26 - Compound Interest Formula & Exponential Growth of Money - Part 1 - Calculate Compound Interest - Free Educational videos for Students in K-12

26 - Compound Interest Formula & Exponential Growth of Money - Part 1 - Calculate Compound Interest - By Math and Science

Transcript


00:00	Hello , Welcome back . My name is Jason with
00:02	math and science dot com and this is honestly one
00:05	of the most important lessons that I've ever taught in
00:08	any subject to any , to anybody watching . It's
00:10	one of the most important topics . The title is
00:13	called Exponential growth of money , also called compound interest
00:18	. You know , probably the number one question that
00:20	I get from students of all ages , especially young
00:23	students is when will I ever use this algebra stuff
00:26	? When will I ever use trig or calculus ?
00:28	When will I ever use it ? I don't care
00:29	about that . That's what I hear a lot .
00:31	Well , this lesson is going to teach you that
00:34	because whenever you put money in the bank and earn
00:36	interest , whenever you take out a loan and pay
00:39	interest on a loan , what you're doing is you're
00:42	kind of trusting them if you don't understand what's going
00:44	on . But really what's happening is there's an exponential
00:46	function that determines how much money you're going to earn
00:50	interest or how much debt you're going to pay in
00:52	terms of , of interest on the loan . That's
00:54	an exponential function . And that exponential function is very
00:58	, very simple to understand , but it does take
01:00	a little bit of of talking about it for me
01:02	to help you understand what it means and I can
01:05	tell you this and you can probably understand this by
01:07	listening to a quote from somebody you've probably heard before
01:11	. His name's Albert Einstein . Einstein said the following
01:14	thing , compound interest is the eighth wonder of the
01:18	world . He who understands it earns it , but
01:22	he who doesn't understand it , pays it . And
01:24	what that means is really , really important for you
01:27	to understand compound interest governs how much money you're going
01:30	to earn when you put money in the bank or
01:32	when you invest money in a retirement account or just
01:35	in the stock market to try try to acquire uh
01:38	wealth . So that compound interest is an exponential function
01:42	. We've been learning about exponential functions . You know
01:44	the shape of that exponential function goes up very rapidly
01:46	. So if you save money , it's the eighth
01:49	wonder of the world because it gives you more and
01:51	more and more money over time . But he ,
01:53	who doesn't understand it , pays it . And what
01:55	that means is when you buy a house , you
01:57	have a alone and that loan acquires , you have
02:01	to pay interest on that loan . And the amount
02:02	of money you owe goes up and up and up
02:05	also exponentially with time , every time you buy a
02:07	house and you take a loan for a car or
02:09	a student loan for university , you have to pay
02:12	interest . And that's exponential also . So it really
02:15	is one of the most practical things you could possibly
02:17	learn . And one of the most important lessons here
02:19	. Now we're gonna start off by asking , I'm
02:21	gonna ask you a question and I'm gonna demonstrate powerfully
02:24	why compound interest and exponential growth of money is so
02:27	important . I'm gonna give you two choices for 30
02:30	days over a month , 30 days . We're going
02:32	to do the following options . I'm gonna give you
02:34	two options option A is I'm gonna give you $10,000
02:38	every single day . So 100,000 . That's how much
02:43	money you're gonna earn . If you choose option A
02:45	option B is I'm gonna give you one penny which
02:48	is one us cent . Or you could you could
02:50	you pick some other currency ? 11 cent , right
02:53	? One tiny little penny on day number one .
02:56	But what I'm going to do is double that every
02:58	day . So to be one cent , then the
03:00	next day you get two cents . Then I would
03:02	double again and I will give you four cents and
03:03	so on . So what would you choose for 30
03:06	days ? Just for one month ? Would you pick
03:08	the $10,000 every single day ? And at the end
03:11	of the 30 days you would have that much money
03:12	? Or would you pick to go with the one
03:14	cent per day ? And I will double that balance
03:17	every day . Only for 30 days though . How
03:19	much money do you think we're gonna have at the
03:20	end of the month for each of those cases ?
03:23	All right . So here we go . Here's what
03:24	the answer is . Here's day number one choice A
03:27	As I give you $10,000 a day , you start
03:29	with $10,000 a day . Choice B . Is I
03:31	give you only one penny , but then on day
03:34	number two , I give you another $10,000 . That's
03:36	how this column works , right ? But Choice B
03:38	. I double the previous , the previous balance ,
03:41	I double that , and I give you two cents
03:43	, right ? Then we go to day three .
03:44	Obviously , every day you go down this column ,
03:46	it goes 60,000 . All the way down to $100,000
03:51	at the end of 10 days . Choice B .
03:53	I keep doubling it , one cent . Uh then
03:55	two cents four cents , eight cents , then 16
03:57	cents . You see right here , you have $50,000
04:00	and only 16 cents in this column . Let's keep
04:02	on going . If you keep doubling at 32 cents
04:04	, 64 cents dollar , 28 $2.56 $5.12 in this
04:10	column , you would have $100,000 . So , if
04:12	you were going to stop this at 10 days ,
04:14	you would obviously want to take the $10,000 a day
04:17	. But let's keep going . All right . What's
04:20	going to happen on day ? Number 11 ? I'm
04:22	gonna have 100 and $10,000 but I doubled that $5.12
04:26	over here to get 10 24 . Now , in
04:28	this column , I keep going down down 10,000 day
04:31	. But in this column , I go from 10
04:32	20 for $20.48 $40.96 81 90 to 163 84 3
04:40	, 27 , 6 55 36 1310 70 to 2621
04:47	and 44 then 5242 and 88 cents . Still in
04:52	this column , I would have $200,000 compared to only
04:55	about $5000 after day 20 . So obviously , if
04:59	you were gonna go only to 20 days , you
05:01	would want to take the column a to get your
05:04	10,000 a day . Let's see what happens when we
05:06	go into the tail end of the month here ,
05:09	here's where we start seeing some interesting things going on
05:12	day 21 . I have $210,000 here , but in
05:15	this column I've doubled that $5000 to 4 10,085 .
