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5-1 Chapter 5 Risk and Return © 2001 Prentice-Hall, Inc. Fundamentals of Financial Management, 11/e Created by: Gregory A. Kuhlemeyer, Ph.D. Carroll College, Waukesha, WI

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5-2 Risk and Return u Defining Risk and Return u Using Probability Distributions to Measure Risk u Attitudes Toward Risk u Risk and Return in a Portfolio Context u Diversification u The Capital Asset Pricing Model (CAPM) u Defining Risk and Return u Using Probability Distributions to Measure Risk u Attitudes Toward Risk u Risk and Return in a Portfolio Context u Diversification u The Capital Asset Pricing Model (CAPM)

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5-3 Defining Return Income received change in market price beginning market price Income received on an investment plus any change in market price, usually expressed as a percent of the beginning market price of the investment. D t P t - P t-1 D t + (P t - P t-1 ) P t-1 R =

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5-4 Return Example $10 $9.50 $1 dividend The stock price for Stock A was $10 per share 1 year ago. The stock is currently trading at $9.50 per share, and shareholders just received a $1 dividend. What return was earned over the past year?

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5-5 Return Example $10 $9.50 $1 dividend The stock price for Stock A was $10 per share 1 year ago. The stock is currently trading at $9.50 per share, and shareholders just received a $1 dividend. What return was earned over the past year? $1.00 $9.50$10.00 $ ($ $10.00 ) $10.00 R R = 5% = 5%

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5-6 Defining Risk What rate of return do you expect on your investment (savings) this year? What rate will you actually earn? Does it matter if it is a bank CD or a share of stock? What rate of return do you expect on your investment (savings) this year? What rate will you actually earn? Does it matter if it is a bank CD or a share of stock? The variability of returns from those that are expected.

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5-7 Determining Expected Return (Discrete Dist.) R = ( R i )( P i ) R is the expected return for the asset, R i is the return for the i th possibility, P i is the probability of that return occurring, n is the total number of possibilities. R = ( R i )( P i ) R is the expected return for the asset, R i is the return for the i th possibility, P i is the probability of that return occurring, n is the total number of possibilities. n i=1

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5-8 How to Determine the Expected Return and Standard Deviation Stock BW R i P i (R i )(P i ) Sum Stock BW R i P i (R i )(P i ) Sum The expected return, R, for Stock BW is.09 or 9%

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5-9 Determining Standard Deviation (Risk Measure) = ( R i - R ) 2 ( P i ) Standard Deviation Standard Deviation, , is a statistical measure of the variability of a distribution around its mean. It is the square root of variance. Note, this is for a discrete distribution. = ( R i - R ) 2 ( P i ) Standard Deviation Standard Deviation, , is a statistical measure of the variability of a distribution around its mean. It is the square root of variance. Note, this is for a discrete distribution. n i=1

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5-10 How to Determine the Expected Return and Standard Deviation Stock BW R i P i (R i )(P i ) (R i - R ) 2 (P i ) Sum Stock BW R i P i (R i )(P i ) (R i - R ) 2 (P i ) Sum

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5-11 Determining Standard Deviation (Risk Measure) = ( R i - R ) 2 ( P i ) = % =.1315 or 13.15% = ( R i - R ) 2 ( P i ) = % =.1315 or 13.15% n i=1

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5-12 Coefficient of Variation standard deviation mean The ratio of the standard deviation of a distribution to the mean of that distribution. RELATIVE It is a measure of RELATIVE risk. R CV = / R CV of BW =.1315 /.09 = 1.46 standard deviation mean The ratio of the standard deviation of a distribution to the mean of that distribution. RELATIVE It is a measure of RELATIVE risk. R CV = / R CV of BW =.1315 /.09 = 1.46

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5-13 Discrete vs. Continuous Distributions Discrete Continuous

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5-14 Determining Expected Return (Continuous Dist.) R = ( R i ) / ( n ) R is the expected return for the asset, R i is the return for the ith observation, n is the total number of observations. R = ( R i ) / ( n ) R is the expected return for the asset, R i is the return for the ith observation, n is the total number of observations. n i=1

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5-15 Determining Standard Deviation (Risk Measure) n i=1 = ( R i - R ) 2 ( n ) Note, this is for a continuous distribution where the distribution is for a population. R represents the population mean in this example. = ( R i - R ) 2 ( n ) Note, this is for a continuous distribution where the distribution is for a population. R represents the population mean in this example.

