Investment - Expected Market Return and Risk-Free Rate - Statistics Project Example

Customer Inserts His/Her Name Customer Inserts Grade Course Customer Inserts Tutor’s Name 13, October, 2012 Outline 1. Introduction 2. Return construction 3. Expected Returns 4. Optimal Portfolio 5. Finance theory and Quantitative analysis relationship 6. Conclusion Introduction Ms. Pretty approached us, Wealth Management Company, to give her professional advice on how she could invest in a well-diversified portfolio after she won a lottery recently to maximize returns at the same time minimize risk. She named ten listed companies in Australian Stock Exchange, yet she needed a portfolio which combined only five shares. To make this selection, we had to construct a return for each of these companies from their monthly returns as listed in ASX for the period August 2007 to August 2012. Afterwards, we had to statistically analyze the data by calculating the descriptive statistics variables that include the variance, correlation coefficients as well as the expected returns since they were useful in selecting the optimal portfolio. The expected return was calculated using the CAPM model where we first calculated the beta and later obtained the expected market rate of return and the risk free rate using the average monthly returns stated above. The client was interested in knowing the percentage weights and the dollar amounts invested in each share. Finally, we had to give Ms. Pretty a professional conclusion since financial statistical analysis has a forecasting error and as such the analytical data cannot be fully reliable for selection of a viable portfolio. This report documents the study conducted by Wealth Management Company in Australia. 1. Return Construction 1.1 Data collection The monthly returns for the five companies and their total return index from 1st August 2007 to 1st August 2012 were collected from DataStream. The returns however had to be adjusted for dividend and capital changes. The monthly share price and the monthly returns are shown in appendix 1 and appendix 2, respectively. The five companies are listed below indicating their Global Industry Classification Standard (GICS) and ASX code. Company GICS ASX Code BHP Billiton Materials BHP National Australia Bank Banks NAB Telstra Ltd Telecommunication services TLS Woolworth Ltd Foods and Staples retailing WOW AGL Utilities AGK 1.2 Return Series The companies’ monthly return were calculated using the holding period return formulated as the difference between the current period returns and the previous period returns divided by the previous period returns (h1-h0)/h0. The holding period return was then converted into a percentage by multiplying by 100 and later standardized by adding 1000 to the percentage for comparison purposes. Comments BHP: This material company had the widest band based on the 1000 margin we used to standardize our data. It had a high of 1017 in May 2008 and a low of 979 in October 2008. This is an indicator of high variability in the company’s sales. The investors therefore, have a risk of making high profits and huge loses at the same time. WOW: The food and staple retailing company had a narrow band from the margin. This indicates that the returns from trading the share were low while the loss incurred is equally minimal. AGK: This utilities company had a negative correlation with BHP. When BHP was trading at a profit, AGK was trading at a loss and viceversa. An investor who purchased the two shares reduced the risk of trading highly. TLS: The telecommunication services company made average profits which were slightly above the standard mark, but its loss was so huge that it could not be compensated against the minimal profits. On average, the total earnings were below 1000. The shares however could be bought at a specific period when they traded at a loss like in December 2008, and sold at a profit when the price had appreciated as was the case in August 2009. NAB: The National Bank shares had the same trend as the BHP from August 2007 to December 2010. Afterwards, the share’s band narrowed, and its earnings remained within the standard mark. 1.3 Variance and Correlation coefficient The variance and expected return of the five companies selected were as follows: Company Variance Expected Return AGK 0.0041578 0.003170884 WOW 0.0025955 0.002599961 TLS 0.0030644 -0.000220254 NAB 0.0068817 -0.002979316 BHP 0.0054835 0.000615485 The correlation coefficients were as shown below: AGK – WOW -0.014929614 AGK –TLS -0.088188191 AGK – NAB 0.155088542 AGK – BHP 0.337289557 WOW – TLS 0.20064077 WOW – NAB 0.305332201 WOW – BHP 0.027227 TLS – NAB 0.04508346 TLS – BHP 0.016059846 NAB – BHP 0.440232519 The correlation between AGK and WOW was negative indicating that investing in the two shares would reduce the risk. A similar case was observed between AGK, and TLS with a correlation of -0.0881. The rest of the coefficients were positive but less than 0.5 indicating a weak correlation between the share prices. The strongest positive correlation was between NAB and BHP with a value of 0.440. This means that the two companies have a similar trend and movement over the period. 