سهم ساده و مرکب

Simple and Compound Interest

Simple and Compound Interest. This is one of my favorite topics here. Let’s begin with simple interest. Well, simple interest is a mathematical fiction, created to teach the idea of interest to grade school children. Absolutely, no one in the real world uses simple interest.

It is unlikely to appear on the test. With simple interest, the interest payment, the dollar amount of the interest payment is exactly the same each time. So let’s think about this. Bob deposits $1,000 in an account that yields 5% simple interest compound annually.

Well, 5% of $1,000 is $50. So what this really means is that every year Bob is going to get another $50 worth of interest. So in the first year, $1,000 plus $50, we got $1,050. Then in the second year, we add $50 again, then we add $50 again. And each year all we’re doing is we’re adding that same amount $50.

And so the account is rising by $50 each year. The exact same amount, dollar amount is added each time. Simple interest pays interest only on the principal and not on the interest. Thus, the amount of interest accrued makes no difference to the interest payment, which is the same in each period. And indeed, as this graph shows.

If we were to graph the amount in the account versus time, it would rise as a straight line. All that’s happening each year is another $50 is being added, so that’s why it rises at a constant slope. Simple interest is not often tested directly, but and this is very important. It can be used for estimation purposes.

And we’ll talk about this more later in this video. The big idea of compound interest is interest on interest. In compound interest, we get the percent of interest paid on the total amount that has already accrued in the account principal plus all previous interest payments. Thus, the more interest that has accrued, the larger the amount of the next interest payment.

No two successive interest payments are ever the same, so the dollar amount is always different. The entire amount experiences the same percentage increase in each period, so that’s what stays the same. It’s not the dollar amount that stays the same, it’s the percent increase that stays the same.

That’s what’s going on in compound interest. So for example, Bob deposits $1,000 in an account that yields 5% interest compounding annually. So that means that every year, the amount in the account is going to experience a 5% increase. So it’s gonna get multiplied by 1.05, which is the multiplier for a 5% increase.

So we start at $1,000. Let’s construct that multiplier 1.05 multiply by 1.05, we get 1,050. Notice after one year, that we get exactly the same amount. Either with simple or compound interest and that will always be, that will always be the case if we’re compounding annually that after one year, one compounding period.

Simple inter, simple interest and compound interest get exactly the same amount. After that, they diverge. So after two years, we multiply 1,050 by this same multiplier, we get 1102.50. Then multiply by that multiplier again and again. So these numbers are starting to get very hairy. So first of all, don’t worry about getting these numbers by hand.

You need a calculator to get these numbers. That’s important thing number one. Important thing number two to notice is that say, after four years, simple answers would get up to $1,200. So here we have $1,215.50. We’ve done a little bit better than simple interest would do for us.

Now you might say, well, $15 what’s the big deal about that? Well, think about it this way. Here we’ve only talked about four years. Four years is a relatively short time in terms of investing and we’ve only talked about a principal of a thousand dollars. A thousand dollars is peanuts in terms of large scale investing.

And so you really have to think about this in terms of a more grand scale. With large amounts of money and or long periods of time, the difference between successive interest payments becomes substantial and also the difference between the total amount gained with compound interest versus the total amount gain with simple interest becomes substantial. So big idea number one, compound interest always outperforms simple interest, as long as there’s more than one year.

That is to say, as long as there is more than one compounding period. And this graph really summarizes it. The green line is the straight line of simple interest. The purple curve is the line of compound interest. Notice that it curves away from simple interest. It accelerates away from simple interest.

So in the short term, it does a little bit better than simple interest. But then the longer we go, the more it diverges and the, it does much, much better than simple interest does. But the end of this graph, notice that the compound interest payment, the compound interest total is over a thousand dollars more. That is the size of the principal.

It’s more than the size of the principal above the simple interest. Also, notice in y years the principal will be multiplied by the percent increase multiplier y times. Let P be the principal and r be the multiplier. And the total amount in the account after n years is P times the multiplier to the power of y.

We can even be a little more formal about this. Of course, the annual percentage interest rate is I, then the multiplier for that. We change the interest to a decimal, that is divide by 100 and then add 1. That’s how we get a percentage increase, so that’s the multiplier for an I percent increase. And of course, we’re gonna be multiplying by that y times.

Now here’s what I’m gonna say, I printed a formula here. Do not, do not, do not memorize this formula. That would be a big mistake. Instead, what I want you to do is understand the logic of how this formula was put together and when you’re doing a problem, rebuild the formula using that logic.

Do not blindly memorize the result. So here’s a formula and I’ll show what I mean. Sheila invests $4,000 in an account that yields 6% compounding annually for eight years. What is the total amount after eight years? So again, don’t worry about getting the dollar amount.

