Probability density functions

WEBVTT
Kind: captions
Language: en

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In the last video, I introduced
you to the notion of-- well,

00:00:04.290 --> 00:00:05.773
really we started with
the random variable.

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And then we moved on to the two
types of random variables.

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You had discrete, that took on
a finite number of values.

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And the these, I was going
to say that they tend to be

00:00:16.480 --> 00:00:18.420
integers, but they don't
always have to be integers.

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You have discrete, so finite
meaning you can't have an

00:00:22.880 --> 00:00:25.800
infinite number of values for
a discrete random variable.

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And then we have the
continuous, which can take

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on an infinite number.

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And the example I gave
for continuous is, let's

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say random variable x.

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And people do tend to use-- let
me change it a little bit, just

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so you can see it can be
something other than an x.

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Let's have the random
variable capital Y.

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They do tend to be
capital letters.

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Is equal to the exact
amount of rain tomorrow.

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And I say rain because I'm
in northern California.

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It's actually raining
quite hard right now.

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We're short right now,
so that's a positive.

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We've been having a drought,
so that's a good thing.

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But the exact amount
of rain tomorrow.

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And let's say I don't know
what the actual probability

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distribution function for this
is, but I'll draw one and

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then we'll interpret it.

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Just so you can kind of think
about how you can think about

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continuous random variables.

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So let me draw a probability
distribution, or they call

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it its probability
density function.

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And we draw like this.

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And let's say that there is--
it looks something like this.

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Like that.

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All right, and then I don't
know what this height is.

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So the x-axis here is
the amount of rain.

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Where this is 0 inches, this
is 1 inch, this is 2 inches,

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this is 3 inches, 4 inches.

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And then this is some height.

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Let's say it peaks out
here at, I don't know,

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let's say this 0.5.

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So the way to think about it,
if you were to look at this and

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I were to ask you, what is the
probability that Y-- because

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that's our random variable--
that Y is exactly

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equal to 2 inches?

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That Y is exactly
equal to two inches.

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What's the probability
of that happening?

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Well, based on how we thought
about the probability

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distribution functions for the
discrete random variable,

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you'd say OK, let's see.

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2 inches, that's the case
we care about right now.

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Let me go up here.

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You'd say it looks
like it's about 0.5.

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And you'd say, I don't
know, is it a 0.5 chance?

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And I would say no, it
is not a 0.5 chance.

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And before we even think about
how we would interpret it

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visually, let's just think
about it logically.

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What is the probability that
tomorrow we have exactly

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2 inches of rain?

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Not 2.01 inches of rain,
not 1.99 inches of rain.

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Not 1.99999 inches of rain,
not 2.000001 inches of rain.

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Exactly 2 inches of rain.

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I mean, there's not a single
extra atom, water molecule

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above the 2 inch mark.

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And not as single water
molecule below the 2 inch mark.

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It's essentially 0, right?

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It might not be obvious to you,
because you've probably heard,

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oh, we had 2 inches
of rain last night.

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But think about it,
exactly 2 inches, right?

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Normally if it's 2.01
people will say that's 2.

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But we're saying no,
this does not count.

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It can't be 2 inches.

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We want exactly 2.

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1.99 does not count.

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Normally our measurements, we
don't even have tools that

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can tell us whether it
is exactly 2 inches.

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No ruler you can even say
is exactly 2 inches long.

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At some point, just the way we
manufacture things, there's

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going to be an extra atom
on it here or there.

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So the odds of actually
anything being exactly a

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certain measurement to the
exact infinite decimal

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point is actually 0.

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The way you would think about a
continuous random variable,

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you could say what is the
probability that Y is almost 2?

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So if we said that the absolute
value of Y minus is 2 is

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less than some tolerance?

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Is less than 0.1.

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And if that doesn't make sense
to you, this is essentially

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just saying what is the
probability that Y is greater

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than 1.9 and less than 2.1?

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These two statements
are equivalent.

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I'll let you think
about it a little bit.

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But now this starts to make
a little bit of sense.

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Now we have an interval here.

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So we want all Y's
between 1.9 and 2.1.

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So we are now talking
about this whole area.

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And area is key.

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So if you want to know the
probability of this occurring,

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you actually want the area
under this curve from this

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point to this point.

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And for those of you who have
studied your calculus, that

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would essentially be the
definite integral of this

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probability density function
from this point to this point.

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So from-- let me see, I've
run out of space down here.

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So let's say if this
graph-- let me draw it

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in a different color.

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If this line was defined
by, I'll call it f of x.

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I could call it p
of x or something.

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The probability of this
happening would be equal to the

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integral, for those of you
who've studied calculus, from

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1.9 to 2.1 of f of x dx.

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Assuming this is the x-axis.

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So it's a very important
thing to realize.

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Because when a random variable
can take on an infinite number

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of values, or it can take on
any value between an interval,

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to get an exact value, to
get exactly 1.999, the

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probability is actually 0.

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It's like asking you what
is the area under a

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curve on just this line.

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Or even more specifically,
it's like asking you

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what's the area of a line?

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An area of a line, if you
were to just draw a line,

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you'd say well, area
is height times base.

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Well the height has some
dimension, but the base,

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what's the width the a line?

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As far as the way we've defined
a line, a line has no with,

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and therefore no area.

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And it should make
intuitive sense.

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That the probability of a very
super-exact thing happening

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is pretty much 0.

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That you really have to say,
OK what's the probably

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that we'll get close to 2?

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And then you can
define an area.

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And if you said oh, what's
the probability that we get

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someplace between 1 and 3
inches of rain, then of course

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the probability is much higher.

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The probability is much higher.

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It would be all of
this kind of stuff.

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You could also say what's
the probability we have

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less than 0.1 of rain?

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Then you would go here and
if this was 0.1, you would

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calculate this area.

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And you could say what's the
probability that we have more

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than 4 inches of rain tomorrow?

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Then you would start here and
you'd calculate the area in the

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curve all the way to infinity,
if the curve has area all

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the way to infinity.

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And hopefully that's not an
infinite number, right?

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Then your probability
won't make any sense.

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But hopefully if you take this
sum it comes to some number.

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And we'll say there's only a
10% chance that you have more

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than 4 inches tomorrow.

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And all of this should
immediately lead to one light

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bulb in your head, is that the
probability of all of the

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events that might occur
can't be more than 100%.

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Right?

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All the events combined--
there's a probability of 1 that

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one of these events will occur.

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So essentially, the whole
area under this curve

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has to be equal to 1.

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So if we took the integral of f
of x from 0 to infinity, this

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thing, at least as I've drawn
it, dx should be equal to 1.

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For those of you who've
studied calculus.

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For those of you who haven't,
an integral is just the

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area under a curve.

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And you can watch the calculus
videos if you want to learn a

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little bit more about
how to do them.

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And this also applies to
the discrete probability

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distributions.

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Let me draw one.

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The sum of all of the
probabilities have

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to be equal to 1.

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And that example with the
dice-- or let's say, since it's

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faster to draw, the coin-- the
two probabilities have

00:09:01.800 --> 00:09:02.850
to be equal to 1.

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So this is 1, 0, where x is
equal to 1 if we're heads

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or 0 if we're tails.

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Each of these have to be 0.5.

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Or they don't have to be 0.5,
but if one was 0.6, the

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other would have to be 0.4.

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They have to add to 1.

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If one of these was-- you can't
have a 60% probability of

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getting a heads and then a 60%
probability of getting

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a tails as well.

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Because then you would have
essentially 120% probability

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of either of the outcomes
happening, which makes

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no sense at all.

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So it's important to realize
that a probability distribution

00:09:37.330 --> 00:09:42.790
function, in this case for a
discrete random variable, they

00:09:42.790 --> 00:09:44.160
all have to add up to 1.

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So 0.5 plus 0.5.

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And in this case the area
under the probability

00:09:48.990 --> 00:09:52.500
density function also
has to be equal to 1.

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Anyway, I'm all
the time for now.

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In the next video I'll
introduce you to the idea

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of an expected value.

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See you soon.


Introduction to Random Variables

WEBVTT
Kind: captions
Language: en

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I'll now introduce you to the
concept of a random variable.

00:00:06.980 --> 00:00:09.390
And for me this is something
that I always had a lot of

00:00:09.390 --> 00:00:12.550
trouble getting my head around,
and that's really because

00:00:12.550 --> 00:00:14.040
it's a byproduct of
what it's called.

00:00:14.040 --> 00:00:17.470
It's called a variable and
we're used to variables as kind

00:00:17.470 --> 00:00:18.770
of an unknown in the equation.

00:00:18.770 --> 00:00:24.830
If I write x plus 3 is equal
to 7, the variable was x.

00:00:24.830 --> 00:00:26.680
Maybe you could solve for it
or maybe you could have an

00:00:26.680 --> 00:00:30.550
equation y is equal
to 3x minus 2.

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And then, here y and x
are both variables.

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If you input 1x you could solve
for the other variable, y.

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And you can change them.

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And variables were kind of
things that could change and

00:00:40.500 --> 00:00:41.990
that you could solve for.

00:00:41.990 --> 00:00:44.540
And they could take on
particular values.

00:00:44.540 --> 00:00:47.160
A random variable is kind of
the same thing in that it can

00:00:47.160 --> 00:00:49.820
take on multiple values, but
it's not something that you

00:00:49.820 --> 00:00:51.570
really ever solve for.

00:00:51.570 --> 00:00:53.910
So just so you get used to the
notation, a random variable

00:00:53.910 --> 00:00:56.180
is usually a capital letter.

00:00:56.180 --> 00:00:58.140
Usually a Capital X, Y, or Z.

00:00:58.140 --> 00:00:59.940
Usually a capital X.

00:00:59.940 --> 00:01:02.530
And where it really defers from
a traditional variable is that

00:01:02.530 --> 00:01:04.850
it can take on a bunch of
different values, like a

00:01:04.850 --> 00:01:06.890
traditional variable but
you never solve for it.

00:01:06.890 --> 00:01:08.750
And really, it's a little
misleading to call it

00:01:08.750 --> 00:01:09.480
a variable at all.

00:01:09.480 --> 00:01:11.110
It's really a function.

00:01:11.110 --> 00:01:15.490
And it's a function that maps
you from the world of random

00:01:15.490 --> 00:01:17.740
processes to an actual number.

00:01:17.740 --> 00:01:20.180
So let's say-- I don't
know-- I wanted to somehow

00:01:20.180 --> 00:01:23.560
quantify a random process.

00:01:23.560 --> 00:01:25.590
Is it going to rain
or not tomorrow?

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So let's see-- rain tomorrow.

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No rain.

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So you could observe that.

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You could wait until tomorrow
and see if it rains or not, but

00:01:38.850 --> 00:01:39.940
then, how do you quantify it?

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Well, we can define a random
variable that will quantify it.

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We can say this random variable
is going to be equal to 1 if it

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rains tomorrow and it equals 0
if it doesn't rain tomorrow.

00:01:50.390 --> 00:01:52.580
We didn't have to assign 1
and 0; those tend to be a

00:01:52.580 --> 00:01:53.500
little bit more useful.

00:01:53.500 --> 00:01:54.460
They make sense.

00:01:54.460 --> 00:01:56.400
But we could have assigned this
as-- I don't know-- we could

00:01:56.400 --> 00:01:59.370
have said that this is 21
and that this is 100.

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It's however you define it.

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So it's important to keep this
distinction in mind, that a

00:02:03.570 --> 00:02:06.470
random variable-- it isn't a
variable in the traditional

00:02:06.470 --> 00:02:07.520
sense of the world.

00:02:07.520 --> 00:02:12.160
It's more of a function that
maps us from the world of a

00:02:12.160 --> 00:02:15.430
random process to a number.

00:02:15.430 --> 00:02:18.760
And then, this number
is going to be random.

00:02:18.760 --> 00:02:20.820
Because we don't know if it's
going to rain or not tomorrow,

00:02:20.820 --> 00:02:23.540
maybe we have some sense
of the probabilities.

00:02:23.540 --> 00:02:27.360
So this variable can
take on either value.

00:02:27.360 --> 00:02:30.780
And that leads us-- let me
draw a couple of more random

00:02:30.780 --> 00:02:35.000
variable definitions just so
you get the intuition that

00:02:35.000 --> 00:02:37.490
these really are functions.

00:02:37.490 --> 00:02:41.460
So we could define a random
variable x as equal to--

00:02:41.460 --> 00:02:42.650
well, actually, let me--

00:02:42.650 --> 00:02:45.940
It could just be a very
obvious numeric mapping.

00:02:45.940 --> 00:02:55.690
It could just be the
number facing up when

00:02:55.690 --> 00:02:58.856
I roll a fair dice.

