Final Papaer

FINC5001 GROUP ASSIGNMENT FINAL REPORT

Executive Summary In this group assignment, by historical data analysis, we evaluate the two approaches Mean-Variance and CAPM specific in the stock risk estimation for minimize risk investor. The two approaches are consistent in the stock risk, but differ in the risk of portfolios we construct. Through our observation and the approach assumption analysis which refer to academic literatures, the former one represents more reasonable result ultimately as we conclude in the last 2 pages in this report body. There are four sections in our report body. After the general conception understanding and discussion about requirement, the first problem we confront is the data collection as we explain in Part 1 of our report. We finally achieve the agreement to use 4 years monthly data via a lot of article reading and group meetings. The second part in our report is about the more accurate method for return calculation -- Continuous Compound formula which refers to “Research design issues in the estimation of Beta” (Brailsford, Tim, 1997). In the third section we present the detail of calculation and outcomes following the two approaches. It consists of tables and graphs with formula and some comments. The last section of our report is the analysis of two approaches’ assumption and the outcomes, ending with our conclusion as the recommendation. The reference, appendixes and coversheet are attached in the end as well.

1. Data Explanation
1) Stocks we choose: ANZ and AXA
2) Sample interval: Our raw-data is based on month both in Mean-Variance and CAPM: Against Yearly data, quarterly data and Daily data, monthly data is preferred: Yearly data was not accurate to analysis the return of a share price for the share price was announced daily. Since macroeconomic data are typically sampled at quarterly or monthly frequency, the standard approach is to match macro data with monthly or quarterly aggregates of financial series to build prediction models. (Should macroeconomic forecasters use daily financial data and how? By Elena Andreou, Eric Ghysels and Andros Kourtellos). However, data of intra-day intervals would result in unstable and unreliable estimate of beta, while using of quarterly data requires us to cover price histories of at least ten years to yield adequate data points (Brailsford, Tim, 1997). Consequently, we’d prefer to use the monthly data.
3) Sample period: We choose four years data which from Mar 31, 2006 to Mar 31, 2010 in both of Mean Variance approach and CAPM approach. To get a reliable beta, around 50 data points are required to be derived from 4 to 5 years' time, in which beta estimate appears reasonable stable(Brailsford, Tim,1997). The time period we chose is a large enough sample to ensure us get a reliable statistical estimate, and because we will use the estimate beta to study the risk and return of current stock portfolio, the recent data is more preferred because it has a larger relevance(Brailsford, Tim,1997).
4) Sample resources (refer to Appendix1) ANZ stock prices & dividends Morningstar DatAnalysis AXA stock prices & dividends Morningstar DatAnalysis S&P/ASX 200 Accumulation Index Reserve Bank of Australia Capital Market Yields Reserve Bank of Australia
2. Return calculation formula (refer to Appendix2.1) In both Mean-Variance approach and CAPM approach, refer to the monograph (Brailsford, Tim, 1997), we calculate return by continuous compound formula. Rt=ln[(Pt+Dt)/Pt-1] Rt – stock/Market return in period t, Pt -- stock price/Market Index at the end of measurement interval t, and Dt – stock/market dividend on period t (ASX200 Accum. Index including dividends) Continuous compound can reduce the effect of outlier data or the data error because the continuous compound data series is more likely to follow a normal distribution which is the assumption of both Mean-Variance and CAPM. It also removes the effect that higher price stock generates greater variation of stock price. (Brailsford, Tim,1997) In Stock Return calculation, as part of the Excel spreadsheets below, we allocate the dividends on the month which the“ex-dividend date” belonged to . [pic]
3. Calculation
1) Mean-Variance Approach Through calculation, the table below is the outcomes of 48 monthly return series in two stocks and their Mean, Standard Deviation and Covariance. (refer to Appendix 2.3) [pic] Mean (ANZ) = 0.0037, Mean(AXA)=0.0048 S.D. (ANZ) = 0.0738, S.D. (AXA) = 0.1058 Covariance of ANZ and AXA = 0.0022 The result shows, AXA has higher return than ANZ while the risk is also higher than ANZ. Their covariance is positive which indicate that they are positively correlated that is an increase in returns of ANZ is associated with an increase in returns of AXA. Formulas in Excel spreadsheets: Mean = AVERAGE(data) S.D = STDEV(data) Cov = COVAR(data)*48/47 (for sample)
Construct portfolios: (refer to Appendix2.4) [pic] The mean and Standard Deviation represents each portfolio’s return and the risk. Although the ANZ has lower return and risk, the portfolios show that the lowest risk appears in the point of 72.5% ANZ and 27.5% AXA. The risk of this portfolio is even lower than ANZ, and further more, with a higher return 0.00401 against 0.0037 of ANZ. Formulas in Excel spreadsheets: Portfolio S.D=SQRT(ANZ%2*ANZS.D2+AXA%2*AXAS.D2+2cov*ANZ%*AXA%) Portfolio Mean= ANZ%*(ANZMean) + AXA%*(AXAMean)
Draw the graph according to the portfolios constructed:
[pic]
The curve illustrates the risk and return from combining ANZ and AXA, the line plots the change in risk and return starting from an investment of 100% in ANZ, to100% in AXA. From this graph, we approximately figure our that the left-most point was about 72.5% investment in ANZ and 27.5% in AXA, which the portfolio will give the investors a minimum investing risk with a certain expected returns.
