The Markowitz Mean Variance Optimization Model Finance Essay - Free Essay Example

Abstract This project examined the optimal allocation of stocks based on the Markowitz Mean Variance Optimization Model. It is mainly based on the comparison between two samples of stock returns. The first sample is referred to as the full data set, and contains the returns of the stocks GE, Baxter, Dow, Caterpillar, Apple, and Procter and Gamble calculated over the period of five years. The second sample takes the 250 most recent daily returns as the base data set. The objective is to find the optimal allocation in a portfolio of risky assets only and the Complete Portfolio that an investor could choose in order to maximize his utility. Before analyzing the results from the optimization process, we started by analyzing the data, and verifying that the assumption of normality the model is valid in both data sets. Then we describe the results given by the different optimizations that we calculated and we compared both samples showing the optimal allocation for risky assets only and the optimal allocation that includes the risk free asset. Then, we graph the efficient frontier along with the capital allocation line, and we make some comments about the results obtained. Finally, we formulate the recommendation to the investor based on the results obtained, and we point out some of the limitations of the model that could explain some of the unrealistic results that we fo und. The methodology of the calculations, the tables and the charts used are all referenced in the appendix of this paper in order to illustrate the results obtained. Introduction This project is based on the Markowitz Mean Variance Optimization Model for defining the optimal weights of assets in a given portfolio based on various investment constraints. The model generally seeks to maximize return for a given level of risk, or minimize risk for a given level of return. Markowitz formulated the portfolio construction problem as a utility maximization problem and used this to develop a framework for selecting a range of optimal portfolios. In his model, Markowitz made several assumptions on which this project is based. These assumptions designed the portfolio selection problem to a mean-variance portfolio optimization problem and are as follows: All investors have a single holding period during which they will maximize their utility function. Investors dont incur any transaction costs or taxes while trading the securities. Investors have a quadratic utility function that they should maximize based on the expected return, variance, and risk aversion. The returns used should be normally distributed. Investors are assumed to be risk averse where they prefer to maximize the returns given a minimum level of risk All the investors share the same economic view of the world, and they analyze the securities in the same way. I/ Analyzing Stock Returns and Standard Deviations According to Table 1 and 2 in Appendix 2, the results show that the expected returns in the full data set obtained were generally lower than the expected returns in the sample of the 250 most recent returns. (Please refer to Appendix 1 for the methodology of calculations.) The major explanation is that the full data set covers the 2007/08 financial crisis in which the U.S stock market has dropped down by more than 50%, which had an implication on the mean expected returns calculated over the period from 2006 until 2011, and shows relatively lower returns. In comparison, the 250 most recent returns sample covers a period of one year starting from the 2nd of March, 2010 until the 31st of January, 2011. During this period, the stock market has recovered from the historic lows of March 2009, and is still facing a market rally that has been consistent until now, which affects the performance of individual stocks at the exception of Baxter, and Procter Gamble, which underperformed due t o some fundamental issues that are related to their business sectors. Regarding standard deviation, we can see that the stocks in the full data set present a higher standard deviation than in the second data set. This shows that during the period from 2006 to 2011, there has been a high volatility in the market, which is illustrated by the stocks demonstrating some extreme returns. This is mainly due the shape of the recovery after the financial crisis, which displayed some significant drops in the share prices and recovering back at a rapid rate of growth, after March 2009. However, the volatility has stabilized during the last year which means that stocks are becoming less risky in comparison with the last three years. II/ Analyzing the Data Sets 1) Testing for normality Before going through the process of selecting the optimal portfolios, we first examine the returns computed and see if they match the assumptions of normality that have been stated earlier