Risk and Return Calculation

4/15/2004 Chapter 5. 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 Strong Normal Weak Probability of this demand occurring 0.30 0.40 0.30 1.00 Rate of Return on stock if this demand occurs Sale.com Basic Foods 100% 40% 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 Sale.com Rate of Return Basic Foods Rate of Return 40% 15% -10% Product 30% 6% -21% Product 12% 6% -3% 15% 0.3 100% 0.4 15% 0.3 -70% 1.0 EXPECTED RATE OF RETURN, r hat = 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 Sale.com's returns. Demand for the company's products Strong Normal Weak Probability of this Deviation from r hat demand occurring 0.3 85% 0.4 0% 0.3 -85% Squared deviation Sale.com 72.25% 0.00% 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 25% 0% -25% Basic Foods 6.25% 0.00% 6.25% Std. Dev. = Square root of sum 1.88% 0.00% 1.88% 3.75% 19.36% 19.36% 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 Sale.com 65.84% 15% 4.39 Basic Foods 19.36% 15% 1.29 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 weight 0.25 0.25 0.25 0.25 Expected Return 12.0% 11.5% 10.0% 9.5% 10.75% Portfolio's Expected Return 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.) 2001 40% -10% 15% 2002 -10% 40% 15% 2003 35% -5% 15% 2004 -5% 35% 15% 2005 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 Stock M returns 2001 -10% 2002 40% 2003 -5% 2004 35% 2005 15% Average return 15% Standard deviation 22.64% Correlation Coefficient 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 2001 2002 2003 2004 2005 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 calculate betas below. Beta Graphs Year 2001 2002 2003 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: 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 1 R Square 1 Adjusted R Square 1 Standard Error 1.38778E-17 Observations 3 ANOVA df Regression Residual Total 1 1 2 Coefficients -0.1 2.00 SS 0.186666667 1.92593E-34 0.186666667 Standard Error 9.08514E-18 6.42417E-17 MS 0.186666667 1.92593E-34 F 9.69229E+32 Significance F 2.04488E-17 Beta Coefficient for Stock H = 2.00 Intercept X Variable 1 t Stat -1.1007E+16 3.11324E+16 P-value 5.78378E-17 2.04488E-17 Lower 95% -0.1 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 ANOVA df Regression Residual Total Coefficients Intercept X Variable 1 0 1.00 1 1 2 SS 0.046666667 0 0.046666667 Standard Error 0 0 MS 0.046666667 0 F #NUM! Significance F #NUM! 1 1 1 0 3 t Stat 65535 65535 P-value #NUM! #NUM! Lower 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 ANOVA df Regression Residual Total 1 1 2 Coefficients 0.05 0.5 SS 0.011666667 4.33334E-34 0.011666667 Standard Error 1.36277E-17 9.63625E-17 MS 0.011666667 4.33334E-34 F 2.6923E+31 Significance F 1.22693E-16 1 1 1 2.08167E-17 3 Intercept X Variable 1 t Stat 3.66899E+15 5.18874E+15 P-value 1.73513E-16 1.22693E-16 Lower 95% 0.05 0.5 CALCULATING THE BETA COEFFICIENT FOR AN ACTUAL COMPANY Now we show how to calculate beta for an actual company, General Electric. Step 1. Acquire Data We downloaded stock prices and dividends from http://finance.yahoo.com for General Electric, using its ticker symbol GE. We also downloaded data for the S&P 500 Index (^SPX), which contains 500 actively traded large stocks. For example, to download the GE data, enter its ticker symbol and click Go. then select Historical Prices from the left side of the page. After the daily prices come up, get monthly prices by entering the start date and the end date. Yahoo provides monthly prices as of the first trading day of the month. Note that these prices are "adjusted" to reflect any dividends or stock splits. Download the data by right-clicking