00:02:58.856 --> 00:03:00.560
That's a random variable.

00:03:00.560 --> 00:03:04.575
I could've also said just the
number most facing east when I

00:03:04.575 --> 00:03:06.890
rolled a fair die, although
that's a little less useful.

00:03:06.890 --> 00:03:11.200
You could also define a random
variable, x is equal to-- you

00:03:11.200 --> 00:03:17.730
could say it's equal to 1 if
heads, and I could

00:03:17.730 --> 00:03:21.000
say 0 if tails.

00:03:21.000 --> 00:03:22.680
Or I could have done it
the other way around.

00:03:22.680 --> 00:03:25.940
Once again, these heads
and tails are outcomes

00:03:25.940 --> 00:03:27.810
of an experiment.

00:03:27.810 --> 00:03:32.770
When I flip a coin that's a
random process, each flip is an

00:03:32.770 --> 00:03:36.090
experiment and then the random
variable is just quantifying

00:03:36.090 --> 00:03:37.100
that experiment.

00:03:37.100 --> 00:03:39.010
So I know it sounds a little
technical with the terminology,

00:03:39.010 --> 00:03:41.540
but what we're really doing
is fairly something simple.

00:03:41.540 --> 00:03:43.380
We're just assigning a
1 if we get a heads, a

00:03:43.380 --> 00:03:44.660
0 if we get a tails.

00:03:44.660 --> 00:03:49.150
And the random variable is
just that function mapping.

00:03:49.150 --> 00:03:52.630
Now I'm going into all of this
random variable business, which

00:03:52.630 --> 00:03:55.500
I didn't do before when we did
probability because it does

00:03:55.500 --> 00:03:58.800
start to become a little
bit useful notationally.

00:03:58.800 --> 00:04:01.090
Because we'll start to talk
about things like probability

00:04:01.090 --> 00:04:03.380
distributions and expected
values, and it really is

00:04:03.380 --> 00:04:06.590
useful to quantify things
as a random variable.

00:04:06.590 --> 00:04:10.270
And with that said, let's talk
about probability distributions

00:04:10.270 --> 00:04:12.130
and expected values.

00:04:12.130 --> 00:04:15.560
So the first thing,
probability-- actually, let

00:04:15.560 --> 00:04:17.570
me define something else.

00:04:17.570 --> 00:04:20.140
There's two types of
random variables.

00:04:20.140 --> 00:04:25.250
You can have discrete random
variables, or continuous.

00:04:28.030 --> 00:04:29.560
So there's two types.

00:04:29.560 --> 00:04:32.150
Discrete would be like, really
all of the definitions

00:04:32.150 --> 00:04:35.210
that we had up until now.

00:04:35.210 --> 00:04:38.200
The flip of a coin, you're
either going to get a 1 or a 0.

00:04:38.200 --> 00:04:40.010
There's two discrete cases.

00:04:40.010 --> 00:04:42.560
When I roll a die there's
1 of 6 different

00:04:42.560 --> 00:04:44.570
outcomes I can have.

00:04:44.570 --> 00:04:46.380
Whether it rains
tomorrow, there's 1 of

00:04:46.380 --> 00:04:49.080
2 outcomes: yes or no.

00:04:49.080 --> 00:04:52.110
So in all of these situations
you could almost say you

00:04:52.110 --> 00:04:53.920
could have accountable
number of outcomes.

00:04:53.920 --> 00:04:57.240
A continuous random variable
could take on an infinite

00:04:57.240 --> 00:04:58.100
number of outcomes.

00:04:58.100 --> 00:05:01.850
So a continuous random variable
would be the exact-- it could

00:05:01.850 --> 00:05:11.660
be x is equal to the exact
amount of rain in

00:05:11.660 --> 00:05:13.610
inches tomorrow.

00:05:13.610 --> 00:05:16.500
Now why is this continuous?

00:05:16.500 --> 00:05:19.640
Well, if you think about it
that could take on any of

00:05:19.640 --> 00:05:21.200
an infinite set of values.

00:05:21.200 --> 00:05:23.940
You might have 1 inch of
rain tomorrow or you might

00:05:23.940 --> 00:05:25.750
have 1.1 inches of rain.

00:05:25.750 --> 00:05:28.620
You might have 1.111
inches of rain.

00:05:28.620 --> 00:05:31.560
You might have 2.1111
inches of rain.

00:05:31.560 --> 00:05:34.080
As you could see, you can come
up with an infinite number

00:05:34.080 --> 00:05:37.610
of combinations of the
amount of rain you have.

00:05:37.610 --> 00:05:39.840
To get exact-- well, I'll go
into more of this when I

00:05:39.840 --> 00:05:41.950
talk about probability
density functions.

00:05:41.950 --> 00:05:44.790
But in general, even though you
might say, oh, well, I can't

00:05:44.790 --> 00:05:47.280
imagine having an infinite
amount of rain because we're

00:05:47.280 --> 00:05:49.560
not going to have 100
gallons of rain tomorrow.

00:05:49.560 --> 00:05:52.180
But if you think about it, you
could take on any value between

00:05:52.180 --> 00:05:53.990
1 and 2 inches of rain.

00:05:53.990 --> 00:05:56.170
There's actually an
infinite number of

00:05:56.170 --> 00:05:59.900
values between 0 and 1.

00:05:59.900 --> 00:06:03.950
For any value you could find
a number that's a little--

00:06:03.950 --> 00:06:04.670
just to think about it.

00:06:04.670 --> 00:06:07.090
Between 0 and 1, 1/2
is in between those.

00:06:07.090 --> 00:06:09.460
And you can always find a
number that's halfway between

00:06:09.460 --> 00:06:11.740
those and then a number that's
halfway between those.

00:06:11.740 --> 00:06:13.770
There's an infinite number
of numbers between

00:06:13.770 --> 00:06:15.640
any two numbers.

00:06:15.640 --> 00:06:19.130
While, with the discrete random
variables, it just took on

00:06:19.130 --> 00:06:21.050
a finite number of values.

00:06:21.050 --> 00:06:23.740
Those ones up here, it
could only take on 1 or 0.

00:06:23.740 --> 00:06:25.400
It didn't take on 1.1.

00:06:25.400 --> 00:06:27.550
Or it couldn't
take on infinity.

00:06:27.550 --> 00:06:31.960
And I make that difference
because actually, how we look

00:06:31.960 --> 00:06:35.190
at it in terms of their
probability distributions

00:06:35.190 --> 00:06:35.990
are a little different.

00:06:35.990 --> 00:06:37.100
But they're very related.

00:06:37.100 --> 00:06:40.330
So a discreet random variable
like the ones that we had

00:06:40.330 --> 00:06:42.800
defined at the beginning of
these videos, they have a

00:06:42.800 --> 00:06:44.980
probability distribution.

00:06:44.980 --> 00:06:51.880
Let's do it for-- let's say
x is equal to the number

00:06:51.880 --> 00:06:59.440
facing up on a fair dice.

00:07:02.960 --> 00:07:06.550
So we already know that
to get each outcome--

00:07:06.550 --> 00:07:07.360
there's 6 outcomes.

00:07:07.360 --> 00:07:11.360
You either can get a 1, 2,
3, 4, 5, or 6 facing up.

00:07:11.360 --> 00:07:15.430
So let me draw the
probability distribution.

00:07:15.430 --> 00:07:16.690
So here are all the outcomes.

00:07:16.690 --> 00:07:19.580
I'll draw that on the
horizontal axis.

00:07:19.580 --> 00:07:30.330
So you get one outcome
is 1, 2, 3, 4, 5, 6.

00:07:30.330 --> 00:07:33.850
And then, in the y-axis or in
the vertical, you plot what is

00:07:33.850 --> 00:07:36.230
the probability of each of
those outcomes occurring?

00:07:38.890 --> 00:07:40.295
What's the probability?

00:07:40.295 --> 00:07:42.510
I'll switch colors.

00:07:42.510 --> 00:07:43.980
So what's the probability
of getting a 1?

00:07:43.980 --> 00:07:45.960
Well, it's 1/6.

00:07:45.960 --> 00:07:47.976
So if I draw 1/6 here.

00:07:50.640 --> 00:07:53.330
So the probability is of each
of these occurring-- oh, I

00:07:53.330 --> 00:07:56.970
didn't want to do it
like that-- are 1/6.

00:07:56.970 --> 00:07:59.425
It's essentially like a
histogram or a bar chart.

00:08:02.570 --> 00:08:04.930
My intention is to draw them
all the exact same height.

00:08:08.690 --> 00:08:11.840
So this is telling us that
each of these there's a 1/6

00:08:11.840 --> 00:08:14.850
chance of it occurring.

00:08:14.850 --> 00:08:19.040
If I were to draw a probability
distribution for-- if I were to

00:08:19.040 --> 00:08:27.080
find the random variable
x is equal to 1 if

00:08:27.080 --> 00:08:31.840
heads, 0 if tails.

00:08:31.840 --> 00:08:34.150
This has a very simple
probability distribution.

00:08:37.420 --> 00:08:39.140
There's only 2 outcomes.

00:08:39.140 --> 00:08:43.730
You're either going
to get a 1 or a 0.

00:08:43.730 --> 00:08:49.360
And then in the y-axis you'll
have the probabilities.

00:08:49.360 --> 00:08:51.550
And each of these, if it's
a fair dice, it's going

00:08:51.550 --> 00:08:53.040
to be a 1/2 probability.

00:08:53.040 --> 00:08:55.410
And you're going to draw a
little box like that that tells

00:08:55.410 --> 00:08:59.160
you each of them are 1/2.

00:08:59.160 --> 00:09:00.880
Now let's say that you had
another-- I don't know--

00:09:00.880 --> 00:09:04.320
probably distribution function
that looks something like this.

00:09:04.320 --> 00:09:07.390
Those are kind of very run of
the mill examples, but what if

00:09:07.390 --> 00:09:10.400
I had something like this?

00:09:10.400 --> 00:09:10.920
Let's see.

00:09:13.510 --> 00:09:16.120
Where I had some dice-- let's
say we had some weird

00:09:16.120 --> 00:09:23.910
weightings on it where it had
an outcome of 1-- I had a

00:09:23.910 --> 00:09:28.680
1/6 chance like that.

00:09:28.680 --> 00:09:31.805
Let's say for some reason
a 2 could never happen.

00:09:34.950 --> 00:09:36.240
Maybe there is no 2.

00:09:36.240 --> 00:09:42.532
Let's say 3 has a 1/6 chance,
a 4 has a 1/6 chance,

00:09:42.532 --> 00:09:44.810
a 5 has a 1/6 chance.

00:09:44.810 --> 00:09:46.710
And let's say there's two 6's.

00:09:46.710 --> 00:09:49.970
Like we erased the 2 and we
put a 6 where the 2 is,

00:09:49.970 --> 00:09:50.690
so there's two 6's.

00:09:50.690 --> 00:09:54.710
So the 6 has a 2/6 chance.

00:09:54.710 --> 00:09:56.810
So the 6 would be a 2/6 chance.

00:09:56.810 --> 00:09:59.720
So now I think you can see
where the probability

00:09:59.720 --> 00:10:02.490
distribution function
starts to become a little

00:10:02.490 --> 00:10:03.620
bit more interesting.

00:10:03.620 --> 00:10:06.040
In the other ones it was kind
of what we would call a

00:10:06.040 --> 00:10:07.410
uniform distribution.

00:10:07.410 --> 00:10:09.290
All of the outcomes
are equally likely.

00:10:09.290 --> 00:10:10.550
That's a uniform distribution.

00:10:10.550 --> 00:10:13.010
Once again, this is a
uniform distribution.

00:10:13.010 --> 00:10:16.410
This one, all of a sudden, is
not a uniform distribution.

00:10:16.410 --> 00:10:19.560
1 is just as likely as 3, 4, 5.

00:10:19.560 --> 00:10:22.740
2 is impossible to happen, but
6 is twice as likely to happen

00:10:22.740 --> 00:10:25.800
as everything else-- is 2/6.

00:10:25.800 --> 00:10:28.610
If someone gives you a
probability density function or

00:10:28.610 --> 00:10:31.460
if they give you a little chart
like this, you can immediately

00:10:31.460 --> 00:10:34.710
say, what's the probability of
different events occurring?

00:10:34.710 --> 00:10:44.570
So if you said that x is equal
to the number on unfair dice

00:10:44.570 --> 00:10:46.730
that's described by
this probability

00:10:46.730 --> 00:10:48.190
distribution function.