2) CAPM approach
Estimation of Beta: (refer to Appendix2.5)
The calculate process begins with the return calculation in market and bond, and then we can get the excess return both of stocks and market. Finally, we figure out the beta through the excess returns’ variance and covariance calculation. Details are listed in below table.
[pic]
Beta of ANZ = 0.90, Beta of AXA = 1.25 which means ANZ has lower risk than AXA. This outcome is consistent with the Mean-Variance approach in risk estimation.
For market return calculation: We choose S&P/ASX 200 Accumulation Index as market indicator. S&P/ASX 200 index measures 200 largest index-eligible stocks listed on ASX and cover approximately 80% of Australia equity market. S&P/ASX 200 Accumulation Index assume the dividends are all reinvested. It becomes the preferred benchmark accumulation index instead of AOAI. (Alex and Zhian Chen, 2006) Formula in Excel spreadsheets: Monthly Market Return = ln (It/It-1)
For risk free return calculation: We choose 10-years government bonds yield which estimated by RBA as risk free indicator. Australian government issues various bond like treasury or cooperates with a different period of 3-months, 6-months, 1-year, 3-year, 5-year, 10-year or even 20-year, 30year, however, to analyses the financial market, 10-year bond was recommended by most analysts. After suffering the enormous loss in the financial crisis, the investors tended to seek a more safety way to deal with their idle money, but not simply gain the low interest return paid by the bank’s saving account. Government bond was acknowledged to be safety and worthy investment. They are high-quality, having fixed return, tax free and most important, they are risk free. Formula in Excel spreadsheets: Monthly Risk free return = yield per an. / 12
For market risk premium calculation: We use the expected value of Rm-Rf to calculate the excess index return which is also known as the market risk premium. We should know that the data period we choose was during the financial crisis and economic recovery while the return of 10-year bond is positive, consequently, the stock return and the market return may be negative or less than the return of 10-year bond, so the market risk premium possibly is negative. We tried to find more ways to measure the market risk premium. The approach we use which comes from the textbook is Expect value of (rm-rf). The other way we referenced is one of forward-looking methods. “We focused on the Macroeconomic Model has the function as follows: [pic]Y[1]-rf But it is only appropriate for the developed countries where public stocks represent a relatively large share of the economy, implying that it is reasonable to believe there should be some relationship between macroeconomic variables and asset prices.”(Richard A. DeFusco, CFA, Dennis W. McLeavey, CFA, Jerald E. Pinto, CFA, and David E. Runkle, CFA, 2010)
The Expect Return of two stocks (refer to Appendix2.6)
[pic]
In CAPM approach, outcomes are contrary against the result in Mean-Variance approach. ANZ turns to get a higher return than AXA.
Formula in Excel spreadsheets:
Stock Expect Return = Mean of Rf + Beta * Mean of (Rm – Rf)
Construct Portfolios: (refer to Appendix2.7)
[pic]
When we add two returns together in CAPM Model, the formula of portfolio reveals the linear relationship between portfolio return and portfolio beta. Whatever the weight increase or decrease, the highest portfolio return equals to return of ANZ while the lowest portfolio risk also equals to the risk of ANZ.
