- Portfolio Management 107
The line on theσ, μplane representing portfolios constructed from one risky
and one risk-free security is shown in Figure 5.5. As usual, the bold line segment
corresponds to portfolios without short selling.
5.3 Several Securities.........................................
5.3.1 Risk and Expected Return on a Portfolio
A portfolio constructed fromndifferent securities can be described in terms of
their weights
wi=xiSi(0)
V(0),i=1,...,n,wherexiis the number of shares of typeiin the portfolio,Si(0) is the initial
price of securityi,andV(0) is the amount initially invested in the portfolio. It
will prove convenient to arrange the weights into a one-row matrix
w=[
w 1 w 2 ··· wn]
.
Just like for two securities, the weights add up to one, which can be written in
matrix form as
1=uwT, (5.14)
where
u=
[
11 ··· 1
]
is a one-row matrix with allnentries equal to 1,wTis a one-column matrix,
the transpose ofw, and the usual matrix multiplication rules apply. Theat-
tainable setconsists of all portfolios with weightswsatisfying (5.14), called
theattainable portfolios.
Suppose that the returns on the securities areK 1 ,...,Kn. The expected
returnsμi=E(Ki)fori=1,...,nwill also be arranged into a one-row matrix
m=[
μ 1 μ 2 ··· μn]
.
The covariances between returns will be denoted bycij=Cov(Ki,Kj). They
are the entries of then×ncovariance matrix
C=
c 11 c 12 ··· c 1 n
c 21 c 22 ··· c 2 n
..
...
.
... ..
.
cn 1 cn 2 ··· cnn