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7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model Ch 03 Tool Kit 1/28/2001 Chapter 3. Tool Kit for Risk and Return: Portfolio Theory and Asset Pricing Models PROBABILITY DISTRIBUTIONS The probability distribution is a listing of all possible outcomes and the corresponding probability. Probability of Occurrence Rate of Return Distribution E F G H 0.10 10% 6% 14% 4% 0.20 10% 8% 12% 6% 0.40 10% 10% 10% 8% 0.20 10% 12% 8% 15% 0.10 10% 14% 6% 22% 1.00 EXPECTED RATE OF RETURN AND STANDARD DEVIATION 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. 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. Calculation of expected return and standard deviation for E Expected rate of return for E Standard deviation for E Probability of Deviation from Squared Occurrence Rate of Return Product k hat deviation Sq Dev * Prob. 10% 10% 1.00% 0% 0.00% 0.00% 20% 10% 2.00% 0% 0.00% 0.00% 40% 10% 4.00% 0% 0.00% 0.00% 20% 10% 2.00% 0% 0.00% 0.00% 10% 10% 1.00% 0% 0.00% 0.00% 100% Sum: 0.00% Expected Std. Dev. = Rate of Return, Square root of k hat = 10% sum = 0.00% If the probabilities are fairly simple, then a short-cut method is to use the excel functions for AVERAGE and ay "weigh STDEVP, but to "trick" them by entering arguments more than once, in a way that "weights" them like ten probability. the probabilities. For example, for stock E we would enter 6% once, since it has only a one innten probability lity. We would enter 8% twice, since it has a two in ten probability. We would enter 10% 4 times, since it has a four in ten probability. We can do the same thing with the standard deviation function. Note that we use STDEVP and not STDEV, since we are measuring the standard deviation for the entire population and not for a sample. We call this the "indirect" method. Indirect method k hat = 10% s= 0.00% Calculation of expected return and standard deviation for F Michael C. Ehrhardt Page 1 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model Expected rate of return for F Standard deviation for F Probability of Deviation from Squared Occurrence Rate of Return Product k hat deviation Sq Dev * Prob. 10% 6% 0.60% -4% 0.16% 0.02% 20% 8% 1.60% -2% 0.04% 0.01% 40% 10% 4.00% 0% 0.00% 0.00% 20% 12% 2.40% 2% 0.04% 0.01% 10% 14% 1.40% 4% 0.16% 0.02% 100% Sum: 0.05% Expected Std. Dev. = Rate of Return, Square root of k hat = 10% sum = 2.19% Indirect method k hat = 10% s= 2.19% Calculation of expected return and standard deviation for G Expected rate of return for G Standard deviation for G Probability of Deviation from Squared Occurrence Rate of Return Product k hat deviation Sq Dev * Prob. 10% 14% 1.40% 4% 0.16% 0.02% 20% 12% 2.40% 2% 0.04% 0.01% 40% 10% 4.00% 0% 0.00% 0.00% 20% 8% 1.60% -2% 0.04% 0.01% 10% 6% 0.60% -4% 0.16% 0.02% 100% Sum: 0.05% Expected Std. Dev. = Rate of Return, Square root of k hat = 10% sum = 2.19% Indirect method k hat = 10.00% s= 2.19% Calculation of expected return and standard deviation for H Expected rate of return for H Standard deviation for H Probability of Deviation from Squared Occurrence Rate of Return Product k hat deviation Sq Dev * Prob. 10% 4% 0.40% -6% 0.36% 0.04% 20% 6% 