The Geometry of Mean Variance Portfolios 275
The covariances among the market systems, with NIC included, are as fol-
lows:
TI LSNT.1 −.0237 .01 0 0
I −.0237 .25 .079 0 0
L .01 .079 .4 0 0
S0 0 0 00
N0 0 0 00Thus, when we include NIC, we are now dealing with five market sys-
tems; therefore, the generalized form of the starting augmented matrix is:
X 1 U 1 +X 2 U 2 +X 3 U 3 +X 4 U 4 X 5 *U 5 =E
X 1 +X 2 +X 3 +X 4 X 5 =S
X (^1) COV1, 1+X (^2) COV1, 2+X (^3) COV1, 3+X (^4) COV1, 4+X 5
COV1, 5+.^5 L^1 U^1 +.^5 L^2 =^0
X (^1) COV2, 1+X (^2) COV2, 2+X (^3) COV2, 3+X (^4) COV2, 4+X 5
COV2, 5+.^5 L^1 U^2 +.^5 L^2 =^0
X 1 COV3, 1+X 2 COV3, 2+X 3 COV3, 3+X 4 COV3, 4+X 5
COV3, 5+.^5 L 1 U 3 +.^5 L 2 =^0
X 1 COV4, 1+X 2 COV4, 2+X 3 COV4, 3+X 4 COV4, 4+X 5
COV4, 5+.^5 L 1 U 4 +.^5 L 2 =^0
X 1 COV5, 1+X 2 COV5, 2+X 3 COV5, 3+X 4 COV5, 4+X 5
COV5, 5+.^5 L 1 U 5 +.^5 L 2 =^0
where: E=The expected return of the portfolio.
S=The sum of the weights constraint.
COVA, B=The covariance between securities A and B.
Xi=The percentage weighting of the ith security.
Ui=The expected return of the ith security.
L 1 =The first Lagrangian multiplier.
L 2 =The second Lagrangian multiplier.