Earnings and EPS

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Kind: captions
Language: en

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In the last video we learned
a little bit about what an

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income statement would look
like for a very kind of

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vanilla company that
sells widgets.

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And in this video what I want
to do is close the loop and

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see how this relates to what we
talked about in the first

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video, in terms of
price per share.

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And we'll talk about earnings
per share in this case.

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And then also how the income
statement relates to the

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balance sheet.

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And I said I would do it in
the last video, but then I

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realized I was running
out of time.

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And just as a bit of a quick
review, and you probably just

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watched that video.

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But just in case you didn't,
revenue was literally the

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widgets that I sold.

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Let's say this was for 2008.

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I sold $3 million worth
of widgets.

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Cost of goods sold, well that's
just the cost of the

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goods sold.

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So let's say I sold 3 million
widgets for $1.

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To produce those 3 million
widgets for $1, it cost me

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maybe $0.30 a widget,
or $0.33 a widget.

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And so I had a total expense or
total cost of goods sold of

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$1 million.

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And so the profit from just
selling those-- say in that

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case that I just made up-- $3
million worth of widgets was

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$2 million.

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But I have other expenses.

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I can't just put that
in my pocket.

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I had marketing, sales, general
and administrative.

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And then I was left
with $1 million.

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And that's how much operating
profit my

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widget company has produced.

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But I'm not done yet.

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We learned in the last
video I didn't

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own the company outright.

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I had some debt.

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In this case, $5 million
worth of debt.

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So I have to pay some interest.
There was 10%

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interest. So you take
the interest out.

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You're left with $500,000
of pre-tax income.

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And of course you have to
pay the government.

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And we can debate on whether
we're paying them too much or

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too little.

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But they're providing me defense
so that other foreign

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countries can't come and
bomb my factories.

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And they're providing,
hopefully, an educated

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workforce for me.

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And nice roads and
infrastructure and

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electricity.

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So who knows?

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So that's what I'm paying
for, arguably.

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And so I'm left with net
income of $350,000.

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And when people talk about
earnings of a company, this is

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what they're talking about.

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Net income is also earnings.

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So when someone says, Google
made $4 billion this past

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quarter-- and quarter just means
a quarter of a year, a

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three month period.

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And they normally are literally
March, June,

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September and December.

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So every fourth of the year.

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They're literally saying,
Google made $3

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billion in net income.

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We talked in the first video
when we started this series

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about how different
companies have

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different numbers of shares.

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So let's make up a number of
shares for this company.

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So let's say this company has--
let me do it in mauve--

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so this company's shares,
let's say it

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has 1 million shares.

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So you've probably heard not
only the term earnings, you've

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probably also heard the term
earnings per share.

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So what they do is they just
take your earnings number and

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you divide by the number
of shares.

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And you have earnings
per share.

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Or EPS.

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Sometime people just say EPS for
this company in 2008 was,

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and in this case you take
$350,000, your total earnings,

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divided by the total number
of shares you

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have. And that's what?

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That's $0.35 per share of EPS.

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Before I go into price to
earnings and all that, I want

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to connect the dots between the
income statement and the

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balance sheet.

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So in this period,
I made $350,000.

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So the way you can think about
what an income statement is,

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is it tells you what
happens between

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balance sheet snapshots.

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So let's say on 12/31/2007,
the balance sheet for the

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company would have
looked like this.

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Let me just draw it.

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I won't use the square tool.

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So it had $10 million
of assets.

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It had $5 million of debt.

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You can kind of view this as the
liabilities of which debt

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is one of them.

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Let's just say it's all of it.

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So $5 million of liabilities.

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So the shareholders' equity,
or you could almost say the

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book value of the equity of the
company, is $5 million.

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$5 million equity.

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And just to kind of tie it a
little bit with what we did in

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the first video, if we have a
million shares, the book value

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of the equity is $5 million,
so if we say the book value

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per share is just going to be
this number divided by the

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million shares I have. So
$5 million divided by

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1 million is $5.

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This is the book value
for a share.

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Remember, there's a difference
between book

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value and market value.

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The stock of this company at
12/31/2007, it could have been

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trading at $10, at $7, at $1.

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That would have been the
market value per share.

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This is the book value.

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And the book value is
essentially saying, OK, if you

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take the book value of the
assets-- and I talked about

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this a little bit in the last
video-- but if you say, oh,

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well we paid $10 million, and
this is how much value there.

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And I'll go into more detail
into how you value

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these types of things.

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And you take out the
liabilities, this is what the

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assets are worth according
to the accountants.

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The market value could
be very different.

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But fair enough.

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I don't want to delve
too much into that.

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But now, over this period of
time, over 2008, 2008 happens.

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So the rest of 2008 happens.

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And we're left at the end of,
let's say 1/1/2009, or we

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could say 12/31/2008.

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It doesn't matter.

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Give or take a day.

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It's a holiday anyway.

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We have a new balance sheet.

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But what happened here?

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Over the course of the year,
we earned some money.

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And I won't go too complicated
into cash and accrual

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accounting and all
that right now.

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We will get into that
eventually.

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But I just want to give you the
gist of what's happening.

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Is that we made $350,000
in this year.

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So our assets will have
increased by $350,000.

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It could have increased
by $350,000 of cash.

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It might have been increased
by just other people saying

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they owe us $350,000.

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Those are accounts
receivables.

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I'll make a whole video,
probably a whole series of

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videos, on accounts
receivables.

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It's an asset that says, someone
owes me that money.

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But any way you think about
it, our assets will have

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increased by this amount
of money over the

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course of the year.

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So now, our new balance
sheet at the end of

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2008-- so 2008 happened.

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Let me draw 2008.

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2008 happens.

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At the end of which, our assets
would have grown a

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little bit.

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Our assets are now
at $10,350,000.

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So I'll put this in thousands.

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This is $10,350 thousands.

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So that's the same thing
as $10.35 million.

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And let's see.

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All we did was, we paid
the interest on the

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debt in this example.

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We just paid interest. And
just so you know, most

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corporate debt actually
works that way.

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We'll talk later about
amortization schedules and how

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can you pay down the debt.

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But unlike mortgages, a lot of
corporate debt, they just pay

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the interest. So we still have
the same amount of debt.

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We didn't pay down any of the
principal of the debt.

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So we still have $5
million of debt.

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And what's left over
for the equity?

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Well all of our earnings,
since our

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debt stayed the same.

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All of our earnings actually
fall down straight to equity.

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So we have $5,350,000
of equity.

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So the way I think about it,
and probably the way you

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should think about it, is the
earnings of a company-- in

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this case, we have $350,000
for the year,

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or $0.35 per share.

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They essentially tell you what
happened from one balance

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sheet to another.

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So the amount that this number
grows by, the amount that our

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equity grows by, is earnings.

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So this is $350,000
of earnings.

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And there's something I might
want to touch on.

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Because in the last video I
talked a little bit about

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return on asset, where we just
took the operating profit, and

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we divide that by the assets.

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We had a 10% return on assets.

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And you might say, oh, well why
shouldn't the assets grow

00:09:01.880 --> 00:09:02.760
by that amount?

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And the reason they didn't grow
by that amount is because

00:09:05.080 --> 00:09:08.300
we actually ended up having to
pay taxes on some of that.

00:09:08.300 --> 00:09:10.970
Even if we had no debt, we would
still have to pay taxes.

00:09:10.970 --> 00:09:12.480
This line would have
disappeared.

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So sometimes when people talk
about return on asset-- which

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I did in that last video-- it
might be interesting to say,

00:09:17.530 --> 00:09:20.280
is that the pre-tax or the
post-tax return on asset.

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In this example that I did in
the previous video, it was the

00:09:23.650 --> 00:09:26.140
pre-tax return on asset.

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But fair enough.

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Balance sheets are just
snapshots in time.

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They're kind of like
your bank account.

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How much are you worth
at any one moment?

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And income statements tell you
how do you get from one

00:09:38.710 --> 00:09:41.200
balance sheet to the
next balance sheet?

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And we'll learn later the cash
flow statement essentially

00:09:45.190 --> 00:09:47.680
reconciles the income statement
with what happens

00:09:47.680 --> 00:09:50.350
with your actual cash account,
or how things actually move

00:09:50.350 --> 00:09:51.440
within the balance sheet.

00:09:51.440 --> 00:09:53.470
And we'll do that later on.

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But anyway, we started off in
this video talking about oh,

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what's cheap?

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What's expensive?

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So the question is, how do you
determine that if you can't

00:10:02.400 --> 00:10:07.180
just go by the actual dollar
price of-- my baby is crying.

00:10:07.180 --> 00:10:09.320
I should probably run to that.

00:10:09.320 --> 00:10:10.010
Let me do that right now.

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I realize I'm already
at 10 minutes.

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I'll see you in the
next video.


Introduction to the income statement

WEBVTT
Kind: captions
Language: en

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I figure now is as good a time
as any to learn about probably

00:00:05.420 --> 00:00:08.420
what most people focus the
most on when they analyze

00:00:08.420 --> 00:00:10.190
companies, and that's the
income statement.

00:00:10.190 --> 00:00:13.600
And the income statement is
one of the three financial

00:00:13.600 --> 00:00:15.620
statements that you'll look at
when you look at a company.

00:00:15.620 --> 00:00:17.360
There's the income statement
and the other two are the

00:00:17.360 --> 00:00:21.560
balance sheet, which I have
drawn a lot in a lot of the

00:00:21.560 --> 00:00:23.760
other explanations I've done
on the financial crisis and

00:00:23.760 --> 00:00:24.330
whatever else.

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And actually, in this video,
we're going to see how the

00:00:25.640 --> 00:00:27.750
income statement relates
to the balance sheet.

00:00:27.750 --> 00:00:29.670
And, of course, the last one--
well, it's not of course if

00:00:29.670 --> 00:00:31.015
you don't know it-- is the
cash flow statement.

00:00:35.480 --> 00:00:38.190
And we'll focus on that a
little bit later because

00:00:38.190 --> 00:00:40.080
that's a little bit more
nuanced relative

00:00:40.080 --> 00:00:41.730
to the income statement.

00:00:41.730 --> 00:00:44.570
So the income statement is
literally just saying how much

00:00:44.570 --> 00:00:47.925
a company might earn in a given
period, and it's always

00:00:47.925 --> 00:00:48.920
related to a period.

00:00:48.920 --> 00:00:51.890
So it could be an annual
income statement.

00:00:51.890 --> 00:00:53.790
It could be for the year 2008.

00:00:53.790 --> 00:00:55.480
It could be a quarterly
income statement.

00:00:55.480 --> 00:00:56.970
Those are usually the two
types that you see, but

00:00:56.970 --> 00:01:01.390
sometimes, there's monthly or
six-month income statements.

00:01:01.390 --> 00:01:05.080
And the general format is pretty
consistent, although

00:01:05.080 --> 00:01:07.410
there is a lot of variation
depending on what a business

00:01:07.410 --> 00:01:10.320
does, but in this video, I
really just want to cover

00:01:10.320 --> 00:01:13.020
almost a plain vanilla income
statement for a company that

00:01:13.020 --> 00:01:14.680
just sells a widget.

00:01:14.680 --> 00:01:17.980
So the first thing when you sell
a widget is you make it

00:01:17.980 --> 00:01:19.440
and you just sell it.

00:01:19.440 --> 00:01:20.290
You sell the widget.

00:01:20.290 --> 00:01:25.630
You give a customer a widget,
and they give you some money.

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And that money that they give
you-- and I'm not going to get

00:01:28.310 --> 00:01:30.040
too technical about the
accounting right now-- is

00:01:30.040 --> 00:01:31.290
considered revenue.

00:01:34.360 --> 00:01:39.080
It's sometimes considered
sales.

00:01:39.080 --> 00:01:41.590
And that's literally the money
that they give you at a

00:01:41.590 --> 00:01:42.620
certain period of time.

00:01:42.620 --> 00:01:44.760
And some of you accountants out
there are like, oh, well,

00:01:44.760 --> 00:01:46.450
no, that's not just the money
that they give you.

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It's the money that you've
earned in a certain period of

00:01:48.680 --> 00:01:49.710
time, and that's true.

00:01:49.710 --> 00:01:53.790
But for our sake, let's just
say that when you give the

00:01:53.790 --> 00:01:56.410
widget, you have earned the
money that they give you, and

00:01:56.410 --> 00:01:57.460
that's revenue sales.

00:01:57.460 --> 00:01:59.750
Later on, we'll talk about
different ways to account

00:01:59.750 --> 00:02:01.200
revenue and sales.