1.20% -4% 0.16% 0.03% 40% 8% 3.20% -2% 0.04% 0.02% 20% 15% 3.00% 5% 0.25% 0.05% 10% 22% 2.20% 12% 1.44% 0.14% 100% Sum: 0.28% Expected Std. Dev. = Rate of Return, Square root of k hat = 10.00% sum = 5.27% Indirect method k hat = 10.00% s= 5.27% Michael C. Ehrhardt Page 2 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model COVARIANCE The covariance is a measure that combines the variance of a stock's return with the tendency of those returns to move up or down at the same time another stock moves up or down. To calculate the covariance, there are a few steps. First find the differences of all the possible returns from the expected return; do this for both stocks. Second, multiply the differences of both stocks. Third, multiply the previous product by the probability of its occurrence. Fourth, find the some of all the weighted products. The result is the covariance. Calculation of covariance between F and G Probability of Deviation of F Deviation of G Product of Product * Occurrence from k hat from k hat deviations Prob. 10% -4% 4% -0.1600% -0.02% 20% -2% 2% -0.0400% -0.01% 40% 0% 0% 0.0000% 0.00% 20% 2% -2% -0.0400% -0.01% 10% 4% -4% -0.1600% -0.02% 100% Covariance = sum = -0.048% Calculation of covariance between F and H Probability of Deviation of F Deviation of H Product of Product * Occurrence from k hat from k hat deviations Prob. 10% -4% -6% 0.2400% 0.02% 20% -2% -4% 0.0800% 0.02% 40% 0% -2% 0.0000% 0.00% 20% 2% 5% 0.1000% 0.02% 10% 4% 12% 0.4800% 0.05% 100% Covariance = sum = 0.108% Calculation of covariance between F and E Probability of Deviation of F Deviation of E Product of Product * Occurrence from k hat from k hat deviations Prob. 10% -4% 0% 0.0000% 0.00% 20% -2% 0% 0.0000% 0.00% 40% 0% 0% 0.0000% 0.00% 20% 2% 0% 0.0000% 0.00% 10% 4% 0% 0.0000% 0.00% 100% Covariance = sum = 0.000% Michael C. Ehrhardt Page 3 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model CORRELATION COEFFICIENT Like covariance, the correlation coefficient also measures the tendency of two stocks to move together, but it is standardized and it is always in the range of -1 to +1. The correlation coefficient is equal to the covariance divided by the product of the standard deviations. Calculation of the correlation between F and G rFG = Covariance FG ÷ SigmaF * SigmaG = -0.048% ÷ 2.19% 2.19% = -0.048% ÷ 0.048% rFG = -1.0 Calculation of the correlation between F and H rFH = Covariance FH ÷ SigmaF * SigmaH = 0.108% ÷ 2.19% 5.27% = 0.108% ÷ 0.116% rFH = 0.935 PORTFOLIO RISK AND RETURN: THE TWO-ASSET CASE Suppose there are two assets, A and B. X is the percent of the portfolio invested in asset A. Since the total percents invested in the asset must add up to 1, (1-X) is the percent of the portfolio invested in asset B. The expected return on the portfolio is the weighted average of the expected returns on asset A and asset B. ^ ^ ^ k p = X k A + (1 - X ) k B The standard deviation of the portfolio, sp, is not a weighted average. It is: sp = X2 s 2 + (1 - X) 2 sB + 2X(1 - X) rAB s A sB A 2 ATTAINABLE PORTFOLIOS: THE TWO ASSET-CASE Asset A Asset B Expected return, k hat 5% 8% Standard deviation, s 4% 10% Using the equations above, we can find the expected return and