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So, I want to do some examples now to, to
try to give you some intuition about

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efficient portfolios and I'm going to do
some, some simple examples.

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And so, the first problem that I'm going
to ask is let's find the efficient

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portfolio that has the same risk as
bowing, okay?

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And if you want to think about this
graphically, let me just draw a picture

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for you.
In our diagram, you know, we have our here

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is bowing, here is Amazon and then we have
our tangency portfolio here and expected

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return, standard deviation.
All efficient portfolios are on this, this

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line here, okay?
So, this is the, this is all efficient

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portfolios.
So, the problem is asking us let's find

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the efficient portfolio that has the same
volatility as, as Boeing, alright?

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And so, if the volatility of bowing is
here, then the efficient portfolio,

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vertically, is going to be the portfolio
that's a combination of the tangency

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portfolio in T-Bills that has the same
volatility as, as bowing.

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Now, because it's an efficient portfolio,
this, this combination is going to have a

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higher expected return cuz the expected
return on bowing is only down over here.

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So, one way we can, we can think of using
efficient portfolios is, you know, here

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we're, if we're 100 percent invested in
bowing, we have a certain risk and a

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certain expected return, the efficient
portfolio will have a higher expected

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return, but the same risk as bowing, okay?
So now, the question becomes, how do you

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find the combination of T-bills in the
tangency portfolio that has the same risk

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as bowing.
And so here, we just use the fact that, or

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efficient portfolio has the same
volatility of bowing and that's 11.4%,

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okay.
Now, to solve this equation, we, we use

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the following fact and that is, efficient
portfolios have the following

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characteristics.
Their combinations of T-Bills and tangency

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portfolio, So, the expected returns of any
efficient portfolio is the T-Bill rate

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plus how much you invest in the tangency
portfolio times the risk premium on the

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tangency portfolio, okay?
And so, in our example, the T-bill rate is

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three%.
The tangency portfolio has an expected

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return of eleven%, and the T-bill rate has
three%.

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Now, the volatility of any efficient
portfolio is how much you invest in the

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tangency portfolio times the volatility of
the tangency portfolio.

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In our example, the volatility of the
tangency portfolio is 12.4%.

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So, if we look at you know, our, our, my
diagram, you know, the tendency portfolio

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is here, you know, sigma t and then the
expected return on the tangency portfolio

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is, is here, okay?
Now, going back to this question, you know

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you know, how, how much to invest in the
tangency portfolio in order for us to have

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a, a volatility of 11.4%?
So, essentially what we do is we, we do

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the following.
We want our efficient portfolio volatility

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between, to be 11.4 percent and we know
that the volatility of the efficient

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portfolio is how much we invest in the
tangency portfolio times the volatility of

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the tangency portfolio.
So, we can we know the volatility of the

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teangency portfolio, it's 12.4%.
So, then we just solve, how much we invest

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in the tangency portfolio is the ratio of
our target volatility to the volatility of

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the tangency portfolio.
That's 0.92.

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So, if we invest 92 percent of our wealth
in the tangency portfolio and eight

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percent in T-Bills, then we get a
portfolio that has the same volatility as

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bowing.
Now, because it's an efficient portfolio,

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this portfolio will have a higher expected
return than bowing, alright?

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So, the expected return is3 percent plus
0.92 times the risk premium on the

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tangency portfolio, that's 10.4%.
So, the advantage of the efficient

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portfolio is that we have a higher
expected return than bowing, but we have

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the same risk.
Now, when we look at you know, what is our

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investment in Amazon, bowing, and T bills
in this efficient portfolio.

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Well, how much to invest in Amazon?
Well, we have 92 percent of 46 percent or

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42 percent in Amazon.
How much is in bowing?

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92 percent of 54 percent or 50 percent in
bowing.

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You add 42 to, you know, 50 and 42 that
gives you your 0.92.

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So, that's how much we invest in the
tangency portfolio, and the remainder is

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in T-Bills, okay?
So, that picture is here.