for the model to be valid. In order to do so, the Jarque-Bera test for normality is chosen, and provides a formal method in which Skewness and Kurtosis are used to analyze the distribution of the returns. (Please refer to the appendix for the description of the Jarque and Bera test for normality) Analysis of the results: The data in tables 3 and 4 from Appendix 2 show different results for the two sample sizes. In the Full data set, the JB test statistics computed are all higher than the 9.21 critical value with 99% confidence level, which means that the distribution of the returns is not normal as we reject the null hypothesis. This is mainly explained by the fact that the kurtosis observed in all the stocks returns is positive and is higher than 3, which shows that the distributions exhibit fat tails. Moreover, Skewness is different from zero for all the distributions showing that they are not perfectly symmetrical, which rejects another condition for the normality of returns. If the returns display positive Skewness, this means that there is a high probability that the stocks involved will show positive returns in the future. By comparison in the second data set, we notice that the distributions of returns of the Stocks GE, Baxter, Procter Gamble, and Apple can all be assumed normal, as the JB statistic is lower than 9.21. One similarity between the stocks that dont have a normal distribution of returns in both data sets is that they display a leptokurtic distribution which means that the distribution is more focused around the center and in the tails than the normal distribution, and this is due to the excess kurtosis. 2) Testing for evidence of Volatility Clustering Volatility clustering is mainly characterized by the historical data of stock returns showing periods of high volatilities giving some extreme returns, which are then followed by periods of relatively low volatilities. Analysis of the results: According to the volatility charts in Appendix 2, we notice that all the stocks in the complete data set showed evidence of volatility clustering with a certain similarity in the clusters observed. In fact, most of the volatility clusters observed can be located between the periods of late 2007 to the mid 2009. This is explained by the high volatility of returns during the period of the financial crisis, where most stocks showed some extreme variations in the returns due to the investors trying to digest the panic that was going on in the market, and then they reacted accordingly by liquidating their positions. This pattern lasted until the mid of 2009 until some of the confidence was restored and investors came back into the market again. In comparison with the sample of the 250 most recent returns and according to the graphs, we cannot detect any significant evidence of volatility clustering for the stocks of Caterpillar and DOW, but we can clearly observe some evidence regarding Apple, PG, GE, and Baxter as the cluster can be clearly identifiable. This is mainly interpreted by the fact that since the information arrives in cluster, investors tend to take more or less time to react relative to the kinds of information that they receive with respect to the stocks. III/ The Portfolio Selection Process The Main objective of the portfolio optimization process is to find the optimal allocation of assets in the two data sets given two kinds of portfolios: one containing risky assets only, and another one that would include a risk free asset. For this, we used the Excel Solver to run three types of optimizations that result in three types of portfolios: the Minimum Variance Portfolio, the Optimal Risky Portfolio, and the Optimal Complete Portfolio. 1) Analyzing the results given by the Minimum Variance Portfolios In the Markowitz model, we have assumed that investors seek to maximize the return at a given level of risk, or minimize the risk at a given level of return. The minimum variance portfolio is the one that gives the optimal allocation of risky assets by minimizing the general risk of the portfolio. In order to do so, we used the Excel Solver and we set the portfolio variance as the target to minimize given both the constrained and unconstrained optimization criteria. The formula for the portfolio variance is given in the Appendix and is derived from the correlation matrixes that were computed for both data sets. The general parameters used in the Excel Solver are also explained for both the constrained and the unconstrained portfolios. Analysis of the results: According to table 5 and 6 in Appendix 2, the minimum variance portfolio for the full data gives two results based on the constrained optimization and the unconstrained optimization. In the Long Only Portfolio, the results suggest that the investor should invest in three stocks which are Baxter, Procter Gamble and, Apple with the respective weights of 32.87%, 62.14% and 4.99%. This