standard deviation of a portfolio with different percents invested in each asset. Michael C. Ehrhardt Page 4 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model Correlation = 1 Proportion of Proportion of Portfolio in Portfolio in Security A Security A (Value of X) (Value of 1-X) kp sp 1.00 0.00 5.00% 4.0% 0.90 0.10 5.30% 4.6% 0.80 0.20 5.60% 5.2% 0.70 0.30 5.90% 5.8% 0.60 0.40 6.20% 6.4% 0.50 0.50 6.50% 7.0% 0.40 0.60 6.80% 7.6% 0.30 0.70 7.10% 8.2% 0.20 0.80 7.40% 8.8% 0.10 0.90 7.70% 9.4% 0.00 1.00 8.00% 10.0% rAB = +1: Attainable Set of Risk/Return Combinations 10% Expected return 5% 0% 0% 5% 10% 15% Risk, sp Correlation = 0 Proportion of Proportion of Portfolio in Portfolio in Security A Security A (Value of X) (Value of 1-X) kp sp 1.00 0.00 5.00% 4.0% 0.90 0.10 5.30% 3.7% 0.80 0.20 5.60% 3.8% 0.70 0.30 5.90% 4.1% 0.60 0.40 6.20% 4.7% 0.50 0.50 6.50% 5.4% 0.40 0.60 6.80% 6.2% 0.30 0.70 7.10% 7.1% 0.20 0.80 7.40% 8.0% 0.10 0.90 7.70% 9.0% 0.00 1.00 8.00% 10.0% Michael C. Ehrhardt Page 5 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model rAB = 0: Attainable Set of Risk/Return Combinations 10% Expected return 5% 0% 0% 5% 10% 15% Risk, sp Correlation = -1 Proportion of Proportion of Portfolio in Portfolio in Security A Security A (Value of X) (Value of 1-X) kp sp 1.00 0.00 5.00% 4.0% 0.90 0.10 5.30% 2.6% 0.80 0.20 5.60% 1.2% 0.70 0.30 5.90% 0.2% 0.60 0.40 6.20% 1.6% 0.50 0.50 6.50% 3.0% 0.40 0.60 6.80% 4.4% 0.30 0.70 7.10% 5.8% 0.20 0.80 7.40% 7.2% 0.10 0.90 7.70% 8.6% 0.00 1.00 8.00% 10.0% rAB = -1: Attainable Set of Risk/Return Combinations 10% Expected return 5% 0% 0% 5% 10% 15% Risk, sp Expected Portfolio Efficient Set Michael C. Ehrhardt Page 6 4/19/2011 Return, kp 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model ATTAINABLE AND EFFICIENT PORTFOLIOS: MANY ASSETS Expected Portfolio Efficient Set Return, kp Feasible Set Risk, sp Feasible and Efficient Portfolios . OPTIMAL PORTFOLIOS Expected IB2 I Return, kp B1 Optimal Portfolio IA2 Investor B IA1 Optimal Portfolio Investor A Risk sp Optimal Portfolios . Efficient Set with a Risk-Free Asset Michael C. Ehrhardt Page 7 4/19/2011 Expected Z Return, kp .B 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model EFFICIENT SET WITH A RISK-FREE ASSET Efficient Set with a Risk-Free Asset Expected Z Return, kp .B ^ kM . M The Capital Market kRF A . Line (CML): New Efficient Set sM Risk, sp . OPTIMAL PORTFOLIO WITH A RISK-FREE ASSET Expected Return, kp CML I2 I1 . . ^ M kM ^ R k R R = Optimal kRF Portfolio sR sM Risk, sp . Michael C. Ehrhardt Page 8 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model CALCULATING BETAS 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 Wilshire 5000 Index (^WIL5), which contains most actively traded stocks, and the Fidelity Magellan mutual fund (FMAGX). We computed returns, as shown in Chapter 2. We also obtained the monthly rates on Treasury bills from the Federal Reserve, http://www.federalreserv.gov/. kRF, Risk-free kM, Market kp, Fidelity rate (Monthly Excess stock Return (Wilshire kS, Wal-Mart Magellan Fund return on T- Excess market return Date 5000 Index) Return Return bill) return (kM-kRF) (ks-kRF) Apr-00 -5.3% -2.0% -4.3% 0.47% -5.7% -2.5% Mar-00 