So, I want to do some examples now to, to
try to give you some intuition about
efficient portfolios and I'm going to do
some, some simple examples.
And so, the first problem that I'm going
to ask is let's find the efficient
portfolio that has the same risk as
bowing, okay?
And if you want to think about this
graphically, let me just draw a picture
for you.
In our diagram, you know, we have our here
is bowing, here is Amazon and then we have
our tangency portfolio here and expected
return, standard deviation.
All efficient portfolios are on this, this
line here, okay?
So, this is the, this is all efficient
portfolios.
So, the problem is asking us let's find
the efficient portfolio that has the same
volatility as, as Boeing, alright?
And so, if the volatility of bowing is
here, then the efficient portfolio,
vertically, is going to be the portfolio
that's a combination of the tangency
portfolio in T-Bills that has the same
volatility as, as bowing.
Now, because it's an efficient portfolio,
this, this combination is going to have a
higher expected return cuz the expected
return on bowing is only down over here.
So, one way we can, we can think of using
efficient portfolios is, you know, here
we're, if we're 100 percent invested in
bowing, we have a certain risk and a
certain expected return, the efficient
portfolio will have a higher expected
return, but the same risk as bowing, okay?
So now, the question becomes, how do you
find the combination of T-bills in the
tangency portfolio that has the same risk
as bowing.
And so here, we just use the fact that, or
efficient portfolio has the same
volatility of bowing and that's 11.4%,
okay.
Now, to solve this equation, we, we use
the following fact and that is, efficient
portfolios have the following
characteristics.
Their combinations of T-Bills and tangency
portfolio, So, the expected returns of any
efficient portfolio is the T-Bill rate
plus how much you invest in the tangency
portfolio times the risk premium on the
tangency portfolio, okay?
And so, in our example, the T-bill rate is
three%.
The tangency portfolio has an expected
return of eleven%, and the T-bill rate has
three%.
Now, the volatility of any efficient
portfolio is how much you invest in the
tangency portfolio times the volatility of
the tangency portfolio.
In our example, the volatility of the
tangency portfolio is 12.4%.
So, if we look at you know, our, our, my
diagram, you know, the tendency portfolio
is here, you know, sigma t and then the
expected return on the tangency portfolio
is, is here, okay?
Now, going back to this question, you know
you know, how, how much to invest in the
tangency portfolio in order for us to have
a, a volatility of 11.4%?
So, essentially what we do is we, we do
the following.
We want our efficient portfolio volatility
between, to be 11.4 percent and we know
that the volatility of the efficient
portfolio is how much we invest in the
tangency portfolio times the volatility of
the tangency portfolio.
So, we can we know the volatility of the
teangency portfolio, it's 12.4%.
So, then we just solve, how much we invest
in the tangency portfolio is the ratio of
our target volatility to the volatility of
the tangency portfolio.
That's 0.92.
So, if we invest 92 percent of our wealth
in the tangency portfolio and eight
percent in T-Bills, then we get a
portfolio that has the same volatility as
bowing.
Now, because it's an efficient portfolio,
this portfolio will have a higher expected
return than bowing, alright?
So, the expected return is3 percent plus
0.92 times the risk premium on the
tangency portfolio, that's 10.4%.
So, the advantage of the efficient
portfolio is that we have a higher
expected return than bowing, but we have
the same risk.
Now, when we look at you know, what is our
investment in Amazon, bowing, and T bills
in this efficient portfolio.
Well, how much to invest in Amazon?
Well, we have 92 percent of 46 percent or
42 percent in Amazon.
How much is in bowing?
92 percent of 54 percent or 50 percent in
bowing.
You add 42 to, you know, 50 and 42 that
gives you your 0.92.
So, that's how much we invest in the
tangency portfolio, and the remainder is
in T-Bills, okay?
So, that picture is here.
So, you know, I have my here is bowing.
Here is the volatility of bowing.
If I want the efficient portfolio of the
same volatility as bowing, I just go
vertically and I find where I am on the
green line.
Here's the tangency portfolio.
Notice, that I'm close to the tangency
portfolio.
At point E1, I'm 92 percent in the
tangency portfolio, eight percent in
T-bills, okay?
So then, we can do a similar problem and
that is, you know, assuming the risk free
rate of three%, find the efficient
portfolio that has the same average return
as bowing, okay?
So now, what we want to do is we want the
expected return on our efficient portfolio
to be the same as the expected return on
bowing and that's 5.5%.
So, that's our target return.
Now because, so we wanna find some
combination of the tangency portfolio and
T-Bills that gives us a 5.5 percent rate
of return.
Now, because this is an efficient
portfolio, this portfolio will have a
lower volatility than bowing, right?
And so now, the question is, how much do
we invest in the tangency and T-Bills to
reach this target return?
So, graphically, we have the following
problem.
So, we just can, you know, draw or diagram
again.
We think about these efficient portfolios.
Here is bowing, here is Amazon, here is
our tangency portfolio now.
Now, the problem is we want our target
return to be the, the same return on
bowing.
So, this is the expected return on bowing,
and here's the volatility of bowing.
The efficient portfolio then, will be at
this point here cuz this point, the green
dot, is the combination of the tangency
portfolio and T-Bills that gives you
exactly the same average return as bowing.
And notice that this portfolio has a much
lower volatility than the, the investment
in bowing.
So again, another way of viewing an
efficient portfolio, we look at a target
return, and then we want to find the
portfolio that achieves that target
return.
And that's going to have the smallest risk
of any portfolio that, that we can invest
in.
So, how we solve for this, you know, we
use the fact that the expected return on
any efficient portfolio is equal to the
risk-free rate plus how much we invest in
the tangency portfolio times the risk
premium on the tangency portfolio.
Now, we want the expected return of our
efficient portfolio to be this target rate
of return, 5.5%.
So, notice there's only one thing to solve
for, that's xt.
So, xt is equal to our target return minus
the T-Bill rate divided by the risk
premium on the tangency portfolio, that
turns out to be 31%.
So, if we invest 31 percent in the
tangency portfolio and 69 percent in
T-bills, then we get a target return of
5.5%, okay?
So, what is our portfolio composition?
Our investment in Amazon is 31 percent of
46%, so we're fourteen percent in Amazon.
Our investment in bowing is 31 percent of
54%, so we're 70 percent in bowing, and
the remainder is in T-Bills.
Now, this portfolio has the same average
return as bowling, but what's is
volatility?
It's 0.31 times the volatility of the
tangancy portfolio and that's 3.8%.
So you know, that's the you know, bowing
has a standard deviation of eleven%, the
efficient portfolio has a standard
deviation of four percent so we have a
volatility reduction by going into the
efficient portfolio, okay?
And so, that picture is illustrated here.
So again, you look at, here's bowing,
here's the expected return on bowing.
I want to find my efficient portfolio that
has the same expected return as bowing.
So, I just draw you know, a horizontal
line and I see where the, where does that
intersect the green, this green line here?
And then, the calculations that I just
showed you said, you know, at this point
we're 31 percent in the tangency portfolio
and 69 percent in T-Bills, okay?
So, this is, you know, a way, you know, if
you want to think of yourself as, you
know, how do you use this theory for asset
allocation, right?
So, once you do this stuff, how do you use
it?
You know, these two examples give you two
different ways of thinking about the
problem.
You might have a target level of risk and
then say, how do I find a portfolio that
achieves my target risk?
Or you might have a target expected return
and then you're going to ask, how do I
find a portfolio to achieve my target
expected return?