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Ch_03_Tool_Kit - Ch 03 Tool Kit Chapter 3 Tool Kit for Risk...

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Unformatted text preview: Ch 03 Tool Kit 5/26/2002 Chapter 3. Tool Kit for Risk and Return The relationship between risk and return is a fundamental axiom in finance. Generally speaking, it is totally logical to assume that investors are only willing to assume additional risk if they are adequately compensated with additional return. This idea is rather fundamental, but the difficulty in finance arises from interpreting the exact nature of this relationship (accepting that risk aversion differs from investor to investor). Risk and return interact to determine security prices, hence its paramount importance in finance. PROBABILITY DISTRIBUTION The probability distribution is a listing of all possible outcomes and the corresponding probability. Demand for the company's products Probability of this demand occurring Strong Normal Weak 0.30 0.40 0.30 1.00 Rate of Return on stock if this demand occurs Martin Products U.S. Water 100% 20% 15% 15% -70% 10% EXPECTED RATE OF RETURN The expected rate of return is the rate of return that is expected to be realized from an investment. It is determined as the weighted average of the probability distribution of returns. Demand for the company's products Strong Normal Weak Probability of this demand occurring Martin Products Rate of Return Product 0.3 100% 0.4 15% 0.3 -70% 1.0 EXPECTED RATE OF RETURN, r hat 30% 6% -21% = U.S. Water Rate of Return 20% 15% 10% 15% Product 6% 6% 3% 15% MEASURING STAND-ALONE RISK: THE STANDARD DEVIATION To calculate the standard deviation, there are a few steps. First find the differences of all the possible returns from the expected return. Second, square that difference. Third, multiply the squared number by the probability of its occurrence. Fourth, find the sum of all the weighted squares. And lastly, take the square root of that number. Let us apply this procedure to find the standard deviation of Martin Products' returns. Demand for the company's products Strong Normal Weak Probability of this demand occurring 0.3 0.4 0.3 Deviation from r hat Squared deviation Martin Products 85% 72.25% 0% 0.00% -85% 72.25% Sum: Std. Dev. = Square root of sum Sq Dev * Prob. 21.68% 0.00% 21.68% 43.35% 65.84% 65.84% Sq. root can be found in two ways Strong Normal Weak Probability of this demand occurring 0.3 0.4 0.3 5% 0% -5% U.S. Water 0.25% 0.00% 0.25% Std. Dev. = Square root of sum 0.08% 0.00% 0.07% 0.15% 3.87% 3.87% Sq. root can be found in two ways MEASURING STAND-ALONE RISK: THE COEFFICIENT OF VARIATION The coefficient of variation indicates the risk per unit of return, and is calculated by dividing the standard deviation by the expected return. Std. Dev. Expected return CV Martin Products 65.84% 15% 4.39 U.S. Water 3.87% 15% 0.26 PORTFOLIO RETURNS The expected return on a portfolio is simply a weighted average of the expected returns of the individual assets in the portfolio. Consider the following portfolio. Stock Microsoft General Electric Pfizer Coca-Cola Portfolio's Expected Return Portfolio weight 0.25 0.25 0.25 0.25 Expected Return 12.0% 11.5% 10.0% 9.5% 10.75% PORTFOLIO RISK Perfect Negative Correlation. The standard deviation of a portfolio is generally not a weighted average of individual standard deviations--usually, it is much lower than the weighted average. The portfolio's SD is a weighted average only if all the securities in it are perfectly positively correlated, which is almost never the case. In the equally rare case where the stocks in a portfolio are perfectly negatively correlated, we can create a portfolio with absolutely no risk. Such is the case for the next example of Portfolio WM, a portfolio composed equally of Stocks W and M. Portfolio WM Year Stock W returns Stock M returns (Equally weighted avg.) 1998 40% -10% 15% 1999 -10% 40% 15% 2000 35% -5% 15% 2001 -5% 35% 15% 2002 15% 15% 15% Average return 15% 15% 15% Standard deviation 22.64% 22.64% 0.00% Correlation Coefficient -1.00 These two stocks are perfectly negatively correlated--when one goes up, the other goes down by the same amount. We could use Excel's correlation function to find the correlation, but when