00:10:48.190 --> 00:10:53.450
Then if I were to ask you the
probability that x is equal to

00:10:53.450 --> 00:10:56.370
6, you'd say, oh, the
probability x equals 6

00:10:56.370 --> 00:10:59.570
is equal to 2/6 or 1/3.

00:10:59.570 --> 00:11:04.660
If I were to say what's the
probability that x, the random

00:11:04.660 --> 00:11:10.390
variable x as given by this
definition, is greater than 5,

00:11:10.390 --> 00:11:13.350
you would take-- well, let's
say greater than or equal to 5.

00:11:13.350 --> 00:11:16.270
Because you want to include
5, and at least, I did.

00:11:16.270 --> 00:11:19.290
So it would include this
situation and this situation.

00:11:19.290 --> 00:11:21.070
So it'd be 1/6 plus 2/6.

00:11:25.680 --> 00:11:30.060
And so you could say
that's 3/6 or 1/2.

00:11:30.060 --> 00:11:32.450
So it gets pretty interesting
once you don't have a

00:11:32.450 --> 00:11:34.130
completely uniform
distribution.

00:11:34.130 --> 00:11:36.220
And I'm doing all of this
because in the next couple of

00:11:36.220 --> 00:11:39.290
videos-- well, one, we'll move
away from discreet and actually

00:11:39.290 --> 00:11:42.990
study continuous probability
distributions, which are called

00:11:42.990 --> 00:11:44.680
probability density functions.

00:11:44.680 --> 00:11:47.260
And then we'll go to study kind
of a lot of distributions

00:11:47.260 --> 00:11:48.560
that show up in nature.

00:11:48.560 --> 00:11:51.490
The uniform distribution
is one of them.

00:11:51.490 --> 00:11:54.520
There's also the binomial
distribution and the normal

00:11:54.520 --> 00:11:56.590
distribution, which is
what some people call the

00:11:56.590 --> 00:11:58.070
Gaussian or the bell curve.

00:11:58.070 --> 00:11:59.590
Which is kind of a
very common thing.

00:11:59.590 --> 00:12:01.500
But anyway, I'm all out
of time and I'll see

00:12:01.500 --> 00:12:03.410
you in the next video.


Repurchase Agreements (Repo transactions)

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:04.180
Let's say that you're in
desperate need of money and I

00:00:04.180 --> 00:00:06.070
have money to lend
to other people.

00:00:06.070 --> 00:00:10.820
So this is me and this
is my gold chain.

00:00:10.820 --> 00:00:16.760
So you come to me and say,
Sal, I need $10,000 for a

00:00:16.760 --> 00:00:18.240
kidney transplant.

00:00:18.240 --> 00:00:19.270
Can you lend me the money?

00:00:19.270 --> 00:00:20.520
I'm in desperate need.

00:00:25.780 --> 00:00:27.530
And I have $10,000.

00:00:27.530 --> 00:00:29.650
Sure, I'm willing to lend it
to you, but it's a tough

00:00:29.650 --> 00:00:32.770
economy and you never know where
that money's going to go

00:00:32.770 --> 00:00:35.130
and I don't know if you're going
to be able to keep your

00:00:35.130 --> 00:00:37.970
job after going through this
kidney surgery and all that.

00:00:37.970 --> 00:00:41.080
So I'm very careful with my
money so I want to make sure

00:00:41.080 --> 00:00:41.900
that you're good for it.

00:00:41.900 --> 00:00:44.700
So we think about it a little
bit and I say, hey, that watch

00:00:44.700 --> 00:00:47.040
you have on your wrist, that
looks pretty nice.

00:00:47.040 --> 00:00:48.550
You say, this watch?

00:00:48.550 --> 00:00:49.800
Let me draw the watch.

00:00:53.430 --> 00:00:54.680
And I say, yeah, that watch.

00:00:54.680 --> 00:00:57.150
You're like, this watch
I got from my

00:00:57.150 --> 00:01:00.440
great-great-grandfather and it's
actually worth-- I don't

00:01:00.440 --> 00:01:04.310
know-- maybe it's a diamond
studded Rolex of some sort and

00:01:04.310 --> 00:01:07.330
it's actually worth
$30,000, right?

00:01:07.330 --> 00:01:09.660
And I know that, clearly because
I've already become

00:01:09.660 --> 00:01:13.260
well acquainted with gold and
diamonds and things like that.

00:01:13.260 --> 00:01:19.300
So I say, I trust you and
you trust yourself.

00:01:19.300 --> 00:01:23.200
I'll lend you the $10,000, but
just so that we all know that

00:01:23.200 --> 00:01:26.370
everything's going to be fine,
why don't you just leave your

00:01:26.370 --> 00:01:29.360
great-grandfather's
watch with me?

00:01:29.360 --> 00:01:33.600
And when you pay back the
$10,000 with a low interest--

00:01:33.600 --> 00:01:43.320
let's say it's 10% a week-- when
you pay back the money,

00:01:43.320 --> 00:01:45.490
the principal, with the
interest, however long you

00:01:45.490 --> 00:01:47.650
borrow it, I'll give you
back your watch.

00:01:47.650 --> 00:01:49.600
You're like, I don't
know about that.

00:01:49.600 --> 00:01:51.440
First of all, 10% a week
sounds like a lot.

00:01:51.440 --> 00:01:53.210
I was like, well, don't you just
need it for the kidney

00:01:53.210 --> 00:01:55.330
transplant tomorrow and then you
can work a couple of weeks

00:01:55.330 --> 00:01:56.030
and then pay it back.

00:01:56.030 --> 00:01:58.550
10% a week's not bad-- and we
all know that that is because

00:01:58.550 --> 00:02:00.520
you compound that over the year
and it becomes some type

00:02:00.520 --> 00:02:02.120
of horrendous interest rate.

00:02:02.120 --> 00:02:03.160
But you're desperate.

00:02:03.160 --> 00:02:06.250
You need your kidney and I'm
like, look, you plan on paying

00:02:06.250 --> 00:02:07.150
me back, right?

00:02:07.150 --> 00:02:09.650
So you're going to get your
$30,000 watch back, so what's

00:02:09.650 --> 00:02:11.090
there to worry, right?

00:02:11.090 --> 00:02:15.460
So why don't you just leave this
with me as collateral?

00:02:15.460 --> 00:02:18.220
And you say, fine and you leave
your watch and you get

00:02:18.220 --> 00:02:21.340
your kidney transplant and then
you try to work really

00:02:21.340 --> 00:02:23.230
hard to pay it back,
but you never can

00:02:23.230 --> 00:02:24.280
and I keep your watch.

00:02:24.280 --> 00:02:26.950
And this is how the pawn process
essentially works.

00:02:26.950 --> 00:02:31.760
You've pawned your watch off to
me, but in a less kind of

00:02:31.760 --> 00:02:34.080
derisive way of talking about
it, you've given it as

00:02:34.080 --> 00:02:35.630
collateral for a loan.

00:02:35.630 --> 00:02:38.600
And I was willing to give you
the loan because I knew that

00:02:38.600 --> 00:02:41.470
if you couldn't pay it back, I
could keep this nice asset.

00:02:41.470 --> 00:02:45.430
And this happens all the
time in less shady

00:02:45.430 --> 00:02:46.790
parts of our economy.

00:02:46.790 --> 00:02:49.030
A bank will give you a loan and
they'll collateralize it

00:02:49.030 --> 00:02:49.600
by the house.

00:02:49.600 --> 00:02:51.820
If you can't pay the loan,
they keep the house.

00:02:51.820 --> 00:02:53.630
They want you to put a down
payment on the house so that

00:02:53.630 --> 00:02:56.400
even if the house devalues, they
still capture back most

00:02:56.400 --> 00:02:57.750
of their money.

00:02:57.750 --> 00:02:59.500
And so this is a pretty
straightforward

00:02:59.500 --> 00:03:00.430
collateralized loan.

00:03:00.430 --> 00:03:02.090
This is the collateral
you give me.

00:03:02.090 --> 00:03:02.820
I give you the loan.

00:03:02.820 --> 00:03:03.390
You pay it back.

00:03:03.390 --> 00:03:05.190
I give you back the
collateral.

00:03:05.190 --> 00:03:09.050
Now what if there was a reality
where I don't-- me as

00:03:09.050 --> 00:03:11.890
the lender, I don't even like
this notion of collateral and

00:03:11.890 --> 00:03:12.660
all of that.

00:03:12.660 --> 00:03:15.970
I actually want to make it
very clear that I have

00:03:15.970 --> 00:03:19.640
ownership of the collateral
when it happens, right?

00:03:19.640 --> 00:03:23.480
I don't want some Feds coming
in and saying, wow, whose

00:03:23.480 --> 00:03:25.620
watch is this that
you're holding?

00:03:25.620 --> 00:03:26.710
Where's the receipt for this?

00:03:26.710 --> 00:03:27.300
Where did you get it?

00:03:27.300 --> 00:03:29.060
Was it stolen from somebody?

00:03:29.060 --> 00:03:32.450
And it looks all shady and I
probably do have some side

00:03:32.450 --> 00:03:36.020
shady operations anyway so I
want to know that I own this

00:03:36.020 --> 00:03:38.050
watch in the event that
you don't come

00:03:38.050 --> 00:03:39.840
back to pay the loan.

00:03:39.840 --> 00:03:44.310
So instead, we could have done
this exact same transaction--

00:03:44.310 --> 00:03:46.290
so this is you again.

00:03:46.290 --> 00:03:47.330
This is me again.

00:03:47.330 --> 00:03:50.490
I'll draw that top hat
and the moustache.

00:03:50.490 --> 00:03:53.410
Those are my differentiating
characteristics, maybe the

00:03:53.410 --> 00:03:55.230
gold chain.

00:03:55.230 --> 00:03:57.850
And now what I can do is--
because I want ownership of

00:03:57.850 --> 00:04:02.130
that watch, what I'll do is I
will buy that watch from you.

00:04:02.130 --> 00:04:05.730
So you'll give the watch-- so
the watch will literally

00:04:05.730 --> 00:04:08.930
change hands.

00:04:08.930 --> 00:04:21.279
So I will buy the watch
from you for $10,000.

00:04:21.279 --> 00:04:23.940
But we'll also have
a side agreement.

00:04:23.940 --> 00:04:25.850
So far it's almost
identical, right?

00:04:25.850 --> 00:04:27.370
The only difference between this
and what we did before

00:04:27.370 --> 00:04:29.050
is, I'm actually selling
the money.

00:04:29.050 --> 00:04:32.590
I'm actually buying the
watch from you.

00:04:32.590 --> 00:04:34.370
This is a cash transaction.

00:04:34.370 --> 00:04:37.600
This wasn't a loan, strictly
speaking, but the same thing

00:04:37.600 --> 00:04:38.450
is happening, right?

00:04:38.450 --> 00:04:41.070
Up here you handed
me the watch and

00:04:41.070 --> 00:04:43.380
I handed you $10,000.

00:04:43.380 --> 00:04:47.940
Here, you handed me the watch
and I handed you $10,000.

00:04:47.940 --> 00:04:51.450
But what we'll add on to this
is an agreement that at some

00:04:51.450 --> 00:04:53.510
future date-- so this is now.

00:04:56.970 --> 00:04:59.620
We also going to have an
agreement-- and both of us are

00:04:59.620 --> 00:05:05.820
parties to this-- that in the
future, I will agree to sell

00:05:05.820 --> 00:05:09.040
and you will agree to buy this
watch for me for something

00:05:09.040 --> 00:05:10.390
more than $10,000.

00:05:10.390 --> 00:05:14.570
So in the future, you're going
to give me back my money.

00:05:14.570 --> 00:05:21.550
So it'll be $10,000 plus
something-- and that's

00:05:21.550 --> 00:05:24.130
something is essentially
interest, right?

00:05:24.130 --> 00:05:27.170
So I'll get my money back
and then I'll give

00:05:27.170 --> 00:05:29.890
you back the watch.

00:05:29.890 --> 00:05:33.060
So if you think about it, this
is completely identical

00:05:33.060 --> 00:05:36.210
economically to what we
did up here, right?

00:05:36.210 --> 00:05:37.930
I gave you $10,000.

00:05:37.930 --> 00:05:39.880
You gave me the watch
as collateral.

00:05:39.880 --> 00:05:44.190
When you pay back the $10,000
plus interest, I give you back

00:05:44.190 --> 00:05:46.130
the watch, right?

00:05:46.130 --> 00:05:49.220
But at no time did I really
have real, legitimate

00:05:49.220 --> 00:05:51.590
ownership of that watch.

00:05:51.590 --> 00:05:55.370
Well, in this situation,
the same thing happens.