2. Expected Return 2.1 Beta estimation Beta is the measure of the sensitivity of the asset’s return to the market return; it measures the systematic risk or the risk that affect the market as a whole (Linden, 2001). If the beta of an asset is less than one, the asset is less risky than the market. The beta can be calculated as Cov(Ri, Rm)/varRm Where Rm used is NAB since it is a government corporation, we assume that it reflects the market risk (treasury risk) when compared to S&P/ASX 200. The beta calculated using the formulation was as shown below: Company Covariance NAB, Stock Variance NAB Beta AGK 0.000815761 0.006767014 0.120549618 WOW 0.001268915 0.006767014 0.187514816 TSL 0.000203582 0.006767014 0.030084496 NAB 0.006767014 0.006767014 1 BHP 0.002659247 0.006767014 0.392972071 Comment AGK: The Company had a beta of 0.12 indicating that its sensitivity to the market was relatively low. This indicated the risk on AGK stock was less than the market risk. WOW: The food and staples company had a Beta of 0.18. This showed that the risk in trading the stock was slightly higher than that of AGK but still less than the market risk. TSL: The Company had a minimal volatility to the market risk. It had a beta of 0.03 indicating the minimal sensitivity of the stock to the market. BHP: This Company had a beta of 0.39, which was the highest amongst the four companies. This shows that the sensitivity of the company was high and as such it traded high values than the other companies. 2.2 Expected Market Return and risk free rate. The expected market return is the broad market’s expected rate of return whole (Linden, 2001). In this valuation, the expected market rate return was calculated by annualizing the average monthly returns as shown below: Company Av. Monthly return Av monthly return *12 Expected market return AGK 0.003170884 0.038050609 0.038050609 WOW 0.002599961 0.031199536 0.031199536 TLS -0.000220254 -0.002643049 -0.002643049 NAB -0.002979316 -0.035751788 -0.035751788 BHP 0.000615485 0.007385825 0.007385825 The risk free rate for the period August 2007 to August 2012 were obtained from a 10-year Australian government note with a coupon of 5.5%. It had a yield of 3.02%, and this is the value that was used as the risk free rate. 2.3 CAPM Model The CAPM model was used to estimate the expected return of a stock by taking into consideration the time value of money and risk in the stock itself. The formula used in the model had three key variables. These variables were; the beta, which measured the individual stock risk of the market risk, the overall stock market risk, and the risk free rate as measured by the investment index. Arithmetically the formula is written as: E(r) = RFR + (Rm – RFR) Beta The expected return for Ms. Pretty’s project using CAPM was as shown below for each respective company. Company RFR Beta Expected market return Expected Return AGK 0.0302 0.120549618 0.038050609 0.031146388 WOW 0.0302 0.187514816 0.031199536 0.030387428 TLS 0.0302 0.030084496 -0.002643049 0.029211933 NAB 0.0302 1 -0.035751788 -0.035751788 BHP 0.0302 0.392972071 0.007385825 0.021234666 3. Optimal portfolio 3.1 Construct an optimal portfolio with or without short-selling An optimal portfolio exists when the returns from an investment are maximized at a certain level of risk, or the risk is minimized given a certain level of portfolio. This is according to the Markowitz approach, which provides specific weighting scheme if the variance, expected return and variance are known with certainty. In our case, we use the Excel add-in solver to design a system for optimizing the weights. When short selling is allowed, the weights of an investment can be less or greater than zero, but when short selling is not allowed, the weights are usually greater than zero. Using the solver, the results were as shown below: First, we had to calculate the sharp ratio by dividing the effective return with the standard deviation, which we found to be 1.838378. Next, we used the solver to calculate the optimal weights, which we found to be: Target Cell (Max) CELL Name Original value Final value $H$69 Expected return 0.003170884 0.003170884 Adjustable cells CELL Name Original Value Final value $A$82 AGK weight 0.2 0.301519 $A$83 WOW weight 0.2 0 $A$84 TLS weight 0 0 $A$85 NAB weight 0.2 0.698481 $A$86 BHP weight 0.4 0 Constraints Beta = 1 Total Weight = 1 3.2 For one to short sell in ASX, the shares had to be 50 million with a capitalization of $100 million and a high liquidity level. Since Ms. Pretty did not specify the amount she won in the lottery, we assumed a dollar amounting to $50 million. This means that she would no short sell her portfolio since it does not meet the ASX requirement. The weights of the optimal portfolio were as shown above: 69% of NAB and 30% of AGK. The portfolio return was shown above while calculating the sharp ratio, and it was estimated to be 2.156%. Thus, the ending period value of the portfolio was estimated using the holding period return, which was estimated to be $51.078million. 