exact positive or negative correlation occurs, an error message is given. We demonstrate correlation in a later section. Perfect Positive Correlation. Now suppose the stocks were perfectly positively correlated, as in the following example: Year 1998 1999 2000 2001 2002 Average return Standard deviation Correlation Coefficient Stock M returns -10% 40% -5% 35% 15% 15% 22.64% Stock M' returns -10% 40% -5% 35% 15% 15% 22.64% Portfolio MM' -10% 40% -5% 35% 15% 15% 22.64% 1.00 With perfect positive correlation, the portfolio is exactly as risky as the individual stocks. Partial Correlation. Now suppose the stocks are positively but not perfectly so, with the following returns. What is the portfolio's expected return, standard deviation, and correlation coefficient? Year 1998 1999 2000 2001 2002 Average return Standard deviation Correlation coefficient Stock W returns 40% -10% 35% -5% 15% 15% 22.64% Stock Y returns 28% 20% 41% -17% 3% 15% 22.57% Portfolio WY 34% 5% 38% -11% 9% 15% 20.63% 0.67 Here the portfolio is less risky than the individual stocks contained in it. We found the correlation coefficient by using Excel's "CORREL" function. Click the wizard, then Statistical, then CORREL, and then use the mouse to select the ranges for stocks W and Y's returns. The correlation here is about what we would expect for two randomly selected stocks. Stocks in the same industry would tend to be more highly correlated than stocks in different industries. THE CONCEPT OF BETA The beta coefficient reflects the tendency of a stock to move up and down with the market. An average-risk stock moves equally up and down with the market and has a beta of 1.0. Beta is found by regressing the stock's returns against returns on some market index. It is also useful to show graphs with individual stocks' returns on the vertical axis and market returns on the horizontal axis. The slopes of the lines represent the stocks betas. We show a graph of the illustrative stocks in the screen to the right, and we use Beta Graphs regression to calculate betas below. We used the scatt and stock returns Returns on The Market and on Stocks L (for Low), A (for Average), and H (for High) rM rL rA rH 10% 10% 10% 10% 20% 15% 20% 30% -10% 0% -10% -30% Regression analysis is performed by following the command path: Tools => Data Analysis => Regression. This will yield the Regression input box. If Data Analysis is not an option in your Tools menu, you will have to load that program. Click on the Add-Ins option in the Tools menu. When the Add-Ins box appears, click on Analysis ToolPak and a check mark will appear next to the Analysis ToolPak. Then, click OK and you will now be able to access Data Analysis. From this point, you must designate the Y input range (stock returns) and the X input range (market returns). You can have the summary output placed in a new worksheet, or you can have it shown directly in the worksheet, as we did here. The Regression dialog box for the regression of Stock H is as follows: 30% Stocks returns Year 2000 2001 2002 0% -30% -10% Note: When you get the menu box on the screen, and the cursor blinking in the Y Range slot, use the mouse to select the Y range, and then click on the X range box. Then fill in the X range the same way. Regression Output of Stock H Returns SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations Beta Coefficient for Stock H = 2.00 1 1 1 0 3 ANOVA df Regression Residual Total SS 1 1 2 Coefficients Intercept X Variable 1 MS 0.19 0 0.19 Standard Error -0.1 2.00 0 0 F 0.19 0 t Stat -1.10E+016 3.11E+016 Significance F 9.69E+032 P-value 0 0 0 Lower 95% Upper 95% -0.1 -0.1 2 2 Regression Output of Stock A Returns SUMMARY OUTPUT Beta Coefficient for Stock A is 1 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 1 1 1 0 3 ANOVA df Regression Residual Total SS 1 1 2 Coefficients Intercept X Variable 1 MS 0.05 0 0.05 Standard Error 0 1.00 F 0.05 0 t Stat 0 0 #NUM! P-value 65535 65535 Significance F #NUM! Lower 95% #NUM! #NUM! 0 1 Upper 95% 0 1 Regression Output of Stock L Returns SUMMARY OUTPUT The beta coefficient for Stock L is .5 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 1 1 1 0 3 ANOVA df Regression Residual Total SS 1 1 2 Coefficients Intercept X Variable 1 MS 0.01 0 0.01 Standard Error 0.05 0.5 0 0 F 