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Ch 11 Return and Risk: The Capital Asset Pricing Model CAPM

Main Idea. What is an appropriate measure of risk?If you hold single asset, standard deviation of the asset is a good measure of the risk.If you hold a widely diversified portfolio, the beta of the asset is a good measure of the risk of the asset.Average return of the asset is a good measure of t

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Ch 11 Return and Risk: The Capital Asset Pricing Model CAPM

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    1. Ch 11 Return and Risk: The Capital Asset Pricing Model (CAPM) 1 Expected Return and Variance for a single asset Portfolios Expected return and variance for a portfolio Efficient set with two assets Efficient set with N assets Riskless borrowing and lending Market equilibrium Diversification and Portfolio Risk Diversification with N assets Systematic Risk and Beta 4 The Security Market Line and CAPM 5 Summary

    2. Main Idea What is an appropriate measure of risk? If you hold single asset, standard deviation of the asset is a good measure of the risk. If you hold a widely diversified portfolio, the beta of the asset is a good measure of the risk of the asset. Average return of the asset is a good measure of the expected return in both cases.

    3. 1. Expected Return and Variance Goal: To derive a relation between return and risk in the form of E(Ri) = Rf + ?i [E(Rm) – Rf ]  where Rf ? risk free rate (e.g., T-bill rate) Rm ? return on market portfolio (e.g., a value-weighted portfolio of all TSE stocks) ?i = Measure of risk of asset i ? Cov (asseti, market portfolio) / Var (market portfolio) In order to measure and develop models for the relation between risk and return, we need some formal statistical measures.

    4. 1.1 Calculating the Expected Return … Example 1: Average of two scores 80 and 60= (80+60)/2 = 70. Example 2: You feel you’d get 80 and 60 in a finance course with equal probability of 0.5 and 0.5. What do you expect to get?

    5. Example: Calculating the Expected Return… Example 3: You feel you’d get 80 and 60 in a finance course with probability of 0.4 and 0.6. What do you expect to get? Qu: What are E(x2), E(x3 - 3x), or in general, E [anything ]? E(x2) = sum (Pi * xi2) E(x3 - 3x) = sum ( Pi * (xi3 - 3xi) ) E (anything) = sum (Pi * possible outcome i of anything), i.e., E [f(x)] = sum [ Pi * f(xi) ] ? ?I [ Pi * f(xi) ]

    6. 1.2 Calculating variance Example 1: You expect to get 80 and 60 in a finance course with probability of 0.4 and 0.6. What is variance of your score? variance(x) ? sx2 ? E [ (x - E(x))2 ]. Why squared? Recall E [anything] = sum (Pi * possible outcome i of anything). sx2 ? E [ (x - E(x) )2] = sum [Pi* possible outcome of (xi - E(x))2

    7. 1.2 Calculating variance… Why standard deviation? Variance isn't in the same units as the mean--it's in (unit)2. It is often useful to work with standard deviation which is in the units as the mean. 1.3 Calculating covariance and correlation A measure of how random variables move together is covariance. If we have two random variables, X and Y, their covariance is defined as: cov (x, y) ? E [ (x - E(x)) (y - E(y)) ]

    8. 1.3 Calculating covariance and correlation… Example 1: You feel you’d get 80 and 60 in a finance and 70 and 90 in an accounting course with probability of 0.4 and 0.6. Find out the mean, variance and standard deviation of finance (x) and accounting (y) scores. E(x)= 68 sx2 = 96 sx = 9.8 E(y) = 82 sy2 = 96 sy = 9.8 What is the covariance of your finance and accounting scores? Recall E [anything] = sum (Pi * possible outcome i of anything). cov (x, y) ? E [ (x - E(x)) (y - E(y)) ] = sum [Pi * possible outcome of (xi - E(x))* (yi - E(y)) ] = ?I [ Pi * (xi - E(x))* (yi - E(y))]

    9. 1.2 Calculating covariance and correlation…

    10. Qu 12.7 Calculation of mean, variance & covariance

    11. Qu Calculation of mean, variance & covariance…

    12. Qu. Calculation of mean, variance & covariance with Probability distribution

    13. Evidence on Covariance and Correlation (1) Stock returns are serially uncorrelated. If stock returns are high one year then, you can't use this information to predict whether returns in the subsequent year will be high or low. This evidence is important to market efficiency. (2) Most stocks are positively correlated to the market portfolio: important evidence for our discussion of asset pricing model. market portfolio: a value-weighted portfolio of all the stocks in the economy (or as a proxy, all stocks on the NYSE or TSE).