Formulas in Excel spreadsheets:
Portfolio Beta = ANZ%*ANZ Beta + AXA%* AXA Beta
Portfolio Return = ANZ%*ANZ Return + AXA%*AXA Return
Draw the graph according to the portfolios Constructed:
[pic]
The line was downward in our graph. If beta was positive and the market risk premium was negative (as we explained above), than the expected return will have an inverse relationship with the beta. Though a higher risk should generate a higher return, it is quite abnormal to have a low return with a high risk, however, in the financial crisis environment, it should be acceptable.

4. Analysis and Recommendation
1) Understanding on two approaches
a. Mean-variance approach
Assumptions:
All investors are risk averse and they prefer less risk to more for the same level of expected return. Expected returns for all assets are known. The variances and covariances of all asset returns are known. Investors need only know the expected returns, variances, and covariances of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes of a distribution.
(Richard A. DeFusco, CFA, Dennis W. McLeavey, CFA, Jerald E. Pinto, CFA, and David E. Runkle, CFA, 2010)
Critiques:
In the period of 1950s, 20th century, an approach proposed by Markowitz which suggested that we can use the expected rate of return (E(R)) and variance ([pic]2) of investment to calculate the return and risk of investment, and that is the first time that the combination of statistics and linear projection was applied on the choice of portfolios (Alex and Zhian Chen, 2006). We all know that the mean-variance theory was established on a series of rigorous assumptions including the efficiency of market, all the investors are risk averse, the unit of assets can be divided infinitely and the rate of return of each asset attributes to the normal distribution and so on. So in the procedure of utilizing the mean-variance approach we found that two stages were involved in the theory, one is making distinction of efficient asset portfolios, another is selecting the assets based on the principle of maximizing the expected utility and recognizing the optimal portfolio. The efficient portfolio is the portfolio which affords the lowest risk on a given return or reaps the highest return on a given level of risk. In the approach of geometrical analysis, the sets of efficient portfolio known as the efficient frontier are derived from a series of critical lines. And the optimal asset portfolio is determined by the tangent point between the efficient frontier and the indifference curve representatives the investors’ utilities. Some constraints of mean-variance theory: The first one is the constraint on the measurement of risk. The theory takes the advantage of the variance of the return to measure the level of the risk, but it has the following flaws: the part excesses the expected return which will benefit the investor has been put into the extent of risk if we measure the risk as the variance or standard deviation of expected return. Additionally, using variance to display the risk only makes sense under the situation that the probability distribution of investing return submits to the normal distribution, yet more and more empirical studies show that the return of securities obeys the asymmetrical and irregular distribution. One more important point is that it is too simple to measure the risk by the parameter variance since the risk of portfolio is made up of multiple factors not only the fluctuation of expected return. The second restrict is about the assumptions. The gap exists between the thermo and reality because of the strict preconditions and which may have negative impact on its actual effect. First, the parameters of the model and estimate on the correlation among different securities have been founded on the historical data. In fact the historical situation is least possible to be repeated and the variable of a single security is going changing as the time passing. Second, the ignorance of transaction costs and taxes should be unreasonable. Investors are going to require higher return to compensate the loss resulted from the transaction costs and taxes and the efficient portfolio may be transferred to the inefficient ones.