05:19	And then I've doubled again to 20,000 . And some
05:21	change 41,000 and some change 83,000 and some change 167,000
05:27	and some change compared to the 250,000 , I would
05:30	have here here in red is where the right hand
05:33	column starts to beat the other column . So actually
05:37	, if you run this thing out to 30 days
05:38	, what's gonna happen is you're gonna run through 635,000
05:42	, 671,000 , here's where you hit a million dollars
05:45	on day 28 $1,342,177 . 28 cents on day 28
05:53	. As compared to the just simply $280,000 we double
05:57	that , we get 3 2,684,054 and some change .
06:01	And on day 30 we double that again to five
06:03	million $368,709.12 . Because compared to $300,000 , it's incredible
06:12	. And it because it comes about because the right
06:15	hand column , it isn't really obvious when you first
06:18	look at it , but the column where I'm doubling
06:20	every day is actually an exponential growth of money ,
06:23	even though it doesn't really seem like it at first
06:25	, it seems like who cares , one sent to
06:27	sent force and whatever . I don't care about that
06:29	. But really when you get down to a certain
06:32	critical number of days in the future , you overtake
06:35	the linear function . This is a linear function because
06:37	I'm gaining the same amount of dollars every day ,
06:39	10,000 , 20,000 . So this is a linear growth
06:43	of money . If I graph it days versus dollars
06:45	is just going to be a straight line because I'm
06:47	gaining a constant number of dollars every day . But
06:50	this column notice what's happening , What I did is
06:53	I took the one cent and on the first day
06:55	I multiplied by two to get two cents . And
06:57	then the next day after that I multiplied by two
06:59	again . And then the next day after that and
07:01	multiplied by two again and again . So if you
07:03	get a calculator and actually just do it , put
07:05	in one cent and then hit times two times two
07:08	times two times two , then you're going to get
07:10	all of these numbers in this column . This is
07:12	not rocket science , what's happening in the right ?
07:14	Just times two times two every day ? And you
07:16	do that 29 times because on day one , the
07:20	first doubling When I multiply by two , this was
07:23	one doubling , then this was another doubling and so
07:25	on . So even though there's 30 days , there's
07:27	actually 29 doubling periods . And so if you multiply
07:31	by 229 times , you'll get the $5 million dollar
07:35	answer . Okay ? So in terms of when you
07:38	look at what you have here , it's one cent
07:40	times two times two times two times 2 29 teus
07:43	what does this look like ? This looks like an
07:45	exponential here too . To the power of how ever
07:48	many days I doubled because remember what an exponent is
07:51	is when you multiply by itself . So I'm taking
07:54	the first number , I'm multiplying by two , which
07:56	means I have to raise to the 29th power .
07:59	So if you don't want to do the times two
08:01	times to deal in your calculator , just grab a
08:03	calculator . Take to raise it to the power of
08:06	29 because that's how many doubling periods we had .
08:10	And you take the answer multiplied by one cent .
08:12	You will get 3 5,000,068 709 decimal 12 . And
08:16	if you were to plot the money that you get
08:18	as a function of the days , in comparison to
08:21	the linear , it's going to look exactly like an
08:23	exponential function in the beginning here it grows very slowly
08:27	, which is what we saw . But then once
08:29	it hits a critical value it starts to take off
08:31	. And in those last what ? 12345 days of
08:35	the month is when you earn incredible amounts of money
08:38	. So what does this mean ? Okay , it
08:40	doesn't mean I'm trying to teach everybody to be greedy
08:42	. It's just that the number one question I get
08:44	in math is what is math useful for ? Why
08:47	do I care about math ? When am I ever
08:50	going to learn and use math in everyday life ?
08:53	That's what I get all the time . And here
08:55	is your most practical benefit and use of math in
08:58	everyday life is your growth of money . And your
09:00	growth of debt is governed by an exponential function called
09:04	compound interest . Now this was a crazy compound interest
09:07	just for illustration purposes . But what I want to
09:09	do now , as I want to um I want
09:13	to give you a more practical example to show you
09:16	what the growth of money does over a realistic time
09:19	period with a realistic interest rate . And you will
09:22	see this nice , beautiful exponential function . And then
09:24	I want to go dive more into the math and
09:26	I wanted to solve a couple of problems and really
09:28	show you what the the formula is for compound interest
09:32	in where it comes from and also that you will
09:34	then know and see that it is an exponential function
09:37	. Any time you have an exponential function , you
09:39	will always always beat a linear function , no matter
09:42	what , As long as you give it enough time
09:44	. So let's go off and do the computer demo
09:46	. We're gonna do a computer demo here where I'm
09:47	going to show you what's going to happen to a
09:49	growth of money . Let's pick an interest rate ,
09:52	8% interest . Let's look over a period of time
09:55	of , let's say 40 years . Let's say you
09:56	start working around 20 years old and then you go
09:59	until you're about 40 more years . It's about 60
10:02	years old . When most people start to retire .
10:03	Let's see how much money you'll have if you start
10:05	saving early , a small amount of money with compound
10:09	exponential growth of money in the form of interest .
10:13	Hello , welcome back . So here is a website
10:15	. It's a U . S . Government website called
10:17	investor dot gov . It's free . Just go to
10:20	investor dot Gov . You can you can do all
10:21	everything I'm about to show you here . What we're
10:23	going to do is look at a realistic example of
10:26	saving money . Okay ? You have an initial investment
10:29	, let's say that you're 20 years old . You
10:31	have your first , you know , I guess good
10:33	job . And you start to try to save some
10:35	money . Let's say you start this account with $100
10:38	or 100 whatever currency that you have in your country
10:41	. Let's say $100 . And let's say every month
10:44	you take your paycheck and you save $100 out of
10:47	your paycheck . Now , you might say $100 is
10:49	a lot of money and it is a lot of
10:50	money , but $100 is something like , you know
10:53	, a few dollars a day . It's like three
10:55	, let's say $3 a day . There's 30 days
10:57	in a month , that's like $3 a day .
