# What is the relationship?

1. Feb 6, 2005

### aisha

This girl was given 1000 dollars to invest and she buys an investment that pays 4% per year simple interest.

I completed a chart to show how much she will have at the end of 5 years calculating the annual interest each time and then calculating the total amount for the end of the year.

I need to determine an equation that best models the relationship between the year and the total amount. Here are the numbers that are needed to make the equation....

Year -->Total amount
1 ------->1040
2-------->1080
3-------->1120
4-------->1160
5-------->1200

I've been trying for a while but cant get anything that works, can someone plz give me a hint!

2. Feb 6, 2005

### aisha

I just got an eqn I think it works

1000+ 40n where n=year

3. Feb 6, 2005

### Andrew Mason

Let P = principal and i = interest rate: The amount after one year is:

$$A(1) = P + Pi = P(1+i)$$

After 2 years, if you leave the interest invested, the amount after 2 years is:

$$A(2) = P(1+i) + P(1+i)i = P(1+i)(1+i) = P(1+i)^2$$

Work it out for 3 years and see if you can see a pattern developing.

AM

4. Feb 6, 2005

### aisha

I already did that my eqn in the last post works are u saying that is wrong?

5. Feb 6, 2005

### Curious3141

Andrew, that's wrong. The question said simple interest not compound.

6. Feb 6, 2005

### Curious3141

This is correct. Do you know the name of the type of progression you have ?

7. Feb 6, 2005

### aisha

um im not sure I think its an arithmetic series?

8. Feb 6, 2005

### Curious3141

Correct, but more properly you should call it an "arithmetic progression" since a series is the summation of terms of a sequence.

Well done.

9. Feb 6, 2005

### dextercioby

I'm sorry Courious,but it doesn't make any GD sense...I mean,your acceptance of "simple interest" would not be applied by any bank in the whole wide world,simply because it's totally unfair to the invester...

Again,i'll have to admit that economy is not even close to my heart,not to mention my brain...

Yet,i've worked with banks in my life and i know pretty well how these things trully work.

Daniel.

10. Feb 6, 2005

### The Bob

I am not sure if this has been said but ignoring the question, people are normally given interest monthly. For the question, however, the numbers are related by 40 e.g. each is in addition to 40.

Now this might be what the question wants but ANY interest in the world will be added from the last figure you had, e.g.
Year 1 --> 1040
Year 2 --> 1081.6
etc.

I think this is a bad question. Students are going to get the wrong idea and it isn't that hard to do more calculations to have normal interest.

11. Feb 6, 2005

### Curious3141

Did you read the question ? Do you know the definition of simple interest ?

It's not *my* fault if it doesn't make any "GD" sense. :grumpy:

Last edited by a moderator: Apr 21, 2017 at 12:23 PM
12. Feb 6, 2005

### Andrew Mason

Well, it says it is simple interest paid each year so the interest accumulates. Why would interest not accrue on the interest that is paid each year. Simple interest means that there is no compounding within the payment period.

AM

13. Feb 6, 2005

### Curious3141

That's just the way simple interest works. Read the definition here : http://www.lse.co.uk/financeglossary.asp?searchTerm=&iArticleID=943&definition=simple_interest

Quoting,
Isn't that very clear ? You do not include the accumulated interest in computing the ensuing interest. You only take the initial principal. I thought this should be obvious.

14. Feb 6, 2005

### Andrew Mason

When interest is actually PAID, it becomes principal. The definition you have provided assumes that interest is not paid within the accrual period.

Compound interest is interest charged on accumulated but unpaid interest. If the accruing, compounding and payment periods coincide, there is no difference between simple and compound interest. For example, there is no difference between simple interest at 1% per month accruing and paid monthly and compound interest at 1% per month, accruing and paid monthly. There is a difference between simple interest at 1% per month paid annually and compound interest at 1% per month, paid annually.

Look at the question this way. What is the point of saying that interest is paid every year? If it is paid, and if the payee leaves it in the bank, are you saying that the bank, in paying interest the next year, will NOT pay 4% on the interest simply because it agreed to pay only simple interest of 4% per annum?