gives a portfolio expected return of 6.60% with a standard deviation of 18.63%. In the Unconstrained Portfolio, the results suggest that the investor should invest in all the stocks by going long on Baxter (33.66%,) Caterpillar (3.90%,) Procter Gamble (65.22%,) and Apple (6.06%,) and shorting GE (-1.08%) and DOW (-7.75%.) The portfolio expected return in this scenario is 7.48% and the standard deviation is 18.43%. In the second data set, for the constrained portfolio, the results suggest that the investor should invest only in stocks Baxter, Procter Gamble, and Apple with the respective proportions of 11.90%, 84%, and 4.10%. The expected return obtained is 2.06%, for a standard deviation of 13.15%. In the unconstrained portfolio, the results suggest that the investor should go long GE (2.30%,) Baxter (12.51%,) Caterpillar (2.96%,) Procter Gamble (88.77%,) and Apple (9.44%,) and short Dow (-15.97%.) The expected return obtained is 2.42% for a standard deviation of 12.51%. Comparing the results obtained in the full data set and the 250 most recent return sample, we can notice that there are differences in the results we get. This difference is mainly explained by the change in the expected returns and standard deviations of the stocks in both data sets, and we can notice that this change has an impact on the performance of the portfolios expected return as well as the allocation of the assets in the portfolio. The best portfolio is chosen by using comparing the Sharpe ratios, which gives a risk adjustment measure that compares the performance of individual portfolios. From table X and Y in the appendix, we can say that best portfolio from the two samples is the Unrestricted Portfolio in the full data set since it gives a higher Sharpe ratio of 0.2432 relative to the other portfolios. 2) The Optimal Risky Portfolio The optimal risky portfolio is the one that maximizes the Sharpe Ratio and allows identifying the optimal weights of the risky assets in the portfolio. The formula for the Sharpe ratio is given in the appendix in the Methodology section. The ratio seeks to know how much additional return an investor would receive for the additional risk of holding the risky assets over a risk-free asset. The higher is this ratio the better is the performance of the portfolio with respect to risk and return. Analysis of the results: According to the table 7 and 8 in Appendix 2, for the full data set, the Long only portfolio suggests that the investor should be fully invested in Apple stock, which gives an expected return that is equivalent to the stocks expected return of 37.97% and a standard deviation also equivalent of 40.28%. By comparison, the unconstrained portfolio suggests that the investors should go long the stocks Baxter (48.77 %,) Caterpillar (91.02%,) and Apple (211.44%,) and short the stocks GE (-161.16%,) Dow (32.51,) and Procter and Gamble (-57.55%.) This gives a portfolio expected return of 102.24% and a standard deviation of 86.71%. In the second data set, the results suggest that the investor should invest in stocks Caterpillar (43.63%,) and Apple (56.37%.) this gives a portfolio expected return of 60.33% and a standard deviation of 24.65%. In the unconstrained scenario, the results suggest that the investor should go long the stocks Caterpillar (100.00%,) Procter Gamble (3.88%,) and App le (100.00%,) and short the stocks GE (-18.38%,) Dow (-41.73%,) and Baxter (-43.77%.) the expected return in this scenario is 110.34% and the standard deviation is 36.57%. By comparing the results of the two data sets, we notice that both optimizations give some unrealistic results in terms of the expected returns and variances of the portfolios, but also in terms of the proportions to be invested in the stocks. In fact, in the full data set, the Long-Only portfolio suggests that the investor should be invested in only one stock which eliminates any benefits from diversification and is a very risky strategy to pursue for long term asset allocation in general. The unconstrained portfolio is also unrealistic in terms of the weights that are attributed to the portfolios as some of them reach 100% proportion to be invested in, which is not possible because it doesnt allow for proper asset diversification. The only portfolio that seems to be realistic is given in the second data set w here the expected return is of 60.33%, which seems quite attractive given the level of risk that is suggested of 24.65%, and that is what one could consider as a realistic result since the proportions to be invested are shared between two stocks as it is the minimum diversification of risky assets that one could consider given the constraint of maximizing the Sharpe ratio. 