108156.pdf

114 Mathematics for Finance

Figure 5.8 Portfolios without short selling

Exercise 5.14

For portfolios constructed with and without short selling from the three

securities in Exercise 5.12 compute the minimum variance line parame-

trised by the expected return and sketch it a) on thew 2 ,w 3 plane and

b) on theσ, μplane. Also sketch the set of all attainable portfolios with

and without short selling.

5.3.2 Efficient Frontier ...................................

Given the choice between two securities a rational investor will, if possible,
choose that with higher expected return and lower standard deviation, that is,
lower risk. This motivates the following definition.

Definition 5.1

We say that a security with expected returnμ 1 and standard deviationσ 1
dominatesanother security with expected returnμ 2 and standard deviationσ 2
whenever
μ 1 ≥μ 2 and σ 1 ≤σ 2.

This definition readily extends to portfolios, which can of course be considered
as securities in their own right.

Remark 5.4

Given two securities such that one dominates the other, the dominated security
may appear quite redundant on first sight. Nevertheless, it can also be of some