Consider a portfolio selection problem with a slight modification. You are now required to take into account the current position and transaction costs associated with buying and selling assets. The objective is to find the minimum variance portfolio. In order to understand the scenario better, consider the following data.

You are given three assets. The current holding of the three assets is denoted by the vector = [200, 300, 500], the amount of asset bought and sold is denoted by and , respectively, and the net investment in each asset is denoted by and is defined by the following relation:

Suppose you pay a transaction fee of 0.01 every time you buy or sell. Let the covariance matrix be defined as

Assume that you hope to obtain at least 12% growth. Let = [1.109048, 1.169048, 1.074286] be the vector of expected return on the three assets, and let =1000 be the available funds. Mathematically, this problem can be written in the following manner:

The QPS-format input data set is as follows:

data potrdata; input field1 $ field2 $ field3 $ field4 field5 $ field6 @; datalines; NAME . POTRAN . . . ROWS . . . . . N OBJ.FUNC . . . . G RETURN . . . . E BUDGET . . . . E BALANC1 . . . . E BALANC2 . . . . E BALANC3 . . . . COLUMNS . . . . . . X1 RETURN 1.109048 BUDGET 1.0 . X1 BALANC1 1.0 . . . X2 RETURN 1.169048 BUDGET 1.0 . X2 BALANC2 1.0 . . . X3 RETURN 1.074286 BUDGET 1.0 . X3 BALANC3 1.0 . . . B1 BUDGET .01 BALANC1 -1.0 . B2 BUDGET .01 BALANC2 -1.0 . B3 BUDGET .01 BALANC3 -1.0 . S1 BUDGET .01 BALANC1 1.0 . S2 BUDGET .01 BALANC2 1.0 . S3 BUDGET .01 BALANC3 1.0 RHS . . . . . . RHS RETURN 1120 . . . RHS BUDGET 1000 . . . RHS BALANC1 200 . . . RHS BALANC2 300 . . . RHS BALANC3 500 . . RANGES . . . . . BOUNDS . . . . . QUADOBJ . . . . . . X1 X1 0.054978 . . . X1 X2 -.01748 . . . X1 X3 -.0003 . . . X2 X2 0.218898 . . . X2 X3 -.00024 . . . X3 X3 0.001532 . . ENDATA . . . . . ;

Use the following SAS statements to solve the problem:

proc optqp data=potrdata primalout = potrpout printlevel = 0 dualout = potrdout; run;

The optimal solution is displayed in Output 12.3.1.

Output 12.3.1: Portfolio Selection with Transactions

The OPTQP Procedure |

Primal Solution |

Obs | Objective Function ID | RHS ID | Variable Name |
Variable Type |
Linear Objective Coefficient |
Lower Bound |
Upper Bound | Variable Value | Variable Status |
---|---|---|---|---|---|---|---|---|---|

1 | OBJ.FUNC | RHS | X1 | N | 0 | 0 | 1.7977E308 | 397.584 | O |

2 | OBJ.FUNC | RHS | X2 | N | 0 | 0 | 1.7977E308 | 406.115 | O |

3 | OBJ.FUNC | RHS | X3 | N | 0 | 0 | 1.7977E308 | 190.165 | O |

4 | OBJ.FUNC | RHS | B1 | N | 0 | 0 | 1.7977E308 | 197.584 | O |

5 | OBJ.FUNC | RHS | B2 | N | 0 | 0 | 1.7977E308 | 106.115 | O |

6 | OBJ.FUNC | RHS | B3 | N | 0 | 0 | 1.7977E308 | 0.000 | O |

7 | OBJ.FUNC | RHS | S1 | N | 0 | 0 | 1.7977E308 | 0.000 | O |

8 | OBJ.FUNC | RHS | S2 | N | 0 | 0 | 1.7977E308 | 0.000 | O |

9 | OBJ.FUNC | RHS | S3 | N | 0 | 0 | 1.7977E308 | 309.835 | O |