3) The Optimal Complete Portfolio The optimal complete portfolio is the portfolio that consists of the optimal risky assets and the risk free asset. The optimal complete portfolio is determined by maximizing the investors utility function which is defined in Appendix 1. The result of the optimization gives the proportions to be invested in the risky assets and the risk free assets, which allows the investors to be properly diversified based on three main components that are, expected return, volatility, and risk aversion. Analysis of the results: According to table 9 and 10 Appendix 2, for the full data set, the Long-Only portfolio suggests that the optimal complete portfolio is the one that provides an expected return of 33.13% and a standard deviation of 34.72%. The proportion to be invested in the risky assets is 86.16%, and the proportion to be invested in the risk free asset is 13.84%. In this scenario, the maximization of the investors utility function suggests that the investor should be diversified between Apple stock and the risk free asset. In the unconstrained scenario, the optimization results in a portfolio expected return of 55.39%, and a standard deviation of 45.78%. The investors portfolio in this case consists of 52.77% in risky assets and 47.23% in the risk free asset. For the positions in the risky assets, the investor should go long the stocks Baxter (48.76 %,) Caterpillar (91.06 %,) and Apple (211.53 %,) and short the stocks GE (-161.24 %,) Dow (-32.53 %,) and Procter and Gamble (-57.58 %.) The Lo ng-Only portfolio in the second data set suggests that the investor should invest 90% of the portfolio in risky asset, and 10% in the risk free asset. The proportions of the risky assets are as follows: 98.84% in Dow, and 1.16% in Caterpillar. In the unconstrained portfolio, the results suggest that the investor should invest 90% in risky assets, and 10% in the risk free asset. The distribution of the asset weights is as follows: the investor should go long the stocks DOW (100%), Caterpillar (100 %,) GE (100 %,) and short Procter Gamble (-64.13 %,) and Baxter (-100 %,) and Apple (-35.87%.) When comparing the two samples, we notice that the unconstrained portfolios in both data sets provide some unrealistic results in terms of the portfolio weights as some of them equal or exceed 100% of the portfolios total weight. In the second data sets when we eliminate some of the constraints that allow us to have more accurate data, the result obtained show very large values that are comple tely unrealistic and prevent us from drawing any meaningful conclusions. In the full data set, the Long-Only portfolio although it suggests that the investor should be mixed between risky assets and the risk free asset does not provide proper diversification as the portfolio of risky assets is concentrated on one stock which is Apple stock, however, it seems that it is the one that provides the most realistic result. 4) The Efficient Frontier and the Capital Allocation Line Drawing the efficient frontier is the last step that allows the investors to visualize the optimal portfolios computed previously and choose the best alternatives that are offered given the results that are produced, and whether they are realistic or not. The Appendix 2 provides 4 graphs for both data sets that show the efficient frontier given the constrained and the unconstrained optimized portfolios. The capital allocation line which is determined by maximizing the Sharpe ratio is represented on each of the graphs and shows the optimal risky portfolios as well as the optimal complete portfolios. The indifference curve drawn display the curve of equally preferred portfolios that will generate the same utility to the investor. In the second data set, it was not possible to draw the Capital Allocation Line as the values obtained were inaccurate and extremely large. Moreover, the points showing the Optimal Risky Portfolio and the Optimal Complete Portfolio are those that were comput ed given the constraints that we have set in Appendix 1, for the purpose of obtaining more accurate results. This shows one of the limitations of the Mean Variance optimization model as it fails to provide meaningful results when using a small number of observations in the sample that we used. 5) Recommendation to the Investor The results obtained from each type of portfolio are summarized in tables 11 and 12. For a matter of convenience, we labeled the portfolios from Portfolio 1 to 12. In the recommendation to the investor we suggest to look first at the portfolios that give some realistic results, and then we can compare which ones provide the best results by using the Sharpe ratio as a measure of performance. Based on the results obtained, the portfolios that have been chosen are Portfolio 1, 2, 4, 7 and 10. Portfolios 1 and 7 do not provide a satisfying performance to the investor as the returns are lower than the risk free asset, and the risks observed are too high. In this case the investor is better off investing all his money into the risk free asset at 3%. Portfolio 2 is based on the Optimal