110 Mathematics for Finance
provided that the determinant in the denominator is non-zero. The weights
depend linearly onμV.
Proof
Here we need to find the minimum of (5.16) subject to two constraints (5.14)
and (5.15). We take
G(w,λ,μ)=wCwT−λuwT−μmwT,whereλandμare Lagrange multipliers. The partial derivatives ofGwith
respect to the weightswiequated to zero give a necessary condition for a
minimum, 2wC−λu−μm= 0, which implies that
w=λ
2uC−^1 +μ
2mC−^1.Substituting this into the constraints (5.14) and (5.15), we obtain a system of
linear equations
1=λ
2 uC− (^1) uT+μ
2 uC
− (^1) mT,
μV=
λ
2
mC−^1 uT+
μ
2
mC−^1 mT,
to be solved forλandμ. The asserted formula follows by substituting the
solution into the expression forw.
Example 5.10
(3 securities) Consider three securities with expected returns, standard devia-
tions of returns and correlations between returns
μ 1 =0. 10 ,σ 1 =0. 28 ,ρ 12 =ρ 21 =− 0. 10 ,
μ 2 =0. 15 ,σ 2 =0. 24 ,ρ 23 =ρ 32 =0. 20 ,
μ 3 =0. 20 ,σ 3 =0. 25 ,ρ 31 =ρ 13 =0. 25.We arrange theμi’s into a one-row matrixmand 1’s into a one-row matrixu,
m=[
0 .10 0.15 0. 20
]
, u=[
111
]
.
Next we compute the entriescij=ρijσiσjof the covariance matrixC,and
find the inverse matrix toC,