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Financial Management Lecture 6 (Ch.11) Return and Risk: The Capital Asset Pricing (CAPM) Lecturer: Sheng-Syan Chen College of Management National Taiwan University 1 Key Concepts and Skills Know how to calculate expected returns Know how to calculate covariances, correlations, and betas Understand the impact of diversification Understand the systematic risk principle Understand the security market line Understand the risk-return tradeoff Be able to use the Capital Asset Pricing Model 2 Part 1: Individual Securities The characteristics of individual securities that are of interest are the: Expected Return This is the return that an individual expects a stock to earn. Of course, because this is only an expectation, the actual return may be either higher or lower. 3 Variance and Standard Deviation Variance is a measure of the squared deviations of a security’s return from its expected return. Standard deviation is the square root of the variance. Covariance and Correlation (to another security or index) Covariance is a statistic measuring the interrelationship between two securities. Alternatively, this relationship can be restated in terms of the correlation between the two securities. Covariance and correlation are building blocks to an understanding of the beta coefficient. 4 Part 2 : Expected Return, Variance, and Covariance Consider the following two risky assets: Supertech Company and Slowpoke Company. There is a 1/4 chance of each state of the economy. The return predictions are as follows: Rate of Return Scenario Probability Supertech(R At ) Slowpoke(R Bt ) Depression 25.0% -20% 5% Recession 25.0% 10% 20% Normal 25.0% 30% -12% Boom 25.0% 50% 9% 5 Variance Variance can be calculated in four steps. An additional step is needed to calculate standard deviation. Step 1: calculate the expected return 6 Step 1: Expected Return Supertech Slowpoke Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Depression -20% 0.1406 5% 0.0000 Recession 10% 0.0056 20% 0.0210 Normal 30% 0.0156 -12% 0.0306 Boom 50% 0.1056 9% 0.0012 Expected return 17.50% 5.50% Variance 0.0669 0.0132 E (rA ) 1 / 4 * (20%) 1 / 4 * (10%) 1 / 4 * (30%) 1 / 4 * (50%) E (rA ) 17.5% 7 Step2: Calculate the deviation of each possible return from the company’s expected return given previously 8 Step 3: Square deviations 9 Step 4: Calculate the average squared deviation (variance) Supertech Slowpoke Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Depression -20% 0.1406 5% 0.0000 Recession 10% 0.0056 20% 0.0210 Normal 30% 0.0156 -12% 0.0306 Boom 50% 0.1056 9% 0.0012 Expected return 17.50% 5.50% Variance 0.0669 0.0132 Standard Deviation 25.9% 11.5% .0669 1 / 4(.140625 .005625 .015625 .105625) 10 Step5: Calculate standard deviation by taking the square root of the variance 11 Covariance Rate of R At -E(R A ) Rate of R Bt -E(R B ) Deviations Scenario Return Return Depression -20% -0.3750 5% -0.0050 0.001875 Recession 10% -0.0750 20% 0.1450 -0.010875 Normal 30% 0.1250 -12% -0.1750 -0.021875 Boom 50% 0.3250 9% 0.0350 0.011375 R A 17.50% R B 5.50% -0.0195 Deviation compares return in each state to the expected return. Weighted takes the product of the deviations multiplied by the probability of that state. 12 Calculate the average value of the four states in the last column. The average is the covariance. 13 Correlation Divide the covariance by the standard deviations of both of the two Securities. Cov(A,B) ρ AB Corr(RA,R B ) σ Aσ B .004875 ρ AB 0.1639 (.2586)(.1150) 14 Examples of Different Correlation Coefficients-Perfect Positive Correlation 15 Perfect Negative Correlation 16 Zero Correlation 17 Part 3: The Return and Risk for Portfolios Supertech Slowpoke Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Depression -20% 0.1406 5% 0.0000 Recession 10% 0.0056 20% 0.0210 Normal 30% 0.0156 -12% 0.0306 Boom 50% 0.1056 9% 0.0012 Expected return 17.50% 5.50% Variance 0.0669 0.0132 Standard Deviation 25.9% 11.5% 18 The expected return on a portfolio is simply a weighted average of the expected returns on the individual securities . Rate of Return Scenario Supertech Slowpoke Portfolio squared deviation Depression -20% 5% -10.0% 0.0515 Recession 10% 20% 14.0% 0.0002 Normal 