10:59	Uh a little over $3 a day . It's like
11:01	a coffee . You know , stop getting your coffee
11:03	you know once a day . All right . And
11:05	so you you save $100 a day . Now let's
11:07	take a look from 20 years till most people retire
11:10	at around 60 years old . So let's say that
11:12	you save this and you do this every single month
11:14	for 40 years And your interest rate , let's take
11:17	a look at and calculated for an 8% interest rate
11:21	. Now when you look at the U . S
11:23	. Stock market over the last 30 or 40 years
11:25	, you've averaged about 8728% . So , you know
11:30	, you may or may not agree with this number
11:31	, but we're gonna calculate based on 8% . And
11:33	that's pretty realistic based on history . Okay . That's
11:36	all you gotta do . You have an initial amount
11:37	of money , how much money you're gonna put in
11:38	every month ? How many years you're gonna save it
11:40	for ? And what is the interest rate ? And
11:42	we're going to compound it annually . That means we're
11:44	going to calculate we're gonna get 8% free money .
11:47	That's the interest every single year on the money that
11:50	we're putting in . Let's calculate and see what the
11:52	results are . All right . Let's take a look
11:55	here is what the graph looks like . All right
11:57	on year . You can see here on year zero
12:00	, we saved $100 in the next year we actually
12:04	saved and we contributed $1300 . We got a little
12:08	bit of free money from the interest at $1308 there
12:11	. You can see what's happening though as you go
12:13	off and off and off into the years . The
12:15	red curve is pulling away exponentially from the green curve
12:19	. The green curve down here is the money that
12:21	I'm putting in every month into that account . The
12:23	$100 . This this green number is only the money
12:26	I'm putting in for my paycheck . The red money
12:29	is all basically the free money I'm getting , it's
12:31	the total value of the growth of that money over
12:34	time . So you can see after year 40 how
12:36	much money do I have ? I have $313,040 in
12:40	this account But I only actually put in $48,100 .
12:45	So I earned something like 250 , something like that
12:48	, $260,000 , totally , completely for free . But
12:52	you see in order to do that , there was
12:53	many , many years here in the beginning where I
12:55	didn't earn very much money but after some critical amount
12:58	of time it starts to pull away and starts to
13:00	go exponential on you . All right now , $100
13:04	a month is you know , real money you have
13:06	to say , I agree with you . But whenever
13:08	you get , you know , you go to university
13:10	, you get a good job . I think it's
13:11	perfectly possible for most people to save around $350 a
13:16	month . You might say that's impossible for me .
13:18	Well maybe it is , but $350 is what a
13:21	new car cost . So instead of getting a new
13:24	car and paying that monthly fee in a in a
13:26	new car , maybe you get a used car or
13:29	maybe you just drive your car for a much longer
13:31	period of time . And instead of buying that new
13:33	car , you just , you take that money and
13:35	you put it into this account . Let's see what
13:37	would happen over 40 years . We start out with
13:39	$100 , we save 350 every month for 40 years
13:43	at this exact same interest rate . Let's recalculate it
13:46	. And what I have at the end of 40
13:49	years is now instead of 300,000 , it comes out
13:52	to $1,090,000 , But the amazing thing is look at
13:57	how much money I put in . I only put
13:59	in $168,100 . So I saved that 350 every month
14:05	. And yes , it was a challenge to save
14:06	that money . But I put in $168,000 but I
14:09	got basically $850,000 for free by doing that . This
14:14	red curve is the exponential growth curve of all the
14:17	free money that has come into my pocket by saving
14:20	early and saving as much as I possibly could .
14:22	And then , you know , the green line is
14:25	Is just the the money that I've put in .
14:28	Now . I know probably some of you are curious
14:30	. So let's go ahead and do something probably at
14:32	the upper end of what you know some people can
14:34	do . Let's say you could save $500 a month
14:36	. Uh , if you're married , you both have
14:38	a job , you save some money and after say
14:41	$500 a month and you recalculate that at the end
14:45	of that 40 years , you're gonna have one million
14:46	, 556,000 . So we can play this game all
14:50	day . I don't want to continue playing this game
14:52	. I want you to go to investor dot gov
14:54	and type these numbers in here . The point is
14:56	, is maybe you can save 500 . Maybe you
14:58	can't maybe you can only save $50 a month .
15:01	All right . Let's see what happens if you save
15:02	$50 a month . What do you get at the
15:04	end of that At the end of 40 years ,
15:06	I only put in $24,100 . But this account is
15:10	worth $157,600 by saving $50 a month . That's something
15:15	like $2 a day , a dollar , $80 a
15:18	day or something like that . To save that much
15:20	money . This is the power of compound interest .
15:23	It's the power of the exponential function and you can
15:26	see that it does look like an exponential function .
15:28	So what I want to do now is go back
15:30	to the board and dive into where this curve comes
15:32	from and how to solve problems that deal with compound
15:35	interest . All right . I hope you've enjoyed the
15:38	computer demo and understand that the shape of the curve
15:42	when you invest money at some sort of annual interest
15:46	rate actually follows an exponential curve . Now why does
15:49	it follow an exponential curve ? That's what we want
15:51	to talk about now . Why is it an exponential
15:53	function in words ? In a nutshell . This is
15:56	the reason why . Let's say you start on day
15:58	one with some money $100 on day number two ,
16:02	let's say you earn some interest very small amount ,
16:05	let's say 8% . So now I have more money
16:07	than I did on day one because I earned a
16:09	little bit of money . But on day three I
16:11	again earned let's say another 8% . Okay . But
16:15	that 8% is calculated based on what I had on
16:17	day two . So every time you compound , that's
16:21	what it's called . When you calculate the the amount
16:23	of money you get , it's called compounding , right
16:25	? So when you calculate the new amount of money
16:27	I have , it's always based on the previous amount
16:30	I had . So I started with some number I
16:32	gained a little bit . But when I calculate the
16:34	next amount of money , it's a percentage of what
16:37	I have on day number two , which is already
16:39	more money than day one . So now I have
16:41	even more money . But then the next time I
16:43	calculate the interest I calculated based on the latest balance
16:46	, which is even more money . So you see
16:48	what's happening is every day I'm earning more money or
16:51	every year I'm earning more money and I'm calculating the
16:54	free money . I'm getting based on the amount of
16:57	money I had the previous period which also had free
16:59	money . So every day I'm gaining money and I'm
17:01	calculating the amount of free money I'm getting based on
17:04	the free money I had yesterday , so it snowballs
17:07	and it kind of gets out of control and that's
17:09	why it goes up like that . So we have
17:11	to define some terms . The money that you start
17:15	with is called the principal . That's how much money
17:17	I start on day number one . So I'm gonna
17:19	write a few terms down and we're going to talk
17:21	and derive specifically why this thing is an uh an
17:24	exponential function . So the principle , that's how ,
17:31	just how much money I start with , let's say
17:33	I start with $1000 and I'm gonna put it in
17:35	the bank and I'm gonna earn some interest on it
17:36	. So this is the initial investment , it's just
17:43	how much money I put in myself . Okay ,
17:48	And um let's say that I earn , let's say
17:53	that I earn 8% uh interest . Now we have
18:02	to talk about something else which is called , Let's
18:05	say that it's compounded annually , whoops annually . You're
18:16	gonna start to see something where it talks about compounding
18:19	annually or compounding quarterly or other kinds of compounding .