below the data where its has "Dowload to Spreadsheet". Step 2. Calculate Returns We used the percentage change in adjusted prices (which already reflect dividends) for GE 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 317. Market Level (S&P 500 Index) Date April 2004 1,128.17 March 2004 1,126.21 February 2004 1,144.94 January 2004 1,131.13 December 2003 1,111.92 November 2003 1,058.20 October 2003 1,050.71 September 2003 995.97 August 2003 1,008.01 July 2003 990.31 June 2003 974.50 May 2003 963.59 April 2003 916.92 March 2003 848.18 February 2003 841.15 January 2003 855.70 GE Adjusted Stock Price 30.48 30.52 32.52 33.43 30.79 28.31 28.65 29.44 29.02 27.91 28.15 27.98 28.71 24.86 23.45 22.38 Market Return 0.2% -1.6% 1.2% 1.7% 5.1% 0.7% 5.5% -1.2% 1.8% 1.6% 1.1% 5.1% 8.1% 0.8% -1.7% -2.7% GE Return -0.1% -6.2% -2.7% 8.6% 8.8% -1.2% -2.7% 1.4% 4.0% -0.9% 0.6% -2.5% 15.5% 6.0% 4.8% -5.0% December 2002 November 2002 October 2002 September 2002 August 2002 July 2002 June 2002 May 2002 April 2002 March 2002 February 2002 January 2002 December 2001 November 2001 October 2001 September 2001 August 2001 July 2001 June 2001 May 2001 April 2001 March 2001 February 2001 January 2001 December 2000 November 2000 October 2000 September 2000 August 2000 July 2000 June 2000 May 2000 April 2000 879.82 936.31 885.76 815.28 916.07 911.62 989.82 1,067.14 1,076.92 1,147.39 1,106.73 1,130.20 1,148.08 1,139.45 1,059.78 1,040.94 1,133.58 1,211.23 1,224.38 1,255.82 1,249.46 1,160.33 1,239.94 1,366.01 1,320.28 1,314.95 1,429.40 1,436.51 1,517.68 1,430.83 1,454.60 1,420.60 1,452.43 -6.0% 5.7% 8.6% -11.0% 0.5% -7.9% -7.2% -0.9% -6.1% 3.7% -2.1% -1.6% 0.8% 7.5% 1.8% -8.2% -6.4% -1.1% -2.5% 0.5% 7.7% -6.4% -9.2% 3.5% 0.4% -8.0% -0.5% -5.3% 6.1% -1.6% 2.4% -2.2% NA -4.9% 16.9% 61.0% 0.908 23.55 26.04 24.24 23.66 28.74 30.70 27.69 29.50 29.89 35.43 36.47 35.03 37.80 36.14 34.18 34.92 38.22 40.65 45.65 45.65 45.21 39.00 43.16 42.68 44.50 45.85 50.71 53.49 54.12 47.71 48.79 48.45 48.26 -9.6% 7.4% 2.5% -17.7% -6.4% 10.9% -6.1% -1.3% -15.6% -2.9% 4.1% -7.3% 4.6% 5.7% -2.1% -8.6% -6.0% -11.0% 0.0% 1.0% 15.9% -9.6% 1.1% -4.1% -2.9% -9.6% -5.2% -1.2% 13.4% -2.2% 0.7% 0.4% NA -8.3% 25.2% Correlation between GE and the market. Beta (using the SLOPE function) Step 3. Examine the Data Using the AVERAGE function and the STDEV function, we found the average historical return and standard deviation for GE 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 shown in the rows above. Notice that GE has a standard deviation about 1.5 times that of the market. We also used the CORREL function to find the correlation between GE and the market. Step 4. Plot the Data and Calculate Beta Using the Chart Wizard, we plotted the GE 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 R 2 on the chart. The chart is shown below. Step 4. Interpret the Results The beta coefficient is about 0.91, as shown by the slope of the coefficient in the regression equation on the chart. The R2 indicates that about 37% 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. Historic Realized Returns on GE, rj(%) rj = -0.0032 + 0.9084 rM R2 = 0.3725 20% 10% 0% -30% -20% -10% 0% 10% 20% 30% -10% Historic Realized Returns on the Market, rM (%) -20% 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. Required Return 8.5% 6.0% 8.5% 11.0% 13.5% 16.0% Beta 0.00 0.50 1.00 1.50 2.00 Security Market Line 18% Required Return 12% 6% 0% 0.00 0.50 1.00 Beta 1.50 2.00 2.50 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 25% Required Returns 20% 15% 10% 5% 0% 0.00 0.50 1.00 Beta 1.50 2.00 Original Scenario #1 Scenario #2 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.