    14. Statistical Review: Conclusion The main thing is to have an intuitive understanding of what these statistics mean. Arithmetic mean is a measure of how much you can expect to receive if you hold a stock for a year. The variance and standard deviation are measures of how variable the returns are likely to be. The higher the variance or standard deviation the greater the variation. Covariance and correlation are measures of whether two variables move together or in opposite directions. Move together: positive Move in opposite directions: negative Independent: zero.

    15. 2. Diversification 2.1 Mean and variance of portfolio Suppose we have $1,000 to invest, and there are two risky assets:     We could invest it all in asset 1 or all in asset 2. However, we may do better off by taking a combination of asset 1 and asset 2, i.e., diversification may provide better result than by taking 1 or 2 alone. Let us see Suppose correlation between asset 1 and 2’s return is +0.5. Let x1 be the proportion of our wealth invested in asset 1. (What is proportion of wealth invested asset 2?)

    16. 2.1 Mean and variance of a portfolio

    17. 2.1 Mean and variance of a portfolio… What will happen in-between? E(Rp) = x1 E(R1) + (1 - x1) E(R2) (1) ?p2 = x12 ?12 + (1 - x1)2 ?22 + 2x1(1 - x1) cov (R1, R2) (2) Note: Recall ?12 ? Corr (R1, R2) = cov (R1, R2)/(?1 ?2). Sometimes, we use ?12 ?1 ?2 instead of cov (R1, R2). Suppose that x1=0.1. We can calculate the mean return and ? of the portfolio using equations (1) and (2). We can do the same calculation for x1=0.2, 0.3,….,x1=1, and fill up the following table. Using these results, we can draw the following chart:

    18. 2.1 Mean and variance of a portfolio…

    19. 2.1 Mean and variance of a portfolio…

    20. 2.2 Relation between ? and investment opportunity set

    21. 2.2 Relation between ? and investment opportunity set…

    22. 2.3 Efficient frontier with N assets

    23. 2.4 Riskless Borrowing and Lending With a risk-free asset available and the efficient frontier identified, an investor chooses the capital allocation line with the steepest slope.

    24. Note that the risk-return relation of a porfolio of risk free asset and a risky asset Q is represented by a straight line between the risk free rate on y-axis and the risk asset Q. (proof: optional material)

    25. 2.4 Riskless Borrowing and Lending With the capital allocation line identified, an investor chooses a point along the line—some combination of the risk-free asset and a risky portfolio M.

    26. The Separation Property The Separation Property states that investors can separate their risk aversion from their choice of the risky portfolio. Implications: portfolio choice can be separated into two tasks: (1) determine the optimal risky portfolio, and (2) selecting a point on the CML.

    27. Market equilibrium In a world with homogeneous expectations, the portfolio of risky asset is the same for all investors. In capital market equilibrium, demand equal supply. The portfolio of risky asset in equilibrium is called the market portfolio. market portfolio: A portfolio of all stocks in the market. Portfolio weight of stock i is equal to the proportion of stock i’s market value to the market value of all stocks in the market portfolio.   If total value of stock 1 is $10 billion and the total value of the market portfolio is $1,000 billion. Then x1=10/1,000=1%. We denote the market portfolio by M.

    28. 3. Diversification and portfolio risk We know how investors behave in a world with risk free asset, and with homogeneous expectations. Investors will hold the same risky portfolio M, and risk free asset in their portfolio. The proportions of the risky and risk free assets are dependent on the investor’s risk aversion. The risky portfolio M is the market portfolio. Next Qu: what is the risk-return relation among assets in portfolio M?

