108156.pdf

108 Mathematics for Finance

It is well known that the covariance matrix is symmetric and positive definite.
The diagonal elements are simply the variances of returns,cii=Var(Ki). In
what follows we shall assume, in addition, thatChas an inverseC−^1.

Proposition 5.8

The expected returnμV=E(KV)andvarianceσV^2 =Var(KV) of a portfolio
with weightsware given by

μV=mwT, (5.15)

σV^2 =wCwT. (5.16)

Proof

The formula forμVfollows by the linearity of expectation,

μV=E(KV)=E

(n

∑

i=1

wiKi

)

=

∑n

i=1

wiμi=mwT.

Forσ^2 Vwe use the linearity of covariance with respect to each of its arguments,

σ^2 V=Var(KV)=Var

(n

∑

i=1

wiKi

)

=Cov





∑n

i=1

wiKi,

∑n

j=1

wjKj



=

∑n

i,j=1

wiwjcij

=wCwT.

Exercise 5.11

Compute the expected returnμVand standard deviationσVof a port-

folio consisting of three securities with weightsw 1 = 40%,w 2 =−20%,

w 3 = 80%, given that the securities have expected returnsμ 1 =8%,

μ 2 = 10%,μ 3 = 6%, standard deviationsσ 1 =1.5,σ 2 =0.5,σ 3 =1. 2

and correlationsρ 12 =0.3,ρ 23 =0.0,ρ 31 =− 0 .2.

We shall solve the following two problems:

1. To find a portfolio with the smallest variance in the attainable set. It will
   be called theminimum variance portfolio.