What I want to do in this

video is go over the math behind a mortgage loan. And this isn’t really going

to be a finance video. It’s actually a lot

more mathematical. But it addresses, at least in

my mind, one of the most basic questions that’s at least been

circling in my head for a long time. You know, we take out these

loans to buy houses. Let’s say you take out a

$200,000 mortgage loan. It’s secured by your house. You’re going to pay it over–

30 years, or you could say that’s 360– months. Because if you normally pay the

payments every month, the interest normally compounds

on a monthly basis. And let’s say you’re

paying 6%– interest. This is annual interest, and

they’re usually compounding on a monthly basis,

so 6% divided by 12. You’re talking about

0.5% per month. Now normally when you get a

loan like this, your mortgage broker or your banker will look

into some type of chart or type in the numbers into some

type of computer program. And they’ll say oh OK,

your payment is going to be $1,200 per month. And if you pay that $1,200 per

month over 360 months, at the end of those 360 months you

will have paid off the $200,000 plus any interest that

might have accrued. But this number it’s not

that easy to come along. Let’s just show an example of

how the actual mortgage works. So on day zero, you

have a $200,000 loan. You don’t pay any

mortgage payments. You’re going to pay your

first mortgage payment a month from today. So this amount is going to be

compounded by the 0.5%, and as a decimal that’s a 0.005. So in a month, with interest,

this will have grown to 200,000 times 1 plus 0.005. Then you’re going

to pay the $1,200. Just going to be minus 1,200

or maybe I should write 1.2K. But I’m just really just

showing you the idea. And then for the next month,

whatever is left over is going to be compounded

again by the 0.5%, 0.005. And then the next month you’re

going to come back and you’re going to pay this $1,200 again. Minus $1,200. And this is going to

happen 360 times. So you’re going to

keep doing this. And you can imagine if you’re

actually trying to solve for this number– at the end of it

you’re going to have this huge expression that’s going to have

you know 360 parentheses over here– and at the end, it’s

all going to be equal to 0. Because after you’ve paid your

final payment, you’re done paying off the house. But in general how did they

figure out this payment? Let’s call that p. Is there any mathematical

way to figure it out? And to do that, let’s get a

little bit more abstract. Let’s say that l is equal

to the loan amount. Let’s say that i is equal

to the monthly interest. Let’s say n is equal to

the number of months that we’re dealing with. And then we’re going to set

p is equal to your monthly payment, your monthly

mortgage payment. Some of which is interest, some

of which is principle, but it’s the same amount you’re going to

pay every month to pay down that loan plus interest. So this is your

monthly payment. So this same expression I just

wrote up there, if I wrote it in abstract terms, you start

off with a loan amount l. After 1 month it

compounds as 1 plus i. So you multiply it times

1 plus i. i in this situation was 0.005. Then you pay a monthly

payment of p, so minus p. So that’s at the

end of one month. Now you have some amount still

left over of your loan. That will now compound

over the next month. Then you’re going to

pay another payment p. And then this process is going

to repeat 300 or n times, because I’m staying abstract. You’re going to have

n parentheses. And after you’ve done this

n times, that is all going to be equal to 0. So my question, the one that

I’m essentially setting up in this video, is how

do we solve for p? You know if we know the loan

amount, if we know the monthly interest rate, if we know the

number of months, how do you solve for p? It doesn’t look like this is

really an easy algebraic equation to solve. Let’s see if we can

make a little headway. Let’s see if we can rearrange

this in a general way. So let’s start with an example

of n being equal to 1. If n is equal to 1, then our

situation looks like this: you take out your loan, you

compound it for one month, 1 plus i, and then you pay

your monthly payment. Now this was a mortgage that

gets paid off in 1 month, so after that 1 payment you are

now done with their loan, you have nothing left over. Now if we solve for p, you

can now swap the sides. You get p is equal to

l times 1 plus i. Or if you divide both sides

by 1 plus i, you get p over 1 plus i is equal to l. And you might say hey you

already solved for p why are you doing this? And I’m doing this, because

I want to show you a pattern that’ll emerge. Let’s see what happens

when n is equal to 2. Well then you start

with your loan amount. It compounds for one month. You make your payment. Then there’s some

amount left over. That will compound

for one month. Then you make your

second payment. Now this mortgage only

needs two payments, so now you are done. You have no loan left over. You’ve paid all the

