working. The format changed when i posted.
I'm in Adelaide, 3.13am Sunday morning.
I will try again tomorrow. let you know how i get on.
 
Well that is a ridiculous time of day.

1159.97 is the correct answer. I can prove to you that it is with a spreadsheet.

We are 13.5 hours apart. It is just past 2 in the afternoon here.

Talking about nine in the evening your time is human for both of us is human.

We can also talk around nine-thirty in the morning your time unless I have been drinking, which I have been known to do from time to time.
 
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You are probably in bed. I have now looked in detail at your latest working. I truly think that if we were using this site at the same time so we could chat and if you were to give me your email address so I could send you a spread sheet, we would soon have you confident that you have got this in a way that it will always come back to you with just a minute's review.

Now as to your working. A formula is like a magic spell: you leave out one ingredient, and you get an ugly toad instead of a sexy sheila (or a toy boy if that's your thing).

[MATH]i = 0.005, \ n = 96, \ p = 60,000 \implies\\ a = 60,000 * \dfrac{0.005(1 + 0.005)^{96}}{(1 + 0.005)^{96} - 1} = \\ 60,000 * \dfrac{0.005 * 1.005^{96}}{1.005^{96} - 1} \approx \\ 60,000 * \dfrac{0.005 * 1.614}{0.614} \approx \\ 788.49.[/MATH]I am not quite sure what you were doing wrong, but I think I understand why you were doing it wrong, namely that the formula seems to have no rhyme or reason to it.

So in an earlier post I showed that the formula gives answers that you can easily confirm by hand to be correct. Do you remember that?

And with a spreadsheet I can prove to you that the answers we got for these much more complex and realistic cases are correct.

To show you why the formula is correct in general requires some algebra a bit beyond the basics. But the critical thing when using the formula is to not deviate a jot from it.
 
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