00:02:01.200 --> 00:02:03.620
So let's say the revenue or the
sales in this case in a

00:02:03.620 --> 00:02:05.060
given period, let's say
that this is an income

00:02:05.060 --> 00:02:07.700
statement for 2008.

00:02:07.700 --> 00:02:13.050
So over 2008, we sold
let's say $3

00:02:13.050 --> 00:02:14.570
million worth of widgets.

00:02:19.180 --> 00:02:24.340
So let's say it's $3 million.

00:02:24.340 --> 00:02:26.970
And a lot of times when you look
at income statements for

00:02:26.970 --> 00:02:28.040
companies, if you go to Yahoo!

00:02:28.040 --> 00:02:30.310
Finance, you could do this right
now, instead of writing

00:02:30.310 --> 00:02:32.520
$3 million, you'll
see $3,000 there.

00:02:32.520 --> 00:02:33.160
It's like, oh, my God!

00:02:33.160 --> 00:02:35.440
This company, they're hardly
selling anything.

00:02:35.440 --> 00:02:39.250
But it's kind of a standard that
they tend to write things

00:02:39.250 --> 00:02:40.050
in thousands.

00:02:40.050 --> 00:02:42.950
So 3,000 would be
3,000 thousands,

00:02:42.950 --> 00:02:44.000
which would be 3 million.

00:02:44.000 --> 00:02:46.060
And for really big companies,
they actually sometimes write

00:02:46.060 --> 00:02:47.330
their numbers in millions.

00:02:47.330 --> 00:02:50.530
So if you saw 3,000 there, it
would actually mean 3 billion.

00:02:50.530 --> 00:02:53.190
But we'll actually look at real
income statements in the

00:02:53.190 --> 00:02:54.310
not-too-far-off future.

00:02:54.310 --> 00:02:57.770
So that's how much money
they give us.

00:02:57.770 --> 00:03:01.910
But that's not how much income
we made, because there was a

00:03:01.910 --> 00:03:04.750
lot of cost that went into
making that widget that we

00:03:04.750 --> 00:03:05.280
have to account for.

00:03:05.280 --> 00:03:07.710
It's not like when someone gives
me $3 million, I can

00:03:07.710 --> 00:03:09.100
just say, oh, I made
$3 million.

00:03:09.100 --> 00:03:10.460
Let me just put it
all in the bank.

00:03:10.460 --> 00:03:11.040
I'm done.

00:03:11.040 --> 00:03:12.980
That was all income.

00:03:12.980 --> 00:03:15.720
So the first thing that you
tend to see on an income

00:03:15.720 --> 00:03:18.930
statement is the cost of those
actual widgets, the cost of

00:03:18.930 --> 00:03:21.510
producing those widgets.

00:03:21.510 --> 00:03:25.550
And I'll put all my expenses
in magenta.

00:03:25.550 --> 00:03:30.870
So it'll sometimes be written
as cost of sales or cost of

00:03:30.870 --> 00:03:32.120
goods sold.

00:03:35.440 --> 00:03:37.570
And this is literally-- well,
there's two things.

00:03:37.570 --> 00:03:41.190
There's a variable cost which
is, each widget, they might

00:03:41.190 --> 00:03:44.800
have used some amount of metal
and some amount of energy to

00:03:44.800 --> 00:03:47.540
produce it and some
amount of paint if

00:03:47.540 --> 00:03:48.960
it's a painted widget.

00:03:48.960 --> 00:03:51.490
And so that the cost of goods
is literally how much did it

00:03:51.490 --> 00:03:54.680
cost to buy the metal and the
paint and provide the

00:03:54.680 --> 00:03:59.000
electricity to make those $3
million worth of widgets.

00:03:59.000 --> 00:04:00.820
That's the variable cost. And
then on top of that you have

00:04:00.820 --> 00:04:03.500
the fixed costs, or the
relatively fixed costs, where

00:04:03.500 --> 00:04:07.280
just to have the factory open,
it costs a certain amount of

00:04:07.280 --> 00:04:09.960
money every year, regardless of
how many widgets you make.

00:04:09.960 --> 00:04:11.960
And we'll go into more detail
on that, But for simplicity,

00:04:11.960 --> 00:04:14.070
let's say all those costs
of making the

00:04:14.070 --> 00:04:17.730
widgets were $1 million.

00:04:17.730 --> 00:04:21.220
So sometimes someone might say
it's a $1 million cost. When I

00:04:21.220 --> 00:04:23.870
make models, I like to put
a minus there, so that I

00:04:23.870 --> 00:04:28.370
remember that that's a cost.
Anything that detracts from

00:04:28.370 --> 00:04:29.550
income I put as a minus.

00:04:29.550 --> 00:04:31.760
Anything that adds is a plus,
although that's not

00:04:31.760 --> 00:04:33.430
necessarily the standard
convention.

00:04:33.430 --> 00:04:36.820
Some people say, oh, it's a
positive $1 million cost,

00:04:36.820 --> 00:04:37.920
which means you subtract.

00:04:37.920 --> 00:04:40.730
But either way I think
you get the point.

00:04:40.730 --> 00:04:46.430
And then if you subtract your
costs from your revenue, or if

00:04:46.430 --> 00:04:48.280
you just add these two numbers,
because this one is

00:04:48.280 --> 00:04:50.490
negative, you have your
gross profit.

00:04:58.980 --> 00:05:00.900
And in this case, it would
be $2 million.

00:05:05.030 --> 00:05:10.260
And this number tells you, how
much money did you make, or

00:05:10.260 --> 00:05:12.350
how much profit did
you make just from

00:05:12.350 --> 00:05:13.680
selling these widgets?

00:05:13.680 --> 00:05:18.530
So the more widgets you sell,
in most circumstances, the

00:05:18.530 --> 00:05:21.560
larger this number
is going to be.

00:05:21.560 --> 00:05:26.720
So this is your profit before
all of the other expenses that

00:05:26.720 --> 00:05:29.290
a company has to incur,
like the taxes

00:05:29.290 --> 00:05:30.730
and the CEO's salary.

00:05:30.730 --> 00:05:32.610
The CEO's salary doesn't
go in here, right?

00:05:32.610 --> 00:05:38.890
Because the CEO doesn't go out
there to the factory in most

00:05:38.890 --> 00:05:40.480
cases and actually help
make the widget.

00:05:40.480 --> 00:05:43.610
So the CEO's salary or the
CFO's salary or the

00:05:43.610 --> 00:05:46.160
headquarters in a nice
skyscraper, that doesn't get

00:05:46.160 --> 00:05:47.140
factored in here.

00:05:47.140 --> 00:05:48.620
Or the marketing
expense, right?

00:05:48.620 --> 00:05:50.490
You have to tell people, hey,
we make good widgets.

00:05:50.490 --> 00:05:52.240
So none of that is
factored in here.

00:05:52.240 --> 00:05:53.680
So that goes into
the next line.

00:05:53.680 --> 00:05:58.410
And oftentimes, you'll see it
broken up, where they'll have

00:05:58.410 --> 00:05:59.660
marketing expense.

00:06:03.260 --> 00:06:07.200
Sometimes you have to pay
salespeople, so you might have

00:06:07.200 --> 00:06:12.380
sales expense, and then the
stuff like the corporate

00:06:12.380 --> 00:06:15.500
office and the CEO's salary, and
you have to hire auditors

00:06:15.500 --> 00:06:17.050
and accountants and
all of that.

00:06:17.050 --> 00:06:19.130
That might be included
as general.

00:06:19.130 --> 00:06:20.770
Actually, I should be doing this
in magenta because it's

00:06:20.770 --> 00:06:22.300
all expenses.

00:06:22.300 --> 00:06:25.900
Marketing, sales, and then
G&A you'll sometimes see.

00:06:25.900 --> 00:06:26.790
Sometimes you'll see SG&A.

00:06:26.790 --> 00:06:30.920
G&A just stands for general and
administrative expenses.

00:06:30.920 --> 00:06:33.450
If you see SG&A-- sometimes
instead of that you'll see

00:06:33.450 --> 00:06:37.420
SG&A-- that mean selling,
general and

00:06:37.420 --> 00:06:38.580
administrative expenses.

00:06:38.580 --> 00:06:41.920
Selling is things like, it could
be the commissions that

00:06:41.920 --> 00:06:43.070
the salespeople get.

00:06:43.070 --> 00:06:45.830
It could be just the cost of
having salespeople travel

00:06:45.830 --> 00:06:49.060
around the country and taking
people out to steak dinners.

00:06:49.060 --> 00:06:51.330
And then the general and
administrative, that's just

00:06:51.330 --> 00:06:53.270
all the stuff that the corporate
office does, and all

00:06:53.270 --> 00:06:55.110
the people who are
at that level.

00:06:55.110 --> 00:06:57.380
So if you subtract these, and
I'm just making up these

00:06:57.380 --> 00:07:00.120
numbers as I go.

00:07:00.120 --> 00:07:05.220
Say, in marketing, the company
is spending $500,000.

00:07:05.220 --> 00:07:07.280
And I'm putting it as a minus
because I like to remember

00:07:07.280 --> 00:07:08.160
it's an expense.

00:07:08.160 --> 00:07:10.180
Some models you'll
see, they'll say

00:07:10.180 --> 00:07:12.340
it's $500,000 expense.

00:07:12.340 --> 00:07:15.870
Sales, let's say, this
is just G&A here.

00:07:15.870 --> 00:07:18.130
I want to make a separate
line for sales.

00:07:18.130 --> 00:07:24.840
So let's say sales, selling
expenses is $200,000.

00:07:24.840 --> 00:07:27.400
And let's say G&A, the corporate
offices and all of

00:07:27.400 --> 00:07:33.840
that, let's see that's
another $300,000.

00:07:33.840 --> 00:07:37.210
And now we're ready to figure
out how much money did the

00:07:37.210 --> 00:07:39.610
operations of this
business make?

00:07:39.610 --> 00:07:41.260
So this is operating profit.

00:07:45.250 --> 00:07:47.500
This is really important to pay
attention to, because so

00:07:47.500 --> 00:07:49.410
many people say, oh, a company
made this much.

00:07:49.410 --> 00:07:51.940
And you'll hear these numbers,
gross and operating profit and

00:07:51.940 --> 00:07:55.570
net profit and pretax profit,
and it's very hard to

00:07:55.570 --> 00:07:57.890
understand that these are
actually very, very different

00:07:57.890 --> 00:08:00.590
things, because they all have
the word "profit," and what

00:08:00.590 --> 00:08:02.250
does gross and operating
and all that mean?

00:08:02.250 --> 00:08:04.300
But here you see it means very,
very different things.

00:08:04.300 --> 00:08:06.900
Let's calculate this number
first before I go off on one

00:08:06.900 --> 00:08:09.110
of my tangents on all the
differences between the

00:08:09.110 --> 00:08:10.460
operating and the
gross profit.

00:08:10.460 --> 00:08:14.450
But let's see, 2 million
minus 1 million.

00:08:14.450 --> 00:08:15.920
My head I think implicitly
made the

00:08:15.920 --> 00:08:16.940
numbers work out nicely.

00:08:16.940 --> 00:08:23.030
So my operating profit
here is $1 million.

00:08:23.030 --> 00:08:25.830
So already we have some
new nuance on profit.

00:08:25.830 --> 00:08:30.150
I made $2 million just from
actual widget sales, but then

00:08:30.150 --> 00:08:32.520
when you take out all of the
overhead of the company, the

00:08:32.520 --> 00:08:35.210
marketing, the sales, the
general and administrative

00:08:35.210 --> 00:08:38.030
expenses, I'm only left
with $1 million.

00:08:38.030 --> 00:08:41.610
And this is the profit from the
operations of the company,

00:08:41.610 --> 00:08:44.390
or you could say from the assets
or from the business or

00:08:44.390 --> 00:08:45.920
from the enterprise
of the company.

00:08:45.920 --> 00:08:47.500
That's what it is generating.

00:08:47.500 --> 00:08:49.890
But we can see-- I've drawn a
bunch of balance sheets before

00:08:49.890 --> 00:08:54.410
and I think this is a good time
to draw a balance sheet.

00:08:54.410 --> 00:08:58.390
So you have kind of the
assets of a company.

00:08:58.390 --> 00:09:00.570
And we'll talk a little bit
more about assets and

00:09:00.570 --> 00:09:03.460
enterprise value, and there's
a little bit of a nuance

00:09:03.460 --> 00:09:05.800
there, but essentially
the company itself.

