108 C.S. Cheung et al. Finance Research Letters 6 2009 106–113
be identical to the frontier based on the N
+
K assets. For all values of the required expected return q, the corresponding values of
σ
2 p
for the two cases must be the same. Thus, in case of spanning, the 2
×
2 matrix M
′
V
− 1
M must remain the same, regardless of whether it is based on the K assets or the
N
+
K assets. As shown in the following, this analytical property ensures that no investment funds be allocated to the N assets and that the allocations among the K assets in an
N
+
K -asset portfolio
be the same as those based on the K assets alone. In order to establish the conditions of spanning for the case where the covariance matrix based on
the expanded set of assets is invertible, we partition the expanded covariance matrix as follows:
V
=
V
N N
V
N K
V
K N
V
K K
,
4
where V
N N
is N
×
N, V
N K
is N
×
K , V
K N
is K
×
N, and V
K K
is K
×
K . We also partition the expanded expected return vector
µ
into column vectors
µ
N
and
µ
K
with N and K elements, respectively. Likewise, the corresponding
N
+
K -element vector
ι
is partitioned into column vectors
ι
N
and
ι
K
. Following the algebra of block matrix inversion, we have
V
− 1
=
E
−
EV
N K
V
− 1
K K
−
V
− 1
K K
V
K N
E V
− 1
K K
+
V
− 1
K K
V
K N
EV
N K
V
− 1
K K
,
5 where
E
=
V
N N
−
V
N K
V
− 1
K K
V
K N −
1
.
6
Partitioning V
− 1
in this manner allows M
′
V
− 1
M to be directly compared with the corresponding 2
×
2 matrix for the K benchmark assets alone. As spanning requires equality of the two cases, it implies
that
µ
′ N
−
µ
′ K
V
− 1
K K
V
K N
E
µ
N
−
V
N K
V
− 1
K K
µ
K
= ,
7
µ
′ N
−
µ
′ K
V
− 1
K K
V
K N
E
ι
N
−
V
N K
V
− 1
K K
ι
K
= ,
and 8
ι
′ N
−
ι
′ K
V
− 1
K K
V
K N
E
ι
N
−
V
N K
V
− 1
K K
ι
K
= .
9 Notice that the left-hand side of each of these three equations is of the form where the square matrix
E is pre-multiplied by an N-element row vector and post-multiplied by an N-element column vector. In the first case as well as the third case, the two vectors are transposes of each other.
Recall that the covariance matrix V for the N
+
K -asset case is invertible.
2
This means that V is positive definite. So is its inverse, V
− 1
. All principal submatrices of a positive definite matrix are
positive definite. As E is a principal submatrix of V
− 1
, E is positive definite as well. For any nonzero column vector u with N elements, we must have u
′
Eu 0. If u
′
Eu
=
0, then the vector u must be a vector of zeros. Thus, as E is positive definite, Eq.