principal and interest. Now let’s solve for p. So I’m going to color the p’s. I’m going to make this p pink. So let’s add p to both

sides and swap sides. So this green p will be

equal to all of this business over here. Is equal to l times 1 plus

i minus that pink p. They’re the same p, I just

want to show you what’s happening algebraically. Minus that pink p

times 1 plus i. Now if you divide both sides

by 1 plus i, you get p over 1 plus i is equal to l times

1 plus i minus that pink p. Now let’s add that pink p to

both sides of this equation. You get the pink p plus this

p plus p over 1 plus i is equal to l times 1 plus i. Now divide both

sides by 1 plus i. You get the pink p over 1 plus

i plus the green p, the same p, times– it already is being

divided by 1 plus i, you’re going to divide it again by 1

plus i, so it’s going to be divided by 1 plus i squared

is equal to the loan. Something interesting

is emerging. You might want to watch the

videos on present value. In this situation, you take

your payment, you discount it by your monthly interest rate,

you get the loan amount. Here you take each of your

payments, you discount it, you divide it by 1 plus your

monthly interest rate to the power of the number of months. So you’re essentially taking

the present value of your payments and once again,

you get your loan amount. You might want to verify this

for yourself if you want a little bit of algebra practice. If you do this with

n is equal to 3. I’m not going to do it just

for the sake of time. If you do n is equal to 3,

you’re going to get that the loan is equal to p over 1 plus

i plus p over 1 plus i squared plus p over 1 plus

i to the third. If you have some time, I

encourage you to prove this for yourself just using the exact

same process that we did here. You’re going to see it’s going

to get little bit harry. There’s going to be a lot of a

manipulating things, but it won’t take you too long. But in general, hopefully, I’ve

shown to you that we can write the loan amount as the present

value of all of the payments. So we could say in general the

loan amount, if we now generalize it to n instead of

and n equals a number, we could say that it’s equal to– I’ll

actually take the p out of the equation, so it’s equal to p,

times 1 plus 1 over 1 plus i plus 1 over 1 plus i squared

plus, and you just keep doing this n times, plus 1

over 1 plus i to the n. Now you might recognize this. This right here is a

geometric series. And there are ways to figure

out the sums of geometric series for arbitrary ends. As I promised at the beginning

of the video this would be an application of a

geometric series. It’s equal to the sum of 1 over

1 plus i to the, well I’ll use some other letter here, to

the j from j is equal to 1. This is to the one power you

could view this is to the first power to j is equal to n. That’s exactly

what that sum is. Let’s see if there’s any simple

way to solve for that sum. You don’t want to

do this 360 times. You could, you’ll get a number,

and then you could divide l by that number, and you

would have solved for p. But there’s got to be simpler

way to do that, so let’s see if we can simplify this. Just to make the math easier,

let me make a definition. Let’s say that r is equal

to 1 over 1 plus i. And let me call

this whole sum s. This sum right here

is equal to s. Then if we say r is equal to

each of these terms then s is going to be equal to this is

going to be r to the first power. I’ll write r to first this is

going to be r squared, because if you square the numerator

you just get a 1 again. So this is plus r squared plus

r to the third, plus all the way this is r to the n. And I’ll show you

a little trick. I always forget the formula,

so this is a good way to figure out the sum of

a geometric series. Actually this could be used to

find a sum of an infinite geometric series if you

like, but we’re dealing with a finite one. Let’s multiply s times r. So r times s is going

to be equal to what? If you multiply each of these

terms by r, you multiply r to the first times r

you get r squared. You multiply r squared times

r you get r to the third. And then you keep doing that

all the way, you multiply r– see there’s an r to the n minus

one here– you multiply that times r, you get r to the n. And then you multiply r to

the n times r, you get plus r to the n plus 1. All this is right here is all

of these terms multiplied by r, and I just put them

under the same exponent. Now what you can do is you

could subtract this green line from this purple line. So if we were to say s

minus rs, what do we get? I’m just subtracting this

line from that line. Well, you get r1 minus 0,

so you get r to the first power minus nothing there. But then you have r squared

minus r squared cancel out r to the third minus r to

the third cancel out. They all cancel out, all the

way up to r to the n minus r to the n cancel out, but then

you’re left with this last term here. And this is why

it’s a neat trick. So you’re left with minus

r to the n plus 1. Now factor out an s. You get s times 1 minus r– all

I did is I factored out the s– is equal to r to the first

power minus r to the n plus 1. And now if you divide

both sides by 1 minus r, you get your sum. Your sum is equal to r minus r

to the n plus 1 over 1 minus r. That’s what our sum is

equal to, where we defined our r in this way. So now we can rewrite this

whole crazy formula. We can say that our loan amount

is equal to our monthly payment times this thing. I’ll write it in green. Times r minus r