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5-16 Continuous Distribution Problem u Assume that the following list represents the continuous distribution of population returns for a particular investment (even though there are only 10 returns). u 9.6%, -15.4%, 26.7%, -0.2%, 20.9%, 28.3%, -5.9%, 3.3%, 12.2%, 10.5% u Calculate the Expected Return and Standard Deviation for the population assuming a continuous distribution.

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5-17 Let’s Use the Calculator! Enter “Data” first. Press: 2 nd Data 2 nd CLR Work 9.6 ENTER ENTER 26.7 ENTER u Note, we are inputting data only for the “X” variable and ignoring entries for the “Y” variable in this case.

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5-18 Let’s Use the Calculator! Enter “Data” first. Press: -0.2 ENTER 20.9 ENTER 28.3 ENTER -5.9 ENTER 3.3 ENTER 12.2 ENTER 10.5 ENTER

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5-19 Let’s Use the Calculator! Examine Results! Press: 2 nd Stat through the results. u Expected return is 9% for the 10 observations. Population standard deviation is 13.32%. u This can be much quicker than calculating by hand, but slower than using a spreadsheet.

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5-20 Certainty Equivalent CE Certainty Equivalent (CE) is the amount of cash someone would require with certainty at a point in time to make the individual indifferent between that certain amount and an amount expected to be received with risk at the same point in time. Risk Attitudes

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5-21 Certainty equivalent > Expected value Risk Preference Certainty equivalent = Expected value Risk Indifference Certainty equivalent < Expected value Risk Aversion Risk Averse Most individuals are Risk Averse. Certainty equivalent > Expected value Risk Preference Certainty equivalent = Expected value Risk Indifference Certainty equivalent < Expected value Risk Aversion Risk Averse Most individuals are Risk Averse. Risk Attitudes

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5-22 Risk Attitude Example You have the choice between (1) a guaranteed dollar reward or (2) a coin-flip gamble of $100,000 (50% chance) or $0 (50% chance). The expected value of the gamble is $50,000. u Mary requires a guaranteed $25,000, or more, to call off the gamble. u Raleigh is just as happy to take $50,000 or take the risky gamble. u Shannon requires at least $52,000 to call off the gamble.

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5-23 What are the Risk Attitude tendencies of each? Risk Attitude Example “risk aversion”. Mary shows “risk aversion” because her “certainty equivalent” < the expected value of the gamble. “risk indifference”. Raleigh exhibits “risk indifference” because her “certainty equivalent” equals the expected value of the gamble. “risk preference”. Shannon reveals a “risk preference” because her “certainty equivalent” > the expected value of the gamble. “risk aversion”. Mary shows “risk aversion” because her “certainty equivalent” < the expected value of the gamble. “risk indifference”. Raleigh exhibits “risk indifference” because her “certainty equivalent” equals the expected value of the gamble. “risk preference”. Shannon reveals a “risk preference” because her “certainty equivalent” > the expected value of the gamble.

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5-24 R P = ( W j )( R j ) R P is the expected return for the portfolio, W j is the weight (investment proportion) for the j th asset in the portfolio, R j is the expected return of the j th asset, m is the total number of assets in the portfolio. R P = ( W j )( R j ) R P is the expected return for the portfolio, W j is the weight (investment proportion) for the j th asset in the portfolio, R j is the expected return of the j th asset, m is the total number of assets in the portfolio. Determining Portfolio Expected Return m j=1

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5-25 Determining Portfolio Standard Deviation m j=1 m k=1 P P = W j W k jk W j is the weight (investment proportion) for the j th asset in the portfolio, W k is the weight (investment proportion) for the k th asset in the portfolio, jk is the covariance between returns for the j th and k th assets in the portfolio. P P = W j W k jk W j is the weight (investment proportion) for the j th asset in the portfolio, W k is the weight (investment proportion) for the k th asset in the portfolio, jk is the covariance between returns for the j th and k th assets in the portfolio.

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5-26 What is Covariance? r jk = j k r jk j is the standard deviation of the j th asset in the portfolio, k is the standard deviation of the k th asset in the portfolio, r jk is the correlation coefficient between the j th and k th assets in the portfolio. r jk = j k r jk j is the standard deviation of the j th asset in the portfolio, k is the standard deviation of the k th asset in the portfolio, r jk is the correlation coefficient between the j th and k th assets in the portfolio.