Risky Portfolio in the second data set, and gives the highest Sharpe ratio of 156.62% with the following proportions: Caterpillar (43.63 %,) and Apple (56.37 %.) In comparison, portfolios 4, and 10 offer lo wer returns, hence, lower Sharpe ratios, but they also allow for more diversification among the stocks, and they give a relatively lower risk compared to portfolio 2. From a fundamental perspective, we recommend that the investors should always stay diversified between risky and the risk free asset in order to protect themselves against any unexpected downturn in the markets. For this, we suggest that the investors should choose to invest 70 percent in Portfolio 2, and 30% in the risk free asset. This will lower, the standard deviation to 17.262% for an Expected return of 42.24%. This gives a pretty high expected performance in comparison with the SP500 performance of 2010 which was of 15.65%, and it allows the investors to be less exposed to risk compared with portfolios 4 and 10. Conclusion The projects main objective was to identify the optimal allocation of assets from a portfolio containing risky assets and one that includes a risk free asset. The allocations obtained were based on the Markowitz Mean Variance Optimizations, and resulted in three kinds of portfolio along with the efficient frontiers for both samples of data that we have used. In the analysis of the results, we have found some unrealistic values that we judge are due to the assumptions that we made earlier, which reflect some of the limitations of the Model. In fact the first limitation that we can state is the assumption of normality on the returns in both data sets which seems to be unrealistic given the distributions that we have obtained. There is strong evidence that most stock returns display asymmetrical returns as well as showing excess Kurtosis which makes the model used hardly applicable and is responsible for the large values obtained. Second, the model assumes that investors will hold their investment on a fixed time horizon, and will never change the asset allocation, which is false given that some times, investors need to rebalance their portfolios shifting from the risky assets to treasury bills and other risk free assets depending on the market. Finally, the model focuses mainly on minimizing the volatility given a specific return or the opposite, and this is not compatible with todays environment in the sense that investors views and objectives are more sophisticated, and therefore require to use enhanced models such as the Black and Litterman Model. Appendix 1: Methodology Section 1. Computing Returns and Standard Deviations: The first step is to process the historical prices into returns and determine the individual stock returns as well as the standard deviations. The daily stock returns from the stocks closing prices are calculated using the oldest historical prices as the base dates. The computation of the daily stock returns is done using the following formula: Daily stock return= (Closing price of the current trading day Previous days closing Price) / Previous days closing Price We then compute the expected returns of the stocks in the portfolio using the simple arithmetic mean of returns formula, which is as follows: AR = (R1 + R2 + R3 + ÃÆ' ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒâ€šÃ‚ ¦ + RN) / N The returns obtained are then annualized by multiplying the expected returns by 250 trading days on average in a given year. In order to calculate the risk of individual stocks, we used the STDEV function in Excel by selecting the daily returns calculated in the two data sets, and annualizing the values obtained by multiplying by the square root of 250 trading days on average per year. 2. The Jarque and Bera Test for Normality The Jarque-Bera test statistic for normality follows a chi-square distribution with 2 degrees of freedom. Concerning the analysis of the stock returns in the two data sets, I used the 99% confidence interval for which the critical value is 9.21 at 2 degrees of freedom. If the test statistic is higher than the critical value, we reject the null hypothesis that the monthly returns are normal, otherwise we accept it, and we can assume the returns are normal. Skewness gives the measure of asymmetry of the distribution, and is defined by the following formula: Where T is the time period or the length of the data used, and . Kurtosis is the measure of the flatness of the distribution, and is defined as follows: This leads to the computation of the Jarque-Bera test statistic which is as follows: 3. Volatility Clustering Volatility clustering is another important factor that can help to understand the patterns of the variations among stock returns. To show evidence of volatility clustering in the two data sets, we plotted the stock returns in a time series graph, and analyzed the different patterns of the individual stocks using the complete data set first, and then analyzing the most recent 250 stock returns to see if the patterns display some periods of high volatilities show some extreme returns, which are followed by periods of relatively low volatilities. 