This definition, for example, makes the point clear:

simple interest:
Interest computed on the principal balance, and disregards previously accumulated (upaid) interest. http://www.prou.net/utilities/glossary/glossary10.html

AM

15. Feb 6, 2005

### The Bob

Very clever............. if you manage the bank that is giving this deal. Would anyone really do this. Interest on the original principles. The only thing I could think of is that someone puts money in the bank for about 2 months, then withdraws most of the money and they get money for it not being in the bank. Most people invest money or put money in the bank to get as much as possible for doing nothing.

It might be good for some people but it can't be something that is used much, surely???

16. Feb 6, 2005

### Curious3141

I agree that the accruing and compounding terms make a difference in compound interest, but disagree that they make a difference in simple interest calculation. To prove me wrong, show me a worked example of simple interest that proves your point, i.e. that the interest paid is added to the principal in computing the next interest payment. I'd be very interested in seeing this.

What exactly is the definition of "paid" in this context ? You say that the investor can reinvest the "paid" interest and accrue simple interest on it, but I hold that it doesn't work that way. You CANNOT touch the principal on simple interest, and that includes reinvesting accumulated interest. This is my understanding of it.

At any rate, I think you're overthinking the question. I believe they just wanted to see the arithmetic progression in action (as opposed to the geometric progression of compound interest).

If you do have any worked examples of simple interest that support your point, please post the link or the text.

17. Feb 6, 2005

### Curious3141

I couldn't elaborate as much as I wanted to in my last post because I was rushing in to work.

Here is yet another link that explicitly supports my contention : http://www.riskglossary.com/articles/compounding.htm [Broken]

The whole point is that in making an investment, whether it be for a fixed deposit or a mutual fund or other instrument, the terms are clearly drawn up upon application and are agreed upon by all parties. In the case of an investment protocol stipulating simple interest for x number of years, this means that the interest is awarded annually, and then *kept separately* from the original principal. For all intents and purposes, the money has been awarded in cash to the investor and the investor disallowed from reinvesting it under those same terms for the period of validity of the contract.

The whole reason the investor cannot reinvest the earned interest into the principal is the binding contract; if that stipulates that the investor cannot do this, then he/she simply cannot. Whether or not there is a provision to actually allow the investor to physically withdraw the interest every year is dependent on the fine print in the agreement, but this point does not concern us.

This model is rarely followed nowadays (and even then only for very short term deposits), which may be the source of confusion for some of you, but this is just the way "simple interest" works, it is an elementary linear growth instrument. It *is* "unfair" to the investor from a modern economic perspective, but you signed the contract, so there you go.

EDIT : Yet another lucid explanation : http://mathforum.org/dr.math/faq/faq.interest.html

Last edited by a moderator: Apr 21, 2017 at 12:24 PM
18. Feb 6, 2005

### Andrew Mason

If the question had asked how much the investor would have after 5 years on an investment that accrues but does not pay 4% simple interest per year, I would agree with you. It specifically says that the investment PAYS interest each year. One cannot ignore the time value of that annual payment if one wants to answer the question: how much would the investor have after 5 years.

AM

19. Feb 6, 2005

### Curious3141

I'm not saying you can *ignore* the interest payment. But what leads you to make the assumption that the interest is redeposited into the same account under the same terms as the original principal ? For all we know, the investor could have withdrawn those payments and kept it under her pillow at home (in which case the answer I got for the total asset is correct), or even taking that interest and put it in a different undefined fund that pays different terms (in which case, the result would be something else altogether).

Aren't you making a huge assumption that the investor is permitted to reinvest the paid interest under identical terms as the original principal ? And in fact, if she is allowed to do this, it *doesn't* come under "simple interest" anymore, so it's plain wrong for this question.

20. Feb 6, 2005

### Andrew Mason

Well you are quite right that an assumption is required. Either we assume that she keeps the interest under her pillow for 4 years or she does something with it to earn income with it. We know that we are to assume that she doesn't spend it because the question asks how much she has after 5 years. The assumption I made was that it would be invested on the same terms as the original principal since there was nothing to indicate that this couldn't be done. You appear to want to assume that she kept it under her pillow.

I don't know about huge. It makes sense to me. I have never heard of investment businesses that won't take your money. There is nothing in the question to indicate that she takes the paid interest and hides it for four years.

I disagree that doesn't come under 'simple interest' for reasons explained above. I would interpret simple interest to mean only that there is no interest earned on accrued but unpaid interest. Some may take a different interpretation, as you appear to do. Unless there is a definition provided in the question, I guess we will just have to disagree.

AM