b. CAPM approach
Assumptions:
“investors need only know the E(R), [pic]2and the covariances of returns to determine which portfolios are optimal for them. All the investors have identical views about risky assets’ mean returns, variances of returns, and correlations. Investors can buy and sell assets in any quantity without affecting price and all assets are marketable (can be traded). Investors are price takers. Investors can borrow and lend at the risk-free rate without limit, and they can sell short any asset in any quantity. Investors pay no taxes on returns and pay no transaction costs on trades.” (Richard A. DeFusco, CFA, Dennis W. McLeavey, CFA, Jerald E. Pinto, CFA, and David E. Runkle, CFA, 2010)
Critiques:
We should discuss something about our understanding on the CAPM cultivated during the assignment. This model is mainly taken advantage in asset pricing has three main parameters. The first one is the risk-free rate (rf). Treasury bond (T-bond) has always been considered the relative safe financial product without the credit risk. And the shorter the T-bond’s maturity, the safer it is. So in most situations the short-term interest rate of Treasury bill is thought to be the optimal choice for rf. Some other people utilize the spot short-term T-bill’s interest rate and the rate of return on market risk premium to calculate the [pic] or directly use the interest rate of long-term T-bonds as rf. In our assignment, rf is the interest rate of 10-year T-note. The second one is the required return of market (rm), we have to consider many factors when we choose this rate, such as the macroeconomics, political risk, the structure of market etc. In fact the index of the stock market is a crucial reference to set [pic]. The third one is the coefficient[pic], which suggests the market risk or the unsystematic risk. The operating leverage ratio, financial leverage ratio and the type of industry all affect[pic]. Generally speaking, [pic] is often influenced by market portfolio and industry portfolio simultaneously, however, investor seldom concern on the industry. Different industries have their own characteristics. [pic] of firms varies after acquiring or selling its subsidiaries. It is prevailing now to estimate[pic] through looking for a comparable corporate and use the corporate’s [pic] as benchmark. Then the benchmark[pic]needs to be adjusted. Another scientific way is regression, but the requirement for the variable is hard to meet. We concerned on the application of CAPM at the same time. For instance, a company has to lower its financing cost and maximize the firm’s value in the financing activities. The financing cost consists by the weighted-average cost of common stock, preferred stock, bond, mortgages. When we calculate the cost of common stock, the CAPM could be used. Another example appears when a firm wants to invest some project. Whether the project has feasibility or not depends on its net present value. Discount rate should be the most important input in calculating the NPV and CAPM could help us select the proper discount rate according to the level of risk. Just like the mean-variance model, CAPM has its constraints as well which are mainly from the hypothesis. Let us look into the first assumption that investors just considered the E(R),[pic]2 when they make investment decision. Actually, the different distribution of returns will result in the same E(R) and [pic]2. However, CAPM did not differentiate so as to make the data imprecise. The problem of second assumption is it is impossible for all investors having the identical expectation as various individuals and not all of them are rational. Anyway, they cannot select the security based on the efficient portfolio. As we know, psychological factors dominate the investment decision in the process of investing stocks always lead investors make the wrong judgment. On the other side, just a little people can catch the efficient portfolio and a large proportion of investors do not understand the theory of investment portfolio who buy or sell entirely depend on their subjective opinion or proposals from others. The last assumption said that there is no friction in the capital market and all the information can be got freely. We all know that in reality this situation should not come into existence. All the parameters of this modal are come from forecasting so as to there is going to be many deviations, even more, get the totally opposite conclusion. In the CAPM, [pic] is the only measurement for risk since CAPM illustrates that unsystematic risk can be diversified by efficient portfolio. This statement should be too ideal. The market risk should not be the only element influences the return, yet, CAPM is a single-factor model which just brings[pic] into asset pricing. These flaws are inevitable.

2) Conclusion &Recommendation combined with our assignment: Although there is a gap between the returns in the two approaches, the two approaches are consistent in risk estimation that ANZ is less risky than AXA. Its S.D. is 0.0738 and Beta is 0.90, both are less than AXA’s 0.1058 and 1.25. But comparing portfolios we constructed in the two approaches, the outcomes shows discrepancy. Following the Mean-variance model, the portfolios’ expected return goes up as well as the weights of AXA’s increase. While as, the standard deviation of portfolios decreases firstly and then increases. Therefore, the most preferred portfolio is investing 72.5% on ANZ and 27.5% on AXA for the investors who chase for minimizing the risk of their investment, where the minimum risk is 0.06762 and return is 0.401%. Contrarily, in CAPM model, this point on the line indicates that when the beta is 1 the return is 0.2566%, which is neither the maximum return nor the lowest risk point. The character of the line of CAMP approach is when the weights of ANZ are going down, the expected returns of portfolios go under the same pattern, and however the betas are rising. Thus the most preferred point is the minimum risk portfolio which comprised of 100% of ANZ and 0% of AXA, its beta is 0.90 and return is 0.277%. Per our understanding and critiques, we prefer choose portfolio from Mean-variance model. There are three main reasons: Firstly, use beta to describe the risk of stock is too ideal. In some degree, beta is more like a correlation parameter between the stock variation and the market variation rather than a risk parameter. To a stock with greater beta, the higher return in the market, the higher return the stock achieve. Oppositely, from our observation under financial crisis environment, the greater the beta is, the less return the stock has. Secondly, the outcome in CAPM model that 100% invest in ANZ is contrary to the CAPM assumption which state that unsystematic risk can be reduced by efficient portfolio. Further more, we can simply demonstrate that, from the CAPM model, the beta of the portfolio is linear with the weight once the stock beta is determined. That is to say, for minimize risk investor, the risks of constructed portfolios in CAPM is meaningless. Following CAPM model, investor should always choose the stock which has the lowest beta and invest all their money on that stock whatever how many stocks the portfolio is constructed by. It is also contrary to CAPM assumption.
