Project - 2014
- 格式：pdf
- 大小：623.51 KB
- 文档页数：11
Security Analysis and Portfolio TheoryProject 2014AbstractModern portfolio theory, developed by Harry Markowitz (1952, 1959), is the main criteria to select an investment portfolio. Markowitz shows that total risk of a portfolio could be decreased when adding new assets and the expected return of portfolio is the weighted average return of each asset. The Markowitz efficient frontier is a set of portfolios with maximum expected return given certain level of risk. We will follow Markowitz optimum portfolio model and single index model, another significant theory, to solve the portfolio optimization problem.IntroductionAllocation of assets is one of the fundamental investment decisions (Bogle, 1994). Through processing with 9 UK companies data, we examine the basic choice of asset allocation: how to construct a portfolio by choosing the weights of risk-free and risky assets.Data AnalysisTable 1 Descriptive StatisticsThrough transferring original price index to returns by using formula r t=100×ln×P t P t−1 , we obtain the monthly data return for 9 UK companies during 2006 February to 2013 January. Table 1 provides descriptive statistics of data returns which depict some features about these 9 UK companies and the FTSE 100 index, including mean, standard deviation, kurtosis, skewness, minimum and maximum value respectively. In terms of expectation, it can be seen that Tobacco has the largest return which is 1.042606 among these 9 companies. Then, Electricity and Utilities are in the second and third place which returns are 0.464033 and 0.446889 respectively. Finance is the lowest one which return is -1.33805. As for standard deviation, Finance is 1.019042 that shows it is the most volatile asset. Due to financial crisis during 2007 to 2008, the largest lost is -41.96847 in October 2008 while the largest profit is 30.247378 in April 2009. At the same time, Lifeins industry has been influenced by financial crisis as well, which has a standard deviation of 0.841449. Industries such as Electricity and Utilities are much less volatile which have standard deviations of 0.37060 and 0.36371. Skewness is a function of returns of a distribution. Positive skewness implies a distribution with an asymmetric tail extending towards more positive values; while, negative skewness implies a distribution with an asymmetric tail extending towards more negative values (Microsoft, 1996). The table shows that Tobacco’s skewness is – 0.27772, which is close to zero. Thus, it could be an acceptable skewness value for a normally distributed set of test scores. However, skewness of Lifeins and Media companies are -0.95882 and -82648, which are a bit far away from the normal distribution. In terms of kurtosis, positive kurtosis implies a relatively peaked distribution, while negative kurtosis implies a relatively flat distribution (Microsoft, 1996). FTSE 100 has a peaked distribution and is highest among these assets. However, the kurtosis of Tobacco is -0.15177, which becomes a flat distribution and it is the most flat one.Figure 1 Volatility ClusterFrom the volatility cluster figure, it can be discovered that Finance is the most volatile industry, particularly during October 2008 to April 2009. The second one is Lifeins, especially during February 2009 and February 2012 and the rest of these companies are relatively stable. In addition, Figure 2 demonstrates the distribution of 9 companies and the proportion of returns by computing histogram.Figure 2 HistogramData ProcessingMean Variance OptimizationTable 2We firstly compute expected value and standard deviation of 9 companies’ return as shown in Table 2, which are used to obtain variance-covariance matrix.Based on the variance covariancematrix (shown in Appendix), asset returns have positive relations since all the covariance are positive. The highest covariance is cov (Finance, Lifeins) =61.71, which shows high relations between the two industries and indicates the high level of risk undiversified of the portfolio consisting of Finance and Lifeins.Then we use the Excel Solver to solve for the optimal portfolio weights which can minimize the variance of the portfolio given certain level of returns. For short sale permitted situation, which refers to selling assets that are borrowed with the expectation of a fall in asset price, there are two constraints: the sum of the weights are 1.0 and the positive expected return. After firstly obtaining monthly return of 0.6101% with condition of positive expected return, we then set the return equal to 0.7, 0.8, 0.9, 1.0 and 1.5 as constraints to obtain the different results in weights in portfolio. The only difference between long constraints and short run is that in the long run, investors must maintain non-negative portfolio weights. Moreover, using Solver to obtainoptimal rick portfolio with maximum Sharpe ratio through the formula of()p fpE r rσ−. Results ofvarious portfolio decisions are shown in Table 3 and Table 4.Table 3 Short Sale PortfolioFinally, the monthly return of risky optimal portfolio with maximum Sharpe ratio in short sale is approximately 2.20%, while in the long run, return of risky optimal portfolio is 0.8835% and the optimal portfolios shown in Table 4 consists only two assets: Electricity and Tobacco.Table 4 Long Run PortfolioAfter obtaining various portfolios we can draw efficient frontier shown in Figure 3, which is aset of portfolios with minimum variance given a certain level of return. As the number of short sells is unrestricted, investors can ceaselessly sell assets with lower returns and invest in higher return assets therefore to obtain an infinite number of expected returns. Compared to the restricted portfolio in the long run, short sale widens the range of investments from the minimum variance portfolio to plus or minus infinity. This explains the graph we get that the efficient frontier in short selling is steeper than in the long constraint.Figure 3 Efficient FrontierTo construct the portfolio consisting of both risk and risk-free assets, we need to compute the proportion of risky asset using formula y= E(r m -r f )/A δm 2. Here we have a monthly risk-free rate of 0.2083% and risk aversion of 2. The portfolio results are shown in Table 5, which is also thesolution of portfolio optimization