    29. 3. Diversification and portfolio risk…

    30. 3.1 diversification… diversification: diversification eliminates some, but not all of the risk. Systematic risk: risk that influences overall stock market, such as GNP, or interest rate. It can't be diversified away Unsystematic risk: risk that influences single industries, or individual firms such R&D results or a change in CEO. A stock thus has two components of risk: systematic and unsystematic risk. One can eliminate most of unsystematic risk with about 15-30 stocks.

    31. 3.2 A principle A principle: The reward for bearing risk depends only on the systematic risk of an investment. The market does not reward bearing unsystematic risk, since these risk can be diversified away in a reasonably large portfolio. Hence it is systematic risk which is important. Suppose stock A and B have the same expected return. A has a higher variance, but lower systematic risk than stock B. The stock A may be much more desirable than stock B with a lower variance.

    32. 3.3 Systematic risk and beta (ß) So, it is systematic risk, not the total risk of a stock, which is important. We can say that a stock has a high risk if it has large systematic risk. But how do we know that a stock has large systematic risk? I.E., how do we measure systematic risk of a stock? Answer: Beta (ß), where ?i ? Cov (Ri, Rm) / Var (Rm), and Rm is return on the market portfolio.

    33. 3.3 Systematic risk and beta (ß) Basic intuition about ß: ß measures how much systematic risk a stock has relative to the market portfolio (or an average asset). By definition, ß of the market portfolio is 1. Recall that systematic risk influences overall stock market. If a stock’s return co-moves a lot with overall stock market, this stock has high systematic risk. That is why you see Cov (Ri, Rm) in numerator of definition of ß.

    34. Interpretation of ß: (1) It is reasonable to say that the market portfolio has (almost) systematic risk only, since diversification eliminates (nearly) all of unsystematic risk. Consider the extent to which the variance of the market portfolio change if we change the amount of the stock in the portfolio. That is ß, i.e., the contribution of the stock to the variance of the market portfolio (or in mathematical term (? ?m2 / ?xi)).

    35. ß measures sensitivity of a stock’s return to movements in overall market. By definition, ßm = Cov (Rm, Rm) / Var (Rm) = 1. That is, ß of market portfolio is 1. Thus stocks with a ß > 1 tend to be sensitive to movements in the market--they magnify these movements. Stocks with a ß < 1 are relatively insensitive to movements in the market.

    36. 3.3 Systematic risk and beta (ß).. High ß stock Low ß stock ß = regression coefficient

    37. 3.3 Systematic risk and beta (ß).. U.S. Co Beta American Electric Power .65 Exxon .80 IBM .95 Wal-Mart 1.15 General Motors 1.05 Harley-Davidson 1.20 Papa Johns 1.45 America Online 1.65

    38. 3.3 Systematic risk and beta (ß).. Portfolio beta is equal to the weighted average of individual stock’s ß. Example: Amount Portfolio Stock Invested Weights Beta (1) (2) (3) (4) (3) ? (4) Haskell Mfg. $ 6,000 50% 0.90 0.450 Cleaver, Inc. 4,000 33% 1.10 0.367 Rutherford Co 2,000 17% 1.30 0.217 Portfolio $12,000 100% 1.034

    39. 4. The security market line 4.1 The security market line In equilibrium, the reward-to-risk ratio is constant for all assets and equal to [E(RA) - Rf ]/ ßA.   To see this, consider two stocks O and U. Stock U gives higher return relative to its level of risk, making it a more attractive asset. People will buy stock U and sell stock O. This (adjustment) process will continue until both stocks have the same reward/risk ratio.

    40. 4.1 The security market line..

    41. 4.1 The Security Market Line (SML)..