0.01 0 t Stat 3.67E+015 5.19E+015 Significance F 2.69E+031 P-value Lower 95% 0 0 CALCULATING THE BETA COEFFICIENT FOR AN ACTUAL COMPANY Now we show how to calculate beta for an actual company, Wal-Mart Stores. Step 1. Acquire Data We downloaded stock prices and dividends from http://finance.yahoo.com for Wal-Mart, using its ticker symbol WMT. We also downloaded data for the S&P 500 Index (^SPX), which contains 500 actively traded large stocks. Step 2. Calculate Returns We used the percentage change in adjusted prices (which already reflect dividends) for Wal-Mart to calculate returns. We used the percentage change for the S&P 500 Index as the market return. Now go to the bottom of the data, row 310. Date August-01 July-01 June-01 May-01 April-01 March-01 February-01 January-01 December-00 November-00 October-00 September-00 August-00 July-00 June-00 May-00 April-00 March-00 February-00 January-00 December-99 Market Level (S&P 500 Index) 1,294.0 1,341.0 1,386.8 1,424.2 1,250.3 1,257.4 1,414.5 1,476.0 1,454.7 1,532.3 1,566.8 1,696.0 1,739.6 1,731.7 1,746.7 1,663.3 1,663.8 1,712.0 1,674.2 1,677.2 1,715.0 Market Return -3.5% -3.3% -2.6% 13.9% -0.6% -11.1% -4.2% 1.5% -5.1% -2.2% -7.6% -2.5% 0.5% -0.9% 5.0% 0.0% -2.8% 2.3% -0.2% -2.2% 4.9% Wal-Mart Adjusted Stock Price 47.976 55.814 48.725 51.596 51.586 50.350 49.868 56.548 52.890 51.891 45.117 47.852 47.302 54.875 57.234 57.172 54.940 56.056 48.306 54.251 68.495 0 Wal-Mart Return -14.0% 14.5% -5.6% 0.0% 2.5% 1.0% -11.8% 6.9% 1.9% 15.0% -5.7% 1.2% -13.8% -4.1% 0.1% 4.1% -2.0% 16.0% -11.0% -20.8% 20.0% 0.05 0.5 Upper 95% 0.05 0.5 November-99 October-99 September-99 August-99 July-99 June-99 May-99 April-99 March-99 February-99 January-99 December-98 November-98 October-98 September-98 August-98 July-98 June-98 May-98 April-98 March-98 February-98 January-98 December-97 November-97 October-97 September-97 August-97 1,635.2 1,521.5 1,550.6 1,547.4 1,614.8 1,538.7 1,533.4 1,527.6 1,456.2 1,470.6 1,451.7 1,367.6 1,309.7 1,114.9 1,117.3 1,134.7 1,300.8 1,252.7 1,254.6 1,255.3 1,200.9 1,145.7 1,058.7 1,066.7 1,052.4 994.0 1,057.3 1,049.4 7.5% -1.9% 0.2% -4.2% 4.9% 0.3% 0.4% 4.9% -1.0% 1.3% 6.2% 4.4% 17.5% -0.2% -1.5% -12.8% 3.8% -0.2% -0.1% 4.5% 4.8% 8.2% -0.7% 1.4% 5.9% -6.0% 0.8% NA 57.057 55.758 47.094 43.829 41.789 47.724 42.111 45.445 45.538 42.499 42.437 40.185 37.125 34.044 26.927 29.048 31.078 29.909 27.103 24.860 24.983 22.736 19.545 19.361 19.635 17.153 17.950 17.368 2.3% 18.4% 7.4% 4.9% -12.4% 13.3% -7.3% -0.2% 7.2% 0.1% 5.6% 8.2% 9.0% 26.4% -7.3% -6.5% 3.9% 10.4% 9.0% -0.5% 9.9% 16.3% 1.0% -1.4% 14.5% -4.4% 3.4% NA Average (Annual) Standard deviation (Annual) Correlation between Wal-Mart and the market. Beta (using the SLOPE function) 6.9% 31.4% 18.7% 27.4% 0.51 34.5% Step 3. Examine the Data Using the AVERAGE function and the STDEV function, we found the average historical return and standard deviation for Wal-Mart and the market. (We converted these from monthly figures to annual figures. Notice that you must multiply the monthly standard deviation by the square root of 12, and not 12, to convert it to an annual basis.) These are show in the rows above. Notice that Wal-Mart has a standard deviation about twice that of the market. We also used the CORREL function to find the correlation between Wal-Mart and the market. Step 4. Plot the Data and Calculate Beta Using the Chart Wizard, we plotted the Wal-Mart returns on the y-axis and the market returns on the x-axis. We also used the menu Chart > Options to add a trend line, and to display the regression equation and R2 on the chart. The chart is shown below. Step 4. Interpret the Results The beta coefficient is about .51, as shown by the slope of the coefficient in the regression equation on the chart. The R2 of about 0.08 indicates that 8% of the variance in the stock return can be explained by the market. The rest of the stock's variance is due to factors other than the market. If we had done this regression for a portfolio of 50 well diversified stocks, we would have gotten an R2 of over 0.9. 