18:22	Don't get so worried about it compounded annually . Just
18:25	means that's when the free money comes in , it's
18:28	every year . So every year I get 8% and
18:30	then the next year I get 8% of what I
18:32	had from the previous year , And then the next
18:35	year I get 8% of what I then have on
18:37	my most latest balance . So I'm growing every year
18:39	, but I'm gaining 8% on the latest balance in
18:42	my account . So the compounded annually means that's happening
18:45	every year . One time per year . That's all
18:47	that , that means . Okay , So how would
18:49	we calculate how much money I actually have in the
18:51	bank if I invest some amount called the principal initial
18:55	investment ? But let's just pretend just for this example
18:58	that it earns 8% compounded every single year , which
19:01	means annually . Okay . So what we have then
19:03	is the value at end of uh year is going
19:12	to equal the following thing . It's going to be
19:15	, let me write it all down then I'll explain
19:16	it to you . It's gonna be one times the
19:19	value at the beginning of year . Okay . Plus
19:31	It's going to be 0.08 of the value at the
19:36	beginning of year . Alright . So basically what happens
19:44	is every year I earn 8% but I still have
19:47	the initial investment I put in , I still have
19:49	the principal . So what happens is the money I
19:51	have at the end of year , number one is
19:53	one times the principal . The value at the beginning
19:56	of the year is just my principle . It's one
19:58	times that , that's just the principal value but I'm
20:00	going to add to that 8% . Now when you
20:03	deal with percentages in math you never ever write a
20:06	percent symbol . You just convert the percent to a
20:08	decimal . So 8% as a decimal just means you
20:11	move the decimal 80.2 spots because remember per cent means
20:16	per 100 . So you're dividing by 100 . So
20:19	this 8% means it's point oh eight . So you're
20:21	adding to my initial amount of money you're adding to
20:24	it . This is the free money .08 times the
20:27	principle which is the value at the beginning of the
20:29	year . Okay , so if I want to kind
20:32	of write this a little bit neatly then I would
20:34	say value at , I'm gonna call this end of
20:40	year , this is E o I End of year
20:42	is one of them equal to one point oh eight
20:45	times value at the beginning of the year . If
20:52	I can write the word , the word at ,
20:55	okay beginning of your end of year . So what's
20:57	happening here ? The value at the end of year
20:59	one is one point oh eight . A lot of
21:01	students wonder where the one point oh eight comes from
21:03	. All this means one point oh a remember is
21:07	just simply one plus 0.8 That's what that means .
21:10	So one plus 0.8 So the reason we multiply by
21:14	1.8 Is because the one means the one times this
21:18	is this number that means that's how much money I
21:20	started with the .08 is the free money . Again
21:23	that's times this . So you can look at this
21:25	as algebra and you can factor out the value at
21:27	the beginning of the year , pull it out and
21:29	then you're gonna basically be able to add these together
21:32	and it's gonna be 1.08 . So the actual free
21:35	money is just .08 times my my initial investment ,
21:39	but I can add to that what I started with
21:41	. So ultimately the number you're multiplying by is 1.08
21:45	. This is gonna be the amount of money I
21:47	have at the very beginning of year , number two
21:50	. All right , So , uh , a chart
21:52	is worth 1000 words . All right , So ,
21:55	I think it's very , very important that we write
21:57	a chart . So here is the time in years
22:03	. Yes , this is compounded annually . So this
22:05	calculation of getting the free money , it's only happening
22:07	uh , you know , every single year . Right
22:11	? So let's say that I'm at the beginning of
22:14	the year . I'm sorry , at the year zero
22:17	, Right ? And then I have your number one
22:19	year , number two , year , number three .
22:22	And I'll go on after that in just a second
22:24	to show you what that is . But basically this
22:25	is your number one , you're number two , you're
22:27	number three . What is this calculation gonna look like
22:29	? Okay , so this is gonna be the value
22:34	in dollars ? All right , So what's the value
22:37	in year zero ? At the beginning , so to
22:39	speak . It's just gonna be whatever the principle is
22:41	, whatever that number is , let's pretend that we've
22:45	invested $1000 . Okay , So let's say that this
22:49	number is actually $1000 . That's what I start with
22:52	. What is going to happen at the Kind of
22:55	the beginning of the next year , after one year
22:56	has passed . Well , what's going to happen is
22:59	I'm going to take this principle and I'm gonna multiply
23:01	by 1.08 . The one means that I'm kind of
23:04	carrying over what I started with . But the .08
23:07	is the free money I'm getting there . So ultimately
23:10	I multiplied by 1.08 . So what do I have
23:12	here ? I have the principal times 1.8 , that's
23:16	how much money I'm gonna earn . So if I
23:17	if I started with $1000 and I multiply by 1.8
23:22	, I'm going to get a number 1080 So at
23:26	the end of year , at the beginning of the
23:27	next year , once the first year has happened ,
23:30	I no longer have $1000 in the bank . I
23:32	have 1000 and $80 in the bank . The 1000
23:35	is the money I started with . And the $80
23:38	is the free money . Right point Oh eight times
23:40	1000 is 80 . So the free money is the
23:42	80 and I add them together and I get 1000
23:44	and $80 . Okay , what is going to happen
23:47	at the beginning of the next year ? Well I'm
23:50	gonna take this balance , the $1,080 . I'm gonna
23:53	multiply it again by 1.08 because the next year I
23:56	turn again 8% interest and I can still keep what
23:59	I had started with there to begin with . So
24:02	that's going to come out to 1166 decimal 40 .
