# e and compound interest | Interest and debt | Finance & Capital Markets | Khan Academy

Let’s say that you are

desperate for a dollar. So you come to me the local loan shark, and you say hey I need

to borrow a dollar for a year. I tell you I’m

in a good mood, I willing to lend you that dollar

that you need for a year. I will lend it to

you for the low interest of 100% per year. 100% per year. How much would you have

to pay me in a year? You’re going to have to pay the original principal what I lent

you plus 100% of that. Plus one other dollar.

Which is clearly going to be equal to $2. You

say oh gee, that’s a lot to have to pay to pay back

twice what I borrowed. There’s a possibility that

I might have the money in 6 months. What kind

of a deal could you get me for that Mr. Loanshark? I say oh gee, if your

willing to pay back in 6 months, then I’ll

just charge you half the interest for half the time. You borrow one dollar,

so in 6 months, I will charge you 50% interest.

50% interest over 6 months. This, of course, was 1

year. How much would you have to pay? Well, you

would have to pay the original principal what you borrowed. The one dollar plus

50% of that one dollar. Plus 0.50, and that of

course is equal to 1.5. That is equal to $1.50. I’ll

just write it like this. $1.50. Now you say well

gee that’s I guess better. What happens if I don’t

have the money then? If I still actually need a year. We actually have a system for that. What I’ll do is just

say that okay, you don’t have the money for me

yet. I’ll essentially … we could think about it.

I will just lend that amount that you need for

you for another 6 months. We’ll lend that out. We’ll lend that out for

another 6 months at the same interest rate at 50%

for the next 6 months. Then you’ll owe me the

principal a $1.50 plus 50% of the principal, plus 75

cents. Plus 75 cents, and that gets us to $2.25. That equals $2.25. Another way of thinking

about it is to go from $1 over the first

period, you just multiply that times 1.5. If your

going to grow something by 50%, you just multiply it times 1.5. If your going to grow it

by another 50%, you can multiply by 1.5 again. One way of thinking about

it that 50% interest is the same thing as multiplying by 1.5. Multiplying by 1.5. If you start with 1 and

multiply by 1.5 twice, this is going to be the same thing. $2.25 is going to be 1

multiplied by 1.5 twice. 1.5 multiplied twice is the

same thing as 1.5 squared. You can see the same

thing right over here. This is the same thing.

100% is the same thing as multiplying by 2. As we

be multiplying 1 plus 1. This is multiplying by

2, so you could do this right over here. You could

do this as 1 times 1. 1 times 1 to the first … I’m sorry

1 times 2 to the first power , because your only doing it over

one period over that year. You say once again

where’s that 2? Well, if someone is asking for

100%, that means over the period you’re going have to pay twice. You’re going have to pay

the principal plus 100%. You’re going have to pay

twice what you originally borrowed. If someone is

charging you 50% over every period, you’re

going have to pay whatever you borrowed. That’s

kind of the one part plus 50% of it. So 1.5 times what

you borrowed. You multiply times 1.5 every time.

If you wanted to see how this actually related to

the interest, you could view this as … this

right over here is equal to 1 times, the interest

part is 1 plus 100% divided by 1 period to the first power. I know this seems like

a crazy way of rewriting what we just wrote over

here. Writing 1 plus 1, but you’ll see that we

can keep writing this as we compound over different periods. This one right over here, we can rewrite. We can write as 1 times 1 plus 100%. Here we took our 100% for the year, and we divided into 2 periods.

Two 6 month periods. Each of them at 50%. 1

plus 100% over 2 is the same thing as 1.5, and

we compounded it over 2 periods. Let me do that

2 periods into a different color. The periods, let

me do in this orange color right over here.

You might start to see a pattern forming. Let’s

say, well gee, I might have the money back in

… and you don’t really like this. This is

$2.25. That was more than the original $2, so you

say, well what if we do this over every 12

months. I say, “Sure. We got a program for that.” After every 12 months

… or after every month I should say, I’m just

going to charge you 100% divided by 12 interest.

This is equal to 8 1/3%. Having to pay back the

principal plus 8 1/3%, that’s the same thing as

multiplying times 1.083 repeating. After 1 month you would

have to pay 1.083 repeating. After 2 months … and

this isn’t the scale that actually looks

more than 2 months, but it’s not completely at

scale. After 2 months your going have to multiply by this again. Times 1.083 repeating,

and so that would get you 1.083 repeating squared.

If you went all the way down 12 months … let

me get myself some space here. If you went all the way

down 12 months … let me just. I should way from the

beginning 12 months, so another 10 months.

