108156.pdf

5. Portfolio Management 101

Exercise 5.8

Using the data in Example 5.6, find the weights in a portfolio with

expected returnμV= 46% and compute the riskσ^2 Vof this portfolio.

The correlation coefficient always satisfies− 1 ≤ρ 12 ≤1. The next propo-
sition is concerned with the two special cases whenρ 12 assumes one of the
extreme values 1 or−1, which means perfect positive or negative correlation
between the securities in the portfolio.

Proposition 5.4

Ifρ 12 =1,thenσV=0whenσ 1 =σ 2 and

w 1 =− σ^2

σ 1 −σ 2

,w 2 = σ^1

σ 1 −σ 2

. (5.9)

(Short sales are necessary, since eitherw 1 orw 2 is negative.)
Ifρ 12 =−1, thenσV=0for

w 1 =

σ 2

σ 1 +σ 2

,w 2 =

σ 1

σ 1 +σ 2

. (5.10)

(No short sales are necessary, since bothw 1 andw 2 are positive.)

Proof

Letρ 12 = 1. Then (5.8) takes the form

σV^2 =w 12 σ^21 +w 22 σ^22 +2w 1 w 2 σ 1 σ 2 =(w 1 σ 1 +w 2 σ 2 )^2

andσ^2 V= 0 if and only ifw 1 σ 1 +w 2 σ 2 = 0. This is equivalent toσ 1 =σ 2 and
(5.9) becausew 1 +w 2 =1.
Now letρ 12 =−1. Then (5.8) becomes

σV^2 =w 12 σ^21 +w 22 σ^22 − 2 w 1 w 2 σ 1 σ 2 =(w 1 σ 1 −w 2 σ 2 )^2

andσ^2 V= 0 if and only ifw 1 σ 1 −w 2 σ 2 = 0. The last equality is equivalent to
(5.10) becausew 1 +w 2 =1.

Each portfolio can be represented by a point with coordinatesσVandμV
on theσ, μplane. Figure 5.1 shows two typical lines representing portfolios
withρ 12 =−1 (left) andρ 12 = 1 (right). The bold segments correspond to
portfolios without short selling.