30% -12% 13.2% 0.0000 Boom 50% 9% 33.6% 0.0437 Expected return 17.50% 5.50% 12.7% Variance 0.0669 0.0132 0.0239 Standard Deviation 25.86% 11.50% 15.44% 19 Suppose the investor with $100 invests $60 in Supertech and $40 in Slowpoke, the expected return on the portfolio: Returnon portfolio : rp w ArA w BrB Expected returnon portfolio : E(rp ) w A E ( rA ) w B E ( rB ) 12.7% 60% (17.5%) 40% (5.5%) 20 Variance of a Portfolio Suppose a portfolio composes of two securities, A and B, the variance of the portfolio: (Given XA=0.6, XB=0.4) Var (Portfolio)= =0.023851=0.36*0.066875+ 2*[0.6*0.4*(-0.004875)]+0.16*0.013225 21 Standard Deviation of a Portfolio 22 The Diversification Effect Itis instructive to compare the standard deviation of the portfolio with the standard deviation of the individual securities. The weighted average of the standard deviations of the individual securities is: 23 (11.6) 24 Suppose ρSuper,Slow 1, the highest possible value for correlation. The variance of the portfolio is: (11.9) 25 Notethat Equations 11.9 and 11.6 are equal. That is, the standard deviation of a portfolio’s return is equal to the weighted average of the standard deviations of the individual returns when ρ = 1. => As long as ρ< 1, the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviations of the individual securities. => In other words, the diversification effect applies as long as ρ< l. 26 Part 4: The Efficient Set for Two Assets The box or “□” in the graph represents a portfolio with 60% invested in Supertech and 40% invested in Slowpoke. 27 Set of Portfolios Composed of Holdings in Supertech and Slowpoke 28 Thereare a few important points concerning this graph: The diversification effect occurs whenever the correlation between the two securities is below 1. There is no diversification effect if ρ= 1. The point MV represents the minimum variance portfolio. This is the portfolio with the lowest possible variance. 29 An individual contemplating an investment in a portfolio of Slowpoke and Supertech faces an opportunity set or feasible set. He can achieve any point on the curve by selecting the appropriate mix between the two securities. However, He cannot achieve any point above the curve because he cannot increase the return on the individual securities, decrease the standard deviations of the securities, or decrease the correlation between the two securities. 30 Note that the curve is backward bending between the Slowpoke point and MV. This indicates that, for a portion of the feasible set, standard deviation actually decreases as we increase expected return. Actually, backward bending always occurs if ρ 0. No investor would want to hold a portfolio with an expected return below that of the minimum variance portfolio. 31 E.g.no investor would choose portfolio 1, because this portfolio has less expected return but more standard deviation than the minimum variance portfolio has. Though the entire curve from Slowpoke to Supertech is called the feasible set, investors consider only the curve from MV to SuPertech. The curve from MV to Supertech is called the efficient set or the efficient frontier. 32 Figure 11.4 (opportunity sets composed of holdings in supertech and slowpoke The lower the correlation , the more bend there is in the curve. This indicates that the diversification effect rises as ρdeclines. 33 Portfolios with Various Correlations return Supertech = -1.0 = 1.0 Slowpoke = -0.1639 Relationship depends on correlation coefficient -1.0 < < +1.0 If = +1.0, no risk reduction is possible If = –1.0, complete risk reduction is possible 34 Part 5: The Efficient Set for Many Securities Figure11.6 (The feasible set of portfolios constructed from many securities) 35 Figure 11.6 represents the opportunity set or feasible set when many securities are considered. The shaded area represents all the possible combinations of expected return and standard deviation for a portfolio. The upper edge between MV and X is called the efficient set. 36 Any point below the efficient set would receive less expected return and the same standard deviation as a point on the efficient set. For example, consider R on the efficient set and W directly below it. If W contains the risk level you desire, you should choose R instead to receive a higher expected return. 