in terms of monthly return that investing 2.37% in risky asset and 97.63% in risk-free assts in the long run. After annualizing, we have the optimal overall portfolios shown in Table 6 which are also our recommendations both in short run and long run. For instance, the overall annual return is 8.76% and standard deviation is 17.70 in short run.Table 5 Overall PortfoliosTable 6 Annualised Overall PortfoliosSingle Index ModelIn order to investigate single-index model, firstly, it is important to regress each industry’s returns with market index which is a reasonable proxy for the common factor from 2006M02 to 2013M01. The regression equation of single-index model is R jt = a j + b j*Rm t + e jt. The intercept is the asset expected excess return and coefficient β reflects the asset sensibility to market index. According to the tendency of points’ distribution in Figure 4, trend lines can be added in the Excel. Consequently, we can get R2 as well as capture Alpha and Beta from equations for all 9 companies. Next, using equation Cov (r i, r m) = βiβiϭm2, a new variance-covariance matrix is generated based on single-index model.Figure 4 Single Index Model RegressionA high level of model fitting is shown in industry like Finance since the return data is highly converged to the trend line. The main reason is that performance in financial industry is highly related to the macroeconomic environment (represented as FTSE100 in this case). In detail, with booming economy, financial activities are more frequent and so that financial institutions have more opportunities to make profit. Oppositely, when economy suffers downturn, financial industries also shrink consequently. As a result, return data in financial industry has similar tendency with market benchmark. However, in industries like Utilities and Tobacco, the stock returns are displayed scatter from the regression equation. The main reason may be more other factors not just macroeconomic situation can affect returns in these industries. For example, government policy and financial support have significant influence on Utility industry. As for Tobacco, related policy and the output of tobacco may have more effects. In addition, the fitness between return data and market index in the rest industries are not as good as that in financialindustry, but better than Utilities and Tobacco industries. Referring to the new variance-covariance matrix generated based on single-index model, it reduces the requirement of computations greatly compared to Markowitz portfolio since only the beta of individuals and market needed to be calculated. In detail, all covariance among returns are affected by the single common factor so that they can be easily estimated. However, accuracy has been sacrificed as the cost of simple procedure.Table 7 Regression DataForecast stock returnsAccording to the data obtained from single index model regression, we can get the new forecasted data set for 9 UK companies’ return to estimate mean variance optimization. In addition, we obtained previous identical historical variance covariance matrix data excluding FTSE100. We separate the scenario into 2 positions which are Long-Short and Long Only positions.Table 8 Optimal Risk PortfoliosSHORT RUN LONG RUNAs the calculation from solver, we obtained the minimum variance portfolio. The return from this portfolio is 0.5790% and the risk of portfolio which represents by standard deviation is 2.5517%. Moreover, we focus on the optimal portfolio which has a maximum Sharpe ratio. This optimal portfolio has a return of 2.1302% and risk around 5.5663%. The investor should short Finance, Telecom and Food companies because their weight is negative and the investor should long other companies particularly Tobacco since they give positive weights.As for long position, the return from minimum variance portfolio is 0.4820% and the risk of portfolio is 2.7740%. In addition, the optimal portfolio which has a maximum Sharpe ratio has the return of 0.8806% and risk around 3.7584%. The investor should long only Electricity and Tobacco companies because they give positive weights. Due to weights of other companies are equal to zero, the investor will not invest in others.The efficient frontier graph (shown in Figure 5) for Long-Short position and Long position in term of forecasting returns can be plotted based on minimum variance portfolios which have different portfolio returns and risks.Figure 5 Forecast Efficient FrontierAccording to the table below, we can obtain the overall portfolio which consists of risk-free asset and risky asset. For the Long-Short position, the investor should also invest in risk free asset which afford average monthly return of 0.2083% and risky asset which has average monthly return 0f 2.1302% with risk of 5.5663%. As a result, the investor will obtain the overall average monthly return equal to 0.2679% and this portfolio has a risk around 0.1726%.Table 9 Optimal PortfolioTable 10 Annualised Optimal PortfolioAccording to the overall portfolio for Long Only position, investors should invest in risk free asset and risky asset which has 0.8806% of average return and 3.7583% of risk. As a consequence, investor will obtain the overall average return equal to 0.2243% with a risk around 0.089%. Annualised portfolio return and level of risk for short run and long run are demonstrated in Table 10.ConclusionFinancial market changes all the time. To obtain the optimal investment decision, a combination of different security selection criteria is significant since it can provide a comprehensive overview. Among these criteria, Markowitz optimization portfolio and single index model play an important role because they examines from different aspects. However, more efforts are still needed since even best estimation may deviate lot from facts.ReferencesBODIE, Z., 2011. Investments and portfolio management / Zvi Bodie, Alex Kane, Alan J. Marcus. New York: New York : McGraw-Hill/Irwin.Chen,W., Chung,H., and Ho, K.Y. et al [no date].Portfolio optimization models and mean-variance spanning tests (s.l.)(s.n.).Microsoft [Computer software]. (1996). Excel. Redmond, WA: Microsoft CorporationAppendix1 Variance- Covariance Matrix for Actual Return2 Variance-Covariance Matrix for Single Index ModelGroup Members: 1329647 1342267 1342971 1366574 1385858。
