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lios containing securities 2 and 3 only lie on the line through (0. 24 , 0 .15) and
(0. 25 , 0 .20). The three points and the lines passing through them correspond
to the vertices of the triangle and the straight lines passing through them in
Figure 5.6. The shaded area (both dark and light), including the boundary,
represents portfolios that can be constructed from the three securities, that is,
all attainable portfolios. The boundary, shown as a bold line, is the minimum
variance line. The shape of it is known as theMarkowitz bullet.Thedarker
part of the shaded area corresponds to the interior of the triangle in Figure 5.6,
that is, it represents portfolios without short selling.
Figure 5.7 Attainable portfolios on theσ, μplaneIt is instructive to imagine how the wholew 2 ,w 3 plane in Figure 5.6 is
mapped onto the shaded area representing all attainable portfolios in Fig-
ure 5.7. Namely, thew 2 ,w 3 plane is folded along the minimum variance line, be-
ing simultaneously warped and stretched to attain the shape of the Markowitz
bullet. This means, in particular, that pairs of points on opposite sides of the
minimum variance line on thew 2 ,w 3 plane are mapped into single points on
theσ, μplane. In other words, each point inside the shaded area in Figure 5.7
corresponds to two different portfolios. However, each point on the minimum
variance line corresponds to a single portfolio.
Example 5.12
(3 securities without short selling) For the same three securities as in Exam-
ples 5.10 and 5.11, Figure 5.8 shows what happens if no short selling is allowed.
All portfolios without short selling are represented by the interior and bound-
ary of the triangle on thew 1 ,w 2 plane and by the shaded area with boundary
on theσ, μplane. The minimum variance line without short selling is shown
as a bold line in both plots. For comparison, the minimum variance line with
short selling is shown as a broken line.