1. 1 Risk Learning Module

2. 2 Measures of Risk Risk reflects the chance that the actual return on an investment may be different than the expected return. Risk reflects the chance that the actual return on an investment may be different than the expected return. One way to measure risk is to calculate the variance and standard deviation of the distribution of returns. One way to measure risk is to calculate the variance and standard deviation of the distribution of returns. We will once again use a probability distribution in our calculations. We will once again use a probability distribution in our calculations. The distribution used earlier is provided again for ease of use. The distribution used earlier is provided again for ease of use.

3. 3 Measures of Risk Probability Distribution: Probability Distribution: State Probability Return On Return On Stock A Stock B Stock A Stock B 1 20% 5% 50% 1 20% 5% 50% 2 30% 10% 30% 2 30% 10% 30% 3 30% 15% 10% 3 30% 15% 10% 4 20% 20% -10% 4 20% 20% -10% E[R] A = 12.5% E[R] A = 12.5% E[R] B = 20% E[R] B = 20%

4. 4 Measures of Risk Given an asset's expected return, its variance can be calculated using the following equation: Given an asset's expected return, its variance can be calculated using the following equation: N Var(R) =  2 =  p i (R i – E[R]) 2 i=1 i=1 Where: Where: N = the number of states N = the number of states p i = the probability of state i p i = the probability of state i R i = the return on the stock in state i R i = the return on the stock in state i E[R] = the expected return on the stock E[R] = the expected return on the stock

5. 5 Measures of Risk The standard deviation is calculated as the positive square root of the variance: The standard deviation is calculated as the positive square root of the variance: SD(R) =  =  2 = (  2 ) 1/2 = (  2 ) 0.5

6. 6 Measures of Risk The variance and standard deviation for stock A is calculated as follows: The variance and standard deviation for stock A is calculated as follows:  2 A =.2(.05 -.125) 2 +.3(.1 -.125) 2 +.3(.15 -.125) 2 +.2(.2 -.125) 2 =.002625      Now you try the variance and standard deviation for stock B! Now you try the variance and standard deviation for stock B! If you got.042 and 20.49% you are correct. If you got.042 and 20.49% you are correct.

7. 7 Measures of Risk If you didn’t get the correct answer, here is how to get it: If you didn’t get the correct answer, here is how to get it:  2 B =.2(.50 -.20) 2 +.3(.30 -.20) 2 +.3(.10 -.20) 2 +.2(-.10 -.20) 2 =.042      Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's. Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's. This, however, is still only part of the picture because most investors choose to hold securities as part of a diversified portfolio. This, however, is still only part of the picture because most investors choose to hold securities as part of a diversified portfolio.

8. 8 Portfolio Risk and Return Most investors do not hold stocks in isolation. Most investors do not hold stocks in isolation. Instead, they choose to hold a portfolio of several stocks. Instead, they choose to hold a portfolio of several stocks. When this is the case, a portion of an individual stock's risk can be eliminated, i.e., diversified away. When this is the case, a portion of an individual stock's risk can be eliminated, i.e., diversified away. From our previous calculations, we know that: From our previous calculations, we know that: the expected return on Stock A is 12.5% the expected return on Stock A is 12.5% the expected return on Stock B is 20% the expected return on Stock B is 20% the variance on Stock A is.00263 the variance on Stock A is.00263 the variance on Stock B is.04200 the variance on Stock B is.04200 the standard deviation on Stock A is 5.12% the standard deviation on Stock A is 5.12% the standard deviation on Stock B is 20.49% the standard deviation on Stock B is 20.49%

9. 9 Portfolio Risk and Return The Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio. The Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio. The weights reflect the proportion of the portfolio invested in the stocks. The weights reflect the proportion of the portfolio invested in the stocks. This can be expressed as follows: This can be expressed as follows: N E[R p ] =  w i E[R i ] i=1 i=1 Where: Where: E[R p ] = the expected return on the portfolio E[R p ] = the expected return on the portfolio N = the number of stocks in the portfolio N = the number of stocks in the portfolio w i = the proportion of the portfolio invested in stock i w i = the proportion of the portfolio invested in stock i E[R i ] = the expected return on stock i E[R i ] = the expected return on stock i

10. 10 Portfolio Risk and Return For a portfolio consisting of two assets, the above equation can be expressed as: For a portfolio consisting of two assets, the above equation can be expressed as: E[R p ] = w 1 E[R 1 ] + w 2 E[R 2 ] E[R p ] = w 1 E[R 1 ] + w 2 E[R 2 ] If we have an equally weighted portfolio of stock A and stock B (50% in each stock), then the expected return of the portfolio is: If we have an equally weighted portfolio of stock A and stock B (50% in each stock), then the expected return of the portfolio is: E[R p ] =.50(.125) +.50(.20) = 16.25% E[R p ] =.50(.125) +.50(.20) = 16.25%

11. 11 Portfolio Risk and Return The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together. The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together. Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient. Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient. Covariance is a measure that combines the variance of a stock’s returns with the tendency of those returns to move up or down at the same time other stocks move up or down. Covariance is a measure that combines the variance of a stock’s returns with the tendency of those returns to move up or down at the same time other stocks move up or down. Since it is difficult to interpret the magnitude of the covariance terms, a related statistic, the correlation coefficient, is often used to measure the degree of co-movement between two variables. The correlation coefficient simply standardizes the covariance. Since it is difficult to interpret the magnitude of the covariance terms, a related statistic, the correlation coefficient, is often used to measure the degree of co-movement between two variables. The correlation coefficient simply standardizes the covariance.