00:05:55.370 --> 00:05:56.330
You come to me.

00:05:56.330 --> 00:05:57.450
I give you money.

00:05:57.450 --> 00:06:00.570
You give me the watch, but I
bought it from you so I have a

00:06:00.570 --> 00:06:02.760
receipt too.

00:06:02.760 --> 00:06:06.650
I am the official owner of the
watch while you have my money,

00:06:06.650 --> 00:06:09.990
but we have an agreement that at
some future date, you will

00:06:09.990 --> 00:06:12.300
repurchase the watch from me.

00:06:18.350 --> 00:06:21.200
for $10,000 plus some amount,
which is essentially the

00:06:21.200 --> 00:06:25.310
interest. So this was a loan
collateralized by a watch, but

00:06:25.310 --> 00:06:27.460
the only difference here is that
instead of it just being

00:06:27.460 --> 00:06:30.620
collateral, you actually sold
it to me and then we had a

00:06:30.620 --> 00:06:33.620
repurchase agreement-- and it
took me six minutes to say

00:06:33.620 --> 00:06:45.480
that, but I think it was worth
it-- where you agree to

00:06:45.480 --> 00:06:47.910
repurchase the watch
at some point.

00:06:47.910 --> 00:06:51.270
It's actually I'm the holder of
the repurchase agreement.

00:06:51.270 --> 00:06:54.590
So the money lender-- so
I get the watch and

00:06:54.590 --> 00:06:56.070
a repurchase agreement.

00:06:56.070 --> 00:06:59.470
We'll call that a repo.

00:06:59.470 --> 00:07:02.530
So these are two assets that I
now have-- the watch plus the

00:07:02.530 --> 00:07:03.650
repurchase agreement.

00:07:03.650 --> 00:07:04.665
You get the money.

00:07:04.665 --> 00:07:08.460
And it's actually called
a reverse repo, from

00:07:08.460 --> 00:07:09.710
your point of view.

00:07:12.130 --> 00:07:16.290
But the whole idea here is this
agreement forces you to

00:07:16.290 --> 00:07:21.030
buy the watch back at the
original amount that you

00:07:21.030 --> 00:07:26.080
essentially borrowed from me
plus some interest. And this

00:07:26.080 --> 00:07:27.890
essentially forces me
to sell it to you.

00:07:27.890 --> 00:07:29.320
So we have a-- it's

00:07:29.320 --> 00:07:31.060
essentially a forward contract.

00:07:31.060 --> 00:07:33.750
A forward contract is just an
agreement to transact in the

00:07:33.750 --> 00:07:36.900
future at some given price.

00:07:36.900 --> 00:07:39.300
And the whole reason why I did
this is because this is how

00:07:39.300 --> 00:07:40.900
the Fed transacts.

00:07:40.900 --> 00:07:42.600
This is how the Fed lends,
especially with

00:07:42.600 --> 00:07:44.520
the discount window.

00:07:44.520 --> 00:07:47.600
Sometimes it's called a
repo transaction or

00:07:47.600 --> 00:07:51.060
a repurchase agreement.

00:07:51.060 --> 00:07:53.340
And so what the Fed does when
someone comes to it at the

00:07:53.340 --> 00:07:56.180
discount window-- let me
actually draw proper balance

00:07:56.180 --> 00:07:57.660
sheets now.

00:07:57.660 --> 00:08:01.320
Let's say that this over
here is a bank in need.

00:08:01.320 --> 00:08:03.300
I won't worry about the
right-hand side of the balance

00:08:03.300 --> 00:08:04.830
sheet too much.

00:08:04.830 --> 00:08:07.640
It has some liabilities,
some equities.

00:08:07.640 --> 00:08:10.780
Let's say it has a
ton of assets.

00:08:10.780 --> 00:08:15.580
Let's say that these right
here are treasuries.

00:08:15.580 --> 00:08:17.330
But it's all out
of cash, right?

00:08:17.330 --> 00:08:19.810
This is the bank.

00:08:19.810 --> 00:08:24.390
It's all out of cash and on the
other hand, we have the

00:08:24.390 --> 00:08:26.470
Federal reserve.

00:08:26.470 --> 00:08:30.425
Let me see if I can draw their
balance sheet properly.

00:08:30.425 --> 00:08:36.409
And the Federal reserve-- well,
for the most part, these

00:08:36.409 --> 00:08:39.669
are going to be treasuries
right now.

00:08:39.669 --> 00:08:40.659
It has some liabilities.

00:08:40.659 --> 00:08:42.740
I won't go into that just yet.

00:08:42.740 --> 00:08:45.660
It has some equity.

00:08:45.660 --> 00:08:47.270
And let's say you're one
of these pariah banks.

00:08:47.270 --> 00:08:48.440
No-one's willing
to lend to you.

00:08:48.440 --> 00:08:50.580
Your depositors are taking
out their money.

00:08:50.580 --> 00:08:53.730
So you need to convert some of
these treasuries into cash.

00:08:53.730 --> 00:08:55.390
You don't want to just dump
them on the market.

00:08:55.390 --> 00:08:57.270
Maybe there are other assets
that are less liquid and if

00:08:57.270 --> 00:08:59.360
you just dump them, you won't
get the value you want.

00:08:59.360 --> 00:09:02.910
So essentially you enter into
a repurchase agreement with

00:09:02.910 --> 00:09:03.860
the Federal reserve.

00:09:03.860 --> 00:09:07.400
So this is the Fed reserve.

00:09:07.400 --> 00:09:09.420
You go to the Federal reserve
discount window.

00:09:09.420 --> 00:09:12.400
The discount rate might be-- I
don't know-- 5% on an annual

00:09:12.400 --> 00:09:14.770
basis, but we won't get into
the technicalities of that.

00:09:14.770 --> 00:09:17.090
You say, hey, Fed, lend
me some money.

00:09:17.090 --> 00:09:20.080
And the Fed says, OK, let me
print some money for you.

00:09:20.080 --> 00:09:22.980
So these are the liabilities
notes outstanding and then he

00:09:22.980 --> 00:09:27.680
prints some Federal reserve
notes, but instead of just

00:09:27.680 --> 00:09:31.940
lending the money to you and
then keeping these treasuries

00:09:31.940 --> 00:09:36.450
as collateral, the Federal
reserve will actually buy

00:09:36.450 --> 00:09:39.610
these-- it'll enter into a
repurchase agreement with you.

00:09:39.610 --> 00:09:42.820
So the mechanics of that is that
the Federal reserve will

00:09:42.820 --> 00:09:45.030
buy these treasuries from you.

00:09:45.030 --> 00:10:02.200
So now all of a sudden, the
treasuries-- the Federal

00:10:02.200 --> 00:10:09.060
reserve notes will go from from
the Federal reserve to

00:10:09.060 --> 00:10:13.310
you and then your cash will go
to-- then your treasury notes

00:10:13.310 --> 00:10:14.770
will go to the Federal
reserve, right?

00:10:14.770 --> 00:10:18.130
This is cash-- Federal reserve
printed cash, gave it to you,

00:10:18.130 --> 00:10:21.100
and then essentially bought
treasury notes from you.

00:10:21.100 --> 00:10:24.160
So now these are
treasury notes.

00:10:24.160 --> 00:10:27.560
These are the treasury notes
that were here before.

00:10:27.560 --> 00:10:30.300
But it has a repurchase
agreement where you agree at

00:10:30.300 --> 00:10:33.500
some future date to unwind this
transaction, where you're

00:10:33.500 --> 00:10:37.630
going to buy back your
treasuries and you're going to

00:10:37.630 --> 00:10:40.960
pay the amount that the treasury
paid you initially

00:10:40.960 --> 00:10:43.490
plus some interest, right?

00:10:43.490 --> 00:10:46.920
So the basic way to view it
is, this was just a loan.

00:10:46.920 --> 00:10:51.630
You just borrowed this much
from the Federal reserve.

00:10:51.630 --> 00:10:54.780
The Federal reserve kept your
treasuries as collateral, but

00:10:54.780 --> 00:10:57.560
it actually had formal
ownership over it.

00:10:57.560 --> 00:11:03.970
And that's what differentiates
a repo agreement from just a

00:11:03.970 --> 00:11:06.810
traditional collateralized loan,
but in the future you're

00:11:06.810 --> 00:11:10.490
going to buy back those
treasuries for the amount the

00:11:10.490 --> 00:11:12.870
Federal reserve had originally
bought them from you for plus

00:11:12.870 --> 00:11:14.330
a little bit of interest. You're
going to pay a little

00:11:14.330 --> 00:11:16.800
interest and that interest is
going to be dictated by the

00:11:16.800 --> 00:11:18.090
discount rate.

00:11:18.090 --> 00:11:19.650
Anyway, all out of time.

00:11:19.650 --> 00:11:21.590
In the next video, we'll
actually look at the Federal

00:11:21.590 --> 00:11:23.780
reserve's balance sheet.


Federal Reserve Balance Sheet

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.900
I think you'll find this video
exciting because until now,

00:00:03.900 --> 00:00:06.420
we've just been talking about
the Federal reserve in the

00:00:06.420 --> 00:00:08.950
abstract and I was drawing
little boxes to represent

00:00:08.950 --> 00:00:11.830
balance sheets, but this is
the actual Federal reserve

00:00:11.830 --> 00:00:14.510
balance sheet and I took a date
that was before all of

00:00:14.510 --> 00:00:16.390
this silliness started happening
in the banking

00:00:16.390 --> 00:00:19.340
sector just so we could kind of
see what a Federal reserve

00:00:19.340 --> 00:00:21.660
balance would've looked like
in a normal environment.

00:00:21.660 --> 00:00:24.780
And then we can actually, in
future videos, compare what

00:00:24.780 --> 00:00:26.580
they've done since then and
then we can get a better

00:00:26.580 --> 00:00:29.500
insight into all of the
different machinations that

00:00:29.500 --> 00:00:33.370
the Federal reserve has done to
kind of try to keep banks

00:00:33.370 --> 00:00:35.900
liquid and solvent and to
keep everything going.

00:00:35.900 --> 00:00:38.320
And we can debate whether
they've been good or bad or

00:00:38.320 --> 00:00:40.260
they're just keeping banks
in kind of zombie mode.

00:00:40.260 --> 00:00:43.225
But anyway, this is the Federal
reserve's balance

00:00:43.225 --> 00:00:48.200
sheet as of February
14th, 2007.

00:00:48.200 --> 00:00:50.140
So before all the craziness
happened, although a little

00:00:50.140 --> 00:00:52.530
bit started-- this is before the
Feds started taking really

00:00:52.530 --> 00:00:56.350
aggressive action to provide
liquidity for the markets.

00:00:56.350 --> 00:00:58.000
But here the assets.

00:00:58.000 --> 00:00:59.900
So first of all, we have the
total assets number.

00:00:59.900 --> 00:01:02.410
That's just interesting
to look at.

00:01:02.410 --> 00:01:03.240
These are all in millions.

00:01:03.240 --> 00:01:07.660
So this is $871 billion
of assets.

00:01:07.660 --> 00:01:08.910
Let's just get the
big picture.

00:01:11.620 --> 00:01:15.830
Let me draw that here in our
traditional box diagram.

00:01:15.830 --> 00:01:23.020
So if I were to draw the
assets-- the sum of all the

00:01:23.020 --> 00:01:30.160
assets over here is $871
billion, and we know that the

00:01:30.160 --> 00:01:33.715
liabilities plus equity better
add up to $871 billion.

00:01:33.715 --> 00:01:34.280
Let's see.

00:01:34.280 --> 00:01:35.540
What's the total liabilities?

00:01:35.540 --> 00:01:39.800
Total liabilities is
$839 billion-- so

00:01:39.800 --> 00:01:42.260
give or take $840.

00:01:42.260 --> 00:01:44.720
So the liabilities-- I'll do
it in a different color.

00:01:44.720 --> 00:01:49.420
The liabilities are $840
billion, give or

00:01:49.420 --> 00:01:51.940
take a little bit.

00:01:51.940 --> 00:01:54.540
They should have the same width,
not that the width

00:01:54.540 --> 00:01:55.790
matters that much.

00:02:04.080 --> 00:02:06.360
And then whatever's left over
should be equity, right?

00:02:06.360 --> 00:02:08.400
Assets minus liabilities.

00:02:08.400 --> 00:02:12.690
What you have minus what you owe
is what you're left with

00:02:12.690 --> 00:02:14.130
for the owners.

00:02:14.130 --> 00:02:16.130
And of course, owners of the
Federal reserve, you kind of

00:02:16.130 --> 00:02:17.620
have to take with
a grain of salt.

