, Finance and Accounting - Statistics Project Example

Statistics Project, Finance and Accounting Portfolio Management Introduction This study is an analysis of the performance of a selected set of stocks for a period of 20 years between 1994 and 2014. The objective of the study is to measure the relationship between the risks in the investments and the expected return. Many models and methods are applied in this study to maximize the return and minimize the risks in the investments. The study assists in the selection of the most efficient method for optimizing the portfolio. The optional methods available include the Classical Mean-Variance, the robust optimization and the Treynor-Black, and the different input parameter for estimation of the expected return such as the mean / covariance, Black-Litterman and the Bayes-Stein methods. 2. Monthly Return of Assets In this project, the prices of the stocks provide the weights of the portfolios for all the stocks provided. The monthly returns for the stocks in the investment pool are calculated with the formula in equation 1 below: Monthly returns = xp = p^-Tx = ∑xipi ---------------EQ 1 2.1. Time series for All stocks Where xp is the monthly expected return, pi is the weight of the portfolio and n is the average number of assets. The values of calculation of the expected monthly returns are presented in the table 1 presented below: Stock Expected Return CSCO 21 INTC 24 NKE 26 PFE 26 MSFT 27 GE 28 KO 28 T 32 UNH 32 DIS 33 VZ 37 HD 38 JPM 40 AXP 42 MRK 46 DD 47 MCD 47 TRV 47 WMT 48 CAT 49 PG 51 UTX 51 JNJ 56 XOM 56 BA 61 CVX 63 MMM 70 IBM 105 Table 1: Monthly Stock Return The time series for all the 28 stocks appear in figure 1 below, showing IBM to be having the highest stock return on investment. Figure 1: Monthly Returns for all the Stocks The returns computed for the years of this study show the expected return increasing from left to right for all the stocks except IBM that drops at the end of the period within the time series. The stock returns values experience wide variance due to the fluctuation in portfolio weights across the period. 2.2. Time series for Efficient Stocks From the set of 28 stocks, the time series was filtered on the criteria of performance to retain the 11 stocks shown in the time series as the most efficient portfolio. Figure 2: Return – The Best Performing Stocks 2.3. Autocorrelation The result of autocorrelation presents the correlation coefficient matrix with the global and minimum variances as seen in figure 2 below. Opt. Risky Global Min. # Portfolio Var. Port. 1 0.172291075 0.200108206   2 0.481011012 0.345566223   3 0.162953922 0.248568069   4 -0.02040551 -0.00281929   5 0.204149502 0.20857679   Correlations 1 2 3 4 5 1 1.00 -0.99 -0.10 -0.80 0.99 2 -0.99 1.00 0.50 0.00 -0.30 3 -0.10 0.50 1.00 0.20 -0.99 4 -0.80 0.00 0.20 1.00 0.80 5 0.99 -0.30 -0.99 0.80 1.00 Table 2: Auto Correlation Test 3. Refining the Investment Pool The process of refining the investment involved ignoring the portfolio with low weights and retaining the high weight portfolio. The selection aimed at picking 3 stocks with the best returns to represent the high efficiency required in the pool decision. The high efficiency stocks were found to be IBM and MMM. The decision was made on the values based on the original currency returns. The time series for the refined investment pool carries the following stock: IBM JNJ MMM The stocks are therefore, presented in the time series in figure 2 below: Figure 3: Efficient Stocks Returns The major reason for reducing the number of stocks in the refined investment is that many assets have caused a wide variation of the portfolio weights and return on investment (Tobin 1958, p. 65). The analysis sets up individual each of the assets independently to as to classify them as either risky assets or risk free assets using the correlation projections. The refinement judges the investment by their return, hence; it operates with the few selected manageable stocks to reduce the portfolio size by ignoring the low return stocks. 