00:09:05.800 --> 00:09:08.415
Before you think about how the
company is paid for or how

00:09:08.415 --> 00:09:11.420
it's funded, if you just think
about the enterprise itself,

00:09:11.420 --> 00:09:13.450
the assets.

00:09:13.450 --> 00:09:16.980
The assets are generating
this.

00:09:16.980 --> 00:09:19.660
They're generating the operating
profit, and that's a

00:09:19.660 --> 00:09:22.400
very important thing to realize
in the future when we

00:09:22.400 --> 00:09:23.590
talk about return on assets.

00:09:23.590 --> 00:09:25.360
Actually, we could talk
about it now.

00:09:25.360 --> 00:09:31.070
Let's say our assets, if we
paid $10 million for these

00:09:31.070 --> 00:09:34.650
assets, and these assets-- this
is the income statement

00:09:34.650 --> 00:09:38.410
for 2008-- are spitting out $1
million a year, or at least $1

00:09:38.410 --> 00:09:41.390
million in this year, our return
on asset-- I wasn't

00:09:41.390 --> 00:09:42.850
planning on introducing this,
but it doesn't hurt to

00:09:42.850 --> 00:09:47.450
introduce it right now-- our
return on asset, often

00:09:47.450 --> 00:09:53.530
acronymed ROA, would be-- well,
the numerator is the

00:09:53.530 --> 00:09:57.485
return, which is $1 million.

00:09:57.485 --> 00:10:01.240
The denominator is the
assets, $10 million.

00:10:01.240 --> 00:10:04.660
So we got a 10% return
on our assets.

00:10:04.660 --> 00:10:06.240
For a $10 million investment,
we're getting

00:10:06.240 --> 00:10:07.090
$1 million a year.

00:10:07.090 --> 00:10:11.580
We're getting 10% of our asset
investment back every year.

00:10:11.580 --> 00:10:13.580
So that's a nice thing to keep
in the back of your mind, this

00:10:13.580 --> 00:10:17.860
return on asset concept, and
it's very closely tied to

00:10:17.860 --> 00:10:22.170
operating profits and the
actual assets of a firm.

00:10:22.170 --> 00:10:23.970
What we've learned, and
especially if you watched some

00:10:23.970 --> 00:10:26.880
of my other economics videos,
that all companies aren't

00:10:26.880 --> 00:10:28.430
financed the same.

00:10:28.430 --> 00:10:30.040
A lot of them might
have some debt.

00:10:30.040 --> 00:10:33.670
So let's say that company had
$10 million of assets, but

00:10:33.670 --> 00:10:37.380
let's say they paid for it
with $5 million of debt.

00:10:40.500 --> 00:10:45.370
And let's say the interest rate
on that debt is-- let me

00:10:45.370 --> 00:10:50.595
think of a good number-- 5%.

00:10:57.670 --> 00:10:58.190
Let's make it easier.

00:10:58.190 --> 00:11:00.720
Let's make it 10% interest.

00:11:00.720 --> 00:11:02.240
So this is the operating
profit.

00:11:02.240 --> 00:11:04.790
This is the money that just
comes out of the asset itself.

00:11:04.790 --> 00:11:06.980
But, of course, that's not the
money that we get to take

00:11:06.980 --> 00:11:09.090
home, because we have to pay
this interest. So let's throw

00:11:09.090 --> 00:11:10.340
that in there as an expense.

00:11:13.475 --> 00:11:14.320
Interest expense.

00:11:14.320 --> 00:11:16.350
And obviously, a company that
has no debt will have no

00:11:16.350 --> 00:11:18.540
interest expense, but
in this case, we do.

00:11:18.540 --> 00:11:21.350
And this is an annual
income statement.

00:11:21.350 --> 00:11:24.990
So let's see, if we have $5
million of debt, and we're

00:11:24.990 --> 00:11:30.240
paying 10% on that, 10% of $5
million is $500,000 a year in

00:11:30.240 --> 00:11:33.970
interest. So we have to
essentially take half of our

00:11:33.970 --> 00:11:38.660
operating profit and give
it back to the bank.

00:11:38.660 --> 00:11:43.150
And now we are left with our--
we're getting close to where

00:11:43.150 --> 00:11:46.650
we need to get to--
pre-tax income.

00:11:51.370 --> 00:11:58.060
And if we do the subtraction,
we're at $500,000.

00:11:58.060 --> 00:11:59.950
And you could guess what the
next line is going to be,

00:11:59.950 --> 00:12:01.290
given that this says pre-tax.

00:12:01.290 --> 00:12:04.770
This is what the owners of the
company get before they pay

00:12:04.770 --> 00:12:06.510
the government.

00:12:06.510 --> 00:12:07.875
So you can guess what
the next line is.

00:12:07.875 --> 00:12:09.125
It's going to be taxes.

00:12:11.500 --> 00:12:15.760
Let's say that it's a 30%
corporate tax rate, and you're

00:12:15.760 --> 00:12:19.020
going to take 30% of this
number right here.

00:12:19.020 --> 00:12:20.520
30% of that number
right there.

00:12:20.520 --> 00:12:28.580
So 30% of $500,000
is $150,000.

00:12:28.580 --> 00:12:30.060
And then we are done.

00:12:30.060 --> 00:12:33.020
We finally have paid off
everybody we need to pay off.

00:12:33.020 --> 00:12:36.060
So we started off with
$3 million up here.

00:12:36.060 --> 00:12:39.500
We kept paying a bunch of
expenses, and then now we're

00:12:39.500 --> 00:12:40.650
left with what?

00:12:40.650 --> 00:12:48.610
This is $350,000
of net income.

00:12:51.370 --> 00:12:55.300
And this is what goes to the
owners of the company.

00:12:55.300 --> 00:12:57.910
This net income right here.

00:12:57.910 --> 00:13:01.000
So going back to our balance
sheet, we had a $10 million

00:13:01.000 --> 00:13:03.540
asset, we had $5 million
of debt.

00:13:03.540 --> 00:13:06.600
We know what's left over
is the equity.

00:13:06.600 --> 00:13:09.640
So let me do that in
a vibrant color.

00:13:09.640 --> 00:13:11.120
Equity is what's left over.

00:13:11.120 --> 00:13:14.100
So let's say this is
all book value.

00:13:14.100 --> 00:13:15.880
So we have $5 million
of equity.

00:13:15.880 --> 00:13:18.430
And when I say book value,
that's just a fancy way of

00:13:18.430 --> 00:13:21.810
saying this is what our
accountants say that we paid

00:13:21.810 --> 00:13:22.830
for the stuff.

00:13:22.830 --> 00:13:24.180
This is what we have
on our books.

00:13:24.180 --> 00:13:26.920
And we'll talk later about
depreciation and amortization

00:13:26.920 --> 00:13:30.620
and how we might change what
these values are, but a very

00:13:30.620 --> 00:13:33.380
simple way is, if you went out
and bought $10 million worth

00:13:33.380 --> 00:13:34.910
of stuff, you'd write on
your books, I have

00:13:34.910 --> 00:13:36.250
a $10 million asset.

00:13:36.250 --> 00:13:40.010
And if you took a $5 million
loan, then what you really

00:13:40.010 --> 00:13:42.260
own, if you were to kind of sell
all of this, you would

00:13:42.260 --> 00:13:46.955
get $5 million of equity.

00:13:46.955 --> 00:13:48.890
And I think this is an
interesting thing.

00:13:48.890 --> 00:13:51.630
When we did return on asset,
we looked at the operating

00:13:51.630 --> 00:13:55.010
profit, because this is what our
company generated before

00:13:55.010 --> 00:13:57.820
we paid the bank or Uncle Sam
or anything like that.

00:13:57.820 --> 00:14:01.340
And so we took this number as
the numerator and we divided

00:14:01.340 --> 00:14:02.840
by the number of assets.

00:14:02.840 --> 00:14:06.285
Now we can do another notion,
and that's return on equity.

00:14:11.630 --> 00:14:14.620
In return on equity, the
numerator is the net income

00:14:14.620 --> 00:14:19.460
that we got, so it's $350,000.

00:14:19.460 --> 00:14:23.120
And the denominator here is the
equity, the book value of

00:14:23.120 --> 00:14:26.040
our equity, so that's
$5 million.

00:14:26.040 --> 00:14:27.300
One, two, three.

00:14:27.300 --> 00:14:28.200
One, two, three.

00:14:28.200 --> 00:14:30.250
Let's cancel some zeroes out.

00:14:30.250 --> 00:14:33.380
So it's like 35/500.

00:14:33.380 --> 00:14:38.230
35/500 is the same thing
is 7/100, so it equals

00:14:38.230 --> 00:14:42.460
7% return on equity.

00:14:42.460 --> 00:14:44.650
And that's interesting because,
well, why that's

00:14:44.650 --> 00:14:48.780
lower is-- well, I don't want
to go into too much depth

00:14:48.780 --> 00:14:51.330
because I realize I'm already
pushing my time limit.

00:14:51.330 --> 00:14:53.730
But at this point, you should
have a good understanding of

00:14:53.730 --> 00:14:57.720
at least a basic income
statement of a company that

00:14:57.720 --> 00:14:58.420
sells widgets.

00:14:58.420 --> 00:15:00.010
And in the future, we're gonna
look at a lot of different

00:15:00.010 --> 00:15:01.830
companies, financial companies,
insurance

00:15:01.830 --> 00:15:05.650
companies, natural gas pipeline
companies, that will

00:15:05.650 --> 00:15:08.100
have very different-looking
income statements, but this

00:15:08.100 --> 00:15:10.210
gives you the general template
for how things work.

00:15:10.210 --> 00:15:13.920
And at least it'll give you a
sense of how revenues, gross

00:15:13.920 --> 00:15:17.300
profit, operating profit,
pre-tax income and net income

00:15:17.300 --> 00:15:18.570
really are different.

00:15:18.570 --> 00:15:19.960
A lot of times in the
popular press.

00:15:19.960 --> 00:15:22.940
They're all jumbled up as just
kind of the company is making

00:15:22.940 --> 00:15:23.930
this much money.

00:15:23.930 --> 00:15:26.490
Anyway, I'll see you
in the next video.


Price and market capitalization

WEBVTT
Kind: captions
Language: en

00:00:01.250 --> 00:00:04.960
I've had enough requests by now
for videos on investing,

00:00:04.960 --> 00:00:07.910
that I thought I would make
some videos on investing.

00:00:10.640 --> 00:00:13.280
And the way I'm going to go
about it is, over the next few

00:00:13.280 --> 00:00:15.580
videos, I'm going to give people
the vocabulary of at

00:00:15.580 --> 00:00:17.830
least how do you think
about investing?

00:00:17.830 --> 00:00:20.950
And what are the terms and
the ratios people use?

00:00:20.950 --> 00:00:22.210
And why do they make sense?

00:00:22.210 --> 00:00:23.540
Or why do they not make sense?

00:00:23.540 --> 00:00:24.680
And when do they apply?

00:00:24.680 --> 00:00:25.950
And when do they not apply?

00:00:25.950 --> 00:00:28.260
And then we're going to use
those tools later on, and then

00:00:28.260 --> 00:00:31.060
hopefully we'll look at some
particular companies.

00:00:31.060 --> 00:00:35.020
And my goal isn't to do what
they do on CNBC and tell you,

00:00:35.020 --> 00:00:36.940
buy, buy, buy, and
sell, sell, sell.

00:00:36.940 --> 00:00:41.510
Because frankly that's
not a thoughtful way

00:00:41.510 --> 00:00:42.460
of going about things.

00:00:42.460 --> 00:00:45.010
What I want to do is really give
you the tools to come to

00:00:45.010 --> 00:00:46.200
the conclusions yourself.

00:00:46.200 --> 00:00:49.130
And maybe through the videos,
we'll come to conclusions.

00:00:49.130 --> 00:00:51.720
But I don't want to be too
strong about them, because if

00:00:51.720 --> 00:00:54.870
I'm wrong I don't want you
to lose your 401(k).

00:00:54.870 --> 00:00:58.970
So the first thing that I guess
you could say bugs me a

00:00:58.970 --> 00:01:02.680
little bit, is I go to these
family gatherings, and some

00:01:02.680 --> 00:01:05.850
uncle or aunt will come
up to me and says, hey

00:01:05.850 --> 00:01:06.800
I just made a killing.

00:01:06.800 --> 00:01:07.710
I bought Citibank.