    42. 4.2 Capital Asset Pricing Model (CAPM)

    43. 4.2 Capital Asset Pricing Model (CAPM)… The Capital Asset Pricing Model (CAPM) - an equilibrium model of the relation between risk and return. E(Ri ) = Rf + ?i ? [E(Rm ) - Rf ] An asset’s expected return has three components. The risk-free rate - the pure time value of money The market risk premium - the reward for bearing systematic risk The beta coefficient - a measure of the amount of systematic risk of asset i relative to the market portfolio.

    44. The Security Market Line (SML)

    45. 5. Summary I. Total risk: the variance (or the standard deviation) of an asset’s return. II. The benefit from diversification: diversification eliminates some but not all of risk via the combination of assets into a portfolio. The lower the correlation among assets, the greater the benefit from diversification. III. Systematic and unsystematic risks: Systematic risks are unanticipated events that have economy-wide effects. Unsystematic risks are unanticipated events that affect single assets or small groups of assets. IV. Diversification eliminates (most) unsystematic risk, but the systematic risk remains. This observation leads to a principle: the reward for bearing risk depends only on its level of systematic risk. Beta measures a stock’s systematic risk. V. In equilibrium, the reward-to-risk ratio is same for all assets, and equal to the slope of SML (security market line). VI. The capital asset pricing model: E(Ri) = Rf + [E(Rm) - Rf] ????i.

    46. 4.1 The security market line..

    47. 4.1 The security market line..

    48. 7. Questions 1. Assume: the historic market risk premium has been about 8.5%. The risk-free rate is currently 5%. GTX Corp. has a beta of .85. What return should you expect from an investment in GTX? E(RGTX) = 5% + _______ ? .85% = 12.225% 2. What is the effect of diversification? 3. What does SML say? The slope of the SML = ______ .

    49. 7. Questions.. Assume: the historic market risk premium has been about 8.5%. The risk-free rate is currently 5%. GTX Corp. has a beta of .85. What return should you expect from an investment in GTX? E(RGTX) = 5% + 8.5 ? .85 = 12.225% What is the effect of diversification? Diversification reduces unsystematic risk. 3. Return-to-risk ratio is same for all assets. The slope of the SML = E(Rm ) - Rf .

    50. 7. Qu. Consider the following information: State of Prob. of State Stock A Stock B Stock C Economy of Economy Return Return Return Boom 0.35 0.14 0.15 0.33 Bust 0.65 0.12 0.03 -0.06 What is the expected return on an equally weighted portfolio of these three stocks? What is the variance of a portfolio invested 15 percent each in A and B, and 70 percent in C?

    51. 7. Qu. … Expected returns on an equal-weighted portfolio a. Boom E[Rp] = (.14 + .15 + .33)/3 = .2067 Bust: E[Rp] = (.12 + .03 - .06)/3 = .0300 so the overall portfolio expected return must be E[Rp] = .35(.2067) + .65(.0300) = .0918

    52. 7. Qu. … b. Boom: E[Rp] = __ (.14) + .15(.15) + .70(.33) = ____ Bust: E[Rp] = .15(.12) + .15(.03) + .70(-.06) = ____ E[Rp] = .35(____) + .65(____) = ____ so 2p = .35(____ - ____)2 + .65(____ - ____)2 = _____

    53. 7. Qu. … b. Boom: E[Rp] = .15(.14) + .15(.15) + .70(.33) = .2745 Bust: E[Rp] = .15(.12) + .15(.03) + .70(-.06) = -.0195 E[Rp] = .35(.2745) + .65(-.0195) = .0834 so 2p = .35(.2745 - .0834)2 + .65(-.0195 - .0834)2 = .01278 + .00688 = .01966

    54. Qu Using information from the previous chapter on capital market history, determine the return on a portfolio that is equally invested in Canadian stocks and long-term bonds. What is the return on a portfolio that is equally invested in small-company stocks and Treasury bills?

    55. Qu … Solution The average annual return on common stocks over the period 1948-1999 was 13.2 percent, and the average annual return on long-term bonds was 7.6 percent. So, the return on a portfolio with half invested in common stocks and half in long-term bonds would have been: E[Rp1] = .50(13.2) + .50(7.6) = 10.4%

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