Just worry about getting the correct formula. The correct expression for the amount that would be in the account after eight years and I’ll urge you to pause the video and try this on your own. So here’s what I’ll say. Here’s how we’ll approach this.

The multiplier for a 6% increase, that’s 1.06. It’s gonna get multiplied by that eight times. So the total amount in the account is gonna be 4,000 times that multiplier 1.06 to the 8th. So that is the mathematical expression for the total amount in the account after eight years.

Now of course, no one’s gonna expect you to calculate this without a calculator. But keep in mind that the test may not be asking for actual dollar amounts, the test may well list answer choices that are in this form. Certain amount times some per, some decimal to a sum power and then you have to recognize this particular one as the right answer. So sometimes, the test actually does that and just knowing how to construct the formula is enough to get the answer.

Things get more interesting when we change the compounding period. Banks always give an annual percentage interest, but they may compound quarterly or monthly or even daily. For any compounding period smaller than a year, we need n, where n is the number of times that compounding period would occur in a year. So for example, quarterly, that means n equals 4.

We’re compounding 4 times a year. Monthly, n equals 12. We’re compounding 12 times a year. Daily is 365, we’re compounding 365 times a year. Technically, we’d be compounding 365 days during a regular year and 366 during a leap year.

Let’s not even worry about the, the test is not going to get into the region where you have to worry about leap years versus regular years. That’s not something the test is gonna do to you. Suppose the bank plays, pays 5% annual interest compounding quarterly. The bank does not pay us 5% each quarter, that’s not what’s going on. That would be unrealistically generous.

Instead, the bank pays us 5% divided by 4, which is 1.25%. That’s the percent each quarter. In general, if there are n compounding periods in the year, we divide the annual percent of interest by n to get the percent for each individual compounding period. So we divided by 4 to, for quarterly compounding, we divided by 12 for monthly compounding.

The correct multiplier now would be the multiplier for a percent increase of I divided by n. So writing this out very algebraically. One plus i over 100n, that’s our multiplier. In y years, there would be n compounding periods each year or n times y in y years. And so it means that we’re multiplying by that multiplier, ny times.

So, the total formula is this. And again, do not, do not, do not, do not memorize this formula. That would be an exceptionally bad idea. Instead, I want you to think through the logic of it and as it were, rebuild this formula each time using the logic of the situation. That is true understanding.

Here’s another problem. If Susan invents $1,000 in an account that yields 5% annual compounding quarterly, then how much does she have after, after six years? Again, don’t worry about getting an exact dollar amount. Just see if you can build the correct formula. You can pause the video here and work on this.

So the first thing I’ll say, is that 5% annual has to get divided by 4. So, the quarterly percentage is 1.25%. The multiplier for that is 1.0125, so that’s our multiplier. The amount in the account experiences that percent increase four times each year or twenty four times in six years, so we’re multiplying by that multiplier twenty four times.

So the final amount will be the, the principal, $1,000 times that multiplier to the power of 24. Okay. So that’s the expression. And again, the problem is simply looking for building the correct formula. That’s the correct formula. I’ll also say that we could approximate this.

Sometimes, the test will want us to approximate. We can approximate using simple interest. So, let’s think about this. 5%, that’s $50. So that means in six years, they’d get $50 six times, $300. That means simple interest gives us $1,300.

So we can estimate that the compounding interest over six years is gonna be something slightly more than $1,300 probably not gonna be as large as 1,400. So that kinda gives us a ballpark range for how much interest we’re going, going to improve here. How does the size of the compounding period determine the interest earned over time?

For this, we’ll need a large principal and a long amount of time. Let’s say that the annual interest rate is 5%, the principle is a $1 million and the time is 20 years. So we’re really pumping up both the amount that we’re dealing with, as well as the time to look at these subtle differences. So, on that scale, simple interest would give us $2 million after 20 year.

Compounding annually would give us 2.6 and then we see the other numbers for compounding quarterly, monthly, daily and hourly. Notice that as we up the compounding, the more and more compounding we do, the amount of total interest we get paid increases. As the compounding period decreases, the overall amount interest earned increases, so that’s a big idea also.

You always do better with more compounding. In summary, we talked about simple interest. It’s not very realistic, not likely to appear by itself. But an excellent approximation tool. We talked about compounding annually, then we talked about compounding other periods, quarterly and monthly.

One big idea is that compound interest always pays more than simple interest, that’s a really important idea. That’s important, especially when you’re using the simple interest approximation. And also, it’s important to know that more compounding periods give us more money.