4.0 Reasons for fluctuations of actual value from the estimated value The finance theory provides us with extremely powerful forecast but the actual values usually vary from the estimates. This is usually due to the forecasting error that arises due to calculation and the unsystematic risk in the listed companies that are unique to each organization. When estimating the expected return of a company using the Market Model, we usually have an error added to the value. This error represents the financial calculation error. However, while using the CAPM model and the SML line we draw the line of best fit using the systematic risk that is known, and use it to estimate the unsystematic risk that is unique to each organization. The forecasting error can also arise when the investor does not seek professional advice in making an investment decision. Some investors rely on media information, which in most cases is exaggerated. Because of this, they end up losing since they forecasted using incorrect information. Finally, the forecasting error can arise due to inefficiency of a market. The Australian market for example is semi strong and as such, the problem of insider trading can cause fluctuation in the actual return from the forecasted returns. It is clear that the quantitative analysis cannot be fully reliable in making an investment decision, and as such, there is a need to consider professional advice in making an investment decision regarding a portfolio. In addition to this, the portfolio error of estimation does exist, and as such, it can cause the forecasted value to differ from the actual value of a return. This forecasting error however can be reduced by diversifying the investor’s portfolio as shown above, as well as by critically analyzing the data using both descriptive statistics and regression analysis as shown above in Ms. Pretty’s study. Reference Linden. M, (2001). A Model for Stock Return Distribution: International Journal of Finance & Economics, 6,2; pp.159-169. Appendix 1 Monthly Share Prices Name AGK WOW TLS NAB BHP Code 923040(P) 322714(P) 871685(P) 901842(P) 906169(P) CURRENCY A$ A$ A$ A$ A$ 8/1/2007 14.619 26.55 4.49 37.08 35.96 9/1/2007 14.871 29.47 4.41 39.92 38.57 10/1/2007 15.248 29.73 4.31 39.8 44.5 11/1/2007 12.257 33.33 4.71 43.5 46.9 12/1/2007 12.586 33.39 4.68 38.8 43.2 1/1/2008 12.896 33.99 4.69 37.79 40.14 2/1/2008 11.811 29.4 4.41 35.4 38.55 3/1/2008 10.407 28.83 4.66 27.18 38.59 4/1/2008 10.495 28.8 4.42 29.13 36.65 5/1/2008 11.705 28.22 4.62 30.1 42.9 6/1/2008 13.864 27.91 4.79 30.43 45.49 7/1/2008 13.651 24 4.23 25.83 44.4 8/1/2008 12.934 25.76 4.52 24.34 39.11 9/1/2008 14.416 27.63 4.3 24.65 41.08 10/1/2008 13.544 28.7 4.28 25.7 32.75 11/1/2008 13.68 29.63 4.2 25.1 29.9 12/1/2008 15.055 27.3 4.05 19.43 29.9 1/1/2009 14.774 26.67 3.83 20.87 30.44 2/1/2009 13.893 27.15 3.73 18.71 30 3/1/2009 12.867 26.25 3.5 17.51 27.95 4/1/2009 14.803 24.89 3.09 20.32 32.1 5/1/2009 14.667 26.42 3.36 20.76 33 6/1/2009 13.351 25.4 3.09 22.52 35.74 7/1/2009 12.954 26.16 3.35 21.69 33.9 8/1/2009 14.106 26.8 3.52 24.95 38.07 9/1/2009 13.438 28.6 3.28 29.19 37.29 10/1/2009 13.215 29.4 3.28 30.37 37.2 11/1/2009 13.36 28.25 3.24 28.9 36.71 12/1/2009 13.544 28.22 3.44 28.55 41.34 1/1/2010 13.612 28 3.43 27.4 43.12 2/1/2010 13.467 25.51 3.32 26.1 39.2 3/1/2010 14.183 27.54 2.94 25.61 40.98 4/1/2010 14.59 28.2 3 27.65 43.95 5/1/2010 14.435 26.99 3.18 27.9 39.53 6/1/2010 13.612 26.71 2.95 24.34 38.19 7/1/2010 14.512 26.64 3.18 23.03 37.11 8/1/2010 14.425 25.77 3.27 25.31 40.3 9/1/2010 14.812 28.23 2.79 23.85 37.91 10/1/2010 15.674 28.95 2.64 25 39.46 11/1/2010 15.597 28.19 2.68 25.89 42.3 12/1/2010 14.842 26.6 2.79 23.3 43.2 1/1/2011 14.745 26.97 2.79 23.7 45.25 2/1/2011 14.319 26.66 2.77 24.59 44.62 3/1/2011 14.203 27.1 2.78 25.57 46.37 4/1/2011 13.902 27.14 2.81 25.98 46.68 5/1/2011 14.028 26.51 2.88 27.27 46.15 6/1/2011 13.757 27.39 3.03 26.07 44.53 7/1/2011 14.057 27.77 2.92 25.48 43.77 8/1/2011 13.854 27.1 3.02 24.49 42.3 9/1/2011 14.948 25.5 3.06 23.85 39.87 10/1/2011 13.67 24.5 3.04 21.63 34.15 11/1/2011 13.68 23.67 3.11 25.1 36.77 12/1/2011 14.086 24.87 3.18 24.17 36.35 1/1/2012 13.873 25.1 3.33 23.36 34.42 2/1/2012 13.999 24.7 3.3 23.66 36.91 3/1/2012 13.147 25.3 3.3 23.58 35.55 4/1/2012 14.357 25.64 3.27 24.49 35.12 5/1/2012 14.619 26.17 3.57 25.21 35.97 6/1/2012 15.2 26.63 3.64 22.47 31.73 7/1/2012 14.68 26.86 3.73 23.76 31.72 8/1/2012 15.64 28.72 4.04 25.11 31.7 Read More