Morningstar - Summer 2019 - 61

EXHIBIT 1

Portfolios and Efficient Frontiers of Three Securities

where the line intersects the bottom side of
the triangle, the allocation to Security 2 is zero.
I call the part of the efficient frontier that
runs from the minimum variance portfolio to this
point Efficient Frontier Part I and label it "E.F.
Part I" in EXHIB IT 1 . This is part of the efficient
frontier whether or not short selling is
allowed. If short selling is allowed, the efficient
frontier continues along the line forever with more
and more negative allocations to Security 1.
This is part of the unconstrained efficient frontier,
so I call it Efficient Frontier Part II Unconstrained
and label it "E.F. Part II Uncon." But if short
selling is not allowed, the efficient frontier
turns to the right at the point labeled "X2 = 0" and
follows the side of the triangle, ending at the
corner labeled "Security 3." Thus, without short
selling, the efficient portfolio with the highest
expected return is 100% Security 3.

Security 2
Security 2

Min. Var. Port.
Min.
Tang.Var.
Port.Port.
Tang. Port.
Investor
1
Investor 1
E.F. Part I
Security 1

E.F. Part I

Security 1

E.F. Part II Uncon.
E.F. Part II Uncon.

Source: Morningstar.

Market
Market

X2 = 0

Security 3

X 2 =Part
0 II Con.
E.F.

3
InvestorSecurity
2

E.F. Part II Con.

Investor 2

EXHIBIT 2

Constrained and Unconstrained Efficient Frontiers of Three Securities
Expected Return
12%
E.F. Part II Uncon.Expected Return
12%
E.F. Part II Uncon.

X2 = 0
Investor 1
Tang. Port.
Investor 1
Tang. Port.
Min. Var. Port.

T
T

G
AN
G
AN

E

EN

T

LI
LI

X2 = 0

Security 3

E.F. Part II Con.

Security
E.F.1Part II Con.

InvestorSecurity
1
3

9

Investor 1

9

Security 1

6
6

Security 2

N

NE

0%
5
Standard
Deviation
0%
5

Source: Morningstar.

Market
E.F. Part I
E.F.2Part I
Security

Min.E Var. Port.
NT

Market

3
3
0
10

15

20

25

30

10

15

20

25

30

0

Markowitz presented the portfolios that make up
the constrained and unconstrained efficient
frontiers in a similar way that I do in EXHIBIT 1 .
However, he did not present them in the usual
risk/return fashion. I do that in EXHIBIT 2 .
In EXHIB IT 2 , I show the standard deviation and
expected return of each of the portfolios labeled
in EXHIB IT 1 , as well as the Efficient Frontier
Part I, the Efficient Frontier Part I Unconstrained,
and the Efficient Frontier Part II Constrained.
Note how both versions of Part II emanate from
the point labeled "X2 = 0" and that the constrained
version is below the unconstrained version,
indicating that the constraint against short selling
makes the tradeoff between risk and expected
return less favorable.

Standard Deviation

Given the assumptions that Markowitz makes
about the expected returns, standard deviations,
and correlations of returns of the three
securities, the set of all portfolios that minimize
standard deviation of returns for a given level
of expected return is given by the straight
line that cuts through the triangle. There is no
short selling along the part of the line that
is within the triangle. One point on the line within
the triangle is the minimum variance portfolio,
which I have labeled "Min. Var. Port." Along

the dotted portion of the line above the minimum
variance portfolio, expected return declines
with risk. Therefore, this portion of the line is not
part of the efficient frontier. Along the solid
portion of the line below the minimum variance
portfolio, expected returns increase with
risk. Hence, this portion of the line is part of the
efficient frontier.
As we move down the line, the allocation to
Security 2 decreases. At the point labeled "X2 = 0,"

One of the efficient portfolios that Markowitz
identifies and shows on his security allocation
chart is the portfolio that can be identified
by drawing a line from the origin of the
risk/return chart that is tangent to the efficient
frontier. In EXHIB IT  2 , I show how to do this. I call
the portfolio identified this way the tangency
portfolio and label it "Tang. Port." In EXHIBIT 1 ,
I show the location of selected portfolios along
the unconstrained efficient frontier line,
including the minimum variance portfolio and
the tangency portfolio. Because two points
determine a line, I only needed to calculate these

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