00:02:17.620 --> 00:02:19.620
They really don't have the
upside of traditional owners.

00:02:19.620 --> 00:02:22.480
They're really just kind
of stakeholders.

00:02:22.480 --> 00:02:24.060
What's left over
is the equity.

00:02:24.060 --> 00:02:29.220
It should be roughly
$31 billion.

00:02:29.220 --> 00:02:30.790
And let's confirm that
by looking at the

00:02:30.790 --> 00:02:33.330
actual balance sheet.

00:02:33.330 --> 00:02:37.890
And here we have it-- total
capital is $31 billion.

00:02:37.890 --> 00:02:41.310
So big picture, so far it's kind
of meeting up with how

00:02:41.310 --> 00:02:43.770
we've envisioned a Federal
reserve balance sheet, but

00:02:43.770 --> 00:02:46.170
let's dig a little deeper and
see if we can find interesting

00:02:46.170 --> 00:02:47.740
things and things we've
talked about.

00:02:47.740 --> 00:02:50.480
And hopefully at this point,
we should actually

00:02:50.480 --> 00:02:53.830
understand all of it.

00:02:53.830 --> 00:02:55.450
Let's focus on the
assets for now.

00:02:55.450 --> 00:02:59.160
So the assets are just
this part of it.

00:02:59.160 --> 00:02:59.670
OK.

00:02:59.670 --> 00:03:01.580
So it has-- this is what?

00:03:01.580 --> 00:03:04.820
This is $11 billion of gold
certificate accounts and

00:03:04.820 --> 00:03:07.000
that's some type of
rights on gold.

00:03:07.000 --> 00:03:07.220
Let's see.

00:03:07.220 --> 00:03:08.380
Gold certificate accounts.

00:03:08.380 --> 00:03:09.960
Let's see if they have any
other gold anywhere.

00:03:09.960 --> 00:03:11.660
Coin-- $1 billion.

00:03:11.660 --> 00:03:13.570
But this is all small
potatoes, right?

00:03:13.570 --> 00:03:15.620
I don't know what special
drawing rights certificate

00:03:15.620 --> 00:03:17.750
account is, but it's very
small relative to

00:03:17.750 --> 00:03:18.630
the big pie, right?

00:03:18.630 --> 00:03:21.830
There's $871 billion
of assets.

00:03:21.830 --> 00:03:24.310
This is just kind of almost
rounding-off error.

00:03:24.310 --> 00:03:26.690
Here we have a big chunk
of something.

00:03:26.690 --> 00:03:29.750
Actually, I think this $11
billion is actual gold because

00:03:29.750 --> 00:03:31.860
I don't see it anywhere
else on their assets.

00:03:31.860 --> 00:03:35.300
So I think if you combine
roughly-- I don't know what

00:03:35.300 --> 00:03:38.680
this thing here is, but if you
combine this and this, it's

00:03:38.680 --> 00:03:40.920
saying that the Federal reserve
is-- because I don't

00:03:40.920 --> 00:03:45.100
see gold anywhere else here--
that it's holding roughly $12

00:03:45.100 --> 00:03:47.360
billion worth of gold, which
really isn't a lot of gold

00:03:47.360 --> 00:03:50.670
when you consider the total
size of its balance sheet.

00:03:50.670 --> 00:03:56.070
The big piece right here-- let
me pick a different color.

00:03:56.070 --> 00:03:57.700
I'll do it in the purple.

00:03:57.700 --> 00:04:04.030
It has securities, repurchase
agreements, and loans-- $808

00:04:04.030 --> 00:04:06.090
billion-- almost $809 billion.

00:04:06.090 --> 00:04:09.060
So this is a big piece of
the Fed's balance sheet.

00:04:09.060 --> 00:04:12.760
Out of the whole $871, $808.

00:04:12.760 --> 00:04:17.149
So pretty much it's almost
like that much of it.

00:04:17.149 --> 00:04:19.100
Actually, maybe a little bit
less of that-- are these

00:04:19.100 --> 00:04:20.459
securities and things
like that.

00:04:20.459 --> 00:04:22.540
Let's see what kind of
securities they have. And they

00:04:22.540 --> 00:04:23.340
break them down.

00:04:23.340 --> 00:04:31.090
So this $808 billion is made up
of these things right here.

00:04:31.090 --> 00:04:34.340
The bulk of them are U.S.
treasury loans, right?

00:04:34.340 --> 00:04:38.140
So these are going to be bills,
notes, and bonds-- OK.

00:04:38.140 --> 00:04:40.620
So just to explain, a treasury
bill-- and I've done videos on

00:04:40.620 --> 00:04:43.820
this-- this is essentially a
loan to the government for a

00:04:43.820 --> 00:04:45.040
year or less.

00:04:45.040 --> 00:04:47.760
So it's just a loan to the
government that matures in a

00:04:47.760 --> 00:04:49.570
year or in three months.

00:04:49.570 --> 00:04:52.560
Notes and bonds-- these are
loans to the U.S. treasury

00:04:52.560 --> 00:04:53.820
that have longer maturities.

00:04:53.820 --> 00:04:55.430
Notes are up to 10 years.

00:04:55.430 --> 00:04:58.650
Bonds are more than 10 years.

00:04:58.650 --> 00:05:00.833
And then inflation index bonds--
I'll do a whole video

00:05:00.833 --> 00:05:02.490
on that in the future, but
these are essentially

00:05:02.490 --> 00:05:05.250
treasuries that are indexed to
inflation so you can kind of

00:05:05.250 --> 00:05:06.770
hedge out a little bit of
your inflation risk.

00:05:06.770 --> 00:05:10.770
But, needless to say, the big
picture is is that $780

00:05:10.770 --> 00:05:14.970
billion of the Fed's assets are
treasuries, loans to the

00:05:14.970 --> 00:05:15.790
Federal government.

00:05:15.790 --> 00:05:16.710
Let me draw that here.

00:05:16.710 --> 00:05:22.420
So a pretty big piece, roughly
that much is treasury.

00:05:22.420 --> 00:05:26.060
So most of what the Federal
reserve owns are treasuries.

00:05:26.060 --> 00:05:28.040
And that's consistent with
everything we've gone over so

00:05:28.040 --> 00:05:30.960
far and that account for
everything up to here.

00:05:30.960 --> 00:05:33.740
And then what's interesting--
what we just talked about--

00:05:33.740 --> 00:05:38.580
repurchase agreements,
$30 billion.

00:05:38.580 --> 00:05:41.940
And I don't know 100%, but I'm
guessing that these are--

00:05:41.940 --> 00:05:44.270
someone came to the discount
window and essentially

00:05:44.270 --> 00:05:47.480
borrowed $30 billion
from the Fed-- and

00:05:47.480 --> 00:05:48.290
it's not just someone.

00:05:48.290 --> 00:05:50.780
It's probably multiple people
came and borrowed $30 billion

00:05:50.780 --> 00:05:53.890
from the Fed-- and they gave
treasuries as collateral, but

00:05:53.890 --> 00:05:56.355
as we know, just the way
repurchase agreements work,

00:05:56.355 --> 00:05:59.330
they actually kind of sold the
treasuries to the Fed and the

00:05:59.330 --> 00:06:00.920
Fed agreed to buy it later,
but it's essentially

00:06:00.920 --> 00:06:01.880
collateral.

00:06:01.880 --> 00:06:05.180
So these repurchase agreements--
they're included

00:06:05.180 --> 00:06:08.090
in these securities because
they're not

00:06:08.090 --> 00:06:09.010
just agreements, right?

00:06:09.010 --> 00:06:11.770
They actually are-- they're
probably treasuries or they

00:06:11.770 --> 00:06:14.070
might be other types of highly
rated securities and we'll

00:06:14.070 --> 00:06:16.700
learn in future videos that
the Fed has lowered its

00:06:16.700 --> 00:06:20.900
standards over the last year
in terms of what type of

00:06:20.900 --> 00:06:24.030
collateral or what type of
securities it's willing to

00:06:24.030 --> 00:06:26.530
trade in these repurchase
agreements, but in the

00:06:26.530 --> 00:06:28.590
situation it looks like
about $30 billion.

00:06:28.590 --> 00:06:31.400
And you can also-- you get
a clue of what repurchase

00:06:31.400 --> 00:06:33.820
agreements are because here,
they say securities held

00:06:33.820 --> 00:06:36.000
outright, right?

00:06:36.000 --> 00:06:39.030
So there's no repurchase
agreement.

00:06:39.030 --> 00:06:40.660
There's not some contract where
they say they're going

00:06:40.660 --> 00:06:43.370
to sell this to someone
else at another price.

00:06:43.370 --> 00:06:44.740
These are repurchase
agreements.

00:06:44.740 --> 00:06:46.900
These are kind of more
collateral for loans.

00:06:46.900 --> 00:06:48.650
And then they have outright
loans-- $39 million.

00:06:48.650 --> 00:06:51.040
That's pretty much peanuts in
the Federal reserve world.

00:06:51.040 --> 00:06:54.420
So the bulk is treasuries, a
little bit of repurchase

00:06:54.420 --> 00:06:56.940
agreements, and then there's
other assets.

00:06:56.940 --> 00:06:58.090
They don't break out
what this is.

00:06:58.090 --> 00:06:59.250
Maybe there's some
gold in that.

00:06:59.250 --> 00:07:00.620
I'm not sure.

00:07:00.620 --> 00:07:03.800
Bank premises, the buildings
of the Federal reserve are

00:07:03.800 --> 00:07:04.810
worth $2 billion.

00:07:04.810 --> 00:07:06.970
I mean, they have 12 banks
around the country and I'm

00:07:06.970 --> 00:07:08.770
sure they have a bunch
of other things.

00:07:08.770 --> 00:07:10.820
And then items in process
collection.

00:07:10.820 --> 00:07:12.030
I don't know what these
things are, but these

00:07:12.030 --> 00:07:12.940
are all small potatoes.

00:07:12.940 --> 00:07:15.600
The big thing is that the
Federal reserve's assets are

00:07:15.600 --> 00:07:17.470
predominantly U.S. treasuries--
at least, they're

00:07:17.470 --> 00:07:19.700
predominantly treasuries
right now.

00:07:19.700 --> 00:07:21.360
Now let's look at
the liabilities.

00:07:21.360 --> 00:07:24.010
And to some degree, this is
much more interesting.

00:07:24.010 --> 00:07:27.220
So Federal reserve notes,
net of Federal

00:07:27.220 --> 00:07:28.830
reserve bank holdings.

00:07:31.460 --> 00:07:36.400
So $769 billion.

00:07:36.400 --> 00:07:38.540
So when I talked about
notes outstanding,

00:07:38.540 --> 00:07:39.820
that's what this is.

00:07:39.820 --> 00:07:43.620
These are Federal reserve notes
that have been printed

00:07:43.620 --> 00:07:44.950
and they're liabilities,
right?

00:07:44.950 --> 00:07:48.670
Because the Federal reserve bank
printed these notes and

00:07:48.670 --> 00:07:51.000
then used them as currency and
so they're liabilities now

00:07:51.000 --> 00:07:53.830
because someone can come back
another time and say, hey,

00:07:53.830 --> 00:07:58.220
give me back the value of these
things and that's a bit

00:07:58.220 --> 00:08:02.880
of an abstract concept, but
roughly $700 something of this

00:08:02.880 --> 00:08:04.020
are notes outstanding.

00:08:04.020 --> 00:08:08.040
This is money that the Federal
reserve had printed.

00:08:08.040 --> 00:08:10.880
And then there's some reverse
repurchase agreements, which

00:08:10.880 --> 00:08:13.310
essentially-- see for some
reason, the Federal reserve

00:08:13.310 --> 00:08:16.850
used repurchase agreements to
borrow from someone else.

00:08:16.850 --> 00:08:19.840
It has a little bit of
deposits, right?

00:08:19.840 --> 00:08:22.370
And these deposits have actually
grown dramatically in

00:08:22.370 --> 00:08:23.080
the past year.

00:08:23.080 --> 00:08:24.730
Has $22 billion of deposits.

00:08:24.730 --> 00:08:27.630
So these are actually deposits
that banks are keeping with

00:08:27.630 --> 00:08:29.320
the Federal reserve.

00:08:29.320 --> 00:08:32.090
The U.S. treasury actually
keeps some money there.

00:08:32.090 --> 00:08:35.390
Depository institutions have
$17 billion-- and actually,

00:08:35.390 --> 00:08:37.480
that's how the Federal
reserve traditionally

00:08:37.480 --> 00:08:39.360
has paid its expenses.