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5-27 Correlation Coefficient A standardized statistical measure of the linear relationship between two variables Its range is from -1.0 (perfect negative correlation), through 0 (no correlation), to +1.0 (perfect positive correlation). A standardized statistical measure of the linear relationship between two variables Its range is from -1.0 (perfect negative correlation), through 0 (no correlation), to +1.0 (perfect positive correlation).

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5-28 Variance - Covariance Matrix A three-asset portfolio: Col 1 Col 2 Col 3 Row 1W 1 W 1 1,1 W 1 W 2 1,2 W 1 W 3 1,3 Row 2W 2 W 1 2,1 W 2 W 2 2,2 W 2 W 3 2,3 Row 3W 3 W 1 3,1 W 3 W 2 3,2 W 3 W 3 3,3 j,k = is the covariance between returns for the j th and k th assets in the portfolio. A three-asset portfolio: Col 1 Col 2 Col 3 Row 1W 1 W 1 1,1 W 1 W 2 1,2 W 1 W 3 1,3 Row 2W 2 W 1 2,1 W 2 W 2 2,2 W 2 W 3 2,3 Row 3W 3 W 1 3,1 W 3 W 2 3,2 W 3 W 3 3,3 j,k = is the covariance between returns for the j th and k th assets in the portfolio.

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5-29 Stock D Stock BW $2,000 Stock BW $3,000Stock D Stock BW 9%13.15%, Stock D 8%10.65%,correlation coefficient 0.75 You are creating a portfolio of Stock D and Stock BW (from earlier). You are investing $2,000 in Stock BW and $3,000 in Stock D. Remember that the expected return and standard deviation of Stock BW is 9% and 13.15%, respectively. The expected return and standard deviation of Stock D is 8% and 10.65%, respectively. The correlation coefficient between BW and D is What is the expected return and standard deviation of the portfolio? Stock D Stock BW $2,000 Stock BW $3,000Stock D Stock BW 9%13.15%, Stock D 8%10.65%,correlation coefficient 0.75 You are creating a portfolio of Stock D and Stock BW (from earlier). You are investing $2,000 in Stock BW and $3,000 in Stock D. Remember that the expected return and standard deviation of Stock BW is 9% and 13.15%, respectively. The expected return and standard deviation of Stock D is 8% and 10.65%, respectively. The correlation coefficient between BW and D is What is the expected return and standard deviation of the portfolio? Portfolio Risk and Expected Return Example

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5-30 Determining Portfolio Expected Return W BW = $2,000 / $5,000 =.4 W D.6 W D = $3,000 / $5,000 =.6 W D R D R P = ( W BW )(R BW ) + ( W D )(R D ).68% R P = (.4)(9%) + (.6)(8%) 4.8%8.4% R P = (3.6%) + (4.8%) = 8.4% W BW = $2,000 / $5,000 =.4 W D.6 W D = $3,000 / $5,000 =.6 W D R D R P = ( W BW )(R BW ) + ( W D )(R D ).68% R P = (.4)(9%) + (.6)(8%) 4.8%8.4% R P = (3.6%) + (4.8%) = 8.4%

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5-31 Two-asset portfolio: Col 1 Col 2 Row 1W BW W BW BW,BW W BW W D BW,D Row 2 W D W BW D,BW W D W D D,D This represents the variance - covariance matrix for the two-asset portfolio. Two-asset portfolio: Col 1 Col 2 Row 1W BW W BW BW,BW W BW W D BW,D Row 2 W D W BW D,BW W D W D D,D This represents the variance - covariance matrix for the two-asset portfolio. Determining Portfolio Standard Deviation

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5-32 Two-asset portfolio: Col 1 Col 2 Row 1 (.4)(.4)(.0173) (.4)(.6)(.0105) Row 2 (.6)(.4)(.0105) (.6)(.6)(.0113) This represents substitution into the variance - covariance matrix. Two-asset portfolio: Col 1 Col 2 Row 1 (.4)(.4)(.0173) (.4)(.6)(.0105) Row 2 (.6)(.4)(.0105) (.6)(.6)(.0113) This represents substitution into the variance - covariance matrix. Determining Portfolio Standard Deviation

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5-33 Two-asset portfolio: Col 1 Col 2 Row 1 (.0028) (.0025) Row 2 (.0025) (.0041) This represents the actual element values in the variance - covariance matrix. Two-asset portfolio: Col 1 Col 2 Row 1 (.0028) (.0025) Row 2 (.0025) (.0041) This represents the actual element values in the variance - covariance matrix. Determining Portfolio Standard Deviation

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5-34 Determining Portfolio Standard Deviation P = (2)(.0025) P = SQRT(.0119) P =.1091 or 10.91% A weighted average of the individual standard deviations is INCORRECT. P = (2)(.0025) P = SQRT(.0119) P =.1091 or 10.91% A weighted average of the individual standard deviations is INCORRECT.

