Travis Wainman partner1 partner2 Modern Portfolio Optimization: A Practical Approach Using an Excel Solver Single-Index Model Travis Wainman partner1 partner2

Modern Portfolio Theory (MPT) Based on the idea that combinations of assets have the potential to provide better returns with less risk than individual assets. Using Markowitz’s efficient portfolio and Sharpe’s single-index model. This article uses TIAA-CREF mutual fund data as an example to illustrate the MPT concept.

Data Singapore equity historical prices Jan 2003 – Nov 2009 Included only the securities that survived throughout the entire period

Sharpe Ratio We compute the objective function by maximizing the Sharpe Ratio, the portfolio with minimum risk and maximum return. We define the Sharpe Ratio in Equation (1) as: It measures the excess return per one unit of risk. We also use the Sharpe Ratio to calculate the tangent portfolio. (1)

The sum of all weighted coefficients to be equal to one Constraints There are 2 main constraints which are: and The sum of all weighted coefficients to be equal to one The weight for each individual asset has to be greater than or equal to zero because short-selling of securities is not allowed.

Calculation Step1: Compute percentage of monthly returns for the TSI Index and for each security Step2: Compute security average returns, variances, covariances (between the security and index), and betas. Compute average as Compute Betas as the covariance of security i’s returns with the returns of a market index divided by the population variance of the market index.

Calculation (con’t) Compute the market variance as Compute the covariances of the individual securities as

Calculation (con’t) Step3: Use the CAPM methodology to compute excess returns, identify individual fund valuations and retain those that are undervalued. Compute the CAPM as Compute excess returns, Determine individual fund valuations as either under, over, or fairly valued using the CAPM methodology. Thus, if the excess return is greater than zero, the fund is selected for inclusion in the optimal portfolio mix. Step4: Compute the individual mutual fund alphas and residual variances. Compute alphas as Compute residual variances as

Calculation (con’t) Step5: Set up the Microsoft Excel Solver by: Establishing the objective function to maximize the modified Sharpe Ratio as presented in the equation. Equally weighting the starting values for Xi and establishing constraints. Step6: The solver was then loaded with the following inputs: “Set Target Cell:” Input the Tangent or Sharpe Ratio in the cell. “Equal to: ‘Max’”. “By Changing Cells” Input Cell Range “Subject to the Constraints:” Set a cell of total weight equal to 1, which is the sum of individual weights, which satisfied the first constraint in the model. Set each cell of individual weight >=0, which satisfied the second constraint in the model.

Calculation (con’t) Step7: Compute the porfolio return, beta, and standard deviations. Step8: The TIAA-CREF portfolio was optimized by maximizing Sharpe Ratio using the computed weights of the undervalued TIAA-CREF funds.

Results The proportional investment percentages of the selected undervalued funds in the tangent portfolio suggest that the TIAA-CREF investor should purchase 87.76% of the TIAA Real estate fund, and 12.24% of CREF Bond fund, and zero invest in CREF Inflation-Linked Bond fund. Even though the CREF Social Choice, Stock, Global, and Equity funds were undervalued according to the CAPM results, but the funds did not optimize due to minimal residual variances. However, the past performance is not necessarily indicative of future performances. Therefore, investors should re-optimize their portfolios quarterly, at a minimum.