37 Diversification and Portfolio Risk Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns. This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another. However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion. 38 Portfolio Risk and Number of Stocks Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Portfolio risk Nondiversifiable risk; Systematic Risk; Market Risk n 39 Systematic Risk Risk factors that affect a large number of assets Also known as non-diversifiable risk or market risk Includes such things as changes in GDP, inflation, interest rates, etc. 40 Unsystematic (Diversifiable) Risk Risk factors that affect a limited number of assets Also known as unique risk and asset-specific risk Includes such things as labor strikes, part shortages, etc. The risk that can be eliminated by combining assets into a portfolio If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away. 41 Total Risk Total risk = systematic risk + unsystematic risk The standard deviation of returns is a measure of total risk. For well-diversified portfolios, unsystematic risk is very small. Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk. 42 Optimal Portfolio with a Risk-Free Asset return 100% stocks rf 100% bonds In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills. 43 Part 6: Riskless Borrowing and Lending return 100% stocks Balanced fund rf 100% bonds Now investors can allocate their money across the T-bills and a balanced mutual fund. 44 Riskless Borrowing and Lending return rf P With a risk-free asset available and the efficient frontier identified, we choose the capital allocation line with the steepest slope. 45 Example Ms. Bagwell is considering investing in the common stock of Merville Enterprises. In addition, Ms. Bagwell will either borrow or lend at the risk- free rate. The relevant parameters are these : Suppose Ms. Bagwell chooses to invest a total of $l,000, $ 350 of which is to be invested in Merville Enterprises and $650 placed in the risk-free asset. The expected return on her total investment is simply a weighted average of the two returns : 0.35*0.14+0.65*0.1=0.114 46 Variance of portfolio composed of one riskless and one risky asset. Thestandard deviation of portfolio composed of one riskless and one risky asset. 47 Relationship between Expected Return and Risk for a Portfolio of One Risky Asset and One Riskless Asset 48 Suppose that, alternatively, Ms. Bagwell borrows $ 200 at the risk-free rate. Combining this with her original sum of $l,000, she invests a total of $ l,200 in Merville. Her expected return would be : 1.2*0.14+(-0.2*0.1)=14.8% The standard deviation is: ( is greater than 0.20, the standard deviation of the Merville investment, because borrowing increases the variability of the investment.) 49 The Optimal Portfolio The previous section concerned a portfolio formed between one riskless asset and one risky asset. In reality, an investor is likely to combine an investment in the riskless asset with a portfolio of risky assets. E.g. Figure 11.9, Point Q is in the interior of the feasible set of risky securities, and the point represents a portfolio of 30 percent in AT&T, 45 percent in General Motors (GM), and 25 percent in IBM. 50 51 point 1 on the line represents a portfolio of 70 percent in the riskless asset and 30 percent in stocks represented by Q. An investor with $ 100 choosing point 1 as his portfolio would put $70 in the risk-free asset and $ 30 in Q. This can be restated as $ 70 in the riskless asset, $9 (= 0.3×$30) in AT&T, $13.50(= 0.45× $30) in GM, and $7.50 (= 0.25×$30) in IBM. Point 3 is obtained by borrowing to invest in Q. An investor with $100 of her own would borrow $40 from the bank or broker to invest $140 in Q. This can be stated as borrowing $40 and contributing $100 of her money to invest $42 = 0.3× $140 ) in AT&T, $63 (= 0.45×$140 ) in GM, and $35(=0.25× $140 ) in IBM. 52 Though any investor can obtain any point on line 1, no point on the line is optimal. Consider line II, Line II represents portfolios formed by combinations of the risk-free asset and the securities in A, and point A represents a portfolio of risky securities. In fact, because line II is tangent to the efficient set of risky assets, it provides the investor with the best possible opportunities. => In other words, line II can be viewed as the efficient set of all assets, both risky and riskless. 