City University of Hong Kong Department of Economics and Finance
Course EF5213 Assignment #3 ( due March 19, 2023 ) 1. Consider the MVO problem that determines the optimal portfolio content w and w0 by minimizing the
portfolio risk as
minimize 1 wT w 2
subject to wT w0 0 p , uTw w0 1 , and a1 w1 b1 , … , an wn bn
given portfolio mean return p, riskfree rate 0, asset mean returns , and their variance-covariance . There are buy and sell limits in the optimization according to the given positive quantities {a1, … , an} and {b1, … , bn}. It should be noted that the optimal portfolio content can be determined through the Kuhn-Tucker conditions as
MVO problem.
Consider the following procedures in your implementation:
L (w10 1 u)i 0 when ai wi bi
Modify the Markowitz algorithm in the lecture and develop a VBA implementation for the current
0 when wi ai
0 when wi bi , for i 0, 1, … , n
(1) Define an OUT subset , and separate into two disjoint subsets A and B. You can use the given subroutine GetSeparation() for these purposes. Consider the MVO problem with wi bi for i B, and wi ai for i A. The optimal solution of this MVO problem is given by
w 1 (1m 0 1um) 1h , where hi { bi , i B , and i jA ij aj + jB ij ( b )
mmmi,i j w0 1uTw
uT1hT1h 1p00m m
Cm20 2Am0 Bm
Here, {m, m, um} refer to the modified versions of {, , u} according to the assets in the OUT
(2) Check that all the entries of w satisfy both the buy and sell limits. If so, proceed to step (3). If this is
not the case, return to step (1) and try another separation of or another OUT subset.
(3) Check that KKT conditions have been satisfied. If so, w0 and w defined in (1) will be an optimal solution. Otherwise, return to step (1) and try another separation of or another OUT subset.
(80 points)
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