24:08	So 1166 decimal 40 on $1000 . Now in terms
24:13	of math , what's going to happen is you're gonna
24:15	take that principle , you're gonna multiply it by one
24:17	point oh eight . But then you're gonna you're gonna
24:19	take in your you're multiplying this by one point oh
24:22	eight . So I take this and multiply by 11
24:25	point oh eight . So what I have is one
24:26	point oh eight squared . Okay . Because what's happening
24:30	is between this step in this step , I multiplied
24:34	by 1.8 . But between this step in this step
24:36	I again multiplied by 1.08 . So this is times
24:40	1.0 and that's why it's squared right here . So
24:42	what do you think is going to happen Over here
24:44	? Well , I'm going to have a year number
24:47	three , I'm gonna have 1.08 . It's gonna be
24:50	cubed . Why is it gonna be cubed ? Because
24:53	what I do is I take the money I had
24:55	in your number two . And again I multiplied by
24:57	one point oh eight . So every year you're multiplying
24:59	by one point oh eight . And if you're going
25:01	to do this kind of pretend on $1000 I would
25:04	take this number and multiply by one point oh eight
25:06	and I would have one too 59 0.70 . I
25:11	could be rounding here . So the exact number .
25:13	Maybe not quite right , but you see I started
25:15	with 1000 then I went to 10 80 then I
25:17	went to 11 66 then I went to 12 59
25:20	. And so I continue this process . Every year
25:22	. On the end of the first year I multiplied
25:24	by one point oh eight I get some number .
25:26	Then the next year I multiplied by 1.8 I get
25:28	some number and then the next year multiplied by one
25:30	point of weight . I get some number just like
25:32	we were multiplying by twos over and over and over
25:34	again . For that ridiculous example where I was giving
25:37	you so much free money . It's crazy before .
25:39	This is a much more realistic thing , but notice
25:41	what's happening . It's p times one point oh eight
25:44	but in your number two because you multiply by one
25:46	point oh eight again , it's really p times one
25:48	point oh eight squared because it's multiplied two times .
25:52	Then the next year it's p times one point oh
25:54	eight cube 123 Because I'm multiplying again three times because
25:58	I'm taking the previous number and multiplying it again there
26:01	. So what is gonna happen if I extend to
26:04	year number four then it's going to be p times
26:06	one point oh eight to the fourth , Power Year
26:08	, number five , p times one point oh eight
26:10	to the fifth power and so on . So instead
26:12	of going in there and talking about year number four
26:14	and number five , let's go down the road to
26:16	year number T . It's generalize it to year number
26:20	T . So then this is going to be p
26:23	times one point oh eight , not to the fourth
26:26	, not to the fifth , not to the six
26:27	. We're generalizing it . So it's going to be
26:29	to the teeth power . That's what that means .
26:32	Because every year you're multiplying by 1.08 . So the
26:35	exponent goes up and up and up and look at
26:37	what this is . P times 1.08 to the power
26:40	of tea . That is an exponential function . Remember
26:43	exponential functions have the variable in the exponent , that's
26:47	what it means . The time in years is a
26:50	variable that is now in the exponent here . That
26:53	is why compound interest is an exponential function . That
26:57	is why it grows so rapidly . Like these exponential
27:00	functions that we have been studying . Okay Now one
27:03	more definition I want to give you before you go
27:05	on to the next board is for this example we
27:07	were saying that we earned 8% interest compounded annually .
27:11	Okay . So what this means the actual , you
27:15	know , um term that we're going to see in
27:17	your problems . This is called an annual interest .
27:24	Wait , It's an interest rate because it's the percentage
27:29	of multiply by and its annual because it means you
27:31	do it every year , every year I get another
27:33	8% every year I get another 8% and so on
27:36	. All right . So we want to talk about
27:39	this in the most general form . So what this
27:42	means is that if I'm compounding annually like this ,
27:44	the amount of money I'm gonna get a this is
27:47	called the amount of money I'm gonna have . It's
27:49	going to be equal to the principle . However much
27:51	money I put into this thing . Times in this
27:53	example 8% The 1.0 it means I'm multiplying by the
27:57	full balance of what I had before . Plus 8%
28:00	free money to the power of T . So it's
28:04	really important to study this . This T . Is
28:07	time in years , right ? This is called the
28:15	interest rate . Specifically the .08 part of it is
28:21	the interest rate . The p is called the principal
28:27	and the a . Is called the amount . That's
28:31	the amount of money I'm going to have total .
28:33	So I can do this as an annual compounding where
28:36	I get 5% a year , 5% a year ,
28:39	5% a year over and over and over . And
28:40	this is gonna be the equation that governs it .
28:42	However in real life this is where it starts to
28:45	get a little bit confusing but I'm going to make
28:47	it very easy to understand for you . Usually we
28:50	don't compound interest every year . Usually interest is compounded
28:55	quarterly , which means four times a year . There's
28:57	four quarters in a year , you know , four
28:59	quarters in a year , just like there's four quarters
29:01	of a peanut butter and jelly sandwich when you cut
29:02	it for places , right ? Sometimes you even have
29:06	loans where the interest is compounded monthly or daily or
29:09	something like that . So we need to talk about
29:11	, what does it mean when you compound something other
29:14	than annually ? So when you have annual compounding ,
29:17	it means every year . That's when you do the
29:18	calculation your number one , you're number two year .
29:20	That's the only time you get the free money is
29:22	every year . But when you compound quarterly , you
29:25	get the free money every quarter , that means four
29:27	times a year . So after three months , that's
29:30	1/4 of the year bam . You calculate the free
29:33	money then after three more months bam you calculate the
29:36	next amount of free money then and so on ,
29:38	and so forth . You do it four times a
29:39	year . That's what it means . But what happens
29:41	is when you compound quarterly , um Let's talk about
29:46	this . Let's just just write it down , Compounds
29:50	quarterly . All right . What does it mean ?