53 Part 7: Market Equilibrium Expected return M rf P Ina world with homogeneous expectations, M is the same for all investors. With the capital allocation line identified, all investors choose a point along the line—some combination of the risk-free asset and the market portfolio M. 54 Expected return M rf P Where the investor chooses along the Capital Market Line (CML) depends on his risk tolerance. The big point is that all investors have the same CML. 55 Risk When Holding the Market Portfolio Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta (b)of the security. Beta measures the responsiveness of a security to movements in the market portfolio (i.e., systematic risk). Cov( Ri , RM ) bi ( RM ) 2 56 Estimating b with Regression Security Returns Slope = bi Return on market % Ri = a i + biRm + ei 57 The Formula for Beta Cov( Ri , RM ) bi ( RM ) 2 58 One useful property is that the average beta across all securities, when weighted by the proportion of each security’s market value to that of the market portfolio, is 1. That is: where X i is the proportion of security i’s market value to that of the entire market and N is the number of securities in the market. 59 Part 8: Relationship between Risk and Expected Return (CAPM) Expected Return on the Market: R M RF Market Risk Premium Expected return on an individual security: Ri RF βi ( R M RF ) Market Risk Premium This applies to individual securities held within well- diversified portfolios. 60 Expected Return on a Security This formula is called the Capital Asset Pricing Model (CAPM): Ri RF βi ( R M RF ) Expected Risk- Beta of the Market risk return on = + × free rate security premium a security Assume bi = 0, then the expected return is RF. Assume bi = 1, then Ri R M 61 Equation 11.16 can be represented graphically by the upward-sloping line in Figure 11.11. Note that the line begins at R F and rises to RM when beta is l. This line is frequently called the security market line (SML). 62 Relationship Between Risk & Return (Equation 11.16) Expected return Ri RF βi ( R M RF ) RM RF 1.0 b 63 Relationship Between Risk & Return 15% Expected return 3% 1.5 b β i 1.5 RF 3% R M 11% R i 3% 1.5 (11% 3%) 15% 64 Example The stock of Aardvark Enterprises has a beta of 1.5 and that of Zebra Enterprises has a beta of 0.7. The risk-free rate is assumed to be 3 percent, and the difference between the expected return on the market and the risk-free rate is assumed to be 8.0 percent. 65 The expected returns on the two securities are: 66 Three additional points concerning the CAPM should be mentioned: Linearity: The relationship between expected return and beta corresponds to a straight line. Figure 11.11, Securities (S,T) lying below the SML are overpriced. Securities lying above the SML are underpriced. 67 Portfolio as well as securities : Example, Aardvark and Zebra, the expected return on the portfolio is: (11.18) The beta of the portfolio is simply a weighted average of the betas of the two securities: (11.19) Because the expected return in Equation 11.18 is the same as the expected return in Equation 11.19, the example shows that the CAPM holds for portfolios as well as for individual securities. 68 A potential confusion: Line II in Figure 11.9 traces the efficient set of portfolios formed from both risky assets and the riskless asset. Each point on the line represents an entire portfolio. Under homogeneous expectations, point A in Figure 11.9 becomes the market portfolio. In this situation , line II is referred to as the capital market line (CML). SML differs from line II in at least two ways: First , beta appears in the horizontal axis of Figure 11.11, but standard deviation appears in the horizontal axis of Figure 11.9. Second, the SML in Figure 11.11 holds both for all individual securities and for all possible portfolios, whereas line II in Figure 11.9 holds only for efficient portfolios. 69

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posted: | 5/23/2012 |

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