4. Measurement of Portfolio Risk and Return After computing the expected returns and risks of individual stocks, we define some of the formulas for computing the portfolio risk and return that will be used in the optimization process. Modeling the Portfolio Risk The portfolio risk is generally referred to as the portfolio variance. In our case of many assets, the variance of the portfolio takes the form of a matrix and has the following general formula: Where w1 to wn is a matrix that refers to the weights of the assets 1 to N in the portfolio, and ÃÆ' Ãƒâ€ Ã¢â‚¬â„¢xy is the covariance between assets x and y. In practice it is more useful to use the correlation matrix to derive the covariance matrix as it is generally unknown, and we have already computed the standard deviations of the individual stocks. In fact, we can derive the covariance matrix using the following formula: ÃÆ' Ãƒâ€šÃ‚ (x,y) = covariance(x,y)/ÃÆ' Ãƒâ€ Ã¢â‚¬â„¢x ÃÆ' Ãƒâ€ Ã¢â‚¬â„¢y Where ÃÆ' Ãƒâ€šÃ‚ xy is the correlation between assets x and y, and  ÃƒÆ' Ãƒâ€ Ã¢â‚¬â„¢n is the standard deviation of the nth asset. Hence we obtained the following: In excel, we used the following formula given that we have already computed the correlation matrix (appendix), the standard deviations and the expected returns of the stocks: {MMULT(MMULT(TRANSPOSE(wnÃÆ' Ãƒâ€ Ã¢â‚¬â„¢n); ÃÆ' Ãƒâ€šÃ‚ (x,y)); wnÃÆ' Ãƒâ€ Ã¢â‚¬â„¢n)} Where wnÃÆ' Ãƒâ€ Ã¢â‚¬â„¢n is the matrix that results from the multiplication of the portfolio weights with the standard deviations of individual stocks, and ÃÆ' Ãƒâ€šÃ‚ (x,y) is the correlation matrix. Determining the Portfolios Expected Return The portfolio expected return is the sum of the product of the holding of the assets by the expected returns of the individual assets, and is represented as follows: Where the sum of the weights = 1, n is the number of securities held in the portfolio which is 6 stocks in our case, wi is the proportion invested in a given stock i, E(ri) is the expected return on the stock i. The Optimization Process The optimizations were run based on some investment constraints and without any investment constraints. In the constrained optimization scenario, we restricted having short positions in the market meaning that all the optimal weights should be positive or equal to 0. Also, we set the sum of the proportions invested in each stock to equate 1, and finally, we select the range of cells that should be changed in order to obtain the optimal weights for each type of portfolio that we want to have. In the unconstrained optimization scenario, we allowed for both short and long positions to occur, but we still keep the restriction on the sum of the optimal weights, setting it equal to 1. In the second data set, in the Optimal Risky Portfolio and Optimal Complete Portfolios, we add more constraints such as limit the individual weights from exceeding 100% or -100% of the portfolio, and setting the proportion invested in the risky assets not to exceed 90% in order to obtain a minimum level of diversification between the risky asset and the risk free asset in both the constrained and unconstrained portfolios. Sharpe Ratio The sharpe ratio is a risk adjusted measure to evaluate portfolio performance and is based on the following formula: Source: Investopedia.com The investors utility function The investors utility function used in this project is a quadratic function that is represented as follows: Where is the utility value, and A is the investors level of risk aversion. The number 0.005 is a scaling factor used by convention to express the expected return and standard deviation as percentages. The optimal complete portfolio would be the one that provides the highest utility by maximizing the formula using the Excel Solver. The result gives the optimal proportion to be invested in the risky asset which is represented by the following formula: The proportion invested in the risk free asset is given by: 1 Y Appendix 2: Tables and Charts Asset Data (Full Data Set) Expected Return Standard Deviation GE -3.62% 38.77% DOW 5.45% 43.86% Baxter 7.68% 24.63% Caterpillar 17.78% 38.93% Procter and Gamble 3.51% 20.51% Apple 37.97% 40.29% Table 1: Stock Return and Standard Deviations Using the Full Data Set Asset Data (250 Most recent returns) Expected Return Standard Deviation GE 22.75% 27.43% DOW 31.99% 36.29% Baxter -14.10% 24.40% Caterpillar 64.69% 29.83% Procter and Gamble 1.68% 13.49% Apple 56.95% 25.49% Table 2: Stock Return and Standard Deviations Using the 250 Most recent returns Full Data Set GE DOW BAX CAT PG AAPL Kurtosis 9.283169 7.042409 7.47999063 5.080399 7.659262 4.260446 Skewness 0.417837 -0.08243 -0.622318643 