00:01:07.710 --> 00:01:09.590
It's so cheap.

00:01:09.590 --> 00:01:12.200
I don't know what it was at the
time, but it's only $1.

00:01:12.200 --> 00:01:13.600
It's a cheap stock.

00:01:13.600 --> 00:01:16.880
As opposed to implicitly there,
there's the assumption

00:01:16.880 --> 00:01:19.340
that a $10 stock would
be expensive.

00:01:19.340 --> 00:01:22.760
And I think this is very obvious
to you, but let me

00:01:22.760 --> 00:01:23.550
write that down.

00:01:23.550 --> 00:01:24.950
So price per share.

00:01:30.750 --> 00:01:35.300
So when someone tells me that
a $1 stock is expensive,

00:01:35.300 --> 00:01:36.780
they're implicitly saying, well
that's just because it's

00:01:36.780 --> 00:01:41.080
a low number, as opposed
to, say, a $10 stock.

00:01:41.080 --> 00:01:42.330
Let's call this Stock A.

00:01:45.930 --> 00:01:48.130
And Stock B.

00:01:48.130 --> 00:01:51.940
And I think this is very obvious
to anyone hopefully

00:01:51.940 --> 00:01:54.630
who's spent any time investing,
or thought about

00:01:54.630 --> 00:01:56.230
what a stock even represents.

00:01:56.230 --> 00:01:57.290
But you'd be surprised.

00:01:57.290 --> 00:02:00.860
I've had family members
who are doctors and

00:02:00.860 --> 00:02:02.920
engineers tell me this.

00:02:02.920 --> 00:02:05.130
So I thought it's a good
place to start,

00:02:05.130 --> 00:02:07.010
to clarify any confusion.

00:02:07.010 --> 00:02:11.480
So my question to you is, is
something that is $1, is that

00:02:11.480 --> 00:02:13.870
cheap relative to something
that is $10.

00:02:13.870 --> 00:02:15.150
In the everyday world it is.

00:02:15.150 --> 00:02:18.010
If I could buy an apple for
$1, that's cheaper than an

00:02:18.010 --> 00:02:22.560
apple that costs $10,
or any good, really.

00:02:22.560 --> 00:02:27.380
And the twist here is that
a share is just a share.

00:02:27.380 --> 00:02:29.650
It doesn't somehow represent
the entire company.

00:02:29.650 --> 00:02:31.650
It's a fraction of
the company.

00:02:31.650 --> 00:02:33.290
It's just a share.

00:02:33.290 --> 00:02:37.230
And all companies don't have
the same amount of shares.

00:02:37.230 --> 00:02:43.810
For example, if I have one
company-- and actually maybe

00:02:43.810 --> 00:02:46.900
that's a good point to introduce
a balance sheet--

00:02:46.900 --> 00:02:49.910
let's say one company whose
assets, because I want to do

00:02:49.910 --> 00:02:52.610
this throughout our
discussion.

00:02:52.610 --> 00:02:58.440
Whose assets are worth,
let's say, $1 million.

00:02:58.440 --> 00:02:59.280
That's its assets.

00:02:59.280 --> 00:03:02.380
All its buildings and its
employees and its brand.

00:03:02.380 --> 00:03:05.630
It's worth $1 million.

00:03:05.630 --> 00:03:07.030
Let's say it doesn't
have any debt.

00:03:07.030 --> 00:03:09.780
And we'll introduce debt later,
because that's another

00:03:09.780 --> 00:03:12.090
variable that a lot of people
don't think about when they

00:03:12.090 --> 00:03:12.800
look at stocks.

00:03:12.800 --> 00:03:16.220
They just look purely at the
equity value or the market

00:03:16.220 --> 00:03:16.990
capitalization.

00:03:16.990 --> 00:03:19.480
And all these terms we'll
hopefully get very familiar

00:03:19.480 --> 00:03:21.040
with over the course
of these videos.

00:03:21.040 --> 00:03:23.690
But let's say its assets
are worth $1 million.

00:03:23.690 --> 00:03:25.380
It has no liabilities.

00:03:25.380 --> 00:03:30.800
So the asset value
is all in equity.

00:03:30.800 --> 00:03:33.860
So this is all equity.

00:03:33.860 --> 00:03:36.210
Sometimes called stockholders'
equity.

00:03:36.210 --> 00:03:39.830
And the equity is really, the
people who own the company.

00:03:39.830 --> 00:03:41.020
what is their stake?

00:03:41.020 --> 00:03:44.100
So in this case, they never
borrowed any money to buy

00:03:44.100 --> 00:03:45.080
these assets.

00:03:45.080 --> 00:03:49.630
So the owners of the company
own all $1 million, if you

00:03:49.630 --> 00:03:52.560
believe that this is really
worth $1 million.

00:03:52.560 --> 00:03:54.260
Now you could have
this scenario.

00:03:54.260 --> 00:03:56.560
Let me actually draw the same
scenario over again.

00:03:56.560 --> 00:03:58.426
Let me copy and paste this.

00:03:58.426 --> 00:04:01.930
soon.

00:04:01.930 --> 00:04:05.450
So these are two equivalent
companies.

00:04:05.450 --> 00:04:07.470
Completely equivalent
companies.

00:04:07.470 --> 00:04:11.960
But this company over here,
they might have decided to

00:04:11.960 --> 00:04:13.600
have 10 shares.

00:04:13.600 --> 00:04:22.570
So if I were to draw that, 1,
2, 3, 4, 5, 6, 7, 8, 9, 10.

00:04:22.570 --> 00:04:23.480
I think that's 10 shares.

00:04:23.480 --> 00:04:24.360
Well that's my intention.

00:04:24.360 --> 00:04:25.840
They have 10 shares.

00:04:25.840 --> 00:04:28.610
So what is the value
per share?

00:04:28.610 --> 00:04:33.070
Each share in this case-- once
again, if you believe that the

00:04:33.070 --> 00:04:35.930
assets are worth $1 million--
are going to be $1 million.

00:04:35.930 --> 00:04:40.080
And the equity's $1 million,
because there's no debt, so

00:04:40.080 --> 00:04:43.600
all of the assets are held
by the equity holders.

00:04:43.600 --> 00:04:47.255
So each share would be worth
$1 million divided by 10,

00:04:47.255 --> 00:04:49.260
which is equal to $100,000.

00:04:49.260 --> 00:04:52.280
Obviously you never see $100,000
shares out there, or

00:04:52.280 --> 00:04:57.380
at least not in the great
majority of examples.

00:04:57.380 --> 00:04:59.520
So this is a bit of an
artificial example.

00:04:59.520 --> 00:05:02.180
More likely a company might
have a million shares, in

00:05:02.180 --> 00:05:03.560
which case this would
be a $1 stock.

00:05:03.560 --> 00:05:05.110
But anyway, this is
a case where they

00:05:05.110 --> 00:05:05.960
only have 10 shares.

00:05:05.960 --> 00:05:06.720
It's $100,000.

00:05:06.720 --> 00:05:09.870
Now let's say this company
right here says, well

00:05:09.870 --> 00:05:11.140
$100,000, that's
kind of a crazy

00:05:11.140 --> 00:05:12.480
number for a share price.

00:05:12.480 --> 00:05:14.740
It'll keep a lot of people
from buying our shares.

00:05:14.740 --> 00:05:17.960
So let's just divide it
into a million shares.

00:05:17.960 --> 00:05:23.220
So they have times one
million shares.

00:05:23.220 --> 00:05:26.750
So in this situation, the
company is worth what?

00:05:26.750 --> 00:05:28.370
Or the shares are worth what?

00:05:28.370 --> 00:05:32.200
They're worth $1 million divided
by one million shares,

00:05:32.200 --> 00:05:34.280
or $1 per share.

00:05:34.280 --> 00:05:38.430
So going back to the idea where
my family member would

00:05:38.430 --> 00:05:41.220
come to me at a party, they
would say oh, look how cheap

00:05:41.220 --> 00:05:44.720
this company is compared
to this company.

00:05:44.720 --> 00:05:47.450
Even more, let's say for
whatever reason, because not

00:05:47.450 --> 00:05:51.650
so many people could afford this
stock, because just to

00:05:51.650 --> 00:05:53.050
get in the game, you've
gotta put up

00:05:53.050 --> 00:05:54.470
$100,000 to buy a share.

00:05:54.470 --> 00:05:56.670
Let's say this stock
trades down, and

00:05:56.670 --> 00:06:03.560
it's trading at $50,000.

00:06:03.560 --> 00:06:05.290
And their assets
are identical.

00:06:05.290 --> 00:06:08.675
So obviously there's no two
companies that are identical

00:06:08.675 --> 00:06:10.150
in this way, but let's
say that they are.

00:06:10.150 --> 00:06:13.100
Let's say in both cases the
assets of this company are

00:06:13.100 --> 00:06:16.300
worth the exact same thing as
the assets of that company.

00:06:16.300 --> 00:06:20.910
So here investors are valuing
this company at $1 times a

00:06:20.910 --> 00:06:21.730
million shares.

00:06:21.730 --> 00:06:24.560
They're valuing these assets
at $1 million.

00:06:24.560 --> 00:06:28.350
In this case, the investor is
saying, OK, I'm willing to pay

00:06:28.350 --> 00:06:30.690
$50,000 per share.

00:06:30.690 --> 00:06:31.940
And there's only 10 shares.

00:06:31.940 --> 00:06:35.720
So they're valuing the
assets at $500,000.

00:06:35.720 --> 00:06:38.330
And this is of course the market
value of the assets.

00:06:38.330 --> 00:06:40.570
And we'll talk more about
market versus

00:06:40.570 --> 00:06:41.560
book value of assets.

00:06:41.560 --> 00:06:44.180
But the market value or assets
is essentially, what is the

00:06:44.180 --> 00:06:46.220
market saying the assets
are worth?

00:06:46.220 --> 00:06:49.320
The book value of the assets, or
what the accountants within

00:06:49.320 --> 00:06:51.710
the company are saying
the assets are worth.

00:06:51.710 --> 00:06:55.070
And there's a whole methodology
to how one would

00:06:55.070 --> 00:06:56.090
account for that.

00:06:56.090 --> 00:06:58.600
But this is the market
value of the assets.

00:06:58.600 --> 00:07:01.340
And I already told you that
these assets are identical.

00:07:01.340 --> 00:07:03.760
They generate the same
earning stream with

00:07:03.760 --> 00:07:05.240
the exact same risk.

00:07:05.240 --> 00:07:09.340
So in this situation, you're
paying $500,000 for the same

00:07:09.340 --> 00:07:12.610
asset that over here you're
paying $1 for.

00:07:12.610 --> 00:07:14.570
I don't care what the actual
share price is.

00:07:14.570 --> 00:07:16.110
This is what you're
valuing it at.

00:07:16.110 --> 00:07:20.260
So the person who says that $1
here is cheap, relative to

00:07:20.260 --> 00:07:22.010
$50,000 here.

00:07:22.010 --> 00:07:23.850
And they might even say, oh well
they're the same company.

00:07:23.850 --> 00:07:26.300
And I get it here for $1 a
share, and I get it there for

00:07:26.300 --> 00:07:29.335
$50,000 a share, this
is a cheap company.

00:07:29.335 --> 00:07:32.700
But that's completely
180 degrees in the

00:07:32.700 --> 00:07:34.370
wrong direction wrong.

00:07:34.370 --> 00:07:37.010
Because you're actually paying
more for this company.

00:07:37.010 --> 00:07:40.080
You're paying $1 for one
millionth of this company,

00:07:40.080 --> 00:07:42.050
while you're paying $50,000
dollars for one

00:07:42.050 --> 00:07:43.190
tenth of this company.

00:07:43.190 --> 00:07:46.030
So this one is actually
the better deal.

00:07:46.030 --> 00:07:49.930
So in general, when you're
trying to figure out relative

00:07:49.930 --> 00:07:52.900
price of a company-- and we're
going to talk a lot more about

00:07:52.900 --> 00:07:55.953
ratios and how do you know if
something is inherently cheap,

00:07:55.953 --> 00:07:58.760
you relative to its earnings or
what it could earn, or its

00:07:58.760 --> 00:08:01.420
growth or anything like that--
but the first cut is, you

00:08:01.420 --> 00:08:03.350
can't just look at the price.

00:08:03.350 --> 00:08:05.350
The price is almost a
meaningless number.

00:08:05.350 --> 00:08:08.330
It matters to some degree
for trading purposes.