to the n plus 1. All of that over 1 minus r. Now if we’re trying to solve

for p you multiply both sides by the inverse of this, and you

get p is equal to your loan amount times the

inverse of that. I’m doing it in pink,

because it’s the inverse. 1 minus r over r minus

r to the n plus 1. Where r is this

thing right there. And we are done. This is how you can actually

solve for your actual mortgage payment. Let’s actually apply it. So let’s say that your loan

is equal to $200,000. Let’s say that your interest

rate is equal to 6% annually, which is 0.5% monthly which

is the same thing as 0.005. This is monthly interest rate. And let’s say it’s a 30 year

loan, so n is going to be equal to 360 months. Let’s figure out what we get. So the first thing we want

to do is we want to figure out what our r value is. So r is 1 over 1 plus i. So let’s take 1 divided by

1 plus i so plus 0.005. That’s what our monthly

interest is, half a percent. So 0.995 that’s what

our r is equal to. Let me write that down, 0.995. Now this calculator doesn’t

store variables, so I’ll just write that down here. So r is equal to 0.995. We just used that right there. I’m losing a little bit

of precision, but I think it will be OK. The main thing is I want to

give you the idea here. So what is our payment amount? Let’s multiply our loan amount

that’s $200,000 times 1 minus r, so 1 minus 0.995 divided by

r which is 0.995 minus 0.995 to the of the– now n is 360

months, so it’s going to be 360 plus 1 to the 361 power,

something I could definitely not do in my head, and then I

close the parentheses, and my final answer is roughly $1,200. Actually if you do it with the

full precision you get a little bit lower than that, but this

is going to be roughly $1,200. So just like that, we were

able to figure out our actual mortgage payment. So p is equal to $1,200. So that was some reasonably

fancy math to figure out something that most people deal

with everyday, but now you know the actual math behind it. You don’t have to play with

some table or spreadsheet to kind of experimentally

get the number.

awesome worth the time.

thanks a lot…its very nice of you to share your knowledge….

Nice man thanks for your time.

This took us about 5 lessons at A Level.

Love Khan!!

Brilliant and magical!

that was so sexy

Where is this clip on the Khanacademy site?

he uses a drawing tablet, not a mouse

Is 6% apr the same as 0.5% monthly rate?

If you mean approximate, it's exactly! there's 12 months in a year 6 divided by 12 is 0.5 !

ohhh ,cool now im know why at the the end we paid 113 % interest more actually load is .how is the real mafia now hahaha 🙂

Khan is an Indian right?

12 months in 1 year. => 6% Apr / 12 months = 0.5% Apr / 1 month

–

Or you could also say : 0.5% Apr = 6% of the monthly payment. (=> $ 1200 in the vid)

So your ''Monthly Interest Rate'' doesn't change, it remain 6% just like your ''Annual Interest Rate''.

–

In fact in this case, your Interest ''Rate'' NEVER changes, whether you're paying annually, monthly or weekly.

Mr. Khan, Thank you so much for such clear explanation. I get it now!

That's a monthly interest of $644.44. What a rip-off!

Amazing!

I'm looking for this over a long period. Thank you very much. Khan. You are great!

/math/precalculus/seq_induction

Thank you! This is what my teacher refused to show us, because it was "too hard" for her to derive the formula.

Thank you for destroying my brain and then regrowing it.

can you please help explain what if it is bi-weekly payment and interest rate compounded semi annually. what does the formula looks like?

Very good explanation and I think it's one of the best for mathematical background for Mortgage payment derivation!

1200 times 360 is 432,000. so in this case youre paying over 100% of your loan in the end?

Thank you

Very helpful for me and my lack of math knowledge

is n always calculated in months? can we use the formula for different periods like days or semi annually etc? thank you

this is poetry

4:54 shouldn’t it be n-1 parentheses bcz the first one doesn’t have a parentheses