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8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Ch 05 Tool Kit 3/5/2005 Chapter 5. 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 r 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, r Square root of 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 STDEVP, but to "trick" them by entering arguments more than once, in a way that "weights" them like the probabilities. For example, for stock E we would enter 6% once, since it has only a one in ten probability. 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 r hat = 10% s= 0.00% Calculation of expected return and standard deviation for F Expected rate of return for F Standard deviation for F Michael C. Ehrhardt Page 1 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Probability of Deviation from Squared Occurrence Rate of Return Product r 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, r Square root of hat = 10% sum = 2.19% Indirect method r 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 r 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, r Square root of hat = 10% sum = 2.19% Indirect method r 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 r 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, r Square root of hat = 10.00% sum = 5.27% Indirect method r hat = 10.00% s= 5.27% Michael C. Ehrhardt Page 2 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.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, multiplythe 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 r hat from r 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 r hat from r 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 r hat from r 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 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.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. w A is the percent of the portfolio invested in asset A. Since the total percents invested in the asset must add up to 1, (1-w A) 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. ^ ^ ^ r p w A r A (1 w A ) r B The standard deviation of the portfolio, sp, is not a weighted average. It is: s p WAs A (1 WA )2 s B 2WA (1 WA ) r AB s A s B 2 2 2 ATTAINABLE PORTFOLIOS: THE TWO ASSET-CASE Asset A Asset B Expected return, r 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 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Correlation = 1 Proportion of Proportion of Portfolio in Portfolio in Security A Security B (Value of wA) (Value of 1-w B) rp 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 B (Value of wA) (Value of 1-w A) rp 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 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.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 B (Value of wA) (Value of 1-w A) rp 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 Michael C. Ehrhardt Page 6 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model ATTAINABLE AND EFFICIENT PORTFOLIOS: MANY ASSETS OPTIMAL PORTFOLIOS Michael C. Ehrhardt Page 7 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model EFFICIENT SET WITH A RISK-FREE ASSET OPTIMAL PORTFOLIO WITH A RISK-FREE ASSET Michael C. Ehrhardt Page 8 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model CALCULATING BETAS 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 (^SPX) which contains most actively traded stocks, and the Fidelity Magellan mutual fund (FMAGX). We computed returns, as shown in Chapter 4. We also obtained the monthly rates on 3-month Treasury bills from the FRED II data base at the St. Louis Federal Reserve, http://research.stlouisfed.org. rRF, Risk-free rM, Market rp, Fidelity rate (Monthly Excess stock Return (S&P Magellan Fund return on 3 Excess market return Date 500 Index) rj, GE Return Return month T-bill) return (rM-rRF) (rj-rRF) March 2003 0.8% 6.0% 1.1% 0.09% 0.7% 5.9% February 2003 -1.7% 4.7% -1.2% 0.10% -1.8% 4.6% January 2003 -2.7% -5.0% -2.8% 0.10% -2.8% -5.1% December 2002 -6.0% -9.5% -7.2% 0.10% -6.1% -9.6% November 2002 5.7% 7.4% 5.2% 0.10% 5.6% 7.3% October 2002 8.6% 2.4% 9.4% 0.13% 8.5% 2.3% September 2002 -11.0% -17.7% -10.8% 0.14% -11.1% -17.8% August 2002 0.5% -6.4% 1.0% 0.14% 0.4% -6.5% July 2002 -7.9% 10.8% -7.2% 0.14% -8.0% 10.7% June 2002 -7.2% -6.1% -7.7% 0.14% -7.4% -6.3% May 2002 -0.9% -1.3% -0.3% 0.14% -1.1% -1.4% April 2002 -6.1% -15.6% -6.4% 0.14% -6.3% -15.8% March 2002 3.7% -2.9% 3.4% 0.15% 3.5% -3.0% February 2002 -2.1% 4.1% -1.7% 0.14% -2.2% 4.0% January 2002 -1.6% -7.3% -3.1% 0.14% -1.7% -7.4% December 2001 0.8% 4.6% 0.7% 0.14% 0.6% 4.4% November 2001 7.5% 5.7% 7.5% 0.16% 7.4% 5.6% October 2001 1.8% -2.1% 2.4% 0.18% 1.6% -2.3% September 2001 -8.2% -8.6% -8.1% 0.22% -8.4% -8.8% August 2001 -6.4% -6.0% -6.5% 