4. Expected Return Estimation for the Stocks The expected return is derived as a function of the standard deviation using the equation below: X’^T * V’ x = xixjσi,j ------------------- EQ 2 The results for estimation of the expected mean and the corresponding standard deviations are shown in table 3 below: Unrestricted Efficient Frontier Restricted Efficient Frontier Mean Return St. Dev Mean Return St. Dev -0.05 0.27015 0.08 0.12 -0.02 0.222987 0.09 0.119583 0 0.193534 0.1 0.12233 0.02 0.166665 0.11 0.12758 0.04 0.143838 0.12 0.135041 0.06 0.127245 0.13 0.144371 0.08 0.119512 0.14 0.155233 0.1 0.12233 0.15 0.16733 0.12 0.135041 0.16 0.180412 0.14 0.155233 0.17 0.196214 0.16 0.180412 0.18 0.223109 0.18 0.208781 0.19 0.258736 0.2 0.239208 0.2 0.3 0.22 0.271 0.25 0.320358 Table 3: Expected mean Return and corresponding Standard Deviation 5. Estimation of the Covariance Matrix for the Stocks The results for the covariance matrix are presented in table 4 below Standard Expected Stock Deviation Return MMM 0.2 0.14 IBM 0.12 0.08 JNJ 0.3 0.2 MMM IBM JNJ St. Dev 0.2 0.12 0.3 Mean 0.14 0.08 0.2 Correlation Matrix MMM IBM JNJ MMM 1 0.5 0.2 IBM 0.5 1 0.4 JNJ 0.2 0.4 1 B. Covariance Matrix MMM IBM JNJ MMM 0.04 0.012 0.012 IBM 0.012 0.0144 0.0144 JNJ 0.012 0.0144 0.09 C. Equally-Weighted Portfolio MMM IBM JNJ Weights 0.3333 0.3333 0.3333 0.3333 0.004444 0.001333 0.001333 0.3333 0.001333 0.0016 0.0016 0.3333 0.001333 0.0016 0.01 1.0000 0.0071 0.0045 0.0129 Variance 0.0246 St. Dev 0.156773 R * weight 0.046667 0.026667 0.066667 Mean 0.14 Table 4: Covariance matrix Estimation In the stock return distribution, the correlation matrix shows the stocks having positive correlation coefficients. 6. Construction of the Optimal (Tangent) Risky Portfolio The tangent portfolio was constructed using the Matlab program. The program uses the data entries from the covariance matrix with the new weights of portfolios. The mean return values and the optimal portfolio variances are shown with the least variance portfolio return averaging approximately 14.6%. After making the entry of the matrix, the exercise is repeated for the reduction of the variance to show the restricted efficient frontier as shown in figure 6 below. The assumption made in the construction of the frontier is that investment can run on negative portfolio weights, even though the current weights are positive. Figure 4: Optimal Portfolio Frontier – Risk Free Assets The pink line is the restricted portfolio, while the blue line is the unrestricted portfolio. The tangent line is shown as the green dotted line labeled “Stock”. The process is repeated for the risky assets in the matrix to show the efficient in figure 5 below. Figure 5: Optimal Portfolio Frontier – Risky Assets The objective function was selected as CCC to assist in the call for optimal portfolio, and the BBB to meet the objective of calling the efficient Frontier. As seen in the results, certain stocks dominate the weight and returns while others have weights close to zero. The constraint applied in this analysis is that no one asset in this portfolio should utilize more than 14.6% of the overall investment funds. The algorithm used in imposing the constraint and the objective function is presented in equation 3 below Wi = (E (ri) – rf) / dp ------------ EQ 3 The constraint is applied by substituting the risk-free rate of return with 14.6%. The outcome of the tangent portfolio weights is presented in table 5 below. Portfolio MMM IBM JNJ Weight -0.7819 2.4743 -0.6924 -0.781933651 0.024457 -0.02322 0.006497 2.474300159 -0.02322 0.088159 -0.02467 -0.692366508 0.006497 -0.02467 0.043143 1.0000 0.0077 0.0403 0.0250 Variance 0.0730 St. Dev 0.27015 R * weight -0.10947 0.197944 -0.13847 Mean 14.58 Table 5: Results for Optimal Portfolio The optimal risk portfolio gives a projected stock return of 13.84% and a standard deviation of 27.015%. The optimal-risk restricted portfolio gives a predicted return of 10.947% with a standard deviation of 7.308%. 7. Construction of an Optimal Portfolio This is done by mixing the Risk-free Asset and the Tangent Portfolio to produce a new efficient frontier. The optima portfolio is selected in consideration of the risk aversion level of the investors against the expected return. The investment desire is to realize the highest return possible and the least risk exposure possible. Calculating the conditional projected return in the Treynor-Black (TB) model shown below: R(t) = α + β * RM (t) + e(t), t Read More