29:55	What it means is if I take that annual uh
29:59	percentage , remember the annual interest rate in this example
30:03	? Here was 8% . That means I get 8%
30:06	a year , the annual interest rate . But when
30:08	I compound quarterly , I'm not gonna give you 8%
30:10	every quarter . That's crazy . That's too much free
30:12	money . What happens is you take the 8% over
30:15	the year . And since I'm doing it four times
30:17	a year , I just divide that 8% by four
30:20	. That's the amount of interest I'm gonna give you
30:22	every quarter . So what happens is I take the
30:24	8% that I'm doing on a on an annual basis
30:27	, but instead of doing it every year , I'm
30:28	going to do it every quarter . That means I'm
30:31	going to take this thing and divide it up Into
30:33	four pieces , and I'm gonna give you 2% interest
30:37	as a quarterly growth rate . And again , this
30:45	is where a lot of students get confused because it
30:46	is confusing . What happens is if I give you
30:49	8% a year , this is easy to understand every
30:51	year , 8% , 8% , But if I compounded
30:54	quarterly , I'm not going to give you 8% because
30:56	that's an annual rate . So I take the 8%
30:59	, I cut it into four pieces , that means
31:01	it's 2% , I'm gonna give you 2% every quarter
31:04	. That will add up to 8% over the year
31:06	. But I'm gonna do the calculation on the free
31:08	money every quarter . So I'm gonna give you 2%
31:11	every quarter four times a year . Okay . So
31:16	then what you would have is that you would have
31:18	the value at end of the quarter . Remember I'm
31:25	doing the calculation now every quarter . Well actually the
31:30	way I want to do it at the end of
31:31	Q . At the end of Q four is going
31:39	to be equal to P times 1.02 to the power
31:45	of Q . Make sure you understand that it's exactly
31:47	the same formula as this . This was the principal
31:50	times the interest rate raised to the power of tea
31:52	and years . But now I'm not compounding yearly ,
31:55	I'm compounding a lot more often quarterly . There's four
31:58	quarters in a year , but still after Q .
32:01	Quarters I'm gonna multiply this balance by one point oh
32:04	two . Then times itself 1.02 then times itself 1.2
32:08	. But I'm gonna do it every quarter every three
32:10	months , I'm gonna do this thing . So it's
32:12	gonna be the exact same table except the numbers up
32:14	here are going to be in quarters and not in
32:16	years and the interest rate has to be smaller because
32:19	I'm doing it more often and you take the annual
32:22	and by four to get that number . Okay ?
32:26	So then you can say the value at the end
32:32	of t years when I'm compounding quarterly . Like this
32:38	is going to be equal to the principal times one
32:41	point oh two . Now instead of Q . You're
32:43	gonna have four times t . Why are you gonna
32:47	have four times T up here ? Why is this
32:49	four times T . Because every year has four quarters
33:00	. So this is where it gets a little confusing
33:02	. You see what happens is that I'm really doing
33:03	an 8% annual growth rate but I'm gonna compounded quarterly
33:07	so I'm really gonna make it 2% . I'm gonna
33:09	give it to you every quarter . So this principle
33:11	is time this interest rate every quarter . So at
33:14	the end of Q . Quarters . This is what
33:15	it is . But the difference between this and this
33:17	is if I want to calculate it in years Like
33:19	let's say one year passes . But in that one
33:22	year if I put one in for tea I've actually
33:24	compounded four times because I'm compounding every quarter . So
33:28	if I put a t . Of one year I'm
33:30	going to have this exponent of four because I've done
33:32	it four times every three months four times . If
33:36	I put a . T . Of two years in
33:37	there then I'm going over a long period of time
33:40	. I've actually compounded this thing eight times because I've
33:43	done it four times a year . So 12345678 every
33:47	knock is every time I multiplied by one point oh
33:50	two like this . So this equation is the value
33:53	at the end of t years compounded quarterly . That's
33:56	why the four pops up there now . Why are
33:59	we doing all this stuff ? Because in every algebra
34:02	book or every math book , finance book , whatever
34:04	that talks about compound interest , you are going to
34:07	come across the compound interest formula . And to be
34:18	honest with you I thought about just writing this equation
34:21	down in the beginning and just being done with it
34:23	. But the the idea the problem is if I
34:25	just throw it at you , you're not gonna understand
34:27	where it comes from now , I'm gonna write it
34:28	down and you're gonna very quickly understand exactly what it's
34:31	saying . The compound interest formula goes like this .
34:34	The value of money that you have some point in
34:36	the future is equal to the principle , which is
34:39	how much money I put in to begin with .
34:41	Times a number one plus are over in , raised
34:47	to the power of in times . T look at
34:51	how this thing compares with something like this , the
34:54	one plus are over in is what's happening here .
34:57	Let me let me write a few more things down
34:59	here . Okay , the a is the amount of
35:03	money that I have , The P is the principal
35:09	, right ? That's easy . The r is the
35:11	annual interest rate , right ? That's the art ,
35:19	this is the number that goes in there for the
35:20	annual interest rate . As a decimal expresses a decimal
35:23	and the end means compounded spot in times per year
35:39	, right ? In times per year . So if
35:43	I'm compounding quarterly , then I'm gonna put a four
35:45	here and so on and then t is the time
35:51	in years . Alright . This equation actually give you
35:56	can get this and you can get these all these
35:58	other ones from it , so I need to show
36:00	you where that basically comes from . Okay , let's
36:03	say that I'm compounding annually . That means once per
36:06	year . Right ? And the interest rate is what
36:08	it is . So let's just say that let me
36:12	go in box . This actually something like this .
36:16	Actually do I want to do this on this page
36:18	? I think I want to do it on the
36:19	next board . Let's say let's first write down the
36:24	compound interest formula . The value is the principal one
36:28	plus are over in to the N . T .
36:34	All right , Let's say we compound annual Annual annually
36:44	. Right . So the interest rate was 8% .
36:49	I can't even write eight . It was 8% .