00:08:39.360 --> 00:08:42.490
People put deposits with the
Federal reserve-- so let's say

00:08:42.490 --> 00:08:43.750
these are deposits from banks.

00:08:43.750 --> 00:08:45.230
It's a very small piece.

00:08:45.230 --> 00:08:47.960
It's like $17 billion.

00:08:47.960 --> 00:08:51.070
These could be deposits from
member banks, but the Federal

00:08:51.070 --> 00:08:53.770
reserve does not pay interest
in these deposits.

00:08:53.770 --> 00:08:56.000
They don't pay interest on these
deposits and then they

00:08:56.000 --> 00:08:59.290
can take these deposits and by
treasuries or other securities

00:08:59.290 --> 00:09:00.630
and get interest on them.

00:09:00.630 --> 00:09:02.740
So they're essentially getting
free interest and that's what

00:09:02.740 --> 00:09:05.030
they used actually to fund
their operations.

00:09:05.030 --> 00:09:07.750
Any excess after funding their
operations goes back to the

00:09:07.750 --> 00:09:11.420
U.S. Treasury so it's not like
Ben Bernanke can fly around or

00:09:11.420 --> 00:09:13.450
drive a Bentley or something.

00:09:13.450 --> 00:09:15.820
And then I don't know what this
is-- foreign official--

00:09:15.820 --> 00:09:19.880
these are nothing items. So the
bulk of it is money that

00:09:19.880 --> 00:09:22.050
had been printed and that's
a liability of the Federal

00:09:22.050 --> 00:09:23.020
reserve now.

00:09:23.020 --> 00:09:25.980
And then there's a little bit
of deposits from depository

00:09:25.980 --> 00:09:26.850
institutions.

00:09:26.850 --> 00:09:28.520
And the treasury has kept
some money with the

00:09:28.520 --> 00:09:30.030
Federal reserve as well.

00:09:30.030 --> 00:09:33.660
And then everything left
over is the equity.

00:09:33.660 --> 00:09:35.880
Anyway, I thought that would be
pretty neat to see that you

00:09:35.880 --> 00:09:37.910
can actually look at what the
Federal reserve's balance

00:09:37.910 --> 00:09:38.520
sheet is right now.

00:09:38.520 --> 00:09:39.870
You can actually just do
a web search for it.

00:09:39.870 --> 00:09:41.790
You'll find it in a bunch
of different sources.

00:09:41.790 --> 00:09:43.420
And you can actually
analyze it.

00:09:43.420 --> 00:09:46.790
You now have the tools to look
at that and make sense of it.

00:09:46.790 --> 00:09:49.970
And what's even more interesting
is to compare this

00:09:49.970 --> 00:09:52.730
balance sheet with the current
Federal reserve balance sheet

00:09:52.730 --> 00:09:53.960
and then you can know
everything that

00:09:53.960 --> 00:09:55.860
they've been up to.


The discount rate

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.040
Let's do a little review of what
the Fed funds rate was.

00:00:08.550 --> 00:00:11.090
And then we can move into
something that you've probably

00:00:11.090 --> 00:00:14.180
heard in the same context, and
they're often confused, and

00:00:14.180 --> 00:00:15.430
that's the discount rate.

00:00:20.580 --> 00:00:24.310
And they are related and they
kind of do move together.

00:00:24.310 --> 00:00:28.020
They are pretty different in
their actual implementation.

00:00:28.020 --> 00:00:31.040
So the Federal funds rate--
this is a target rate.

00:00:31.040 --> 00:00:34.370
This is the target rate at
which the Federal reserve

00:00:34.370 --> 00:00:36.940
wants banks to lend
to each other.

00:00:36.940 --> 00:00:39.500
So let's say that I have-- and
I won't draw the balance

00:00:39.500 --> 00:00:46.400
sheets every time now-- let's
say I have have Bank One and

00:00:46.400 --> 00:00:49.470
this is Bank Number Two.

00:00:49.470 --> 00:00:51.630
And let's say this bank
over here has

00:00:51.630 --> 00:00:52.910
a surplus of reserves.

00:00:52.910 --> 00:00:57.230
I was already using green, but
I'll do that in gold just so

00:00:57.230 --> 00:00:59.570
we can reminisce about
the gold standard.

00:00:59.570 --> 00:01:03.560
So let's say it has a surplus of
reserves and Bank Two needs

00:01:03.560 --> 00:01:04.870
them, right?

00:01:04.870 --> 00:01:08.650
And let's say right now that
Bank One is willing to lend it

00:01:08.650 --> 00:01:16.380
to Bank Two if Bank Two pays
Bank One a 6% overnight rate.

00:01:16.380 --> 00:01:18.310
And let's say that the
Federal reserve, they

00:01:18.310 --> 00:01:19.120
say, you know what?

00:01:19.120 --> 00:01:21.000
That's above our target rate.

00:01:21.000 --> 00:01:23.860
We want banks to lend to each
other for a lower interest

00:01:23.860 --> 00:01:27.730
rate, so we want to do open
market transactions or open

00:01:27.730 --> 00:01:30.670
market operations to
lower this rate.

00:01:30.670 --> 00:01:34.080
And the mechanics that they do
it by-- let's draw the Fed's

00:01:34.080 --> 00:01:35.330
balance sheet.

00:01:37.550 --> 00:01:38.800
I'll do them in magenta.

00:01:50.120 --> 00:01:53.245
That's half of it and then
this is the other half.

00:01:56.700 --> 00:02:00.610
So let's say that this is the
Fed's current assets-- and in

00:02:00.610 --> 00:02:02.650
a couple of videos, I'll
actually show you what the

00:02:02.650 --> 00:02:05.220
Fed's balance sheet looked like
before all this craziness

00:02:05.220 --> 00:02:10.050
started and what it looks like
now, but that's the Federal

00:02:10.050 --> 00:02:11.510
reserve's assets.

00:02:11.510 --> 00:02:13.530
This is their liabilities.

00:02:13.530 --> 00:02:15.740
And then their liabilities are
going to be a little bit

00:02:15.740 --> 00:02:19.170
smaller then their assets and
they have a little equity.

00:02:19.170 --> 00:02:20.220
Their equity's a little
different

00:02:20.220 --> 00:02:21.640
than traditional equity.

00:02:21.640 --> 00:02:22.890
There really isn't
a lot of upside.

00:02:22.890 --> 00:02:24.340
You just get a dividend
on it, but we won't

00:02:24.340 --> 00:02:25.330
go to details there.

00:02:25.330 --> 00:02:29.400
But the mechanism that the Fed
uses to do these open market

00:02:29.400 --> 00:02:32.110
operations is they essentially
print money.

00:02:32.110 --> 00:02:35.980
So what the Federal reserve will
do is, they will create

00:02:35.980 --> 00:02:39.680
some notes or some
actual reserves.

00:02:39.680 --> 00:02:43.740
So these are Federal reserve
notes-- or as we know them,

00:02:43.740 --> 00:02:47.370
the dollar bills that are
sitting in your wallet or the

00:02:47.370 --> 00:02:49.660
things that could be converted
to dollar bills that are

00:02:49.660 --> 00:02:52.540
sitting in your bank account,
the bits and bytes in some

00:02:52.540 --> 00:02:54.120
computer database someplace.

00:02:54.120 --> 00:02:55.730
And they can't create
it out of thin air.

00:02:55.730 --> 00:02:57.930
They have to have an offsetting
liability and their

00:02:57.930 --> 00:03:01.560
offsetting liability are
Federal reserve notes

00:03:01.560 --> 00:03:02.810
outstanding.

00:03:07.180 --> 00:03:08.720
This is just saying, hey,
we issued this.

00:03:08.720 --> 00:03:11.310
If someone comes back to us,
we have this liability.

00:03:11.310 --> 00:03:14.620
And this is issued even though
these Federal reserve notes--

00:03:14.620 --> 00:03:17.190
I'll circle it in yellow--
are issued by the

00:03:17.190 --> 00:03:19.610
Federal reserve bank.

00:03:19.610 --> 00:03:23.300
They're backed by the full faith
and credit of the U.S.

00:03:23.300 --> 00:03:23.880
government.

00:03:23.880 --> 00:03:26.210
We've talked a lot about what
that means, but needless to

00:03:26.210 --> 00:03:27.550
say, we're just going
over the mechanics.

00:03:27.550 --> 00:03:31.110
So what they'll do is they'll
take these dollars now and

00:03:31.110 --> 00:03:34.240
they'll use these dollars to go
buy treasuries from people

00:03:34.240 --> 00:03:35.010
out in the world.

00:03:35.010 --> 00:03:35.780
It could be me.

00:03:35.780 --> 00:03:38.320
It could be my grandfather.

00:03:38.320 --> 00:03:41.640
It could be even some of these
banks and so let's say that

00:03:41.640 --> 00:03:45.280
there's-- right now somebody
is holding a treasury.

00:03:48.760 --> 00:03:50.180
I hold a T-bill.

00:03:54.960 --> 00:03:57.350
The Federal reserve will use
that money-- let's say I own a

00:03:57.350 --> 00:03:58.420
ton of T-bills.

00:03:58.420 --> 00:04:00.380
I'm the richest man
in the country.

00:04:00.380 --> 00:04:02.340
I could even be China.

00:04:02.340 --> 00:04:04.240
China holds a lot of T-bills.

00:04:04.240 --> 00:04:05.240
They buy the T-bill.

00:04:05.240 --> 00:04:08.720
So then this becomes-- this
asset is no longer Federal

00:04:08.720 --> 00:04:09.460
reserve notes.

00:04:09.460 --> 00:04:10.710
It's now a T-bill.

00:04:13.400 --> 00:04:16.300
And then I'm no longer holding
a T-bill, right, because I

00:04:16.300 --> 00:04:18.990
sold it to the Federal
reserve bank.

00:04:18.990 --> 00:04:20.760
I don't know I sold it to the
Federal reserve bank.

00:04:20.760 --> 00:04:21.790
I just sold it in the market.

00:04:21.790 --> 00:04:22.540
I don't know who bought it.

00:04:22.540 --> 00:04:24.070
It might have been
another guy.

00:04:24.070 --> 00:04:25.190
It might have been
another country.

00:04:25.190 --> 00:04:27.950
But it happened to be in this
case the Federal reserve bank.

00:04:27.950 --> 00:04:30.660
And now I'm holding reserves.

00:04:30.660 --> 00:04:32.950
I'm holding money
as we know it.

00:04:32.950 --> 00:04:34.700
I'm holding a Federal
reserve note.

00:04:34.700 --> 00:04:37.430
And what am I going to do with
that Federal reserve note?

00:04:37.430 --> 00:04:39.910
I'm going to deposit
it in banks, right?

00:04:39.910 --> 00:04:43.130
And so I'm going to take this
Federal reserve note.

00:04:43.130 --> 00:04:45.830
Let's say I have a couple
of bank accounts.

00:04:45.830 --> 00:04:48.220
Just for the sake of simplicity,
I deposit some of

00:04:48.220 --> 00:04:52.430
it in this bank account and
let's say I deposit some of it

00:04:52.430 --> 00:04:55.210
in this bank account,
just for simplicity.

00:04:55.210 --> 00:04:56.620
So what happens now?

00:04:56.620 --> 00:05:00.940
Now this guy has more notes
to lend out and

00:05:00.940 --> 00:05:02.680
this guy needs less.

00:05:02.680 --> 00:05:05.680
So demand has gone down.

00:05:05.680 --> 00:05:07.140
This guy needs less.

00:05:07.140 --> 00:05:10.890
So demand has gone down
from this guy.

00:05:10.890 --> 00:05:13.790
And supply has gone
up from this guy.

00:05:13.790 --> 00:05:17.000
We know that if you need
something less, but people

00:05:17.000 --> 00:05:20.710
have more of it, the price of
buying it or borrowing it is

00:05:20.710 --> 00:05:21.990
going to go down.

00:05:21.990 --> 00:05:25.290
So this guy has more of it and
this guy needs it less, all of

00:05:25.290 --> 00:05:27.670
a sudden this guy's
not willing to pay

00:05:27.670 --> 00:05:29.330
6% to borrow it.

00:05:29.330 --> 00:05:31.730
And this guy's actually more
desperate to offload some of

00:05:31.730 --> 00:05:33.660
these reserves and get
some interest on it.

00:05:33.660 --> 00:05:36.110
So this guy's going to lower the
rate he'll charge and this

00:05:36.110 --> 00:05:38.500
guy's going to lower the rate
he's willing to pay and maybe

00:05:38.500 --> 00:05:40.490
it goes down to 5%.

00:05:40.490 --> 00:05:43.110
And the Federal reserve can
keep buying or selling

00:05:43.110 --> 00:05:45.150
treasuries to adjust
what this happens.