Conclusion
For minimize risk investor, we recommend the portfolio which is 72.5% on ANZ and 27.5% on AXA. Based on historical statistics collected, the Mean-variance approach is popular. From the historical price statistics, the expected return, variance and covariance of all asset returns are easily calculated. In addition, the portfolio decision bases only on expected return and variance. Investors’ preferences are assumed to induce the favoring of higher returns and lower variances, who can only think about the portfolios on the efficient frontier, rather than the entire portfolio curve. Additionally, it quantifies the notion that diversification reduces risk. On the other side, the CAPM analysis has higher volatility and represents the historical returns poorly. More and more groups of interested reader question these findings.

Reference For Monographs: Harry M. Markowiz, Mar., 1991, Portfolio Selection: Efficient Diversification of Investment, Second Edition (Wiley Blackwell); Brailsford, Tim,1997, Research design issues in the estimation of Beta / Timothy J. Brailsford, Robert W. Faff, Barry Oliver(Sydney: McGraw-Hill). For Contributions to collective works: Alex frino, Zhian Chen, Amelia Hill,Carole Comerton-Forde and Simone kelly, 2006, Introduction to corporate finance(Pearson, Prentice Hall press,3); Richard A. DeFusco, CFA, Dennis W. McLeavey, CFA, Jerald E. Pinto, CFA, and David E. Runkle, CFA, 2010, Quantitative Methods for Investment Analysis, Second Edition: CFA® Program Curriculum, Level Ⅱ,Volume 6,Derivatives and Portfolio Management (Pearson Custom Publishing); Jerald Pinto, CFA, Elaine Henry, CFA, Thomas Robinson, CFA, and John Stowe, CFA, 2010, Equity Asset Valuation, Second Edition: CFA® Program Curriculum, Level Ⅱ, Volume 4,Equity (Pearson Custom Publishing). For Periodicals: M. S. Feldstein, Jan., 1969, Mean-Variance Analysis in the Theory of Liquidity Preference and Portfolio Selection, Review of Economic Studies, Volume 36, Issue 1, 5-12; G. Hanoch, and H. levy, Jul., 1969, The Efficiency Analysis of Choices Involving Risk, Review of Economics Studies, Volume 36, Issue 3, 335-346; William. F. Sharpe, Jun., 1967, Portfolio Analysis, Journal of Financial and Quantitative Analysis, Volume 2, Issue 2; Peter Bossaerts, and Charles Plott, 2002, The CAPM in thin experimental financial Markets, Journal of Economic Dynamics &Control, 26.

Appendixes
1. RawData segments
1.1 ANZ prices history
[pic]
1.2 ANZ dividends history
[pic]
1.3 AXA prices history
[pic]
1.4 AXA dividends history
[pic]
1.5 S&P/ASX 200 Accumulation Index history
[pic]
1.6 Capital Market Yields—10 years government bonds
[pic]
2. Calculation functions in Excel
2.1 Stock Return calculation
[pic]
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2.2 Market Return calculation
[pic]
2.3 Mean, Standard Deviation and Covariance calculation in Mean-Variance approach
[pic]
2.4 Portfolios calculation in Mean-Variance approach
[pic]
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2.5 CAPM calculation
[pic]
2.6 CAPM stock returns calculation
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2.7 CAPM portfolios calculation
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Faculty of Economics and Business
The University of Sydney
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[1] Y =Expected yield on the index