36:54	Which means the rate that you put into this equation
36:56	is 0.08 . But since I'm compounding annually , remember
36:59	how it goes in means it's compounded in times per
37:02	year . So if I'm only compounding one time per
37:04	year then n is one . Right ? So that
37:07	means that N is equal to one . This is
37:09	what we started the whole lecture with . Right ?
37:11	So if all this is true , how would this
37:13	equation ? Look the rate is going to be the
37:15	principal times one plus one plus The rate is 8%
37:20	which is 0.08 . Always put it as a decimal
37:22	divided by in times per year I've only compounding once
37:26	. So that's divided by one and then raised to
37:28	the power of in times T in times T .
37:31	What does this equation come out to ? It's p
37:34	times when you divide this , you get point oh
37:36	eight , it's one point oh eight to the power
37:39	of thi this is just thi this thing is exactly
37:42	what we started the whole lecture with . You see
37:45	what I mean ? Where is it out here ?
37:46	One point oh eight times this . So then yeah
37:49	, here it is , it's p times one point
37:51	oh eight to the power of tea , that's what
37:52	it is . So all that's happening in this equation
37:55	, when you look at this in your textbook ,
37:57	a lot of students do not know what this is
37:59	saying . All this is saying is I'm taking that
38:01	annual interest rate and I'm dividing it by however many
38:04	times a year I'm compounding because just as I showed
38:07	you right here , if I'm compounding four times a
38:08	year , I'm not going to give you 8% every
38:11	three months every four times a year . I mean
38:13	they only give you 2% . So this division here
38:15	is telling you what this quarterly growth rate is ,
38:19	then I'm adding that here to get the new ,
38:21	the new multiplier factor , the 1.02 in this example
38:24	. And then the exponent is telling me how many
38:27	compounding periods I have . If I'm putting years in
38:30	and I'm doing it quarterly , then what's gonna happen
38:32	is I'm going to have four compounding period . So
38:35	it has to be raised to the power of four
38:37	. Mhm . So let's go over here and see
38:39	what happens if I compound quarterly , Right , it's
38:48	the same sort of thing . The rate is 0.08
38:51	, but now I'm compounding four times a year .
38:53	So I'm compounding we're in is equal to four ,
38:56	it's compounding in times per year . So it's gonna
38:59	be is the principal times one plus what do I
39:02	have here are the rate 0.08 . That's the annual
39:05	rate compounded four times a year . This is doing
39:08	the division giving me that smaller percentage . And then
39:11	the exponent here is in times t this is four
39:14	times T notice this is exactly what we said ,
39:16	it would be . This is p times one point
39:18	oh two . Then you divide this out here ,
39:21	you get 1.2 to the four to the power of
39:23	14 . That's exactly what we talked about . Where
39:26	was that compounded quarterly ? Mhm . What was it
39:30	that ? Yeah . Right here , compounded quarterly .
39:33	This is if you're just going to talk about it
39:34	in terms of number of quarters , this is gonna
39:36	be what happens if you talk about it in terms
39:38	of the number of years . Okay , so when
39:40	you see the compound interest formula , the way you
39:42	should read it as follows . The exponent up here
39:45	has the time variable . That means it's an exponential
39:48	function . That's going to grow like an exponential function
39:51	. The division here is just because if I'm not
39:53	compounding annually , I need to figure out what the
39:56	new rate is that I'm giving you quarterly . So
39:59	I divided by the number of compounding periods to get
40:01	a smaller interest rate . Then this one plus ends
40:04	up giving me the multiplier factor in here . The
40:07	exponent has to have the number of compounding periods per
40:10	year times the number of years . So this exponent
40:13	is just how many compounding periods I have altogether .
40:17	All right . That's it . That's all you have
40:19	. So what I want to do is at least
40:20	work One problem here to give you a little bit
40:23	of practice . Okay , here's the problem . You
40:28	invest $1,000 at an annual rate of 12% . Compounded
40:33	quarterly . Come down a quarter . There should be
40:35	a period right here . How long does it take
40:37	to triple this thing in value ? So what you
40:40	have to do when you're confronted with a compound interest
40:44	problem is to first write down the compound interest equation
40:47	and then identify what in the problem you're given and
40:50	then the rest of it will fall out . So
40:52	let's start by doing that . The value of my
40:55	money in the future is going to be equal to
40:57	the value of the money that I start with time
41:00	. Some factor . The factor is one plus whatever
41:04	the interest rate is divided by however many compounding periods
41:08	I have to give me kind of a new lower
41:09	rate raised to the power of in times T this
41:13	expo is the total number of compounding periods per year
41:17	, times a number of years . So when you
41:18	multiply you get the total number of compounding periods I
41:21	have now in this case we are given that that
41:27	the principal is 1000 but we're trying to figure out
41:32	how long it takes to triple in value . So
41:34	the amount down the road that I care about is
41:36	three times down . It's given to you by telling
41:39	you it's trying to triple in value The annual rate
41:42	which is always our is 12% . But you never
41:45	ever want to put a percentage in a math equation
41:47	ever . You never ever , ever ever ever put
41:50	a percentage in like that . You always convert it
41:52	to a decimal 0.12 and then it tells you it's
41:56	compounded quarterly . That means in times per year ,
41:59	so n is equal to four . So now at
42:01	this point we have to go and put everything into
42:03	this thing . Let's see what it looks like .
42:05	The amount that I'm gonna have down the road is
42:07	3000 . The amount that us start with is 1000
42:11	. The principle Inside of here I have one plus
42:14	the rate which is 12% . But I never ever
42:16	want to use this , I always want to do
42:18	0.12 . I'm dividing it by four because I'm actually
42:21	compounding this four times a year . So I'm going
42:23	to actually have a smaller interest rate that I'm compounding
42:26	more frequently . Okay ? And then the exponent in
42:30	is four times teeth . Okay . And I'm trying
42:34	to figure out how many years down the road does
42:36	it take to do this ? Okay , so then
42:40	what happens I divide by 1000 and over on the
42:42	right hand side , I'm going to have a three
42:44	. And then inside of here it's gonna be uh
42:48	one Plus when you divide this you're going to get
42:51	0.03 to the power of four T . So what
42:56	you're going to have when you flip everything over here
42:58	, a better way to write it is 1.03 .
