What is the Relationship Between Time and Total Amount in Simple Interest?

[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 can't get anything that works, can someone please give me a hint!

 

[aisha]

I just got an eqn I think it works

1000+ 40n where n=year

 

[Andrew Mason]

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

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

 

[aisha]

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

 

[Curious3141]

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

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

 

[Curious3141]

I just got an eqn I think it works

1000+ 40n where n=year

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

 

[aisha]

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

um I am not sure I think its an arithmetic series?

 

[Curious3141]

um I am not sure I think its an arithmetic series?

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

Well done.

 

[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.

 

[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.
Start with --> 1000
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.

The Bob (2004 ©)

 

[Curious3141]

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.

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

Read this : http://www.datalife.com/mall/pages/examples/EXMP_INT.HTM

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

 

[Andrew Mason]

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

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

 

[Curious3141]

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

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,

Simple Interest
Interest is paid on the original principal: no interest is paid on the accumulated interest payments.

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.

 

[Andrew Mason]

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

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.

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

 

[The Bob]

Isn't that very clear ?

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?

The Bob (2004 ©)

 

[Curious3141]

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

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.

 

[Curious3141]

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

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

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

 

[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

 

[Curious3141]

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

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.

 

[Andrew Mason]

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).

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.

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.

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

 

[Curious3141]

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

Before we agree to disagree, could you please post a link to a source of established reliability that shows a worked example for "simple interest" that supports your scenario ? I've posted several that support mine, and I want to see one from your perspective.

EDIT : If you can actually produce a worked example posted by another site or author that shows a simple interest scheme where interest ("paid", as you define it) can be reinvested together with the principal to earn more interest, then I will be convinced that you are correct.

 

[Andrew Mason]

Before we agree to disagree, could you please post a link to a source of established reliability that shows a worked example for "simple interest" that supports your scenario ? I've posted several that support mine, and I want to see one from your perspective.

EDIT : If you can actually produce a worked example posted by another site or author that shows a simple interest scheme where interest ("paid", as you define it) can be reinvested together with the principal to earn more interest, then I will be convinced that you are correct.

I don't think we disagree on what simple interest means. Principal x interest rate/calculation period x no periods until paid would be simple interest. If I then take the interest that is paid and reinvest it, I am still getting simple interest - it is just that my account balance (principal) has increased. This is how every savings account works. No bank will say that it is paying compound interest. It is always simple interest paid or credited annually (or some other period).

If interest is earned before it is paid, and the accrued interest bears interest until paid, this is not simple interest. It is compound interest. Term deposits often work this way.

If the question had stated that interest would accrue at 4% per annum and paid after 5 years, the methods that you have provided are correct. But I interpret the question as saying that interest is paid annually. If it is paid annually, the investor would be able to reinvest it. Whether she gets 4% on the interest or not, is not given. I don't see why we have to assume it would be 0.

AM

 

[Curious3141]

I don't think we disagree on what simple interest means. Principal x interest rate/calculation period x no periods until paid would be simple interest. If I then take the interest that is paid and reinvest it, I am still getting simple interest - it is just that my account balance (principal) has increased. This is how every savings account works. No bank will say that it is paying compound interest. It is always simple interest paid or credited annually (or some other period).

If interest is earned before it is paid, and the accrued interest bears interest until paid, this is not simple interest. It is compound interest. Term deposits often work this way.

If the question had stated that interest would accrue at 4% per annum and paid after 5 years, the methods that you have provided are correct. But I interpret the question as saying that interest is paid annually. If it is paid annually, the investor would be able to reinvest it. Whether she gets 4% on the interest or not, is not given. I don't see why we have to assume it would be 0.

AM

I see where you're coming from now. Just to be crystal clear, consider this example and tell me if you agree :

Say I have invested 1000 dollars in a policy that accrues simple interest daily at the rate of 0.01 %. The interest is payable annually, following which the policy is automatically renewed to a maximum of a 3 year term. Partial or total withdrawals are permitted annually without penalty.

If I decide to take my money out at the end of 3 years of 365 days each, I would have $1113.54. Is this correct ? Essentially, within each year, I'm making 10 cents a day in interest. This is accrued, and I cannot take out this money until the end of the 1st year comes, by which time I would've made $36.50 in interest alone. I can choose to take all or part of principal, interest or both out or to let it renew automatically in full. I choose to let it renew in full. But once this is renewed, the full $36.50 goes back into the principal for a new principal of $1036.50 at the start of the second term. This means that during the second year of investment, I'm earning a daily interest of 0.01 % of $1036.50, which works out to 10.365 cents. And so on.

Isn't this what you're talking about ?

This is also exactly equivalent to the case when the deposit is compounded annually at the same daily rate for 3 years, correct ?

In the case where the deposit is compounded daily at the rate of 0.01 %, the amount at the end of 3 years is of course higher, around $1115.70. Agreed ?

I think at the end of the day, we were just quibbling about semantics. I didn't realize that "payable" had such a profound impact on changing the mechanics of the accumulation of money. So when a bank uses "simple interest" it means only that accrued interest earns nothing until it is "paid", after which it earns interest just like the principal ? And when a bank uses "compound interest" it is implied that even if the money is accrued and not paid within a number of compounding periods, it will gather more interest in ensuing compoudings before it is actually paid ? This I can accept, and if you're happy with my statements, we understand each other.

But do you honestly think that this particular question expects this level of semantic complexity ?