Investment Portfolio Management Strategies - Essay Example

1. Introduction The objective of this paper is to compare the performance of two portfolios: A U.S portfolio consisting of six Stocks (Citigroup Inc., American Express Company, Motorola Inc., Boeing Co., McDonalds Corp., and Coca-Cola Co.); and an Australian Portfolio consisting of Westpac Banking Corp, ANZ Banking Group Ltd., Rio Tinto Ltd., BHP Billiton Ltd., QBE Insurance Group Ltd, and Qantas Airways Ltd. To achieve this objective, daily stock prices for the stocks included in each portfolio over the period 08/08/08 to 09/05/08 are downloaded from yahoo finance and are used to calculated average daily returns and standard deviations for each stock. The average daily returns for each stock are then used to calculate an average return for each portfolio over the period under study. The correlation coefficient between the daily stock return in each portfolio is also calculated and combined with the standard deviations in a covariance matrix to calculate the variance and standard deviation for each portfolio. Sensitivity analysis is also carried out by changing the weights of each stock in the portfolio to see how the performance changes. 2. Return and Standard Deviation of each Portfolio According to Bodie et al. (2002) the expected return on a portfolio of stocks is simply a weighted average of the individual stock returns in the portfolio. (1) Where represents the return on the portfolio; is the return on stock i, is the proportion of the portfolio invested in stock i; n is the total number of stocks in the portfolio. The variance and thus the standard deviation are far more difficult to calculate because the variance of a portfolio is not simply a weighted average of the individual variances of the stocks included in the portfolio except for the special case where the individual stock returns are uncorrelated with one another. The variance for a two stock portfolio is given by: (2) We need to set up a covariance matrix to bordered covariance matrix to be able to calculate the 1variance and standard deviation for a portfolio of more than two stocks. For a three stock portfolio the bordered covariance matrix can be written as follows (Bodie et al., 2002): The portfolio variance is calculated from the above nine terms by multiplying the bordered weights by the corresponding covariance and then summing the different terms. The standard deviation is calculated by taking the square root of the variance. From the table above, the variance is given by: (3) and the standard deviation of the portfolio is given by: (4) The covariance matrix in table 1 above can be extended to any number of stocks. Haven discussed how to calculate the expected return on portfolio, the variance and standard deviation; we now apply the above models to the U.S and Australian portfolios. Std. Dev (%) Ave Ret (%) Citi Group Inc. 3.58 -0.03 American Express 2.99 0.26 Motorola Inc 2.25 -0.30 Boeing Co. 1.85 -0.38 McDonalds Corp 1.45 -0.44 Coca Cola 0.91 -0.34 Table two above show the standard deviation and average return of each of the stocks included in the U.S portfolio over the period under study (details of the calculations are found in the attached excel file). To be able to calculate the covariance between the different stocks, we need to know the correlation coefficients between the different stocks. We use the correlation function in Excel to calculate the correlation coefficient between the different stocks. Doing this we obtain the following correlation matrix for the different stocks: Table 2: Correlation matrix (U.S Portfolio) Citigroup Inc. American Express Motorola Inc Boeing Co. McDonalds Corp Coca Cola Citi Group Inc. 1.00 0.90 0.54 0.74 0.78 0.25 American Express 0.90 1.00 0.64 0.78 0.87 0.34 Motorola Inc 0.54 0.64 1.00 0.37 0.53 0.23 Boeing Co. 0.74 0.78 0.37 1.00 0.72 0.16 McDonalds Corp 0.78 0.87 0.53 0.72 1.00 0.31 Coca Cola Co. 0.25 0.34 0.23 0.16 0.31 1.00 To obtain the covariance matrix we use the following formula formula to calculate the covariance between the different stocks (Bodie et al., 2002): (5) Thus (6) Applying equation (6) to the U.S portfolio, that is, multiplying the correlation between two stocks by the product