vba代写 Course EF5213 Assignment #3

City University of Hong Kong Department of Economics and Finance

Course EF5213 Assignment #3 ( due March 18, 2018 ) 1. Consider the MVO problem that determines the optimal portfolio content w and w0 by minimizing the

portfolio risk subject to an expected return P as minimize 1 wT w

2
subject to wT  w0 0  P , uTw  w0  1 , and a1  w1  b1 , … , an  wn  bn ,

given 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

L (w0 u)i 0 when ai wi bi

wi
Consider the following procedures in your implementation :

 0 when wi  ai
 0 when wi  bi , for i  0, 1, … , n

(1) Define an OUT subset , and separate  into two disjoint subsets L and U.
Consider the MVO problem with wi  bi for i  L, and wi  ai for i  U. The optimal solution of this MVO problem is given by

w0 1(AR 0 CR)uTR R1, wR (R1R 0 R1uR)R1,   (1) uT1T1

P020RRRR (CR0  2AR0  BR)

where i  jL ij (bj)  jU ij (aj) ,   iL (bi)i  iU (ai)i ,   iLui (bi)  iU ui (ai)

Here, {R , R , uR , wR} refer to the reduced forms of { ,  , u , w} by ignoring the rows and columns corresponding to those assets in the OUT subset. The terms { ,  ,} are defined according to the contents in L and U using { ,  , u}.

(2) Check that all the entries of wR 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 given portfolio return P. Otherwise, return to step (1) and try another separation of  or another OUT subset.

(80 points)