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

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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