00:08:08.330 --> 00:08:11.270
Where a lot of institutional
funds won't look at a stock

00:08:11.270 --> 00:08:12.900
that's below $5.

00:08:12.900 --> 00:08:15.055
A lot of stocks that go
into the penny stocks.

00:08:19.290 --> 00:08:22.470
There's a lot of frictions in
investing in penny stocks,

00:08:22.470 --> 00:08:25.790
because obviously if you have a
$10 stock, a $10 stock could

00:08:25.790 --> 00:08:29.410
go from $10 to $10.01.

00:08:29.410 --> 00:08:33.130
Or if could go to $9.99.

00:08:33.130 --> 00:08:37.140
And this is only a 0.1% move.

00:08:37.140 --> 00:08:39.789
But let's say you have
a penny stock.

00:08:39.789 --> 00:08:44.410
Let's say you have a stock
that's at $0.05, it can only

00:08:44.410 --> 00:08:46.980
move by a penny in
one direction.

00:08:46.980 --> 00:08:49.690
And before it was actually
an eighth.

00:08:49.690 --> 00:08:54.390
So if you move only by a penny
you can only go to $0.06 or

00:08:54.390 --> 00:08:57.010
you could go to $0.04.

00:08:57.010 --> 00:08:59.670
And so in either direction
you're looking at a 20% move,

00:08:59.670 --> 00:09:05.470
while here you're looking at a
1/1000th move or a 0.1% move.

00:09:05.470 --> 00:09:07.250
So that's one reason why
price might matter

00:09:07.250 --> 00:09:08.450
a little bit here.

00:09:08.450 --> 00:09:11.740
There's huge frictions if you
were to buy and sell, there's

00:09:11.740 --> 00:09:13.180
20% every time.

00:09:13.180 --> 00:09:15.350
And we'll talk about things
like bid-ask spreads and

00:09:15.350 --> 00:09:17.220
liquidity and things like
that in the future.

00:09:17.220 --> 00:09:19.250
But that's where the price
really starts to matter.

00:09:19.250 --> 00:09:21.710
And obviously if you have a
really huge price, like

00:09:21.710 --> 00:09:24.850
$100,000, that makes it
difficult for people to buy

00:09:24.850 --> 00:09:26.480
even one share.

00:09:26.480 --> 00:09:29.020
But that's the only place
where price matters.

00:09:29.020 --> 00:09:31.770
Inherently, when you're talking
about value, you have

00:09:31.770 --> 00:09:35.630
to take the price per share and
you multiply it times the

00:09:35.630 --> 00:09:36.880
number of shares.

00:09:39.970 --> 00:09:42.760
So these Stock A and Stock B,
these are different than the

00:09:42.760 --> 00:09:44.000
ones I drew down here.

00:09:44.000 --> 00:09:45.940
So let me draw a little dividing
line right there.

00:09:45.940 --> 00:09:48.060
And let me erase some of this.

00:09:52.380 --> 00:09:55.590
So if in this example-- let
me actually erase more.

00:10:06.160 --> 00:10:09.320
So in this example, it's
important to write down the

00:10:09.320 --> 00:10:12.260
number of shares you have. And
you could look this up on

00:10:12.260 --> 00:10:13.070
Yahoo Finance.

00:10:13.070 --> 00:10:14.060
And we'll do this in the future

00:10:14.060 --> 00:10:14.910
with a bunch of companies.

00:10:14.910 --> 00:10:16.840
We'll just go through the
motions of figuring out all of

00:10:16.840 --> 00:10:17.660
the metrics.

00:10:17.660 --> 00:10:20.290
Just because that's a good
starting point just to get a

00:10:20.290 --> 00:10:22.100
sense of what the company's
all about.

00:10:22.100 --> 00:10:27.060
So let's say that this company
has ten million shares.

00:10:27.060 --> 00:10:34.745
And this company right here
has 500,000 shares.

00:10:34.745 --> 00:10:37.940
So what you do is you multiply
these numbers to figure out,

00:10:37.940 --> 00:10:41.480
what is the market value or the
market capitalization of

00:10:41.480 --> 00:10:41.950
the company?

00:10:41.950 --> 00:10:44.190
So that's a word.

00:10:44.190 --> 00:10:47.320
Let me write that down.

00:10:47.320 --> 00:10:49.740
Sometimes called
the market cap.

00:10:49.740 --> 00:10:51.143
Market capitalization.

00:10:58.450 --> 00:11:01.380
And it's essentially, what
is the market saying

00:11:01.380 --> 00:11:03.060
the equity is worth?

00:11:03.060 --> 00:11:07.650
In this case where you don't
have debt, they're actually

00:11:07.650 --> 00:11:10.450
saying what is the
asset worth?

00:11:10.450 --> 00:11:16.140
So if I took this, $1 per share
times 10 million shares.

00:11:16.140 --> 00:11:19.270
The market is saying, this
company's worth $10 million.

00:11:19.270 --> 00:11:22.360
Or that the equity of this
company's worth $10 million.

00:11:22.360 --> 00:11:25.470
In this case, they're
saying $10 per share

00:11:25.470 --> 00:11:28.330
times 500,000 shares.

00:11:28.330 --> 00:11:31.960
They're saying it's
worth $5 million.

00:11:31.960 --> 00:11:34.260
That's the market capitalization
of the company.

00:11:34.260 --> 00:11:40.020
That's what the people are
saying the company is worth.

00:11:40.020 --> 00:11:42.610
And actually I'm running
out of time from there.

00:11:42.610 --> 00:11:45.220
So in the next video we'll talk
a little bit about how do

00:11:45.220 --> 00:11:47.710
people actually determine
what something is worth?

00:11:47.710 --> 00:11:51.250
And that obviously can be an
infinitely deep discussion.

00:11:51.250 --> 00:11:53.480
But we'll try to get our
feet wet a little bit.

00:11:53.480 --> 00:11:54.730
See


3D path counting brain teaser

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.720
Let's see if we can extend the
path counting brain teaser to

00:00:03.720 --> 00:00:05.330
three dimensions.

00:00:05.330 --> 00:00:09.310
So let's say that I had
a three by three cube.

00:00:09.310 --> 00:00:11.280
I'll keep it at three by
three to keep the math

00:00:11.280 --> 00:00:12.905
from getting too hairy.

00:00:12.905 --> 00:00:14.320
So let me draw it like that.

00:00:14.320 --> 00:00:18.130
I won't use a line tool just
because-- well, maybe I should

00:00:18.130 --> 00:00:19.610
have. So let's see.

00:00:19.610 --> 00:00:23.080
The front of the cube looks
something like that.

00:00:23.080 --> 00:00:24.910
That's the front of the cube.

00:00:24.910 --> 00:00:28.400
And the cube goes backwards
like that.

00:00:28.400 --> 00:00:30.570
Comes down.

00:00:30.570 --> 00:00:31.230
Goes like that.

00:00:31.230 --> 00:00:35.640
It's a three by three,
like a Rubik's Cube.

00:00:35.640 --> 00:00:37.430
I could have drawn this little
bit better, but I think this

00:00:37.430 --> 00:00:41.000
will meet our needs.

00:00:41.000 --> 00:00:42.430
OK.

00:00:42.430 --> 00:00:42.990
There you go.

00:00:42.990 --> 00:00:47.050
Three by three cube.

00:00:47.050 --> 00:00:52.820
And so our goal is to get from
this back left cube.

00:00:52.820 --> 00:00:56.430
This top corner back
left cube.

00:00:56.430 --> 00:01:00.840
And get to this front,
bottom right cube.

00:01:00.840 --> 00:01:01.820
So this is our goal.

00:01:01.820 --> 00:01:03.690
I'll do it in this yellow.

00:01:03.690 --> 00:01:05.640
This is our goal right there.

00:01:05.640 --> 00:01:08.220
And we're allowed to
either go forward.

00:01:08.220 --> 00:01:11.120
From any cube, these are our
three operations or our three

00:01:11.120 --> 00:01:12.590
movements we can do.

00:01:12.590 --> 00:01:17.200
We can go forward, or I guess
towards the front.

00:01:17.200 --> 00:01:18.665
We could go down.

00:01:21.680 --> 00:01:23.240
Or we can go to the right.

00:01:23.240 --> 00:01:24.790
So I can draw it here.

00:01:24.790 --> 00:01:26.720
We can go from that
cube to that cube.

00:01:26.720 --> 00:01:29.330
So just like the two dimensional
problem, you're

00:01:29.330 --> 00:01:31.200
only allowed to make
forward progress.

00:01:31.200 --> 00:01:33.170
You're not allowed to
come down here, and

00:01:33.170 --> 00:01:34.540
then go to over here.

00:01:34.540 --> 00:01:36.280
You're allowed to go down here,
then here, but then

00:01:36.280 --> 00:01:37.630
you're not allowed to go up.

00:01:37.630 --> 00:01:41.180
So every step you're getting a
little bit closer, from this

00:01:41.180 --> 00:01:47.800
back left top cube, to this
front bottom, right cube.

00:01:47.800 --> 00:01:50.790
And so the same question
applies, how many different

00:01:50.790 --> 00:01:54.730
ways are there to get
from there to there?

00:01:54.730 --> 00:01:57.480
And you can pause it now and try
it yourself, because I'm

00:01:57.480 --> 00:01:59.540
about to explain how to do it.

00:01:59.540 --> 00:02:02.480
And the first thing, when you
tried to do it yourself, is to

00:02:02.480 --> 00:02:04.410
realize, boy this is
hard to visualize.

00:02:04.410 --> 00:02:06.830
Even if I had to draw this
out, I'd have to

00:02:06.830 --> 00:02:07.280
go in and then out.

00:02:07.280 --> 00:02:10.180
I mean, how do I even visualize
a three dimensional

00:02:10.180 --> 00:02:10.949
cube like this?

00:02:10.949 --> 00:02:14.360
And the best way to do is to
separately visualize each of

00:02:14.360 --> 00:02:15.310
the separate layers.

00:02:15.310 --> 00:02:15.930
So let's do that.

00:02:15.930 --> 00:02:19.550
Let's make this the magenta
layer up here.

00:02:19.550 --> 00:02:22.050
We'll call that layer one.

00:02:22.050 --> 00:02:26.340
So this is the magenta
layer up here.

00:02:26.340 --> 00:02:28.540
And you'll see what I'm
doing in a second.

00:02:28.540 --> 00:02:35.570
This is the mauve layer
right there.

00:02:38.920 --> 00:02:43.650
And then finally I'll
do the orange layer.

00:02:43.650 --> 00:02:47.060
The orange layer is that
one right there.

00:02:47.060 --> 00:02:51.180
What we can do is separately
draw each of these layers.

00:02:51.180 --> 00:02:52.930
So first, let's do the
magenta layer.

00:02:52.930 --> 00:02:55.580
So the magenta layer will
look like this.

00:02:55.580 --> 00:02:58.610
And now I'll use things that
help-- nope, not like that.

00:02:58.610 --> 00:03:01.190
I want to use the other tool.

00:03:01.190 --> 00:03:03.270
The magenta layer.

00:03:03.270 --> 00:03:07.974
Let me draw some squares
in here.

00:03:07.974 --> 00:03:14.670
It's like that, and like that,
and like that, and like that.

00:03:14.670 --> 00:03:17.075
And then, the middle one
was the mauve layer.

00:03:17.075 --> 00:03:19.740
We'll draw that.

00:03:19.740 --> 00:03:23.320
The mauve layer looks
something like that.

00:03:23.320 --> 00:03:26.350
And you can imagine I'm slicing
it, and just looking

00:03:26.350 --> 00:03:27.280
at it from above.

00:03:27.280 --> 00:03:28.300
That's the idea here.

00:03:28.300 --> 00:03:31.500
And it's going to help us
visualize this problem.

00:03:31.500 --> 00:03:33.850
So the mauve layer looks
something like that.

00:03:36.990 --> 00:03:39.350
And then finally the
orange layer.

00:03:44.400 --> 00:03:46.490
Looks like this.

00:03:46.490 --> 00:03:47.960
And we're almost ready
to actually

00:03:47.960 --> 00:03:49.210
start doing the problem.

00:03:52.520 --> 00:03:53.500
Good enough.

00:03:53.500 --> 00:03:56.430
So just to make sure we
understand our visualization,

00:03:56.430 --> 00:03:59.030
this layer up here-- we
call that layer one.

00:03:59.030 --> 00:04:01.400
We could but this as box one.

00:04:01.400 --> 00:04:05.130
This layer is layer two.