43:01	When you add these together at 1.03 , that's kind
43:03	of like the new effective number . You're multiplying by
43:05	every quarter to the power of fourty is equal to
43:09	three . I just wrote this side on the other
43:11	and move the three to the other side . This
43:13	is an exponential equation . We know how to solve
43:15	exponential equations . Now there's several different ways but the
43:18	easiest way is to take the law algorithm of both
43:20	sides , right ? The law algorithm of both sides
43:23	. Because when I take the log then I can
43:25	pull the exponent out . So you can do whatever
43:27	base log you want . But I'm gonna do a
43:29	base 10 log Of 103 to the power of four
43:34	times . T . Is equal to the log of
43:37	three . Take the log rhythm of both sides .
43:39	Now why do I do that ? Because when I
43:40	take the log of this the exponent can come out
43:42	four times T times the log of 1 , 3
43:48	is equal to the log of three . So now
43:52	I can solve for T . So what's going to
43:54	happen here ? Let's go up over here . T
43:56	. As an exact number when I divide by the
44:00	four and I also divided by the log here ,
44:02	it's going to be the log The base 10 logarithms
44:05	of three , two by four , Divide also by
44:09	the log of 1 3 . Yes . So in
44:14	your calculator you can go do a base 10 log
44:16	of three and a base 10 log of this and
44:18	you know what the number four is . So then
44:19	you can calculate this . So what you're gonna have
44:21	then What is the base 10 log of 3 ?
44:24	You're gonna get 0.4771 . Yeah and then log of
44:29	13 you'll get zero 0128 like this . And then
44:37	whenever you do the division this divide by the multiplication
44:40	of those two things you're going to get When you
44:42	around the decimals here 9.3 years now you're tempted to
44:48	circle this because that's how long it's gonna take to
44:50	triple in value . However if you actually circle this
44:53	I'd probably give you full credit . However this thing
44:56	is telling you that the interest is compounded quarterly .
44:59	That means you only get the money every three months
45:02	. That's when the new calculation happens , it comes
45:04	into your account every three months . But the answer
45:07	that I said this is going to triple in value
45:09	in 9.3 years . But a quarter of the year
45:12	is actually 9.25 and then the next time I get
45:15	money is another quarter which is 9.5 . So the
45:18	9.25-9.5 and 9.75 and 10 years . That's when I
45:23	get the money . So it really I'm not going
45:25	to actually have tripled the money in 9.3 years because
45:28	I don't if I if you can really only get
45:31	money either 9.25 or 9.5 . But if I pick
45:35	9.25 is the answer , I'm not gonna quite have
45:38	triple the money , I'll have a little bit less
45:40	, So I'm actually gonna say that it's going to
45:42	be basically in 9.5 years , I think this is
45:47	the best answer you could say this is the exact
45:49	answer , but since it's compounded quarterly , if I
45:51	wait till 9.5 years , I'm gonna have slightly more
45:54	than triple the value of this money . If I
45:56	were to wait to just nine and a quarter years
45:58	, which is actually closer to this number , I
46:00	wouldn't quite have triple the value , so I think
46:02	this is the better answer , so we have covered
46:06	one of the most important Ideas and all of Math
46:08	Einstein called it the eighth wonder of the world ,
46:11	because if you understand how compound interest works , you
46:13	can use it to amass great wealth and prosperity for
46:17	yourself and your family . If you don't understand how
46:19	it works , then you take out a bunch of
46:20	credit cards and say I'll just pay them later and
46:23	then 10 years down the road , you have so
46:24	much crippling debt , you can never ever get out
46:26	from under it . So what I want to understand
46:28	and make sure you understand is that the growth of
46:30	money , like this is not a straight line ,
46:33	it is not a straight line , it is an
46:35	exponential curve that goes up right . And the reason
46:38	it's an exponential curve that goes up is because when
46:41	you take a principal and you multiply every year with
46:45	some multiplier factor that includes your interest rate . What's
46:48	happening is you're increasing the exponent every year and that
46:51	means it's an exponential , has an exponential curve which
46:54	means it's going to go up as an exponential .
46:56	Now that's called an annual interest rate with annual compounding
46:59	, but oftentimes interest is not compounded annually , it's
47:03	compounded quarterly , but I'm still not going to give
47:05	you that 8% every quarter , I'm gonna take that
47:07	8% divided by four and I'm gonna give you 2%
47:10	every quarter . The formula is exactly the same ,
47:13	but the number in here is now smaller , but
47:16	I'm doing it more often because I'm not doing it
47:18	every year , I'm doing it four times per year
47:20	. So the exponents four times bigger because I'm doing
47:23	it four times more often . This is what I
47:25	tried to do in terms of numbers to make you
47:27	understand what the compound interest formula is doing your principal
47:31	times this multiplier factor . This is the new interest
47:34	rate that comes from taking the annual rate divided by
47:37	however , times I'm compounding per year , the exponents
47:40	is again , how many times I'm compounding per year
47:42	, times a number of years . And that is
47:44	how you can , if you put , if you're
47:47	doing an annual thing , you can get exactly the
47:49	same equation as we derived already before . If you're
47:52	doing a quarterly thing , you get exactly the same
47:54	equation as we're deriving before . And then of course
47:56	we did our problem where we had to put in
47:58	the value of money , the value we started with
48:01	and backwards calculate so often times you have to use
48:03	algorithms to get to the final answer . I would
48:06	encourage you to watch this video a couple of times
48:09	because if I could put one piece of knowledge in
48:11	people's brains that could help them in their everyday life
48:13	, it would be the knowledge in this lesson .
48:15	You don't know how many people out there , have
48:17	no idea how money actually works . And so when
48:20	you go out and you buy a car or a
48:22	house that's way more expensive than you can afford and
48:25	you're paying that interest on a home that you can't
48:27	afford , then you may not understand how in deep
48:31	trouble you can be until much , much later down
48:33	down the line . So I want you to understand
48:35	solve these problems , make sure you conquer this and
48:38	understand it with me and then go on to the
48:40	next lesson where we'll continue working compound interest problems dealing
48:44	with the exponential growth of money .

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