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So, you know, I have my here is bowing.
Here is the volatility of bowing.

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If I want the efficient portfolio of the
same volatility as bowing, I just go

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vertically and I find where I am on the
green line.

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Here's the tangency portfolio.
Notice, that I'm close to the tangency

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portfolio.
At point E1, I'm 92 percent in the

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tangency portfolio, eight percent in
T-bills, okay?

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So then, we can do a similar problem and
that is, you know, assuming the risk free

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rate of three%, find the efficient
portfolio that has the same average return

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as bowing, okay?
So now, what we want to do is we want the

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expected return on our efficient portfolio
to be the same as the expected return on

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bowing and that's 5.5%.
So, that's our target return.

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Now because, so we wanna find some
combination of the tangency portfolio and

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T-Bills that gives us a 5.5 percent rate
of return.

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Now, because this is an efficient
portfolio, this portfolio will have a

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lower volatility than bowing, right?
And so now, the question is, how much do

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we invest in the tangency and T-Bills to
reach this target return?

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So, graphically, we have the following
problem.

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So, we just can, you know, draw or diagram
again.

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We think about these efficient portfolios.
Here is bowing, here is Amazon, here is

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our tangency portfolio now.
Now, the problem is we want our target

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return to be the, the same return on
bowing.

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So, this is the expected return on bowing,
and here's the volatility of bowing.

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The efficient portfolio then, will be at
this point here cuz this point, the green

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dot, is the combination of the tangency
portfolio and T-Bills that gives you

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exactly the same average return as bowing.
And notice that this portfolio has a much

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lower volatility than the, the investment
in bowing.

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So again, another way of viewing an
efficient portfolio, we look at a target

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return, and then we want to find the
portfolio that achieves that target

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return.
And that's going to have the smallest risk

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of any portfolio that, that we can invest
in.

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So, how we solve for this, you know, we
use the fact that the expected return on

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any efficient portfolio is equal to the
risk-free rate plus how much we invest in

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the tangency portfolio times the risk
premium on the tangency portfolio.

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Now, we want the expected return of our
efficient portfolio to be this target rate

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of return, 5.5%.
So, notice there's only one thing to solve

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for, that's xt.
So, xt is equal to our target return minus

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the T-Bill rate divided by the risk
premium on the tangency portfolio, that

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turns out to be 31%.
So, if we invest 31 percent in the

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tangency portfolio and 69 percent in
T-bills, then we get a target return of

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5.5%, okay?
So, what is our portfolio composition?

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Our investment in Amazon is 31 percent of
46%, so we're fourteen percent in Amazon.

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Our investment in bowing is 31 percent of
54%, so we're 70 percent in bowing, and

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the remainder is in T-Bills.
Now, this portfolio has the same average

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return as bowling, but what's is
volatility?

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It's 0.31 times the volatility of the
tangancy portfolio and that's 3.8%.

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So you know, that's the you know, bowing
has a standard deviation of eleven%, the

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efficient portfolio has a standard
deviation of four percent so we have a

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volatility reduction by going into the
efficient portfolio, okay?

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And so, that picture is illustrated here.
So again, you look at, here's bowing,

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here's the expected return on bowing.
I want to find my efficient portfolio that

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has the same expected return as bowing.
So, I just draw you know, a horizontal

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line and I see where the, where does that
intersect the green, this green line here?

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And then, the calculations that I just
showed you said, you know, at this point

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we're 31 percent in the tangency portfolio
and 69 percent in T-Bills, okay?

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So, this is, you know, a way, you know, if
you want to think of yourself as, you

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know, how do you use this theory for asset
allocation, right?

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So, once you do this stuff, how do you use
it?

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You know, these two examples give you two
different ways of thinking about the

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problem.
You might have a target level of risk and

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then say, how do I find a portfolio that
achieves my target risk?

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Or you might have a target expected return
and then you're going to ask, how do I

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find a portfolio to achieve my target
expected return?