Introduction: the selection of our portfolio 3
mean-Variance optimization: set of statistics of our portfolio. 3
The efficient frontier….…………………………………………………………..4
• Mean-variance frontier of our stocks
• Optimal combination of our portfolio
• Equity returns as a linear relation of their
Active Equity MANAGEMENT ……………………………………………………..6
• Forecasting of the returns
• Optimal active positions for a maximized Information Ratio
• Maximization of the value-added function
The first step required in building our portfolio was to choose the constituents of the portfolio. We decided only to choose UK listed stocks and our initial objective was to try and build a well-diversified portfolio therefore we chose stocks in different industries. The industries chosen were technology; food; defense; banking; mining; medicine; manufacturing and retail, and we felt that this was a well diversified group of industries that reflected the UK economy well. We decided initially to weight them all equally (10% weighting per stock). After diversification our secondary criteria was choosing stocks that we believed had a high probability for abnormal growth. We also believed that it would be beneficial to use companies who had been listed on the London stock exchange for a long time as the more sample data available the stronger our results would be.
The stocks chosen were: ARM Holdings (ARM:LN); Associated British Foods (ABF:LN); BAE Systems (BA:LN); Barclays (BARC:LN); BHP Billiton (BLT:LN); British Petroleum (BP:LN); GlaxoSmithKline (GSK:LN); Rolls Royce (RR:LN); SAB Miller (SAB:LN); J Sainsbury (SBRY:LN).
From the chosen stocks we collected historic monthly share prices for the last 10 years, (please see excel file for details) and from these historic prices we calculated the: mean return; variance; covariance and correlation, presented in table 1 the mean return is expressed in terms of monthly return. From these numbers we also calculated the Sharpe ratio which is the risk premium of each share divided by the standard deviation of excess returns, this is based upon the 10 year risk free rate of 2%. Also note that table 1 is constructed on the assumption that each stock is equally weighted.
When looking at the data the firm with the highest average return is BHP with a mean return of 1.81% per month, in contrast the lowest return is held by Barclays who over the last 10 years have an average return of -0.08% per month. Alongside this Barclays also has the highest variance, whilst it does have some significant outliers such as -45% in October 2008 at the peak of the financial crisis we felt that we should include them still as these outliers are representative of the stock. All our stocks have a positive value for covariance against the benchmark FTSE 100; this indicates that our stocks have a direct or increasing linear relationship between their performance and that of the FTSE 100 showing that each firm does contain some systematic risk. Whilst covariance on its own is not particularly useful the correlation coefficient tells us the strength of relationship between the individual stock and the benchmark FTSE 100. Our stocks vary from BHP Billiton with a correlation coefficient of 0.68 to Arm Holdings with a correlation coefficient of 0.33. Finally we calculated the Sharpe ratio as defined above.
• Efficient frontier and optimal combination of the portfolio:
Now using mean variance analysis we plotted the efficient frontier, to do this we used the solver tool in excel to help us calculate what weightings of stocks would give us the minimum variance for the same return. Table 2 shows the optimum weightings of stocks that would allow us to still to get the same expected return as if the stocks were equally weighted.