Consider a portfolio optimization example. The two competing goals of investment are (1) long-term growth of capital and (2) low risk. A good portfolio grows steadily without wild fluctuations in value. The Markowitz model is an optimization model for balancing the return and risk of a portfolio. The decision variables are the amounts invested in each asset. The objective is to minimize the variance of the portfolio’s total return, subject to the constraints that (1) the expected growth of the portfolio reaches at least some target level and (2) you do not invest more capital than you have.

Let be the amount invested in each asset, be the amount of capital you have, be the random vector of asset returns over some period, and be the expected value of . Let G be the minimum growth you hope to obtain, and be the covariance matrix of . The objective function is , which can be equivalently denoted as .

Assume, for example, n = 4. Let = 10,000, G = 1000, , and

The QP formulation can be written as follows:

The corresponding QPS-format input data set is as follows:

data portdata; input field1 $ field2 $ field3 $ field4 field5 $ field6 @; datalines; NAME . PORT . . . ROWS . . . . . N OBJ.FUNC . . . . L BUDGET . . . . G GROWTH . . . . COLUMNS . . . . . . X1 BUDGET 1.0 GROWTH 0.05 . X2 BUDGET 1.0 GROWTH -.20 . X3 BUDGET 1.0 GROWTH 0.15 . X4 BUDGET 1.0 GROWTH 0.30 RHS . . . . . . RHS BUDGET 10000 . . . RHS GROWTH 1000 . . RANGES . . . . . BOUNDS . . . . . QUADOBJ . . . . . . X1 X1 0.16 . . . X1 X2 -.10 . . . X1 X3 -.10 . . . X1 X4 -.10 . . . X2 X2 0.32 . . . X2 X3 -.04 . . . X2 X4 -.04 . . . X3 X3 0.70 . . . X3 X4 0.12 . . . X4 X4 0.70 . . ENDATA . . . . . ;

Use the following SAS statements to solve the problem:

proc optqp data=portdata primalout = portpout printlevel = 0 dualout = portdout; run;

The optimal solution is shown in Output 12.2.1.

Output 12.2.1: Portfolio Optimization

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 | 3452.86 | O |

2 | OBJ.FUNC | RHS | X2 | N | 0 | 0 | 1.7977E308 | 0.00 | O |

3 | OBJ.FUNC | RHS | X3 | N | 0 | 0 | 1.7977E308 | 1068.81 | O |

4 | OBJ.FUNC | RHS | X4 | N | 0 | 0 | 1.7977E308 | 2223.45 | O |

Thus, the minimum variance portfolio that earns an expected return of at least 10% is , , , . Asset 2 gets nothing, because its expected return is 20% and its covariance with the other assets is not sufficiently negative for it to bring any diversification benefits. What if you drop the nonnegativity assumption? You need to update the BOUNDS section in the existing QPS-format data set to indicate that the decision variables are free.

... RANGES . . . . . BOUNDS . . . . . FR BND1 X1 . . . FR BND1 X2 . . . FR BND1 X3 . . . FR BND1 X4 . . . QUADOBJ . . . . . ...

Financially, that means you are allowed to short-sell—that is, sell low-mean-return assets and use the proceeds to invest in high-mean-return assets. In other words, you put a negative portfolio weight in low-mean assets and “more than 100%” in high-mean assets. You can see in the optimal solution displayed in Output 12.2.2 that the decision variable , denoting Asset 2, is equal to 1563.61, which means short sale of that asset.

Output 12.2.2: Portfolio Optimization with Short-Sale Option

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 | F | 0 | -1.7977E308 | 1.7977E308 | 1684.35 | O |

2 | OBJ.FUNC | RHS | X2 | F | 0 | -1.7977E308 | 1.7977E308 | -1563.61 | O |

3 | OBJ.FUNC | RHS | X3 | F | 0 | -1.7977E308 | 1.7977E308 | 682.51 | O |

4 | OBJ.FUNC | RHS | X4 | F | 0 | -1.7977E308 | 1.7977E308 | 1668.95 | O |