程序代写代做代考 AI finance Assume the following single period model

Assume the following single period model

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

2
w

T
 w

subject to w
T
  w0 0  P , u

T
w  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

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

 0 when wi  ai

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

Consider the following procedures in your implementation :

(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)  uR
T R

1 , wR   (R
1R  0 R

1uR)  R
1 ,

 

P

   
0

(1  )  
0
uR
T
R
1
  

T
R
1

R

(CR0
2  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)