Learning outcomes.
After watching this video you will be able to, understand the difference
between the mean variance frontier and the efficient frontier.
Define the minimum variance portfolio, that is MVP.
Diversification and efficient frontier.
In the last video,
we saw what happens when we combine two risky assets into a portfolio.
We saw specifically, that for some combinations of x and
y the standard deviation of the portfolio
is lower than the individual standard deviations of x and y.
This is the idea of diversification.
In this video, we will discuss the idea of diversification in a little more detail,
and introduce the concept of efficient frontier.
Let's revisit the picture from last class.
Look at the black line when the correlation is plus one.
In this case, the standard deviation of all combinations of x and
y are always greater than that of asset x.
This shows that diversification is not possible when
the asset returns are perfectly positively correlated.
Diversification is possible only when asset returns are less than
perfectly positively correlated, that is, row is less than plus one.
Next, look at the pink curve when the correlation between asset returns is
a negative one.
There is one point where the curve touches the vertical axis,
at this point, the risk is zero.
This says that when there is perfect negative correlation between two
assets returns, one particular combination of two risky assets is riskless,
in other words, this combination completely diversifies away all risk.
Let's extend the two risky asset case to three
risky assets by adding another risky asset z.
The expected returns and standard deviation for x are still 10% and
7%, respectively.
The numbers for y are 20% and 10%, respectively, and
those for z are now 15% and 12%, respectively.
We're also given that the correlation between x and y's returns is 0.1.
That between x and z is 0, and that between y and z is 0.9.
First, let's plot the investment opportunity set for
each pair of investments.
The blue curve is for x and y, the green curve is for
y and z, and the pink one is for x and z.
With each pair of risky assets,
the investment opportunity set is just a curve or a line.
What if we look at investing in all three assets at the same time?
Then the investment opportunity set is a whole area.
In the picture, this is the entire area within the black curve.
Each point within this area,
represents a possible combination of x, y, and z.
The black curve is called a mean variance frontier and
plots the optimal combinations of x, y, and z.
What does optimal combination x, y, and z mean?
The mean millions frontier joins portfolios of x,
y, z that yield the least risk at each level of expected return.
Another way of looking at optimal combinations is what is
the maximum expected return at each level of risk.
Looking at the optimal combinations this way, at each level of risk,
the expected return on the top part of the mean variance frontier is always
greater than the expected return on the bottom part of the mean variance frontier.
In other words, the top part always dominates the bottom part.
If we exclude the bottom part of the mean variance frontier
the the top part is called the efficient frontier.
It maximizes the expected return at each level of risk, and
minimizes risk at each level of expected return.
In terms of risk return combinations, no risk return combination
to the left of, or above the efficient frontier, is feasible.
An important point to note is the portfolio called MVP,
it stands for the global minimum-variance portfolio.
Using x, y, and z, this is the one portfolio that has the least variance
of risk across all portfolios in the investment opportunity set.
Given risky assets x, y and z,
no combination of these assets will help us achieve a lower risk.
In our example of three risky assets, how do we determine the efficient frontier?
We will talk about two-fund separation and
how it may be used to draw the efficient frontier next time.
Ramabhadran ThirumalaiAssistant Professor