00:04:05.130 --> 00:04:07.100
So I'll put a little two here.

00:04:07.100 --> 00:04:08.980
And I don't want to get these
confused with the paths and

00:04:08.980 --> 00:04:10.920
all that, so I'm writing
it really small.

00:04:10.920 --> 00:04:14.065
And this is layer three,
or level three.

00:04:14.065 --> 00:04:15.500
And that's right there.

00:04:15.500 --> 00:04:18.130
And just to make sure you
understand, this corner right

00:04:18.130 --> 00:04:20.150
there, this is our
start point.

00:04:20.150 --> 00:04:22.500
And that's right there.

00:04:22.500 --> 00:04:22.770
Right?

00:04:22.770 --> 00:04:24.030
Because this is a whole top.

00:04:24.030 --> 00:04:26.380
So this is the back
left of the top.

00:04:26.380 --> 00:04:31.410
And are finish point, the bottom
right, is right here.

00:04:31.410 --> 00:04:35.140
So, essentially, our problem
goes from, how many ways to

00:04:35.140 --> 00:04:36.930
get from there to there, to
how many ways to get from

00:04:36.930 --> 00:04:37.940
there to there?

00:04:37.940 --> 00:04:40.150
So let's just stay
within a layer.

00:04:40.150 --> 00:04:43.020
So how many ways can I get
to this point right here?

00:04:43.020 --> 00:04:45.990
Well I can only go from this
point, and go straight in the

00:04:45.990 --> 00:04:46.765
layer like that.

00:04:46.765 --> 00:04:48.260
So there's only one
way to get there.

00:04:48.260 --> 00:04:50.110
That movement is the
exact same movement

00:04:50.110 --> 00:04:51.650
as this right here.

00:04:51.650 --> 00:04:53.360
Going from this box
to this box.

00:04:53.360 --> 00:04:54.330
So there's one way
to get there.

00:04:54.330 --> 00:04:55.470
That's the same thing
as there.

00:04:55.470 --> 00:04:56.610
And similarly, I
could go there.

00:04:56.610 --> 00:04:57.860
And I can just go
one more step.

00:04:57.860 --> 00:04:59.360
So there's only one
way to get there.

00:04:59.360 --> 00:05:02.230
And that's like going there
and then there.

00:05:02.230 --> 00:05:04.580
And by the same logic, I could
go one to the right here.

00:05:04.580 --> 00:05:06.160
That's the only way
to get there.

00:05:06.160 --> 00:05:07.555
Or I could go two to
the right there.

00:05:07.555 --> 00:05:09.820
And that's the only
way to get there.

00:05:09.820 --> 00:05:14.900
And now if you watched the two
dimensional path counting

00:05:14.900 --> 00:05:18.590
brain teaser, you know that
there's two ways to get here.

00:05:18.590 --> 00:05:20.600
And the logic is, well you
could draw it out.

00:05:20.600 --> 00:05:21.670
You could go like that.

00:05:21.670 --> 00:05:22.780
One, two.

00:05:22.780 --> 00:05:27.615
And that's the same thing
as saying, one, two.

00:05:27.615 --> 00:05:29.750
Though it's easier to
visualize here.

00:05:29.750 --> 00:05:33.060
But the general logic was, well,
to know how many ways to

00:05:33.060 --> 00:05:36.520
get to any square, think about
the squares that lead to it,

00:05:36.520 --> 00:05:39.090
and how many ways can I get
to those two squares?

00:05:39.090 --> 00:05:41.240
And then sum them up.

00:05:41.240 --> 00:05:43.810
And by the same logic-- so
there's two ways to get here.

00:05:43.810 --> 00:05:45.020
That's that cell.

00:05:45.020 --> 00:05:46.720
Three ways to get here, right?

00:05:46.720 --> 00:05:48.240
Two plus one is three.

00:05:48.240 --> 00:05:50.430
One plus two is three.

00:05:50.430 --> 00:05:51.770
And three plus three is six.

00:05:51.770 --> 00:05:54.930
So there were six ways to
get to this cube right

00:05:54.930 --> 00:05:56.610
there, from that one.

00:05:56.610 --> 00:05:58.690
So this isn't too different
from the two dimensional

00:05:58.690 --> 00:05:59.520
problem so far.

00:05:59.520 --> 00:06:01.610
But now it gets interesting.

00:06:01.610 --> 00:06:05.245
So how many ways-- I did this
in yellow, but I should have

00:06:05.245 --> 00:06:06.810
done it in the color
of that layer.

00:06:06.810 --> 00:06:08.810
How many ways are there to get
to this cell right here?

00:06:08.810 --> 00:06:12.170
This cell is that
one right there.

00:06:12.170 --> 00:06:13.660
Well, I start here.

00:06:13.660 --> 00:06:16.490
And I can just go
straight down.

00:06:16.490 --> 00:06:17.560
There's only one way
to be there.

00:06:17.560 --> 00:06:18.510
But I go straight down.

00:06:18.510 --> 00:06:21.170
So there's only one
way to get there.

00:06:21.170 --> 00:06:22.030
Actually let's extend.

00:06:22.030 --> 00:06:23.740
There's only one way
to get here, if I'm

00:06:23.740 --> 00:06:26.050
going straight down.

00:06:26.050 --> 00:06:28.190
And so there's only one way
to get to this cell too.

00:06:28.190 --> 00:06:30.100
I'd have to go straight
down again.

00:06:30.100 --> 00:06:31.910
So there's only one
way to get there.

00:06:31.910 --> 00:06:34.650
Hopefully you understand the
way we're visualizing it.

00:06:34.650 --> 00:06:36.050
This is the bottom row.

00:06:36.050 --> 00:06:37.220
And there's only one way.

00:06:37.220 --> 00:06:40.010
You go from here, straight
down to there,

00:06:40.010 --> 00:06:40.790
straight down to there.

00:06:40.790 --> 00:06:42.710
And that's the only
way to get there.

00:06:42.710 --> 00:06:43.750
Fair enough.

00:06:43.750 --> 00:06:45.260
Now this is where it
gets interesting.

00:06:45.260 --> 00:06:49.700
How many ways are there
to get to this cell?

00:06:49.700 --> 00:06:52.460
Well in our old example, there
was only one way in two

00:06:52.460 --> 00:06:53.900
dimensions to go
from this cell.

00:06:53.900 --> 00:06:56.380
But now we can go from
this cell, and we

00:06:56.380 --> 00:06:57.500
could come from above.

00:06:57.500 --> 00:06:58.450
And where's above?

00:06:58.450 --> 00:07:00.380
Above is right there.

00:07:00.380 --> 00:07:03.490
So now we add this cell
to that cell.

00:07:03.490 --> 00:07:07.090
So one plus one is, there are
two ways to get there.

00:07:07.090 --> 00:07:09.120
How many ways to get to this?

00:07:09.120 --> 00:07:09.803
You can't even see it.

00:07:09.803 --> 00:07:12.760
This is kind of in the back
middle of this cube.

00:07:12.760 --> 00:07:13.640
How many ways to get there?

00:07:13.640 --> 00:07:15.690
Well, there's two ways to come
from this direction.

00:07:15.690 --> 00:07:18.450
And I can also come from
above right there.

00:07:18.450 --> 00:07:21.570
So two plus one is three.

00:07:21.570 --> 00:07:22.730
How many ways to get here?

00:07:22.730 --> 00:07:24.900
Well, one from behind.

00:07:24.900 --> 00:07:26.470
And then one from above.

00:07:26.470 --> 00:07:27.270
So that's two.

00:07:27.270 --> 00:07:28.900
And you see a little
bit of symmetry.

00:07:28.900 --> 00:07:30.520
And how many ways
to come here.

00:07:30.520 --> 00:07:31.860
Well there's two here.

00:07:31.860 --> 00:07:34.360
From going straight forward.

00:07:34.360 --> 00:07:35.840
Two ways to go that way.

00:07:35.840 --> 00:07:37.340
And then one way to
come from above.

00:07:44.650 --> 00:07:46.650
This is two, and we're
on this cell.

00:07:46.650 --> 00:07:48.380
So if we wanted to know how many
ways to get to this cell?

00:07:48.380 --> 00:07:50.780
There's two ways to
go from there.

00:07:50.780 --> 00:07:52.235
And then one way from above.

00:07:52.235 --> 00:07:53.990
So that's three.

00:07:53.990 --> 00:07:57.660
And now, right here, how many
ways to get to this cell?

00:07:57.660 --> 00:07:58.370
There's three ways.

00:07:58.370 --> 00:08:04.360
I could come from here, from
here, or from above.

00:08:04.360 --> 00:08:08.870
So I have the two plus
two plus two is six.

00:08:08.870 --> 00:08:13.290
Likewise, here, I can come from
six plus three is nine.

00:08:13.290 --> 00:08:15.190
But I can also come
from above.

00:08:15.190 --> 00:08:16.260
From here.

00:08:16.260 --> 00:08:19.620
So there's 12 ways
to get there.

00:08:19.620 --> 00:08:21.160
And you can do the same logic.

00:08:21.160 --> 00:08:22.020
How many ways to get here?

00:08:22.020 --> 00:08:25.000
Well, within the same row
there are nine ways.

00:08:25.000 --> 00:08:27.370
Six plus three is nine.

00:08:27.370 --> 00:08:29.170
And then you could come
from above as well.

00:08:29.170 --> 00:08:30.710
That's 12.

00:08:30.710 --> 00:08:33.230
And finally, how many ways to
get to this cell right here,

00:08:33.230 --> 00:08:35.210
which is this one right there?

00:08:35.210 --> 00:08:38.260
Well, there's 12 ways
to get here.

00:08:38.260 --> 00:08:39.740
So I could go all
of those ways.

00:08:39.740 --> 00:08:42.159
12 ways to come from behind
it, so that's 24.

00:08:42.159 --> 00:08:45.520
And then six ways to
come from above.

00:08:45.520 --> 00:08:48.740
So 12 plus 12 is 24
plus six is 30.

00:08:48.740 --> 00:08:50.690
I think you're seeing
the pattern.

00:08:50.690 --> 00:08:52.530
So how many ways to get here?

00:08:52.530 --> 00:08:55.830
Well it's one plus two,
which is three.

00:08:55.830 --> 00:08:57.460
How many ways to get here?

00:08:57.460 --> 00:09:01.410
Well it's three plus three,
which is six.

00:09:01.410 --> 00:09:02.560
How many ways to get here?

00:09:02.560 --> 00:09:04.920
It's one way here, and
two from above.

00:09:04.920 --> 00:09:06.370
So it's three.

00:09:06.370 --> 00:09:07.940
How many ways to get here?

00:09:07.940 --> 00:09:11.010
Well, three from behind
and three from above.

00:09:11.010 --> 00:09:12.380
That's six.

00:09:12.380 --> 00:09:14.490
Here, it's three plus
three is six.

00:09:14.490 --> 00:09:15.970
But you could also
come from above.

00:09:15.970 --> 00:09:17.520
So six again.

00:09:17.520 --> 00:09:19.670
So that is 12.

00:09:19.670 --> 00:09:20.740
How many ways to get here?

00:09:20.740 --> 00:09:23.280
12 plus six is 18.

00:09:23.280 --> 00:09:25.940
But there's 12 ways to come
from above as well.

00:09:25.940 --> 00:09:29.150
So 18 plus 12 is 30.

00:09:29.150 --> 00:09:31.550
And by the same logic, there's
18 ways to get here

00:09:31.550 --> 00:09:32.380
from these two cells.

00:09:32.380 --> 00:09:34.830
But I could also come
from above.

00:09:34.830 --> 00:09:36.890
So that is 30.

00:09:36.890 --> 00:09:38.900
So how many ways to get
to this last cell?

00:09:38.900 --> 00:09:41.040
Well there's 30 from
this direction.

00:09:41.040 --> 00:09:42.690
30 ways from there.

00:09:42.690 --> 00:09:44.000
30 ways from behind it.

00:09:44.000 --> 00:09:45.820
That's 60.

00:09:45.820 --> 00:09:48.810
And then there's another 30
ways to come from above.

00:09:48.810 --> 00:09:52.040
So there are 90 ways.

00:09:52.040 --> 00:09:53.490
I could write that there,
but you can't see it.

00:09:53.490 --> 00:09:58.170
90 ways to get from that
cell over there to

00:09:58.170 --> 00:09:59.530
this cell over here.

00:09:59.530 --> 00:10:01.560
And then the last video, I
made the analogy to the

00:10:01.560 --> 00:10:02.880
binomial theorem.