0.28% -6.7% -6.3% July 2001 -1.1% -10.9% -1.5% 0.29% -1.4% -11.2% June 2001 -2.5% 0.0% -2.5% 0.29% -2.8% -0.3% May 2001 0.5% 1.0% 0.9% 0.30% 0.2% 0.7% April 2001 7.7% 15.9% 8.8% 0.32% 7.4% 15.6% March 2001 -6.4% -9.6% -6.4% 0.37% -6.8% -10.0% February 2001 -9.2% 1.1% -9.4% 0.41% -9.6% 0.7% January 2001 3.5% -4.1% 3.3% 0.43% 3.0% -4.5% December 2000 0.4% -3.0% 0.7% 0.48% -0.1% -3.4% November 2000 -8.0% -9.6% -8.5% 0.51% -8.5% -10.1% October 2000 -0.5% -5.2% -1.8% 0.51% -1.0% -5.7% September 2000 -5.3% -1.2% -5.4% 0.50% -5.8% -1.7% August 2000 6.1% 13.4% 6.4% 0.51% 5.6% 12.9% July 2000 -1.6% -2.2% -1.2% 0.50% -2.1% -2.7% June 2000 2.4% 0.7% 3.9% 0.47% 1.9% 0.2% May 2000 -2.2% 0.4% -3.4% 0.48% -2.7% -0.1% April 2000 -3.1% 1.0% -4.3% 0.47% -3.6% 0.6% March 2000 9.7% 17.9% 9.0% 0.47% 9.2% 17.4% February 2000 -2.0% -1.2% 1.0% 0.46% -2.5% -1.7% January 2000 -5.1% -13.4% -4.8% 0.44% -5.5% -13.8% December 1999 5.8% 19.2% 6.8% 0.43% 5.4% 18.8% November 1999 1.9% -4.0% 2.1% 0.42% 1.5% -4.4% October 1999 6.3% 14.3% 6.0% 0.41% 5.8% 13.9% September 1999 -2.9% 5.9% -1.5% 0.39% -3.2% 5.5% August 1999 -0.6% 3.0% -1.2% 0.39% -1.0% 2.6% July 1999 -3.2% -3.2% -3.4% 0.38% -3.6% -3.6% June 1999 5.4% 11.1% 6.5% 0.38% 5.1% 10.7% Michael C. Ehrhardt Page 9 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model May 1999 -2.5% -3.5% -3.0% 0.38% -2.9% -3.9% April 1999 3.8% -4.8% 2.4% 0.36% 3.4% -5.1% Average return (annual) -8.8% -3.4% -8.2% 3.7% -12.5% -7.0% Standard deviation (annual) 17.6% 29.2% 18.2% 0.5% 17.6% 29.1% Correlation with market return, r 0.66 0.99 0.09 1.00 0.66 R-square 0.44 0.98 0.01 1.00 0.44 Slope 1.09 1.02 0.00 1.00 1.09 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 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 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. GE Analysis The beta coefficient is about 1.10, as shown by the slope coefficient in the regression equation on the chart. The beta coefficient has a t statistic of 7.07, 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 0.78 to 1.41, which is very wide. The R2 of about 0.44 indicates that 44% 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. GE Regression Results (See columns J-N) Historic Realized Beta Returns Coefficient 1.09 rj = 1.0924 rM + 0.0053 on GE, rj(%) R2 = 0.4358 t statistic 5.96 30% Probability of t stat. 0.0% Lower 95% confidence interval 0.72 Upper 95% confidence interval 1.46 20% Intercept Coefficient 0.01 10% t statistic 0.56 Probability of t stat. 57.6% 0% Lower 95% confidence interval -0.01 -30% -20% -10% 0% 10% 20% 30% Upper 95% confidence interval 0.02 -10% Historic Realized Returns on the Market, rM(%) -20% Michael C. Ehrhardt Page 10 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model -20% Magellan Analysis The beta coefficient is about 1.02, as shown by the slope coefficient in the regression equation on the chart. The beta coefficient has a t statistic of 42.86, 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 0.97 to 1.07, 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 57.0%. 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. Magellan Regression Results (See columns J-N) rp = 1.0206 rM + 0.0007 Historic Realized Beta R2 = 0.9756 Returns Coefficient 1.02 on Magellan, t statistic 42.86 rP(%) 20% Probability of t stat. 0.0% Lower 95% confidence interval 0.97 Upper 95% confidence interval 1.07 10% Intercept Coefficient 0.00 t statistic 0.57 Probability of t stat. 57.0% Lower 95% confidence interval 0.00 0% Upper 95% confidence interval 0.00 -30% -20% -10% 0% 10% 20% 30% Historic Realized Returns on the Market, rM(%) -10% -20% Michael C. Ehrhardt Page 11 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.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 GE. 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. (See columns J-N) CAPM (excess return) Model Regression Results Beta Excess Returns on GE, rS-rRF Coefficient 1.09 y = 1.0881x + 0.0055 t statistic 5.92 R2 = 0.4326 30% Probability of t stat. 0.0% Lower 95% confidence interval 0.72 Upper 95% confidence interval 1.46 20% Intercept Coefficient 0.01 10% t statistic 0.58 Probability of t stat. 56.3% Lower 95% confidence interval -0.01 Upper 95% confidence interval 0.02 0% -30% -20% -10% 0% 10% 20% 30% Excess Returns -10% on the Market, rM-rRF -20% Table 5-4 Regression Probability of Lower 95% Upper 95% Coefficient t Statistic t Statistic Confidence Confidence Panel a: General Electric (Market model) Intercept 0.01 0.56 0.58 -0.01 0.02 Slope 1.09 5.96 0.00 0.72 1.46 Panel b: Magellan Fund (Market model) Intercept 0.00 0.57 0.57 0.00 0.00 Slope 1.02 42.86 0.00 0.97 1.07 Panel c: General Electric (CAPM: Excess returns) Intercept 0.01 0.58 0.56 -0.01 0.02 Slope 1.09 5.92 0.00 0.72 1.46 Note: