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use. Employing the techniques of Section 5.2, it may be possible to construct
portfolios consisting of the two securities with smaller risk than either of the
securities, as in Figure 5.9, in which the security withσ 2 ,μ 2 is dominated by
that withσ 1 ,μ 1.
Figure 5.9 Reduction of risk using a dominated securityDefinition 5.2
A portfolio is calledefficientif there is no other portfolio, except itself, that
dominates it. The set of efficient portfolios among all attainable portfolios is
called theefficient frontier.
Every rational investor will choose an efficient portfolio, always preferring
a dominating portfolio to a dominated one. However, different investors may
select different portfolios on the efficient frontier, depending on their individual
preferences. Given two efficient portfolios withμ 1 ≤μ 2 andσ 1 ≤σ 2 , a cautious
person may prefer that with lower riskσ 1 and lower expected returnμ 1 , while
others may choose a portfolio with higher riskσ 2 , regarding the higher expected
returnμ 2 as compensation for increased risk.
In particular, an efficient portfolio has the highest expected return among
all attainable portfolios with the same standard deviation (the same risk),
and has the lowest standard deviation (the lowest risk) among all attainable
portfolios with the same expected return. As a result, the efficient frontier must
be a subset of the minimum variance line. To understand the structure of the
efficient frontier we shall first study the minimum variance line in more detail
and then select a suitable subset.
Proposition 5.11
Take any two different portfolios on the minimum variance line, with weights
w′andw′′. Then the minimum variance line consists of portfolios with weights