0.160727 -0.03425 -0.02987 JB Test Statistic 2121.731 871.6094 1151.235074 235.9717 1156.239 84.78961 Table 3: The Jarque-Bera Test for Normality Using the Full Data Set 250 Most Recent Returns GE DOW BAX CAT PG AAPL Kurtosis 2.414595 2.058731 21.2954 1.601479884 2.562847 2.396152 Skewness 0.170671 -0.32919 -2.61641 0.112397247 -0.20665 0.316714 JB Test Statistic 4.783472 13.74422 4.783472 20.89990707 3.769913 7.977748 Table 4: The Jarque-Bera Test for Normality Using 250 Most recent returns Full Data Set Long Constraint Unconstrained Portfolio Expected return= 6.60% 7.48% Portfolio Variance = 3.47% 3.40% Portfolio SD = 18.63% 18.43% Risk Free Asset = 3.00% 3.00% Sharpe Ratio = 0.1933 0.2432 Portfolio Weights GE 0.00% -1.08% DOW 0.00% -7.75% BAXTER 32.87% 33.66% CAT 0.00% 3.90% PG 62.14% 65.22% AAPL 4.99% 6.06% Table 5: Minimum Variance Portfolio using the Full Data Set 250 Most Recent Returns Long Constraint Unconstrained Portfolio Expected Return= 9.84% 1.45% Portfolio Variance = 2.28% 1.57% Portfolio SD = 15.11% 12.52% Risk Free Asset = 3.00% 3.00% Sharpe Ratio = 45.31% -12.37% Portfolio Weights GE 7.27% 3.05% DOW 0.00% -15.57% BAXTER 32.14% 12.52% CAT 0.00% 2.38% PG 39.40% 89.78% AAPL 21.18% 7.82% Table 6: Minimum Variance Portfolio using the 250 Most Recent Returns Full Data Set Long Constraint Unconstrained Sharpe Ratio = 86.79% 114.44% Portfolio Expected Return= 37.97% 102.24% Portfolio Variance = 16.23% 75.20% Portfolio SD = 40.29% 86.72% Risk Free Asset = 3.00% 3.00% Portfolio Weights GE 0.00% -161.16% DOW 0.00% -32.51% BAXTER 0.00% 48.77% CAT 0.00% 91.02% PG 0.00% -57.55% AAPL 100.00% 211.44% Table 7: Optimal Risky Portfolio using the Full Data Sets 250 Most Recent Returns Long Constraint Unconstrained Sharpe Ratio = 232.49% 293.41% Portfolio Expected Return= 60.33% 110.34% Portfolio Variance = 6.08% 13.38% Portfolio SD = 24.66% 36.58% Risk Free Asset = 3.00% 3.00% Portfolio Weights GE 0.00% -18.38% DOW 0.00% -41.73% BAXTER 0.00% -43.77% CAT 43.63% 100.00% PG 0.00% 3.88% AAPL 56.37% 100.00% Table 8: Optimal Risky Portfolio using the 250 Most Recent Returns   Full Data Set Long Constraint Unconstrained Investor Utility Function to Maximize 0.329779696 0.551279771 Expected Return on the Complete Portfolio 33.13% 55.39% Variance of the Complete Portfolio 0.120514451 0.209559707 Standard Deviation of the Complete Portfolio 34.72% 45.78% Coefficient of Risk Aversion (A) 2.50 2.50 Y (% investment in risky assets) 86.16% 52.77% Portfolio Expected Return= 37.97% 102.29% Portfolio Variance = 16.23% 75.26% Portfolio SD = 40.29% 86.75% Risk Free Rate = 3.00% 3.00% Sharpe Ratio = 86.79% 114.44% Portfolio Weights   GE 0.00% -161.24% DOW 0.00% -32.53% BAXTER 0.00% 48.76% CAT 0.00% 91.06% PG 0.00% -57.58% AAPL 100.00% 211.53% Table 9: Optimal Complete Portfolio Using the Full Data Set  Long Constraint Unconstrained Investor Utility Function to Maximize 0.293024078 1.006325758 Expected Return on the Complete Portfolio 29.43% 101.12% Variance of the Complete Portfolio 10.57% 39.25% Standard Deviation of the Complete Portfolio 32.52% 62.65% Coefficient of Risk Aversion (A) 2.50 2.50 Y (% investment in risky assets) 90.00% 90.00% Portfolio Expected Return= 32.37% 112.03% Portfolio Variance = 13.05% 48.46% Portfolio SD = 36.13% 69.61% Risk Free Rate = 3.00% 3.00% Sharpe Ratio = 81.29% 156.62% Portfolio Weights GE 0.00% 100.00% DOW 98.84% 100.00% BAXTER 0.00% -100.00% CAT 1.16% 100.00% PG 0.00% -64.13% AAPL 0.00% -35.87% Table 10: Optimal Complete Portfolio Using the 250 Most Recent Returns Full Data Set Long Only Portfolio (Constrained)  Protfolio Expected Return Portfolio Standard Deviation Portfolio 1: Minimum Variance Portfolio 2.06% 13.15% Portfolio 2: Optimal Risky Portfolio 60.33% 24.66% Portfolio 3: Optimal Complete Portfolio 29.43% 32.52%  Long Only Portfolio (Constrained) 250 Most Recent Returns Protfolio Expected Return Portfolio Standard Deviation Portfolio 4: Minimum Variance Portfolio 6.60% 18.63% Portfolio 5: Optimal Risky Portfolio 37.97% 40.29% Portfolio 6: Optimal Complete Portfolio 33.13% 34.72% Table 11: Summary Table of the Results Obtained From the Optimizations Full Data Set Long-Short Portfolio (Unconstrained)  Portfolio Expected Return Portfolio Standard Deviation Portfolio 7: Minimum Variance Portfolio 2.42% 12.51% Portfolio 8: Optimal Risky Portfolio 110.34% 36.58% Portfolio 9: Optimal Complete Portfolio 101.12% 62.65%  Long-Short Portfolio (Unconstrained) 250 Most Recent Returns Portfolio Expected Return Portfolio Standard Deviation Portfolio 10: Minimum Variance Portfolio 7.48% 18.43% Portfolio 11: Optimal Risky Portfolio 102.24% 86.72% Portfolio 12: Optimal Complete Portfolio 55.39% 45.78% Table 11: Summary Table of the Results Obtained From the Optimizations Efficient Frontiers: Full Data Set Efficient Frontiers: 250 Most Recent Returns Volatility of Returns: Full Data Set Volatility of Returns: 250 Most Recent Returns