30% 20% 10% 30% 20% 10% Historic Realized Returns on Wal-Mart, rS 0% -10% -20% -30% -30% -20% -10% 0% 10% 20% 30% Historic Realized Returns on the Market, rM THE SECURITY MARKET LINE The Security Market Line shows the relationship between a stock's beta and its expected return. Risk-free rate (Varies over time) Market return (Also varies over time) Beta (Varies by company) 6% 11% 0.5 Required Return for a stock with beta = 0.50 8.5% With the above data, we can generate a Security Market Line that will be flexible enough to allow for changes in any of the input factors. We generate a table of values for beta and expected returns, and then plot the graph as a scatter diagram. Beta 0.00 0.50 1.00 1.50 2.00 Required Return 8.5% 6.0% 8.5% 11.0% 13.5% 16.0% In order to create To create a data t variable changing changes. Therefo cell that is diagon is originally perfo Then, we highligh (B371 to C376, in the menu choices. of variables is arr input cell, like thi Security Market Line Required Return 18% 12% 6% 0% 0.00 Click on OK, and 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Beta The Security Market Line shows the projected changes in expected return, due to changes in the beta coefficient. However, we can also look at the potential changes in the required return due to variation of other factors, namely the market return and risk-free rate. In other words, we can see how required returns can be influenced by changing inflation and risk aversion. The level of investor risk aversion is measured by the market risk premium (rm-rrf), which is also the slope of the SML. Hence, an increase in the market return results in an increase in the maturity risk premium, other things held constant. We will look at two potential conditions as shown in the following columns: OR Scenario 1. Inflation Increases: Risk-free Rate Change in inflation Old Market Return New Market Return Beta Required Return 6% 2% 11% 13% 0.50 10.5% Scenario 2. Investors become more risk averse: Risk-free Rate 6% Old Market Return 11% Increase in MRP 2.5% New Market Return 13.5% Beta 0.50 Required Return 9.75% Now, we can see how these two factors can affect a Security Market Line, by creating a data table for the required return with different beta coefficients. Required Return Beta Original Situation New Scenario 1 New Scenario 2 8.5% 10.5% 9.75% 0.00 6.00% 8.00% 6.00% 0.50 8.50% 10.50% 9.75% 1.00 11.00% 13.00% 13.50% 1.50 13.50% 15.50% 17.25% 2.00 16.00% 18.00% 21.00% The SML Under Different Conditions 23% Required Returns 20% 18% 15% Column B Column C Column D 13% 10% 8% 5% 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Beta The graph shows that as risk as measured by beta increases, so does the required rate of return on securities. However, the required return for any given beta varies depending on the position and slope of the SML. We used the scatter diagram procedure to calculate the graphs, with Market returns on the horizontal axis and stock returns on the vertical axis. Beta Graph Stocks returns 30% Column C 0% -30% -10% Column D Column E 0% 10% 20% Market returns It is easy to see from the graph that H has the steepest slope, hence is most volatile, which means most risky. This is confirmed by the regression analysis, where betas are calculated. Lower 95.0% Upper 95.0% -0.1 -0.1 2 2 Lower 95.0% Upper 95.0% 0 0 1 1 Lower 95.0% Upper 95.0% 0.05 0.05 0.5 0.5 In order to create the data table that appears to the left, we used the "Data Table" feature in Excel. To create a data table, you must determine a calculation you would like to see simulated with a variable changing. In this case, we want to see the different calculations of expected return if beta changes. Therefore, we set up a vertical column of possible beta values (or, a "beta range"). In the cell that is diagonally up and to the right of this column, we reference the cell in which the calculation is originally performed (F364, in this case). Then, we highlight all of the cells that encompass the domain of the beta range and the calculation cell (B371 to C376, in this case). Then we click on "Data" from the menus above, and select "Table" from the menu choices. A dialog box will appear asking for the column or row input cell. In this case, our of variables is arranged in column format, so we will enter the variable we want to change into the column input cell, like this: Click on OK, and Excel will fill out the rest of the data table for you. ...
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