12. 12 Portfolio Risk and Return The Covariance between the returns on two stocks can be calculated as follows: The Covariance between the returns on two stocks can be calculated as follows: N Cov(R A,R B ) =  A,B =  p i (R Ai - E[R A ])(R Bi - E[R B ]) i=1 i=1 Where: Where:   = the covariance between the returns on stocks A and B   = the covariance between the returns on stocks A and B N = the number of states N = the number of states p i = the probability of state i p i = the probability of state i R Ai = the return on stock A in state i R Ai = the return on stock A in state i E[R A ] = the expected return on stock A E[R A ] = the expected return on stock A R Bi = the return on stock B in state i R Bi = the return on stock B in state i E[R B ] = the expected return on stock B E[R B ] = the expected return on stock B

13. 13 Portfolio Risk and Return The Correlation Coefficient between the returns on two stocks can be calculated as follows: The Correlation Coefficient between the returns on two stocks can be calculated as follows:  A,B Cov(R A,R B ) Corr(R A,R B ) =  A,B =  A  B = SD(R A )SD(R B ) Where: Where:  A,B =the correlation coefficient between the returns on stocks A and B  A,B =the correlation coefficient between the returns on stocks A and B  A,B =the covariance between the returns on stocks A and B,  A,B =the covariance between the returns on stocks A and B,  A =the standard deviation on stock A, and  A =the standard deviation on stock A, and  B =the standard deviation on stock B  B =the standard deviation on stock B

14. 14 Portfolio Risk and Return The covariance between stock A and stock B is as follows: The covariance between stock A and stock B is as follows:  A,B =.2(.05-.125)(.5-.2) +.3(.1-.125)(.3-.2) +.3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105.3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105 The correlation coefficient between stock A and stock B is as follows: The correlation coefficient between stock A and stock B is as follows: -.0105 -.0105  A,B = (.0512)(.2049) = -1.00

15. 15 Portfolio Risk and Return Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows: Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows:  2 p = (w A ) 2  2 A + (w B ) 2  2 B + 2w A w B  A,B  A  B OR OR  2 p = (w A ) 2  2 A + (w B ) 2  2 B + 2w A w B  A,B The Standard Deviation of the Portfolio equals the positive square root of the the variance. The Standard Deviation of the Portfolio equals the positive square root of the the variance.

16. 16 Portfolio Risk and Return Let’s calculate the variance and standard deviation of a portfolio comprised of 75% stock A and 25% stock B: Let’s calculate the variance and standard deviation of a portfolio comprised of 75% stock A and 25% stock B:  2 p =(.75) 2  2 +(.25) 2 (.2049) 2 +2(.75)(.25)(-1)(.0512)(.2049)=.00016  p =.00016 =.0128 = 1.28% Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A. Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A. This is the purpose of diversification; by forming portfolios, some of the risk inherent in the individual stocks can be eliminated. This is the purpose of diversification; by forming portfolios, some of the risk inherent in the individual stocks can be eliminated.

17. 17 Capital Asset Pricing Model (CAPM) If investors are mainly concerned with the risk of their portfolio rather than the risk of the individual securities in the portfolio, how should the risk of an individual stock be measured? If investors are mainly concerned with the risk of their portfolio rather than the risk of the individual securities in the portfolio, how should the risk of an individual stock be measured? In important tool is the CAPM. In important tool is the CAPM. CAPM concludes that the relevant risk of an individual stock is its contribution to the risk of a well-diversified portfolio. CAPM concludes that the relevant risk of an individual stock is its contribution to the risk of a well-diversified portfolio. CAPM specifies a linear relationship between risk and required return. CAPM specifies a linear relationship between risk and required return. The equation used for CAPM is as follows: The equation used for CAPM is as follows: K i = K rf +  i (K m - K rf ) K i = K rf +  i (K m - K rf ) Where: Where: K i = the required return for the individual security K i = the required return for the individual security K rf = the risk-free rate of return K rf = the risk-free rate of return  i = the beta of the individual security  i = the beta of the individual security K m = the expected return on the market portfolio K m = the expected return on the market portfolio (K m - K rf ) is called the market risk premium (K m - K rf ) is called the market risk premium This equation can be used to find any of the variables listed above, given the rest of the variables are known. This equation can be used to find any of the variables listed above, given the rest of the variables are known.

18. 18 CAPM Example Find the required return on a stock given that the risk-free rate is 8%, the expected return on the market portfolio is 12%, and the beta of the stock is 2. Find the required return on a stock given that the risk-free rate is 8%, the expected return on the market portfolio is 12%, and the beta of the stock is 2. K i = K rf +  i (K m - K rf ) K i = K rf +  i (K m - K rf ) K i = 8% + 2(12% - 8%) K i = 8% + 2(12% - 8%) K i = 16% K i = 16% Note that you can then compare the required rate of return to the expected rate of return. You would only invest in stocks where the expected rate of return exceeded the required rate of return. Note that you can then compare the required rate of return to the expected rate of return. You would only invest in stocks where the expected rate of return exceeded the required rate of return.

19. 19 Another CAPM Example Find the beta on a stock given that its expected return is 12%, the risk-free rate is 4%, and the expected return on the market portfolio is 10%. Find the beta on a stock given that its expected return is 12%, the risk-free rate is 4%, and the expected return on the market portfolio is 10%. 12% = 4% +  i (10% - 4%) 12% = 4% +  i (10% - 4%)  i = 12% - 4%  i = 12% - 4% 10% - 4% 10% - 4%  i = 1.33  i = 1.33 Note that beta measures the stock’s volatility (or risk) relative to the market. Note that beta measures the stock’s volatility (or risk) relative to the market.