7.5% 16.2% 9.1% 0.47% 7.1% 15.7% Feb-00 0.5% -11.0% 1.0% 0.46% 0.0% -11.4% Jan-00 -4.2% -20.8% -4.8% 0.44% -4.7% -21.2% Dec-99 5.6% 20.1% 6.9% 0.43% 5.2% 19.7% Nov-99 5.0% 2.3% 2.0% 0.42% 4.6% 1.9% Oct-99 6.3% 18.4% 6.0% 0.41% 5.9% 18.0% Sep-99 -2.7% 7.6% -1.5% 0.39% -3.1% 7.2% Aug-99 -3.4% 4.9% -1.2% 0.39% -3.8% 4.5% Jul-99 -0.9% -12.4% -3.4% 0.38% -1.3% -12.8% Jun-99 5.1% 13.4% 6.5% 0.38% 4.7% 13.1% May-99 -3.1% -7.3% -3.0% 0.38% -3.5% -7.7% Apr-99 5.6% -0.2% 2.4% 0.36% 5.2% -0.6% Mar-99 3.7% 7.3% 5.4% 0.37% 3.4% 6.9% Feb-99 -2.9% 0.1% -3.2% 0.37% -3.3% -0.2% Jan-99 2.7% 5.6% 5.6% 0.36% 2.3% 5.2% Dec-98 6.3% 8.4% 6.4% 0.37% 5.9% 8.0% Nov-98 5.7% 9.0% 7.8% 0.37% 5.4% 8.7% Oct-98 7.8% 26.4% 7.7% 0.33% 7.4% 26.1% Sep-98 6.4% -7.2% 6.0% 0.38% 6.0% -7.5% Aug-98 -15.7% -6.5% -15.5% 0.41% -16.1% -6.9% Jul-98 -2.3% 3.9% -0.7% 0.41% -2.7% 3.5% Jun-98 3.4% 10.5% 4.3% 0.42% 3.0% 10.1% May-98 -2.8% 9.0% -2.1% 0.42% -3.2% 8.6% Apr-98 1.1% -0.5% 1.2% 0.41% 0.7% -0.9% Mar-98 4.9% 10.1% 5.0% 0.42% 4.5% 9.6% Feb-98 7.1% 16.3% 7.6% 0.42% 6.7% 15.9% Jan-98 0.5% 1.0% 1.1% 0.42% 0.0% 0.5% Dec-97 2.1% -1.2% 1.2% 0.43% 1.7% -1.6% Nov-97 2.7% 14.5% 1.9% 0.43% 2.3% 14.0% Oct-97 -3.4% -4.4% -3.4% 0.41% -3.8% -4.9% Sep-97 5.8% 3.6% 5.9% 0.41% 5.4% 3.1% Aug-97 -3.9% -5.3% -4.4% 0.43% -4.3% -5.8% Jul-97 7.6% 10.9% 8.4% 0.42% 7.1% 10.5% Jun-97 4.5% 13.6% 4.1% 0.41% 4.0% 13.2% May-97 6.9% 6.2% 7.2% 0.42% 6.5% 5.8% Apr-97 4.2% 0.9% 4.5% 0.43% 3.8% 0.5% Mar-97 -4.5% 6.2% -3.4% 0.43% -5.0% 5.8% Feb-97 -0.3% 11.1% -1.3% 0.42% -0.7% 10.6% Jan-97 5.2% 4.4% 4.4% 0.42% 4.8% 4.0% Dec-96 -1.3% -10.4% -2.0% 0.41% -1.7% -10.8% Nov-96 6.4% -3.8% 6.5% 0.42% 6.0% -4.2% Oct-96 1.3% 0.5% 2.4% 0.42% 0.9% 0.1% Sep-96 5.2% 0.0% 4.0% 0.42% 4.7% -0.4% Aug-96 3.0% 10.3% 2.5% 0.42% 2.6% 9.9% Jul-96 -5.5% -5.4% -4.7% 0.43% -6.0% -5.8% Michael C. Ehrhardt Page 9 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model Jun-96 -1.0% -1.5% -0.1% 0.42% -1.4% -2.0% May-96 2.5% 8.4% 0.4% 0.42% 2.1% 8.0% Average (Annual) 19.8% 45.3% 21.5% 4.9% 14.9% 40.3% Standard deviation (Annual) 16.6% 32.3% 16.8% 0.1% 16.7% 32.3% Correlation with market. 0.54 0.97 -0.15 1.00 0.54 Slope 1.05 0.98 0.00 1.00 1.05 R-square 0.29 0.94 0.02 1.00 0.29 Using the AVERAGE function and the STDEV function, we found the average historical returns and standard deviations. (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. We also use the CORREL function to find the correlation of the market with the other assets Using the function Wizard for SLOPE, we found the slope of the regression line, which is the beta coefficient. We also use the function Wizard and the RSQ function to find the R-Squared of the regression. 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. We also used the regression feature to get more detailed data. These results are also shown below. Wal-Mart Analysis The beta coefficient is about 1.05, as shown by the slope coefficient in the regression equation on the chart. The beta coefficient has a t statistic of 4.39, and there is virtually a zero chance of getting this if the true beta coefficient is equal to zero. Therefore, this is a statistically significant coefficient. However, the confidence interval