The market model uses unadjusted returns, the CAPM model uses returns in excess of the risk-free rate. Michael C. Ehrhardt Page 12 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Peformance Measures for Magellan Jensens's Alpha Intercept from CAPM regression 0.91% per year 0.62 t statistic 54.020% Probability that the intercept is not zero Sharpe's Reward-to-Variability Ratio Average annual return in excess of risk-free rate divided by standard deviation Magellan -11.8% divided by 18.2% -0.65 S&P 500 -12.5% divided by 17.6% -0.71 Treynor's Reward-to-Volatility Ratio Average annual return in excess of risk-free rate divided by beta Magellan -11.8% divided by 1.02 -0.116 S&P 500 -12.5% divided by 1.00 -0.125 Michael C. Ehrhardt Page 13 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 14 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 15 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 16 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 17 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 18 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 19 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 20 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 21 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Excess portfolio return (rp-rRF) 1.0% -1.3% -2.9% -7.3% 5.1% 9.3% -10.9% 0.8% -7.3% -7.8% -0.5% -6.5% 3.2% -1.9% -3.3% 0.6% 7.3% 2.2% -8.3% -6.8% -1.8% -2.8% 0.6% 8.4% -6.8% -9.8% 2.8% 0.2% -9.0% -2.3% -5.9% 5.9% -1.7% 3.4% -3.9% -4.8% 8.6% 0.5% -5.3% 6.4% 1.6% 5.6% -1.9% -1.6% -3.7% 6.1% Michael C. Ehrhardt Page 22 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model -3.4% 2.0% -11.8% 18.2% 0.99 0.97 1.02 (See columns J-N) SUMMARY OUTPUT 1.09 Regression Statistics Multiple R 0.660151 R Square 0.4358 0.423535 Adjusted R Square 0.063974 Standard Error Observations 48 ANOVA df SS MS F Significance F Regression 1 0.145419213 0.145419 35.53134 3.31E-07 Residual 46 0.188264329 0.004093 Total 47 0.333683542 CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%Lower 95.0% Intercept 0.005252 0.009331973 0.562764 0.576328 -0.01353 0.024036 -0.01353 X Variable 11.092423 0.183267374 5.960817 3.31E-07 0.723526 1.461321 0.723526 Michael C. Ehrhardt Page 23 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model (See columns J-N) SUMMARY OUTPUT 1.02 Regression Statistics Multiple R 0.987709 R Square 0.975568 0.975037 Adjusted R Square 0.008313 Standard Error Observations 48 ANOVA df SS MS F Significance F Regression 1 0.126920945 0.126921 1836.783 9.92E-39 Residual 46 0.003178581 6.91E-05 Total 47 0.130099526 CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%Lower 95.0% Intercept 0.000694 0.001212568 0.572575 0.569721 -0.00175 0.003135 -0.00175 X Variable 11.020579 0.0238132 42.85771 9.92E-39 0.972646 1.068513 0.972646 Michael C. Ehrhardt Page 24 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model (See columns J-N) SUMMARY OUTPUT 1.09 Regression Statistics Multiple R 0.657757 R Square 0.432644 0.42031 Adjusted R Square 0.063991 Standard Error Observations 48 ANOVA df SS MS F Significance F Regression 1 0.143638947 0.143639 35.07783 3.78E-07 Residual 46 0.188363734 0.004095 Total 47 0.33200268 CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%Lower 95.0% Intercept 0.005488 0.009432336 0.58187 0.563495 -0.0135 0.024475 -0.0135 X Variable 11.088092 0.183716906 5.922654 3.78E-07 0.718289 1.457894 0.718289 Michael C. Ehrhardt Page 25 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model LINEST Results: y=mx+b Read me. Slope (m) 1.02050123 0.000756 Intercept (b) m Std. Error of 0.023867615 0.001225 Std. Error of b R2 0.975455386 0.008313 Std. Error of y F 1828.138238 46 Degrees of freedom SS Regression 0.126347973 0.003179 SS Residual 42.75673325 0.617118 t-stat for intercept t-stat for slope Prob of t 1.10246E-38 0.540199 Prob of t Michael C. Ehrhardt Page 26 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 27 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 28 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 29 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 30 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 31 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 32 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 33 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 34 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Michael C. Ehrhardt Page 35 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Upper 95.0% 0.024036 1.461321 Michael C. Ehrhardt Page 36 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Upper 95.0% 0.003135 1.068513 Michael C. Ehrhardt Page 37 4/30/2012 8d5577e6-e498-474c-8b93-4cd9bac3deb0.xls Model Upper 95.0% 0.024475 1.457894 Michael C. Ehrhardt Page 38 4/30/2012