Portfolio Optimization:(Click on the icon at left to go to the demo)
Every investor knows that there is a tradeoff between risk and reward: To obtain greater expected returns on investments, one must be willing to take on greater risk.
In solving the Portfolio Selection problem, we aim to use quantitative measures of risk and reward to obtain a balance between these two factors that suits the individual investor. No one combination of securites is optimal for all investors. The best portfolio for any one investor depends on their own tolerance for risk.
Each investment instrument has its own expected monthly return, and its own propensity for these returns to fluctuate from month to month. However, the returns from different instruments are not in general independent. In some cases they tend to move "in sync;" for instance the stocks of gold mining companies tend to follow the price of gold. In other cases they tend to move in opposite directions from each other. These joint tendencies are quantified by covariances.
In this case study, we show how the portfolio selection problem is formulated as a convex quadratic programming problem. This problem depends on a parameter specified by the user that quantifies their risk tolerance. Often, the portfolio generated by this process has an expected return that is close to the best expected return from any individual security, but at a much lower degree of risk.
A detailed introduction is available for the reader not familiar with the problem.
A detailed explanation of the mathematical formulation of the portfolio selection problem is also provided.
Our demonstration program takes 30 large industrial stocks, and calculates their expected returns, variances, and covariances by examining their historical performance over the six-year period from 1/1/86 to 12/31/91.
In using the demo, the user selects a subset of the 30 stocks that they are willing to include in their portfolio. They input the current annual interest rate on U.S. Treasury Bills, which are considered to be a risk-free investment and which often have a place in the portfolio of conservative investors. Finally, they input a parameter that quantifies their own risk tolerance. (Of course, they can experiment with different values of this parameter.)
The demo program calculates the optimal mix of securities for the given inputs, and returns this information, along with the expected return and variance of the optimal portfolio.
DISCLAIMER: Remember that these results are for educational purposes only. The return/covariance data is based purely on historical considerations; actual working programs take numerous other factors into account.
We're always keen to hear your feedback.
The historical stock returns which include dividends was obtained from the CRISPA database of the University of Chicago. The pie charts are created using routines from the gd 1.2 library. This case study was written by Joe Czyzyk, Rob Stubbs, Tim Wisniewski, and Steve Wright.
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