ranges from 0.57 to 1.54, which is very wide. The R2 of about .29 indicates that 29% of the variance of the stock return can be explained by the market. The rest of the stock's variance is due to factors other than the market. This is consistent with the wide scatter of points in the graph. Historic Wal-Mart Regression Results (See columns J-N) Realized Beta y = 1.054x + 0.0203 Returns Coefficient 1.05 R² = 0.2948 on Wal-Mart, kS t statistic 4.39 30% Probability of t stat. 0.0% Lower 95% confidence interval 0.57 20% Upper 95% confidence interval 1.54 Intercept 10% Coefficient 0.02 t statistic 1.68 0% Probability of t stat. 10.0% -30% -20% -10% 0% 10% 20% 30% Lower 95% confidence interval 0.00 Historic Realized Returns Upper 95% confidence interval 0.04 -10% on the Market, kM -20% -30% Michael C. Ehrhardt Page 10 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model Magellan Analysis The beta coefficient is about 0.98, as shown by the slope coefficient in the regression equation on the chart. The beta coefficient has a t statistic of 25.67, and there is virtually a zero chance of getting this if the true beta coefficient is equal to zero. Therefore, this is a statistically significant coefficient. The confidence interval ranges from 0.90 to 1.05, which is very small compared to the confidence interval for a single stock. The R2 of about 0.98 indicates that 98% of the variance of the portfolio return can be explained by the market. This is consistent with the very narrow scatter of points in the graph. The estimate of the intercept is equal to 0.00, and has a t statistic with a probability of 34.3%. Since this is greater than 5%, we would say that the coefficient is not statistically significant-- in other words, the true intercept might well be equal to zero. Wal-Mart Regression Results (See columns J-N) Historic Realized Beta y = 0.9757x + 0.0018 Returns Coefficient 0.98 R² = 0.9347 on Magellan, t statistic 25.67 kP 20% Probability of t stat. 0.0% Lower 95% confidence interval 0.90 Upper 95% confidence interval 1.05 10% Intercept Coefficient 0.00 t statistic 0.96 Probability of t stat. 34.3% Lower 95% confidence interval 0.00 0% Upper 95% confidence interval 0.01 -30% -20% -10% 0% 10% 20% 30% Historic Realized Returns on the Market, kM -10% -20% Michael C. Ehrhardt Page 11 4/19/2011 7c5f7cd0-61fa-49bc-bf49-3351e34b9f5d.xls Model The Market Model vs. CAPM We have been regressing the stock (or portfolio) returns against the market returns. However, CAPM actually states that we should regress the excess stock returns (the stock return minus the short-term risk free rate) against the excess market returns (the market return minus the short-term risk free rate). We show the graph for such a regression below. Notice that it is virtually identical to the market model regression we used earlier for Wal-Mart. Since it usually doesn't change the results whether we use the market model to estimate beta instead of the CAPM model, we usually use the market model. Excess Returns y = 1.0549x + 0.0205 on Wal-Mart, R² = 0.2956 kS-kRF 30% 20% 10% 0% -30% -20% -10% 0% 10% 20% 30% Excess Returns -10% on the Market, kM-kRF -20% -30% Michael C. Ehrhardt Page 12 4/19/2011