00:10:02.880 --> 00:10:06.060
And I'll leave you to think
about what the three

00:10:06.060 --> 00:10:07.240
dimensional analogy is.

00:10:07.240 --> 00:10:09.910
And I'll throw out a word
which is never really

00:10:09.910 --> 00:10:11.690
mentioned in math class, because
it's normally too

00:10:11.690 --> 00:10:12.690
hairy to deal with.

00:10:12.690 --> 00:10:16.540
Think about formulating a
trinomial theorem, to help you

00:10:16.540 --> 00:10:22.110
multiply things like x plus
y plus z to nth power.

00:10:22.110 --> 00:10:24.150
And think about how this cube,
or an extension of it.

00:10:24.150 --> 00:10:27.460
This is a three by three
by three cube.

00:10:30.480 --> 00:10:34.010
But imagine if it was, you know,
an n by n by n cube.

00:10:34.010 --> 00:10:37.160
Then you can start taking things
to arbitrary powers.

00:10:37.160 --> 00:10:39.520
So I'll leave you to
think about this.

00:10:39.520 --> 00:10:42.250
But I just thought this was a
neat visualization problem,

00:10:42.250 --> 00:10:44.870
which is really not any more
difficult than the last one.

00:10:44.870 --> 00:10:46.680
Actually, before I leave you
I'll leave you with just a

00:10:46.680 --> 00:10:47.560
general principle.

00:10:47.560 --> 00:10:50.150
And this is actually really
useful for some standardized

00:10:50.150 --> 00:10:52.500
tests, or just logic games.

00:10:52.500 --> 00:10:55.125
If I'm trying to get
to this cell.

00:11:00.720 --> 00:11:05.160
And let's say there's a bunch
of ways to get here.

00:11:05.160 --> 00:11:07.240
And it has to have direction.

00:11:07.240 --> 00:11:10.160
So I won't go into a whole
graph theory thing.

00:11:10.160 --> 00:11:11.820
But it has to have direction.

00:11:11.820 --> 00:11:13.330
And you can't have cycles.

00:11:13.330 --> 00:11:15.090
Because then you can have
infinite ways to get to a

00:11:15.090 --> 00:11:16.060
certain point.

00:11:16.060 --> 00:11:20.020
And let's say that there are
x ways to get there.

00:11:20.020 --> 00:11:22.640
Y ways to get there. z
ways to get there.

00:11:22.640 --> 00:11:23.660
And a ways to get there.

00:11:23.660 --> 00:11:24.880
I'll cycle through
the alphabet.

00:11:24.880 --> 00:11:27.560
And this is just a subset
of the larger graph.

00:11:27.560 --> 00:11:31.860
This graph could have a
bunch of connections.

00:11:31.860 --> 00:11:33.850
This is just where we are.

00:11:33.850 --> 00:11:35.950
These are all of the nodes
that connect to this.

00:11:35.950 --> 00:11:38.040
The general rule that we've
learned in the last two brain

00:11:38.040 --> 00:11:41.530
teasers is, you say, OK, how
can I get to this node?

00:11:41.530 --> 00:11:44.700
Well I can go from here,
here, here, or here.

00:11:44.700 --> 00:11:46.100
And I just have to add up.

00:11:46.100 --> 00:11:48.240
So if I can get from here,
there's x ways to

00:11:48.240 --> 00:11:49.600
come via this node.

00:11:49.600 --> 00:11:51.590
Y ways to come via that note.

00:11:51.590 --> 00:11:53.460
z ways to come via that node.

00:11:53.460 --> 00:11:54.970
a ways to come via that node.

00:11:54.970 --> 00:11:58.780
So the total ways to get
to that node is x plus

00:11:58.780 --> 00:12:01.320
y plus z plus a.

00:12:01.320 --> 00:12:03.580
And actually, you'll see
problems like this if you ever

00:12:03.580 --> 00:12:05.430
plan on becoming a lawyer.

00:12:05.430 --> 00:12:07.720
They actually do have problems
like this on the LSAT, that

00:12:07.720 --> 00:12:09.790
aren't maybe as complicated
as what we did here.

00:12:09.790 --> 00:12:12.610
But you need understand
this principle.

00:12:12.610 --> 00:12:15.060
Anyway hope you enjoyed that.


Path counting brain teaser

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.770
Let's say you have a
six by six grid.

00:00:02.770 --> 00:00:04.170
And that's what I've
drawn right here.

00:00:04.170 --> 00:00:06.980
One, two, three, four,
five, six.

00:00:06.980 --> 00:00:09.720
One, two, three, four,
five, six.

00:00:09.720 --> 00:00:11.970
And you were to start in
the top left corner.

00:00:11.970 --> 00:00:14.410
So, you were to start
right here.

00:00:14.410 --> 00:00:17.720
And your goal is to get to
the bottom right corner.

00:00:17.720 --> 00:00:19.990
You want to get right there,
where the star is.

00:00:19.990 --> 00:00:21.690
And you can only move
in two ways.

00:00:21.690 --> 00:00:24.640
You can only move
to the right.

00:00:24.640 --> 00:00:26.760
Or you can only move down.

00:00:26.760 --> 00:00:28.420
You can't move diagonally.

00:00:28.420 --> 00:00:29.910
You can't move up.

00:00:29.910 --> 00:00:31.500
You can't move to the left.

00:00:31.500 --> 00:00:33.355
So in every step, you can't
move diagonally.

00:00:33.355 --> 00:00:35.020
And every step will get
you a little bit

00:00:35.020 --> 00:00:36.110
closer to this star.

00:00:36.110 --> 00:00:38.360
So you can't have
backward moves.

00:00:38.360 --> 00:00:42.770
And my question to you is, how
many different ways are there

00:00:42.770 --> 00:00:46.750
to get from here to here?

00:00:46.750 --> 00:00:48.230
That's the problem.

00:00:48.230 --> 00:00:50.780
So you can pause it now
and try to solve it.

00:00:50.780 --> 00:00:54.400
And I'll try-- when you unpause
it --slowly work to

00:00:54.400 --> 00:00:54.970
the solution.

00:00:54.970 --> 00:00:55.920
And you can pause it anywhere.

00:00:55.920 --> 00:00:59.180
You can kind of view all the
intermediary steps as hints.

00:00:59.180 --> 00:01:01.400
So the way to think
about it is, well

00:01:01.400 --> 00:01:02.640
let's do the easy ones.

00:01:02.640 --> 00:01:05.550
Let's figure out how many ways
can I go from here to each

00:01:05.550 --> 00:01:08.070
intermediate cell, and maybe
that'll help us to figure out

00:01:08.070 --> 00:01:11.020
how many ways can we get
to this end point.

00:01:11.020 --> 00:01:13.540
So let me pick a
suitable color.

00:01:13.540 --> 00:01:17.180
How many ways can I get from
this starting point to this

00:01:17.180 --> 00:01:18.720
cell right there?

00:01:18.720 --> 00:01:20.010
I can only go one way, right?

00:01:20.010 --> 00:01:22.220
I can only go from here,
straight to the right.

00:01:22.220 --> 00:01:23.110
So there's only one way.

00:01:23.110 --> 00:01:25.740
Let me write that right there.

00:01:25.740 --> 00:01:27.150
One way to get there.

00:01:27.150 --> 00:01:29.620
And how many ways can I get
to this cell right here?

00:01:29.620 --> 00:01:31.610
There's only one way.

00:01:31.610 --> 00:01:33.210
I can only go straight down.

00:01:33.210 --> 00:01:36.000
Remember, I can't go
to the right, down,

00:01:36.000 --> 00:01:36.900
and then the left.

00:01:36.900 --> 00:01:38.100
Because left is not allowed.

00:01:38.100 --> 00:01:41.380
Every step has to give me some
progress to my goal.

00:01:41.380 --> 00:01:45.190
So it looks like we've made
a little bit of progress.

00:01:45.190 --> 00:01:50.210
So how many ways can I get
to this cell right here?

00:01:50.210 --> 00:01:52.060
I could go to the right here.

00:01:52.060 --> 00:01:53.580
And then I could go one more
step to the right.

00:01:53.580 --> 00:01:54.860
And that's pretty much
the only way I

00:01:54.860 --> 00:01:55.480
can get there, right?

00:01:55.480 --> 00:01:57.060
If I go down, I can't
go back up.

00:01:57.060 --> 00:01:59.510
So there's only one
way to go there.

00:01:59.510 --> 00:02:01.400
By the same logic, only
one way to go there.

00:02:01.400 --> 00:02:02.280
Just straight out.

00:02:02.280 --> 00:02:03.270
One way to go there.

00:02:03.270 --> 00:02:04.530
One way to go there.

00:02:04.530 --> 00:02:07.710
Similarly, you can kind of
view these as symmetric.

00:02:07.710 --> 00:02:10.070
I mean, when you go straight to
the right, this is if you

00:02:10.070 --> 00:02:11.220
go straight down.

00:02:11.220 --> 00:02:12.680
So this is one way.

00:02:12.680 --> 00:02:13.870
Just straight all the way down.

00:02:13.870 --> 00:02:14.550
One way.

00:02:14.550 --> 00:02:15.050
One way.

00:02:15.050 --> 00:02:16.080
One way.

00:02:16.080 --> 00:02:17.470
Fair enough.

00:02:17.470 --> 00:02:20.380
Now let's do a slightly
more interesting cell.

00:02:20.380 --> 00:02:22.820
How many ways are there to get
to this cell right here?

00:02:25.930 --> 00:02:27.370
Well, this first time,
let's draw it out.

00:02:27.370 --> 00:02:30.920
We could either go down
and to the right.

00:02:30.920 --> 00:02:34.310
And we could go right
and down.

00:02:34.310 --> 00:02:38.350
Right, so there's two ways
to get to this cell.

00:02:38.350 --> 00:02:38.990
That's fair enough.

00:02:38.990 --> 00:02:40.870
And this was easy to do to work
out all the ways, because

00:02:40.870 --> 00:02:41.860
there aren't that many.

00:02:41.860 --> 00:02:44.130
But you might already see
an interesting pattern.

00:02:44.130 --> 00:02:46.400
If I have a cell here.

00:02:46.400 --> 00:02:48.055
Let me draw a couple of cells.

00:02:53.546 --> 00:02:57.200
Let me draw it like that.

00:02:57.200 --> 00:03:00.590
And draw it like that.

00:03:00.590 --> 00:03:03.970
And let's just say this is some
arbitrary cell someplace

00:03:03.970 --> 00:03:04.430
in this grid.

00:03:04.430 --> 00:03:06.010
It doesn't even have to
be a six by six grid.

00:03:06.010 --> 00:03:07.120
It can be an n by n grid.

00:03:07.120 --> 00:03:08.880
We could do this problem
with 100 by 100.

00:03:08.880 --> 00:03:11.670
But then the video would take
long to do all of the math.

00:03:11.670 --> 00:03:14.210
But let's say I want to figure
out how many ways can I get to

00:03:14.210 --> 00:03:15.740
this cell right here?

00:03:15.740 --> 00:03:17.550
And let's say I know that
there are n ways

00:03:17.550 --> 00:03:18.730
to get to this cell.

00:03:18.730 --> 00:03:23.300
And m ways-- not n --m ways
to get to that cell.

00:03:23.300 --> 00:03:25.000
So how many ways can
I get there?

00:03:25.000 --> 00:03:28.200
Well the only way to get to this
cell-- and I'll do it in

00:03:28.200 --> 00:03:31.870
another color --the only way to
get to this cell, based on

00:03:31.870 --> 00:03:34.470
the rules I gave you, is either
to go straight down or

00:03:34.470 --> 00:03:35.320
go straight to the right.

00:03:35.320 --> 00:03:38.575
So you can either go from this
cell, straight down.

00:03:38.575 --> 00:03:43.080
Or you can go from this cell
and straight to the right.

00:03:43.080 --> 00:03:45.070
So how many total ways can
you get to this cell?

00:03:45.070 --> 00:03:47.960
Well the only ways you could
have gotten to this cell is by

00:03:47.960 --> 00:03:49.410
going down from here.

00:03:49.410 --> 00:03:52.420
But there was n ways you could
have gotten there.

00:03:52.420 --> 00:03:55.450
So there's n total ways that
you could get to this cell,

00:03:55.450 --> 00:03:56.950
and then go to this cell.

00:03:56.950 --> 00:03:59.650
And then there's m ways that
you could get to this cell,

00:03:59.650 --> 00:04:01.970
and then go to this cell.

