1. 1 Multi-Objective Portfolio Optimization Jeremy Eckhause AMSC 698S Professor S. Gabriel 6 December 2004

2. 2 Portfolio Optimization: Overview Objective: “I want to maximize a quantity that measures historical portfolio performance subject to constraints such as: 1. Not too much concentration in any one sector, industry, or individual stock. 2. Reasonable capital usage 3. Low correlation with market indices, Etc. The quantities that change will be the weights of the portfolio constituents.”

3. 3 Portfolio Optimization: Overview Constraints: Not too much concentration in any one sector, industry, or individual stock Reasonable capital usage Low correlation with market indices (Change to covariance to get convex constraints. This constraint will be added, but not in time for the results of this presentation. These constraints are not difficult to add, despite being nonlinear, as they are convex.) (Linear Constraints)

4. 4 Portfolio Optimization: Overview Multi-Objective Optimization: “I want to maximize a quantity that measures historical portfolio performance subject to constraints.” “Quantity” is the Sharpe Ratio. Sharpe Ratio = mean / standard deviation

5. 5 Portfolio Optimization: Overview Multi-Objective Optimization: “I want to maximize a quantity that measures historical portfolio performance subject to constraints.” “Quantity: is the Sharpe Ratio. Sharpe Ratio = mean / standard deviation Maximize return while minimizing risk. Using Markowitz’s method, we can generate a Pareto curve!

6. 6 Portfolio Optimization: Definitions Return: The monthly historic returns over the past five years for each possible investment. In reality, the optimization uses historic data to show the best portfolio over the past five years. The hope is that these trends will continue in the future.

7. 7 Portfolio Optimization: Notation Data: return of investment i in month j “capital usage” of investment i (inverse of standard deviation over five year returns) maximum capital usage allowed if investment i is in in sector k threshold level for sector k “directional bias” within some range no. of months vector of 1 (long) and –1 (short) for each investment

8. 8 Portfolio Optimization: Formulation s.t. min max Make second objective function a constraint! (1)

9. 9 s.t. min mean must be above some return (2) Portfolio Optimization: Approach

10. 10 We know that from the constraint method that: If is a binding constraint, then the optimal solution to (2) is along the efficient Pareto frontier for (1). What are those values for where the constraint is binding? Objective function changes for 3.23 < l < 6.702 Portfolio Optimization: Approach

11. 11 Portfolio Optimization: Model Optimization model built in LINGO Change value for lower bound return Quickly calculates the global optimum for even large- scale problems Output to Excel to generate graphs of Pareto optimal values

12. 12 Portfolio Optimization: Model LINGO Code !Objective function; MIN = SHARPEVAR; !Min variance; SHARPEVAR = (@SUM(PD(J): ((Y(J) - MEANY)^2))/N); !Constraints; @FOR(PD(I): @SUM(ST(J): M5(I,J)*X(J) ) = Y(I)); !Assigns y_i to the sum of each row; MEANY = (@SUM(PD(I): Y(I))/N); !Mean of y is the average return; MEANY >= LB; !Return above some lower bound; @SUM(ST(I): CU(I)*XX(I)) <= MAXCU; !Below some maximum capital usage; CU(1)*X(1) + CU(4)*X(4) <= THR(1); !Threshold limits for each sector; CU(2)*X(2) <= THR(2); CU(3)*X(3) <= THR(3); @SUM(ST(I): CU(I)*X(I)) <= DBMAX; @SUM(ST(I): CU(I)*X(I)) >= DBMIN; ! Within some directional bias range; Etc.

13. 13 Portfolio Optimization: Data

14. 14 Portfolio Optimization: Results

15. 15 Portfolio Optimization: Conclusions Multi-objective optimization tool for maximizing return while minimizing variance works well. Relatively easy to minimize variance and find the “knee” of the Pareto frontier when changing the minimum return. Allows the decision maker on Wall Street to pick his or her risk tolerance vs. expected payoff.

16. 16 Portfolio Optimization: Future Work Add additional constraints, such as considering some minimum on recent return. Some may require using integer variables, although this is expected to be a small number. Perform analysis on actual data (in progress). will be in the 100 variable range. Matrix should be about 100x100. Additional constraints may add another 100-500 variables. All easily handled by a LINGO or similar solver. Enhance model to generate Pareto frontier efficiently, automatically providing each strategy (i.e., ) during the output phase.