of their standard deviations we obtain the following covariance matrix for the U.S portfolio: Table 3: Covariance Matrix (U.S Portfolio) Citigroup Inc. American Express Motorola Inc Boeing Co. McDonalds Corp Coca Cola Citi Group Inc. 12.81 9.61 4.34 4.91 4.02 0.81 American Express 9.61 8.93 4.32 4.29 3.76 0.91 Motorola Inc 4.34 4.32 5.08 1.54 1.73 0.46 Boeing Co. 4.91 4.29 1.54 3.43 1.94 0.26 McDonalds Corp 4.02 3.76 1.73 1.94 2.10 0.40 Coca Cola Co. 0.81 0.91 0.46 0.26 0.40 0.82 We then calculate a weighted bordered covariance matrix by multiplying the different weights of the stocks in the portfolio by their covariances as follows: Table 4 Border-Multiplied Covariance Matrix. Citi Group Inc. American Express Motorola Inc Boeing Co. McDonalds Corp Coca Cola Portfolio weights 0.20 0.20 0.15 0.15 0.15 0.15 0.20 0.512 0.384 0.130 0.147 0.121 0.024 0.20 0.384 0.357 0.129 0.129 0.113 0.027 0.15 0.130 0.129 0.114 0.035 0.039 0.010 0.15 0.147 0.129 0.035 0.077 0.044 0.006 0.15 0.121 0.113 0.039 0.044 0.047 0.009 0.15 0.024 0.027 0.010 0.006 0.009 0.018 Sum 1.319 1.140 0.458 0.437 0.372 0.095 Variance is calculated by summing the different border multiplied covariances. The variance is given by: Portfolio Variance = 1.319+1.140+0.458+0.437+0.372+0.095 =3.823 Portfolio SD = 3.823^1/2=1.96% Portfolio Return = -0.17% Similar analysis are applied to the stock returns for the Australian Portfolio. Table 5: Expected return and Standard Deviation of Individual Stocks (Australian Portfolio). Std. Dev (%) Ave Ret (%) Westpac Banking Corp 2.50 -0.02 Anz Banking Group Ltd 2.02 -0.32 Rio Tinto Ltd 3.22 -0.23 BHP BILITON Ltd 2.82 0.02 QBE Insurance Group Ltd 2.32 0.08 Qantas Airways Ltd 2.90 0.34 Table 6: Correlation Matrix (Australian Portfolio) Westpac Banking Corp Anz Banking Group Ltd Rio Tinto Ltd BHP BILITON Ltd QBE Insurance Group Ltd Qantas Airways Ltd Westpac Banking Corp 1.00 0.79 -0.18 -0.14 0.51 0.17 Anz Banking Group Ltd 0.79 1.00 -0.09 -0.11 0.21 -0.01 Rio Tinto Ltd -0.18 -0.09 1.00 0.93 0.02 -0.32 BHP BILITON Ltd -0.14 -0.11 0.93 1.00 0.05 -0.34 QBE Insurance Group Ltd 0.51 0.21 0.02 0.05 1.00 0.03 Qantas Airways Ltd 0.17 -0.01 -0.32 -0.34 0.03 1.00 Table 7: Covariance Matrix (Australian Portfolio) Westpac Banking Corp Anz Banking Group Ltd Rio Tinto Ltd BHP BILITON Ltd QBE Insurance Group Ltd Qantas Airways Ltd Westpac Banking Corp 6.27 4.00 -1.46 -1.02 2.99 1.23 Anz Banking Group Ltd 4.00 4.09 -0.56 -0.62 0.98 -0.06 Rio Tinto Ltd -1.46 -0.56 10.34 8.42 0.15 -2.96 BHP BILITON Ltd -1.02 -0.62 8.42 7.96 0.32 -2.76 QBE Insurance Group Ltd 2.99 0.98 0.15 0.32 5.37 0.18 Qantas Airways Ltd 1.23 -0.06 -2.96 -2.76 0.18 8.41 Table 8: Border-Multiplied Covariance Matrix (Australian Portfolio). Westpac Banking Corp Anz Banking Group Ltd Rio Tinto Ltd BHP BILITON Ltd QBE Insurance Group Ltd Qantas Airways Ltd Portfolio weights 0.20 0.20 0.15 0.15 0.15 0.15 0.20 0.251 0.160 -0.044 -0.031 0.090 0.037 0.20 0.160 0.164 -0.017 -0.019 0.029 -0.002 0.15 -0.044 -0.017 0.233 0.189 0.003 -0.067 0.15 -0.031 -0.019 0.189 0.179 0.007 -0.062 0.15 0.090 0.029 0.003 0.007 0.121 0.004 0.15 0.037 -0.002 -0.067 -0.062 0.004 0.189 Sum 0.463 0.316 0.298 0.264 0.255 0.100 Portfolio Variance 1.696 Portfolio SD 1.30 Portfolio Return -0.04 We see from above that the variance return and standard deviation for the U.S portfolio are -0.17% and 1.96% respectively, while the return and standard deviation for the Australian portfolio are -0.04 and 1.30% respectively. The Australian portfolio has a lower standard deviation and a higher expected return while the U.S portfolio has a higher standard deviation and a lower expected return. Therefore the Australian portfolio performs better than the U.S portfolio. Therefore I will recommend the Australian portfolio to an investor with high degree of risk aversion as it returns a higher return at a lower risk. 3. The Variance and covariance of the portfolios can be observed from the tables for the covariance matrices above. 