00:04:01.970 --> 00:04:06.260
So this cell-- or this box --I
keep using the word cell,

00:04:06.260 --> 00:04:09.410
maybe because my brain
is fixated on Excel.

00:04:09.410 --> 00:04:13.850
There are n plus m ways
to get to it.

00:04:13.850 --> 00:04:15.470
And hopefully you understand
that intuition.

00:04:15.470 --> 00:04:17.670
Because there's two ways to
get to this cell directly,

00:04:17.670 --> 00:04:19.940
from that one and
from that one.

00:04:19.940 --> 00:04:22.000
And I'm saying that there's
n ways to get there.

00:04:22.000 --> 00:04:25.640
So of all of the ways to get
here, there's n ways to get

00:04:25.640 --> 00:04:26.710
here before that.

00:04:26.710 --> 00:04:28.100
And same logic for that.

00:04:28.100 --> 00:04:29.210
And you saw that right here.

00:04:29.210 --> 00:04:31.230
With the two, is one plus one.

00:04:31.230 --> 00:04:33.430
So there's two ways
to get there.

00:04:33.430 --> 00:04:36.070
And let's see if that logic
holds one more time.

00:04:36.070 --> 00:04:37.280
Just because, I don't want
you to just take this

00:04:37.280 --> 00:04:38.860
as a leap of faith.

00:04:38.860 --> 00:04:41.110
It should make sense to you.

00:04:41.110 --> 00:04:42.740
So how many ways are
there to get here?

00:04:42.740 --> 00:04:44.620
So based on what I just told
you, I should just be able to

00:04:44.620 --> 00:04:47.330
add two plus one,
and say three.

00:04:47.330 --> 00:04:49.120
But let's see if
that works out.

00:04:49.120 --> 00:04:51.060
So I could go all the
way to the right.

00:04:51.060 --> 00:04:52.160
That's one.

00:04:52.160 --> 00:04:54.990
I could go two.

00:04:54.990 --> 00:04:58.850
Then I could go down
and that way.

00:04:58.850 --> 00:04:59.550
Three.

00:04:59.550 --> 00:05:03.460
And notice, out of all of the
three ways to get here, one of

00:05:03.460 --> 00:05:04.980
them comes from this cell.

00:05:04.980 --> 00:05:07.720
And two of them comes
from this cell.

00:05:07.720 --> 00:05:09.410
So that makes sense,
relative to the

00:05:09.410 --> 00:05:10.390
intuition I just gave you.

00:05:10.390 --> 00:05:12.415
So let's just fill
out this box.

00:05:12.415 --> 00:05:13.580
Let's just fill out the box.

00:05:13.580 --> 00:05:18.030
This is going to be one plus
two is three as well.

00:05:18.030 --> 00:05:20.530
Let's fill out-- let me switch
colors just to make it

00:05:20.530 --> 00:05:21.640
interesting.

00:05:21.640 --> 00:05:24.640
This is one plus
three is four.

00:05:24.640 --> 00:05:26.840
Three plus three is six.

00:05:26.840 --> 00:05:29.640
Three plus one is four.

00:05:29.640 --> 00:05:33.400
And you can kind of see a
symmetry along this diagonal.

00:05:33.400 --> 00:05:33.710
Right?

00:05:33.710 --> 00:05:34.700
There's a three and a three.

00:05:34.700 --> 00:05:36.130
A four and a four.

00:05:36.130 --> 00:05:38.790
And if you've seen the videos
on the binomia coefficients,

00:05:38.790 --> 00:05:41.900
or Pascal's triangle, you should
hopefully start seeing

00:05:41.900 --> 00:05:44.400
a pattern, or some recognizable
numbers here.

00:05:44.400 --> 00:05:44.720
So what's this?

00:05:44.720 --> 00:05:45.810
This is five.

00:05:45.810 --> 00:05:47.880
And this square has all sorts
of neat things here.

00:05:47.880 --> 00:05:48.630
You have all ones.

00:05:48.630 --> 00:05:50.960
And you have one, two, three,
four, five, six.

00:05:50.960 --> 00:05:51.790
Well, six right?

00:05:51.790 --> 00:05:53.570
One plus five.

00:05:53.570 --> 00:05:56.030
Let's see, four, five, six.

00:05:56.030 --> 00:05:57.710
Six plus four is 10.

00:05:57.710 --> 00:05:59.100
Six plus four is 10.

00:05:59.100 --> 00:06:00.940
10 plus five is 15.

00:06:00.940 --> 00:06:03.140
15 plus six is 21.

00:06:03.140 --> 00:06:05.130
10 plus 10 is 20.

00:06:05.130 --> 00:06:07.900
10 plus five is 15.

00:06:07.900 --> 00:06:09.440
21.

00:06:09.440 --> 00:06:11.110
See, this is 35.

00:06:11.110 --> 00:06:12.810
35.

00:06:12.810 --> 00:06:13.990
70.

00:06:13.990 --> 00:06:20.240
35 plus 21 is 56.

00:06:20.240 --> 00:06:23.490
35 plus 21 is 56.

00:06:23.490 --> 00:06:31.030
56 plus 70 is 126.

00:06:31.030 --> 00:06:35.160
And then 126 plus 126 is 252.

00:06:35.160 --> 00:06:36.970
So let me do that in
a special color.

00:06:36.970 --> 00:06:38.260
There's 200, right?

00:06:38.260 --> 00:06:41.970
yeah, 252 ways from getting
from there to there.

00:06:41.970 --> 00:06:44.430
And that was just a neat kind
of pattern problem, to

00:06:44.430 --> 00:06:45.180
understand this.

00:06:45.180 --> 00:06:46.030
And you could just
fill it out.

00:06:46.030 --> 00:06:48.485
And you can actually do
this for any n by m.

00:06:48.485 --> 00:06:49.710
It doesn't even have
to be n by m.

00:06:49.710 --> 00:06:51.290
You could figure out, I could
have said the problem, how

00:06:51.290 --> 00:06:52.880
many ways are there to
get to this square?

00:06:52.880 --> 00:06:53.740
And you would have solved it.

00:06:53.740 --> 00:06:55.770
Once you see that pattern, you
can just do simple addition

00:06:55.770 --> 00:06:56.700
and figure it out.

00:06:56.700 --> 00:06:59.820
To do it by hand, to figure out
the 252 ways of getting

00:06:59.820 --> 00:07:01.090
there would have taken
you forever.

00:07:01.090 --> 00:07:03.940
It would have wasted
all of your time.

00:07:03.940 --> 00:07:06.150
But there's something
interesting here.

00:07:06.150 --> 00:07:08.240
If you did watch the video on
binomial coefficients, you'll

00:07:08.240 --> 00:07:12.700
recognize that these are the
binomial coefficients.

00:07:12.700 --> 00:07:14.860
Actually let me draw something
for you and maybe you'll

00:07:14.860 --> 00:07:17.080
recognize it there.

00:07:17.080 --> 00:07:18.940
And this is all bonus relative
to the problem.

00:07:18.940 --> 00:07:20.110
You have now solved it.

00:07:20.110 --> 00:07:23.180
If your whole goal was just to
solve this problem, and not to

00:07:23.180 --> 00:07:26.090
see how it connects to other
mathematics, then you can stop

00:07:26.090 --> 00:07:26.750
watching it now.

00:07:26.750 --> 00:07:30.130
But now I'll show you something
neat about it.

00:07:30.130 --> 00:07:35.390
So, if you wanted to figure out
x plus y to the nth power.

00:07:35.390 --> 00:07:38.240
And we do this a lot in the
videos on Pascal's triangle

00:07:38.240 --> 00:07:39.420
and binomial coefficients.

00:07:39.420 --> 00:07:41.240
But I want to show you it's
the same exact thing here.

00:07:41.240 --> 00:07:43.620
This might actually be even an
easier way for you to do it,

00:07:43.620 --> 00:07:46.170
than using Pascal's triangle.

00:07:46.170 --> 00:07:48.270
Although it's essentially
the same thing.

00:07:48.270 --> 00:08:01.730
If you write one, x, x, x
squared, x to the third, x to

00:08:01.730 --> 00:08:04.100
the fourth, x to the fifth.

00:08:04.100 --> 00:08:05.430
These are supposed to go
on top of the square.

00:08:05.430 --> 00:08:07.170
But I realize I was running
out of space.

00:08:07.170 --> 00:08:13.330
You write one and y, y squared,
y to the third, y to

00:08:13.330 --> 00:08:16.630
the fourth, y to the fifth.

00:08:16.630 --> 00:08:22.460
And if I were to say, given
this, what is x plus y, I

00:08:22.460 --> 00:08:26.820
don't know, to the
fourth power?

00:08:26.820 --> 00:08:29.280
So what you can do is, you say
OK, well, x plus y to the

00:08:29.280 --> 00:08:32.380
fourth is going to be, you know,
x to the fourth plus

00:08:32.380 --> 00:08:34.150
something, something, something,
a bunch of terms,

00:08:34.150 --> 00:08:35.650
all the way to y to
the fourth, and

00:08:35.650 --> 00:08:36.640
everything in between.

00:08:36.640 --> 00:08:40.409
So as you go here, where you
say, x to the fourth all the

00:08:40.409 --> 00:08:41.730
way down to y to the fourth.

00:08:41.730 --> 00:08:43.750
So it'll be this diagonal
right here.

00:08:43.750 --> 00:08:46.580
So let me draw this diagonal.

00:08:46.580 --> 00:08:49.630
So this diagonal right here.

00:08:49.630 --> 00:08:51.990
And these will essentially tell
you to the coefficients.

00:08:51.990 --> 00:08:53.530
And not only will it tell
you the coefficients.

00:08:53.530 --> 00:08:56.430
It will tell you the mix
of the coefficients.

00:08:56.430 --> 00:08:58.540
So what do I mean by that?

00:08:58.540 --> 00:09:00.230
So let me erase some
stuff here.

00:09:00.230 --> 00:09:02.720
This is all bonus to
the actual problem.

00:09:02.720 --> 00:09:04.140
We've solved the problem.

00:09:04.140 --> 00:09:06.520
So I can erase all of this.

00:09:06.520 --> 00:09:09.580
So we want to figure out
x plus y to the fourth.

00:09:09.580 --> 00:09:11.910
All we have to do-- and I'll
leave you the play with this

00:09:11.910 --> 00:09:15.600
cube a little bit more just to
see all the patterns in it and

00:09:15.600 --> 00:09:18.560
what do the numbers do along
each row and each column and

00:09:18.560 --> 00:09:19.740
each diagonal?

00:09:19.740 --> 00:09:22.240
But if you want to figure out
x plus y to the fourth, or

00:09:22.240 --> 00:09:25.050
actually just this is
x to the fourth.

00:09:25.050 --> 00:09:27.670
Or you could say one
x to the fourth.

00:09:27.670 --> 00:09:32.320
One x to the fourth times one,
which is just one x to the

00:09:32.320 --> 00:09:41.030
fourth, plus four x to
the third, times y.

00:09:41.030 --> 00:09:41.260
Right?

00:09:41.260 --> 00:09:43.430
That's just this one
right there.

00:09:43.430 --> 00:09:48.015
Then you say, plus six
x squared y squared.

00:09:51.110 --> 00:09:57.910
And then we're at plus four
x y to the third.

00:09:57.910 --> 00:10:01.910
And then finally plus one
y to the fourth times

00:10:01.910 --> 00:10:03.670
just a one or an x.

00:10:03.670 --> 00:10:05.000
And you might say, that's
amazing Sal.

00:10:05.000 --> 00:10:06.060
How did that work?

00:10:06.060 --> 00:10:11.640
And the way to think about it
is, when you multiply this x

00:10:11.640 --> 00:10:15.410
plus y to the fourth-- and I
encourage you to try it if

00:10:15.410 --> 00:10:20.210
you're bored one day --there's
actually six ways to get an x

00:10:20.210 --> 00:10:21.690
squared y squared term.

00:10:21.690 --> 00:10:23.240
You'll see that when you
multiply it all out.

00:10:23.240 --> 00:10:26.710
And we did that in kind of the
intuition behind the binomial

00:10:26.710 --> 00:10:27.760
theorem videos.

00:10:27.760 --> 00:10:30.610
But there's six ways to get
there, which is the exact same

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problem as the brain teaser
that I gave you before.

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How many ways can you
get to this square?

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Well, we figured out
there are six ways.

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Anyway hope that you enjoyed
that a little bit.