What’s the total interest you would have to pay

over a year if you weren’t able to keep coming up with the money? If you had to keep re-borrowing it. I kept compounding that interest. Well, you’re going have

to pay 1.083 to the … this is for 1 month. You

could view this as to the first power. This is

for 2 months, so you’re going have to pay this to the 12th power. We have compounded over

12 periods, 8 1/3% over 12 periods. If you wanted

to write it in this form right over here,

this would be the same thing as the original principal. Our original principal

times 1 plus 100% divided by 12. Now we’ve divided

our 100% into 12 periods, and we’re going to compound that 12 times. We’re going to take

that to the 12th power. What is this going to

equal to? This buisness over here. We can get a

calculator out for that. I’ll get my TI-85 out.

What is this going to be equal to? We could do it a couple of ways. This is 1.083 repeating.

Let’s get our calculator out. We could do it a couple of

ways. Let me write it this way. Your going to get the same value.

I don’t have to rewrite this one. I just did that there to

kind of hopefully you’d see the kind of structure

in this expression. 1 plus … 100% is the same thing as 1. 1 divided by 12 to the 12th

power. 2.613, I’ll just round. So approximately 2.613.

You say well this is an interesting game you all most forgot about your financial troubles,

and you’re just intrigued by what happens if we keep going this. Here we compounded just

… we have 100% over here. Here we do 50%

every 6 months. Here we do a 12th of 100%, 8 1/3% every 12 months until we get to this number. What happens if we did every day? Every day. If I borrowed a one

dollar, and I’d say well gee I’m just going to … each

day I’m going to charge you charge you one three hundred

sixty-fifth of a 100%. So, 100% divided 365,

and I’m going to compound that 365 times. You’re

curious mathematically. You say well, what do we get then? What do we get after a year? You have your original

principal. Let me scroll over a little bit more to the

right, so we have more space. You’re going to have

your original principal times 1 plus 100% divided

by not 12. Now we’ve divided the 100% into 365 periods. 365 periods. We’re going to compound it. Every time we have to

multiply by 1 plus 100% over 365 everyday that

the loan is not paid. 365th power. You say oh gee taking

somebody to 365th power that’s going to give me some huge number. Then you say well maybe

not so bad, because 100% divided by 365 is

going to be a small number. This thing is going to

be reasonable close to 1. Obviously, we can raise

1 to whatever power we want, and we don’t get anything crazy. Let’s see where this one goes. Let’s see where this one goes. This is the same thing

as 1 plus. 100% is the same thing as 1 divided

by 365 to the 365th power. We get 2.71456. Let me put

it over here. Then we get … This is approximately equal

to … this approximate is a very precise

approximation, but 2.7 … but my calculators

precision only goes so far. 2.7145675 and it keeps going on and on. This is really really

interesting. It looks as if we take larger

and larger numbers here, it just doesn’t just

balloon into some crazy ginormous number. It seems

to be approaching some magical and mystical

number. It is, in fact, the case. That if you

would just take larger and larger … if you were

to take your 100% and divide by larger and

larger numbers, but take it to that power, you’re going to approach perhaps the most magical

and mystical number of all. The number E.

You can see it right over here in your calculator.

They have this E to the X. I can do that, so E to

the … I’ll raise it to the first power so you can look at the calculators internal representation of it. You see all ready raising

some … doing 1 plus 1 over 365 to the 365th power, we got pretty … we’re starting

to get really really close to E. I encourage

you try this with larger and larger numbers, and your going to get closer and closer to this magical mystery. You almost wouldn’t mind

paying the loan shark E dollars, because it’s

such a beautiful number.

Could you do a video on various limits involving e, not just this one or the factorial one? It would be pretty interesting to see a proof of e = lim x->inf of x/(x!^(1/x))

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im studying actuarial science. we use the force of interest quite a lot.

Masturbate in california…

thnx this helped and i don't even study this because I'm in 6th grade, but it still helped me understand it =)

Anyone else find yourself getting frustrated at the dope who borrows 1 dollar, and then spends 50 dollars worth of the loan shark's time going through hypothetical repayment scenarios? LOL

been learning economics independently , and your vids are totally awesome

Yeah, but after x periods, you owe (1+1/n)^(n*x), and that's no good. The rate is 1/n, which becomes increasingly appealing for borrowers, but boy does it hurt after compounding. It's basically the question of: "would you rather pay $2 or $e?" Here's a vivid application.

Since the interest per period falls like 1/n, people that are desperate for cash buy things as debt and have to pay back money with interest. These long-period 1/n type loans cause such people to pay back more than they otherwise would. An example of this are 'subprime auto loans', which systematically target low income citizens with bad FICO scores and financial problems. A financially astute individual would never follow a low-interest loan.

Fun fact: in 2015 over $1 trillion was borrowed for US subprime auto loans, for the first time ever (accounting for over 1/3rd of all loan transactions). Yes, ladies and gentlemen, this is how people pay 2, maybe 3 times the value of their asset.

I got excited learning about this. Which really makes sal he BEST teacher I could have.

hahah Sal was in a good mood and agreed to lend you money for a "low interest": 100% a year. So funny when you think it as how you were fooled by some loan guy.

Fascinating, with this number e!