00:05:45.150 --> 00:05:46.350
They could do the opposite.

00:05:46.350 --> 00:05:48.450
If they said, wow, rates are
a little bit too low.

00:05:48.450 --> 00:05:51.410
Let's say whatever happens,
rates are at 3% and the

00:05:51.410 --> 00:05:54.050
Federal reserve doesn't like
that and wants to raise the

00:05:54.050 --> 00:05:56.830
Federal funds rate, which is
the target rate that banks

00:05:56.830 --> 00:05:59.740
lend to each other, then they
can do the opposite thing.

00:05:59.740 --> 00:06:01.500
They could take this
T-bill, right?

00:06:01.500 --> 00:06:04.410
This was a T-bill.

00:06:04.410 --> 00:06:06.170
And they'll sell it, right?

00:06:06.170 --> 00:06:08.240
So they'll take this T-bill and
they'll sell it to someone

00:06:08.240 --> 00:06:12.470
else-- maybe this
guy right here.

00:06:12.470 --> 00:06:16.070
So this guy, he's got
a dollar bill.

00:06:16.070 --> 00:06:19.190
So his dollar bills are
going to be sitting at

00:06:19.190 --> 00:06:20.700
one of these banks.

00:06:20.700 --> 00:06:23.240
Let's say his dollar bills are
sitting at one of-- he's got a

00:06:23.240 --> 00:06:24.980
couple there and couple there.

00:06:24.980 --> 00:06:29.390
So when the Federal reserve
sells this T-bill to this guy,

00:06:29.390 --> 00:06:31.490
this guy might do a wire
transfer to that party-- or a

00:06:31.490 --> 00:06:34.450
check or it doesn't matter, but
either way you look at it,

00:06:34.450 --> 00:06:37.680
these reserves disappear
and they go back

00:06:37.680 --> 00:06:38.740
into the Federal reserve.

00:06:38.740 --> 00:06:41.150
When they go back into the
Federal reserve, they offset

00:06:41.150 --> 00:06:43.410
this liability and then the
currency essentially

00:06:43.410 --> 00:06:46.970
disappears, but the real result
is that all of a sudden

00:06:46.970 --> 00:06:49.900
then demand would have gone up
because there'd be fewer

00:06:49.900 --> 00:06:52.110
reserves in the system.

00:06:52.110 --> 00:06:53.170
Demand goes up.

00:06:53.170 --> 00:06:55.840
And then the supply would have
gone down because there's also

00:06:55.840 --> 00:06:57.440
fewer reserves in the system.

00:06:57.440 --> 00:06:59.960
And now this guy, he's like,
wow, I have less to lend out.

00:06:59.960 --> 00:07:02.400
I need more interest in order
for me lend it out.

00:07:02.400 --> 00:07:04.580
And this guy says, wow, I'm more
desperate than ever to

00:07:04.580 --> 00:07:05.710
borrow some reserves.

00:07:05.710 --> 00:07:10.330
I'm willing to pay more and so
the rates will go up to 4%.

00:07:10.330 --> 00:07:14.080
Now all of this works well
assuming a world where banks

00:07:14.080 --> 00:07:16.240
are willing to lend to each
other at some rate.

00:07:16.240 --> 00:07:19.610
There's some rate at which this
guy says, I'm willing to

00:07:19.610 --> 00:07:21.460
lend to this guy because I know
he's going to pay me the

00:07:21.460 --> 00:07:22.700
next day and it's
just a matter of

00:07:22.700 --> 00:07:23.850
just supply or demand.

00:07:23.850 --> 00:07:25.430
These tend to be overnight
loans.

00:07:25.430 --> 00:07:27.780
They tend to be very short term
loans so they tend to be

00:07:27.780 --> 00:07:32.230
very, very safe, but what
happens in a world-- let me

00:07:32.230 --> 00:07:33.280
draw the same two banks.

00:07:33.280 --> 00:07:35.720
I think I overdrew.

00:07:35.720 --> 00:07:40.600
So this is Bank Number One and
this is Bank Number Two-- and

00:07:40.600 --> 00:07:43.800
Bank Number One had
more reserves.

00:07:43.800 --> 00:07:44.940
Bank Number Two has fewer.

00:07:44.940 --> 00:07:46.250
Bank Number Two needs
reserves.

00:07:46.250 --> 00:07:48.680
Let's say people are worried
about Bank Number Two.

00:07:48.680 --> 00:07:51.420
All of their depositors are
starting to get scared and

00:07:51.420 --> 00:07:54.460
they're starting to pull their
reserves out, right?

00:07:54.460 --> 00:07:57.320
And we all know that these
banks don't keep enough

00:07:57.320 --> 00:07:59.160
reserves to fulfill all
of their deposits.

00:07:59.160 --> 00:08:01.310
Actually, let me draw Bank
Number Two's balance sheet.

00:08:10.690 --> 00:08:14.050
They have equity, hopefully.

00:08:14.050 --> 00:08:16.050
They'll have some deposits.

00:08:16.050 --> 00:08:17.300
Let's say all of these
are deposits.

00:08:21.210 --> 00:08:24.780
They have to keep some
reserves, right,

00:08:24.780 --> 00:08:26.280
so that's an asset.

00:08:26.280 --> 00:08:29.550
Depending on their reserve
requirements, but they'll have

00:08:29.550 --> 00:08:35.179
some reserves in case people
want to take out their money

00:08:35.179 --> 00:08:37.200
from their checking accounts--
and then the rest of these are

00:08:37.200 --> 00:08:39.460
assets that they invested in
and the bank makes money by

00:08:39.460 --> 00:08:41.820
making more money on these
assets than it has to pay out

00:08:41.820 --> 00:08:44.790
in interest. It makes money
on that spread.

00:08:44.790 --> 00:08:47.020
Now what happens if this bank--
its condition starts to

00:08:47.020 --> 00:08:49.800
get a little bit weak, people
start to get afraid, and the

00:08:49.800 --> 00:08:53.160
deposits start to-- people go
to their ATM, start pulling

00:08:53.160 --> 00:08:55.180
their money out, and if anything
maybe they'll start

00:08:55.180 --> 00:08:57.470
depositing it into a safer bank
or just stuffing it under

00:08:57.470 --> 00:08:59.070
their mattresses, right?

00:08:59.070 --> 00:09:01.150
This bank says, all of a sudden
I have a liquidity

00:09:01.150 --> 00:09:04.640
issue because, sure, maybe
that much people withdraw

00:09:04.640 --> 00:09:04.950
their money.

00:09:04.950 --> 00:09:08.880
I have enough reserves to pay
that, but then if another guy

00:09:08.880 --> 00:09:11.030
comes along, that's going to
deplete my reserves and then

00:09:11.030 --> 00:09:13.870
when the next guy comes along,
I'm not going to have any

00:09:13.870 --> 00:09:16.150
reserves left and it's going
to be a full all-out panic

00:09:16.150 --> 00:09:19.490
when I-- I told this guy that I
could give him his money on

00:09:19.490 --> 00:09:22.220
demand and all of a sudden, if
I can't give him on demand,

00:09:22.220 --> 00:09:24.690
then we're going to have this
huge banking panic and then

00:09:24.690 --> 00:09:26.300
everyone else is going to want
their deposits and then I'm

00:09:26.300 --> 00:09:28.960
going to have this huge
liquidity crisis.

00:09:28.960 --> 00:09:31.110
In a normal situation like
that, I'd say, hey, Bank

00:09:31.110 --> 00:09:33.680
Number One, I need some reserves
and just like I did

00:09:33.680 --> 00:09:37.850
in the first half of this video,
this guy would lend the

00:09:37.850 --> 00:09:39.740
reserves and then this guy would
give this guy interest.

00:09:39.740 --> 00:09:42.660
But what if this guy is scared
of Bank Number Two too?

00:09:42.660 --> 00:09:45.000
He's like, wow, that guy's
in a tough situation.

00:09:45.000 --> 00:09:46.900
He's facing a liquidity
crisis.

00:09:46.900 --> 00:09:48.790
I don't even know what
his assets are worth.

00:09:48.790 --> 00:09:50.960
Maybe his assets are
actually shrinking.

00:09:50.960 --> 00:09:52.010
And that's been happening
lately.

00:09:52.010 --> 00:09:54.480
Maybe he made a bunch of
bad mortgage loans.

00:09:54.480 --> 00:09:56.110
I don't want to lend
to this guy.

00:09:56.110 --> 00:09:58.530
And this guy becomes a pariah
of the banking community.

00:09:58.530 --> 00:10:01.700
No-one wants to lend
to this guy.

00:10:01.700 --> 00:10:03.840
But at the same time, it's in
no-one's interest for there to

00:10:03.840 --> 00:10:04.790
be a run on this bank.

00:10:04.790 --> 00:10:07.550
Because if this guy can't pay
one of his depositors-- and

00:10:07.550 --> 00:10:10.590
this is kind of a prime weakness
of a fractional

00:10:10.590 --> 00:10:11.450
reserve system.

00:10:11.450 --> 00:10:15.040
If there's just one weak link
in the banking system and

00:10:15.040 --> 00:10:17.650
people lose confidence-- maybe
this guy was the only bad bank

00:10:17.650 --> 00:10:20.330
out there and people start
taking all their money out.

00:10:20.330 --> 00:10:23.600
The first guy who can't get his
money back, he's going to

00:10:23.600 --> 00:10:25.900
call up the press and say,
my God, the banks aren't

00:10:25.900 --> 00:10:26.410
good for the money.

00:10:26.410 --> 00:10:28.660
Maybe there's a run on all the
banks because people don't

00:10:28.660 --> 00:10:31.020
know which banks are good,
which ones are bad.

00:10:31.020 --> 00:10:33.850
So to prevent this, the Federal
reserve has something

00:10:33.850 --> 00:10:36.180
called the discount window.

00:10:36.180 --> 00:10:38.900
So let me draw the Federal
reserve's balance sheet again.

00:10:38.900 --> 00:10:42.880
And the discount window is
essentially a lender of last

00:10:42.880 --> 00:10:46.970
resort to the banks.

00:10:46.970 --> 00:10:49.630
So there's some type of
Federal funds rate.

00:10:49.630 --> 00:10:51.670
Let's say the Federal
funds rate is at 6%.

00:10:51.670 --> 00:10:53.240
In a normal environment,
this guy would lend

00:10:53.240 --> 00:10:54.570
to get back at 6%.

00:10:54.570 --> 00:10:56.380
But let's say that's
broken and this

00:10:56.380 --> 00:10:57.700
guy is really desperate.

00:10:57.700 --> 00:11:04.570
He can actually go to the
Federal reserve and borrow

00:11:04.570 --> 00:11:07.650
directly from the
Federal reserve.

00:11:07.650 --> 00:11:09.830
So once again, these are the
assets of the Federal reserve.

00:11:12.540 --> 00:11:14.250
These are the liabilities.

00:11:14.250 --> 00:11:17.020
This is the equity of
the Federal reserve.

00:11:17.020 --> 00:11:20.270
And the Federal reserve in this
situation now, they'll

00:11:20.270 --> 00:11:24.790
print notes-- so Federal reserve
notes or reserves,

00:11:24.790 --> 00:11:28.810
either way-- and these are the
notes outstanding liability.

00:11:28.810 --> 00:11:31.910
And they will lend
it to this guy.

00:11:31.910 --> 00:11:36.480
They'll lend these notes to this
guy and in exchange, this

00:11:36.480 --> 00:11:38.030
guy has to give some collateral

00:11:38.030 --> 00:11:39.510
to the Federal reserve.

00:11:39.510 --> 00:11:41.330
So let's say he had some
other assets here

00:11:41.330 --> 00:11:43.210
that are hard to sell.

00:11:43.210 --> 00:11:45.900
He didn't want to sell
them in a hurry.

00:11:45.900 --> 00:11:47.300
So he'll just keep
it as collateral

00:11:47.300 --> 00:11:48.540
with the Federal reserve.

00:11:48.540 --> 00:11:51.000
And these are called repurchased
transactions.

00:11:51.000 --> 00:11:53.120
It's essentially just-- you're
collateralizing a loan-- and

00:11:53.120 --> 00:11:55.910
I'll do a whole video on what a
repo transaction is, but the

00:11:55.910 --> 00:11:58.150
big picture is, this
guy is desperate.

00:11:58.150 --> 00:12:01.000
No-one else is willing to lend
him money so the Federal funds

00:12:01.000 --> 00:12:03.020
rate is now a non-issue.