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5-35 Determining Portfolio Standard Deviation The WRONG way to calculate is a weighted average like: P =.4 (13.15%) +.6(10.65%) P = = 11.65% 10.91% = 11.65% This is INCORRECT. The WRONG way to calculate is a weighted average like: P =.4 (13.15%) +.6(10.65%) P = = 11.65% 10.91% = 11.65% This is INCORRECT.

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5-36 Stock C Stock D Portfolio Return Return 9.00% 8.00% 8.64%Stand. Dev. Dev.13.15% 10.65% 10.91% CV CV The portfolio has the LOWEST coefficient of variation due to diversification. Stock C Stock D Portfolio Return Return 9.00% 8.00% 8.64%Stand. Dev. Dev.13.15% 10.65% 10.91% CV CV The portfolio has the LOWEST coefficient of variation due to diversification. Summary of the Portfolio Return and Risk Calculation

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5-37 Combining securities that are not perfectly, positively correlated reduces risk. Diversification and the Correlation Coefficient INVESTMENT RETURN TIME SECURITY E SECURITY F Combination E and F

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5-38 Systematic Risk Systematic Risk is the variability of return on stocks or portfolios associated with changes in return on the market as a whole. Unsystematic Risk Unsystematic Risk is the variability of return on stocks or portfolios not explained by general market movements. It is avoidable through diversification. Systematic Risk Systematic Risk is the variability of return on stocks or portfolios associated with changes in return on the market as a whole. Unsystematic Risk Unsystematic Risk is the variability of return on stocks or portfolios not explained by general market movements. It is avoidable through diversification. Total Risk = Systematic Risk + Unsystematic Risk Total Risk SystematicRisk UnsystematicRisk Total Risk = Systematic Risk + Unsystematic Risk

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5-39 Total Risk = Systematic Risk + Unsystematic Risk TotalRisk Unsystematic risk Systematic risk STD DEV OF PORTFOLIO RETURN NUMBER OF SECURITIES IN THE PORTFOLIO Factors such as changes in nation’s economy, tax reform by the Congress, or a change in the world situation.

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5-40 Total Risk = Systematic Risk + Unsystematic Risk TotalRisk Unsystematic risk Systematic risk STD DEV OF PORTFOLIO RETURN NUMBER OF SECURITIES IN THE PORTFOLIO Factors unique to a particular company or industry. For example, the death of a key executive or loss of a governmental defense contract.

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5-41 risk-free rate a premium systematic risk CAPM is a model that describes the relationship between risk and expected (required) return; in this model, a security’s expected (required) return is the risk-free rate plus a premium based on the systematic risk of the security. Capital Asset Pricing Model (CAPM)

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Capital markets are efficient. 2.Homogeneous investor expectations over a given period. Risk-free 3.Risk-free asset return is certain (use short- to intermediate-term Treasuries as a proxy). systematic risk 4.Market portfolio contains only systematic risk (use S&P 500 Index or similar as a proxy). 1.Capital markets are efficient. 2.Homogeneous investor expectations over a given period. Risk-free 3.Risk-free asset return is certain (use short- to intermediate-term Treasuries as a proxy). systematic risk 4.Market portfolio contains only systematic risk (use S&P 500 Index or similar as a proxy). CAPM Assumptions

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5-43 Characteristic Line EXCESS RETURN ON STOCK EXCESS RETURN ON MARKET PORTFOLIO Beta Beta = RiseRun Narrower spread is higher correlation Characteristic Line

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5-44 Calculating “Beta” on Your Calculator Time Pd.MarketMy Stock 19.6%12% %-5% 326.7%19% 4-.2%3% 520.9%13% 628.3%14% 7-5.9%-9% 83.3%-1% 912.2%12% %10% The Market and My Stock returns are “excess returns” and have the riskless rate already subtracted.

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5-45 Calculating “Beta” on Your Calculator u Assume that the previous continuous distribution problem represents the “excess returns” of the market portfolio (it may still be in your calculator data worksheet -- 2 nd Data ). u Enter the excess market returns as “X” observations of: 9.6%, -15.4%, 26.7%, -0.2%, 20.9%, 28.3%, -5.9%, 3.3%, 12.2%, and 10.5%. u Enter the excess stock returns as “Y” observations of: 12%, -5%, 19%, 3%, 13%, 14%, -9%, -1%, 12%, and 10%.

