108156.pdf

5. Portfolio Management 109
   2. To find a portfolio with the smallest variance among all portfolios in the
   attainable set whose expected return is equal to a given numberμV.The
   family of such portfolios, parametrised byμV, is called theminimum vari-
   ance line.

Since the variance is a continuous function of the weights, bounded below by 0,
the minimum clearly exists in both cases.

Proposition 5.9 (Minimum Variance Portfolio)

The portfolio with the smallest variance in the attainable set has weights

w=

uC−^1

uC−^1 uT

,

provided that the denominator is non-zero.

Proof

We need to find the minimum of (5.16) subject to the constraint (5.14). To this
end we can use the method of Lagrange multipliers. Let us put

F(w,λ)=wCwT−λuwT,

whereλis a Lagrange multiplier. Equating to zero the partial derivatives ofF
with respect to the weightswiwe obtain 2wC−λu=0,thatis,

w=λ

2

uC−^1 ,

which is a necessary condition for a minimum. Substituting this into con-
straint (5.14) we obtain

1=

λ

2

uC−^1 uT,

where we use the fact thatC−^1 is a symmetric matrix becauseCis. Solving
this forλand substituting the result into the expression forwwill give the
asserted formula.

Proposition 5.10 (Minimum Variance Line)

The portfolio with the smallest variance among attainable portfolios with ex-
pected returnμVhas weights

w=

∣∣

∣∣^1 uC

− (^1) mT
μV mC−^1 mT

∣∣

∣∣uC−^1 +

∣∣

∣∣ uC

− (^1) uT 1
mC−^1 uT μV

∣∣

∣∣mC−^1

∣∣

∣∣ uC

− (^1) uT uC− (^1) mT
mC−^1 uT mC−^1 mT

∣∣

∣∣

,