00:12:03.020 --> 00:12:05.790
So he goes to the discount
window and borrows directly

00:12:05.790 --> 00:12:08.750
from the Federal reserve as
a lender of last resort.

00:12:08.750 --> 00:12:11.640
And the rate at which he
borrows-- the interest that he

00:12:11.640 --> 00:12:14.960
pays this guy, that is
the discount rate.

00:12:14.960 --> 00:12:17.950
So that's the rate that a bank
pays to the Federal reserve

00:12:17.950 --> 00:12:20.950
when it can't borrow from
another bank overnight.

00:12:20.950 --> 00:12:27.770
And in general, the discount
rate tends to be higher than

00:12:27.770 --> 00:12:28.490
the Federal funds rate.

00:12:28.490 --> 00:12:29.640
In fact, it always is, right?

00:12:29.640 --> 00:12:33.230
Because if the discount rate
was less than the Federal

00:12:33.230 --> 00:12:35.880
funds rate, then you'd always
have people using the discount

00:12:35.880 --> 00:12:39.810
window all of the time instead
of borrowing from each other.

00:12:39.810 --> 00:12:42.030
But we'll see in future videos,
when times get tough,

00:12:42.030 --> 00:12:43.760
this gets used a lot more.

00:12:43.760 --> 00:12:46.090
Historically the discount rate
was about a percent higher

00:12:46.090 --> 00:12:48.930
than the Federal funds rate to
encourage people to lend to

00:12:48.930 --> 00:12:52.320
each other or borrow from each
other, but in the recent past

00:12:52.320 --> 00:12:54.380
that spread is gone down and
now all of the rates are

00:12:54.380 --> 00:12:56.650
almost zero, but we'll go into
that in more detail, but it's

00:12:56.650 --> 00:12:58.190
a key differential.

00:12:58.190 --> 00:13:01.280
When the Federal reserve talks
about setting rates, they're

00:13:01.280 --> 00:13:04.740
usually talking about setting
the Federal funds rate and the

00:13:04.740 --> 00:13:07.260
discount rate usually moves down
with it, but it's always

00:13:07.260 --> 00:13:08.470
going to be a little
bit higher than the

00:13:08.470 --> 00:13:09.680
Federal funds rate.

00:13:09.680 --> 00:13:11.950
This is for lending
of last resort.

00:13:11.950 --> 00:13:16.010
This is for everyday borrowing
between banks to make sure

00:13:16.010 --> 00:13:17.910
that everyone has the reserves
they need or they don't have

00:13:17.910 --> 00:13:19.640
too much and they can
get interest on it.

00:13:19.640 --> 00:13:20.890
Anyway, see you in
the next video.


Another u-subsitution example

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:04.050
Evan from Norway has asked me
to do another u substitution

00:00:04.050 --> 00:00:06.350
problem, and I like these
because it gets my momentum

00:00:06.350 --> 00:00:08.920
going for doing other things
that maybe take a little

00:00:08.920 --> 00:00:09.810
bit more preparation.

00:00:09.810 --> 00:00:12.050
And the problem he sent me--
and I hope I pronounced his

00:00:12.050 --> 00:00:21.250
name right-- was the indefinite
integral of sin of x over

00:00:21.250 --> 00:00:25.850
the cosine of x squared dx.

00:00:25.850 --> 00:00:28.270
This it also be written as-- he
wrote it in email, so I don't

00:00:28.270 --> 00:00:31.600
know how he exactly saw it, but
it can also be written as sin

00:00:31.600 --> 00:00:36.130
of x over cosine squared of x.

00:00:36.130 --> 00:00:37.710
Sometimes it's
written like that.

00:00:37.710 --> 00:00:39.270
Either way, I like looking
at this a bit better.

00:00:39.270 --> 00:00:41.350
It's a little bit
less ambiguous.

00:00:41.350 --> 00:00:45.740
But in general you know to do u
substitution, or integration by

00:00:45.740 --> 00:00:47.930
substitution when you see
something and you see its

00:00:47.930 --> 00:00:49.580
derivative sitting there.

00:00:49.580 --> 00:00:49.820
Right.

00:00:49.820 --> 00:00:53.190
You're like wow, this cosine of
x was just an x, or if it was

00:00:53.190 --> 00:00:56.920
just a u, this would be a
really easy integral to do.

00:00:56.920 --> 00:00:58.010
We know how to do
this integral.

00:00:58.010 --> 00:01:00.450
Let me do it on the side.

00:01:00.450 --> 00:01:01.680
This integral would be easy.

00:01:01.680 --> 00:01:05.850
1 over x squared dx.

00:01:05.850 --> 00:01:06.690
We know how to do that.

00:01:06.690 --> 00:01:09.210
That would just be the
antiderivative of x squared.

00:01:09.210 --> 00:01:12.580
This is the same thing as the
antiderivative of x to the

00:01:12.580 --> 00:01:16.500
minus 2 dx, and we know how to
take the antiderivative

00:01:16.500 --> 00:01:17.500
of something like that.

00:01:17.500 --> 00:01:20.810
You increase the exponent by
1 and then you multiply by

00:01:20.810 --> 00:01:22.730
what your new exponent is.

00:01:22.730 --> 00:01:27.510
So it would be minus
x-- I'm sorry.

00:01:27.510 --> 00:01:30.510
You increase your exponent by
1, and then you divide by

00:01:30.510 --> 00:01:32.040
whatever your new exponent is.

00:01:32.040 --> 00:01:34.160
So you would,
[? let me do the, ?]

00:01:34.160 --> 00:01:36.450
increasing the exponent x to
the minus 2, you'd increase

00:01:36.450 --> 00:01:38.420
the exponent, you'd
get x to the minus 1.

00:01:38.420 --> 00:01:41.100
And then when you divide this
by minus 1 you get this minus

00:01:41.100 --> 00:01:43.860
out front, and then of course
you'd have the plus c.

00:01:43.860 --> 00:01:46.070
If you don't believe it
take the derivative.

00:01:46.070 --> 00:01:49.190
Negative 1 times minus 1.

00:01:49.190 --> 00:01:50.340
That's a positive.

00:01:50.340 --> 00:01:52.360
And then you'd decrease
the exponent by 1, you

00:01:52.360 --> 00:01:53.980
get x to the minus 2.

00:01:53.980 --> 00:01:55.700
So if we could get it in
a form that looks like

00:01:55.700 --> 00:01:57.530
this, we'd be all set.

00:01:57.530 --> 00:01:59.250
And you kind of do
see a pattern.

00:01:59.250 --> 00:02:02.330
Where this x is you have a
cosine there, and then we have

00:02:02.330 --> 00:02:03.650
cosines derivative there.

00:02:03.650 --> 00:02:06.950
So that's the big clue that we
should be using u substitution.

00:02:06.950 --> 00:02:07.815
So let's do that.

00:02:07.815 --> 00:02:09.920
And what we're going to do is
we're going to substitute

00:02:09.920 --> 00:02:11.670
u for cosine of x.

00:02:11.670 --> 00:02:20.170
So if we say u is equal to
cosine of x, and let's take

00:02:20.170 --> 00:02:22.750
the derivative of u
with respect to x.

00:02:22.750 --> 00:02:27.740
So du/dx is equal to what?

00:02:27.740 --> 00:02:29.790
What's the derivative of
cosine of x? it's not

00:02:29.790 --> 00:02:31.690
quite sin of x, right?

00:02:31.690 --> 00:02:33.302
It's minus sin of x.

00:02:37.820 --> 00:02:42.370
And then we can multiply both
sides by dx, and you get du is

00:02:42.370 --> 00:02:48.720
equal to minus sin of x dx.

00:02:48.720 --> 00:02:51.090
I just multiplied
both sides by dx.

00:02:51.090 --> 00:02:54.120
And then up here we
have sin of x dx.

00:02:54.120 --> 00:02:57.380
We don't have minus
sin of x dx.

00:02:57.380 --> 00:02:59.710
There we have sin of x dx.

00:02:59.710 --> 00:03:01.980
We could have rewritten this
top integral, we could have

00:03:01.980 --> 00:03:03.500
rewritten it like this.

00:03:03.500 --> 00:03:07.930
Sin of x dx.

00:03:07.930 --> 00:03:12.680
All of that over
cosine of x squared.

00:03:12.680 --> 00:03:14.780
So if we want to substitute for
this, here we have a minus.

00:03:14.780 --> 00:03:17.750
Let's multiply both sides of
this by a negative 1, and you

00:03:17.750 --> 00:03:25.080
get minus du is equal
to sin of x dx.

00:03:25.080 --> 00:03:26.370
And let's see.

00:03:26.370 --> 00:03:28.650
So let's rewrite this
original problem.

00:03:28.650 --> 00:03:30.690
Now I know I'm running
out of space.

00:03:30.690 --> 00:03:32.000
Let me rewrite it.

00:03:32.000 --> 00:03:35.780
So we know that u is equal to
cosine of x, so let's do that.

00:03:35.780 --> 00:03:38.720
So now this integral becomes--
and the denominator, instead of

00:03:38.720 --> 00:03:42.170
cosine of x squared,
u is cosine of x.

00:03:42.170 --> 00:03:42.930
That's u, right?

00:03:42.930 --> 00:03:43.900
We made that definition.

00:03:43.900 --> 00:03:46.240
So that's over u squared.

00:03:46.240 --> 00:03:48.260
Cosine of x becomes u.

00:03:48.260 --> 00:03:55.240
And then sin of x dx right up
there, what is that equal to?

00:03:55.240 --> 00:03:56.490
Well we just solved
for it here.

00:03:56.490 --> 00:03:57.930
That's equal to minus du.

00:03:57.930 --> 00:04:00.120
Sin of x dx is
equal to minus du.

00:04:00.120 --> 00:04:03.810
So that we can replace
with this, minus du.

00:04:03.810 --> 00:04:06.330
And then of course this has
the exact same form as

00:04:06.330 --> 00:04:07.200
this thing right here.

00:04:07.200 --> 00:04:11.420
You could rewrite this,
this is equal to let's say

00:04:11.420 --> 00:04:17.940
minus 1 over u squared du.

00:04:17.940 --> 00:04:19.280
I'm just writing it a
bunch of different ways.

00:04:19.280 --> 00:04:21.455
Whatever is easier for
you to conceptualize.

00:04:21.455 --> 00:04:27.550
The same thing as minus
u to the minus 2 du.

00:04:27.550 --> 00:04:30.180
And then here we do the same
thing we did up here, although

00:04:30.180 --> 00:04:31.870
now we have a minus out front,
that actually makes it

00:04:31.870 --> 00:04:33.300
a little bit cleaner.

00:04:33.300 --> 00:04:37.530
To take the antiderivative, we
raise u-- it was to the minus 2

00:04:37.530 --> 00:04:41.230
power, let's raise it to 1
power higher than that-- so

00:04:41.230 --> 00:04:43.960
minus 2 plus 1 is minus 1.

00:04:43.960 --> 00:04:49.760
So it's u to the minus 1 power,
and then you want to divide by

00:04:49.760 --> 00:04:52.810
minus 1, and I'll do
it explicitly here.

00:04:52.810 --> 00:04:53.760
Minus one.

00:04:53.760 --> 00:04:55.640
And then you had this negative
that was sitting out there

00:04:55.640 --> 00:04:59.200
before, so that negative is
still going to be there.

00:04:59.200 --> 00:05:02.110
And of course you're
going to have a plus c.

00:05:02.110 --> 00:05:05.790
You can view this as a negative
1 or, this negative divided

00:05:05.790 --> 00:05:07.870
by a negative, they're
going to cancel out.

00:05:07.870 --> 00:05:14.540
And so you're just left with u
to the minus 1 plus c, or 1

00:05:14.540 --> 00:05:18.970
over u plus c is the
antiderivative-- oh sorry,

00:05:18.970 --> 00:05:19.950
we're not done yet.

00:05:19.950 --> 00:05:21.400
That's just the
antiderivative of this.

00:05:21.400 --> 00:05:23.330
And now we have our
substitution to deal with.

00:05:23.330 --> 00:05:25.850
What was our substitution
that we started with?

00:05:25.850 --> 00:05:28.780
u is equal to cosine of x.

00:05:28.780 --> 00:05:34.700
So if u is equal to cosine of
x, this thing is equal to 1

00:05:34.700 --> 00:05:41.830
over cosine of x plus c is
equal to the antiderivative of

00:05:41.830 --> 00:05:50.200
our original problem, which was
sin of x over cosine

00:05:50.200 --> 00:05:53.450
of x squared dx.

00:05:53.450 --> 00:05:54.502
There you go.

00:05:54.502 --> 00:05:55.310
See you in the next video.