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5-46 Calculating “Beta” on Your Calculator u Let us examine again the statistical results (Press 2 nd and then Stat ) u The market expected return and standard deviation is 9% and 13.32%. Your stock expected return and standard deviation is 6.8% and 8.76%. u The regression equation is Y=a+bX. Thus, our characteristic line is Y = X and indicates that our stock has a beta of

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5-47 systematic risk An index of systematic risk. It measures the sensitivity of a stock’s returns to changes in returns on the market portfolio. beta The beta for a portfolio is simply a weighted average of the individual stock betas in the portfolio. systematic risk An index of systematic risk. It measures the sensitivity of a stock’s returns to changes in returns on the market portfolio. beta The beta for a portfolio is simply a weighted average of the individual stock betas in the portfolio. What is Beta?

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5-48 Characteristic Lines and Different Betas EXCESS RETURN ON STOCK EXCESS RETURN ON MARKET PORTFOLIO Beta < 1 (defensive) Beta = 1 Beta > 1 (aggressive) characteristic Each characteristic line line has a different slope.

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5-49 R j R j is the required rate of return for stock j, R f R f is the risk-free rate of return, j j is the beta of stock j (measures systematic risk of stock j), R M R M is the expected return for the market portfolio. R j R j is the required rate of return for stock j, R f R f is the risk-free rate of return, j j is the beta of stock j (measures systematic risk of stock j), R M R M is the expected return for the market portfolio. Security Market Line R j R f R M R f R j = R f + j (R M - R f )

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5-50 Security Market Line R j R f R M R f R j = R f + j (R M - R f ) M 1.0 M = 1.0 Systematic Risk (Beta) RfRfRfRf RMRMRMRM Required Return RiskPremium Risk-freeReturn

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5-51 6% R f market expected rate of return 10% beta1.2 required rate of return Lisa Miller at Basket Wonders is attempting to determine the rate of return required by their stock investors. Lisa is using a 6% R f and a long-term market expected rate of return of 10%. A stock analyst following the firm has calculated that the firm beta is 1.2. What is the required rate of return on the stock of Basket Wonders? Determination of the Required Rate of Return

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5-52 R BW R f R M R f R BW = R f + j (R M - R f ) R BW 6%1.210%6% R BW = 6% + 1.2(10% - 6%) R BW 10.8% R BW = 10.8% The required rate of return exceeds the market rate of return as BW’s beta exceeds the market beta (1.0). R BW R f R M R f R BW = R f + j (R M - R f ) R BW 6%1.210%6% R BW = 6% + 1.2(10% - 6%) R BW 10.8% R BW = 10.8% The required rate of return exceeds the market rate of return as BW’s beta exceeds the market beta (1.0). BWs Required Rate of Return

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5-53 intrinsic value dividend next period $0.50grow 5.8% Lisa Miller at BW is also attempting to determine the intrinsic value of the stock. She is using the constant growth model. Lisa estimates that the dividend next period will be $0.50 and that BW will grow at a constant rate of 5.8%. The stock is currently selling for $15. intrinsic value overunderpriced What is the intrinsic value of the stock? Is the stock over or underpriced? intrinsic value dividend next period $0.50grow 5.8% Lisa Miller at BW is also attempting to determine the intrinsic value of the stock. She is using the constant growth model. Lisa estimates that the dividend next period will be $0.50 and that BW will grow at a constant rate of 5.8%. The stock is currently selling for $15. intrinsic value overunderpriced What is the intrinsic value of the stock? Is the stock over or underpriced? Determination of the Intrinsic Value of BW

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5-54 intrinsic value $10 The stock is OVERVALUED as the market price ($15) exceeds the intrinsic value ($10). Determination of the Intrinsic Value of BW $ %5.8% 10.8% - 5.8% IntrinsicValue = = $10

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5-55 Security Market Line Systematic Risk (Beta) RfRfRfRf Required Return Direction of Movement Direction of Movement Stock Y Stock Y (Overpriced) Stock X (Underpriced)

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5-56 Small-firm Effect Price / Earnings Effect January Effect These anomalies have presented serious challenges to the CAPM theory. Small-firm Effect Price / Earnings Effect January Effect These anomalies have presented serious challenges to the CAPM theory. Determination of the Required Rate of Return

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