4. Regression analysis. SUMMARY OUTPUT Regression Statistics Multiple R 0.914252 R Square 0.835856 Adjusted R Square 0.8262 Standard Error 0.004686 Observations 19 ANOVA df SS MS F Regression 1 0.001901 0.001901 86.56756 Residual 17 0.000373 2.2E-05 Total 18 0.002274 Coefficients Standard Error t Stat P-value Alpha U.S.A -0.00111 0.001079 -1.03217 0.316454 Beta U.S.A 0.525556 0.056486 9.304169 4.41E-08 The tables above show the summary outputs after running the following regression: Daily return of S & P 500 = alpha (USA) + Beta (USA) x daily return of USA portfolio + E One can observe that the R-Square is 91% and the adjusted R-Square of the model is 82.6% indicating that the portfolio explains the variability of the daily return of the U.S portfolio explains more than 80% of the variability of the daily return of the S&P 500. The beta of the U.S portfolio is 0.52 and it is statistically significant at the 5% level of significance. The alpha is -.00111 and it is not significant at the 5% level of significance. SUMMARY OUTPUT Regression Statistics Multiple R 0.413956 R Square 0.171359 Adjusted R Square 0.122616 Standard Error 0.012982 Observations 19 ANOVA df SS MS F Regression 1 0.000592 0.000592 3.515528 Residual 17 0.002865 0.000169 Total 18 0.003458 Coefficients Standard Error t Stat P-value Alpha Australia -0.00013 0.002979 -0.04357 0.965754 Beta Australia 0.440574 0.234976 1.874974 0.078082 The tables above show the regression output for the following model: Daily return of ASX 200 = alpha (AU) + Beta (AU) x daily return of Australia portfolio + E In this case it can be observed that the R-Square is quite low only 41.3% and the adjusted R-Square is far more lower than the R-Square. This indicates that the variability of the returns on the portfolio do not explain a significant portion of the variability of the returns on the ASX 200. The beta of is 0.44 and it is not statistically significant at the 5% level of significance. It is however, significant at the 10% level of significance. The alpha is -.000013 and it is neither significant at the 5 nor 10% levels of significance. 5. Critique of Portfolio Both portfolios performed poorly although the U.S portfolio performed worse than the Australian portfolio. The poor performance of the portfolios may be attributable to the proportion invested in each portfolio. In the case of the U.S portfolio a very high proportion is invested in Citigroup which has a very low expected return and a high standard deviation. A similar situation is observed for the Australian portfolio where a high proportion is invested in Westpac Banking Corp with a high standard deviation and low return. 6 Redesign of Portfolio Citi Group Inc. American Express Motorola Inc Boeing Co. McDonalds Corp Coca Cola Portfolio weights 0.10 0.30 0.15 0.15 0.15 0.15 0.10 0.128 0.288 0.065 0.074 0.060 0.012 0.30 0.288 0.804 0.194 0.193 0.169 0.041 0.15 0.065 0.194 0.114 0.035 0.039 0.010 0.15 0.074 0.193 0.035 0.077 0.044 0.006 0.15 0.060 0.169 0.039 0.044 0.047 0.009 0.15 0.012 0.041 0.010 0.006 0.009 0.018 Sum 0.628 1.690 0.457 0.428 0.368 0.097 Portfolio Variance 3.669 Portfolio SD 1.92% Portfolio Return -0.14% To redesign the U.S portfolio, we reduce the investment in Citigroup by 10% and increase the investment in American express by 10% because it has a higher expected return. Doing so the standard deviation reduces from 1.96% to 1.92% and the expected return increases from -0.17% to -0.14%, which reflects a slight improvement from earlier results. Westpac Banking Corp Anz Banking Group Ltd Rio Tinto Ltd BHP BILITON Ltd QBE Insurance Group Ltd Qantas Airways Ltd Portfolio weights 0.10 0.20 0.15 0.15 0.15 0.25 0.10 0.063 0.080 -0.022 -0.015 0.045 0.043 0.20 0.080 0.164 -0.017 -0.019 0.029 -0.004 0.15 -0.022 -0.017 0.233 0.189 0.003 -0.155 0.15 -0.015 -0.019 0.189 0.179 0.007 -0.145 0.15 0.045 0.029 0.003 0.007 0.121 0.010 0.25 0.043 -0.004 -0.155 -0.145 0.010 1.030 Sum 0.193 0.233 0.232 0.197 0.215 0.778 Portfolio Variance 1.849 Portfolio SD 1.22% Portfolio Return 0.00% For the Australian portfolio we increase investment in Qantas Airways limited by 10% to 25% and reduce investment in Westpac Banking Corp by 10% to 10% as shown in the table above. We can observe that the portfolio standard deviation reduces from 1.30% to 1.22% while the portfolio expected return increases from -.04% to 0.00%. References Bodie Z., Alex K., Marcus J. M. (2002). Investments. 5th Edition. McGraw-Hill Irwin. www.yahoofinance.com Read More