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MPT Modern Portfolio Theory

2. . 2. K. Hartviksen. Key Terms. Harry MarkowitzMPTExpected returnrequired returnportfoliosystematic riskunsystematic riskdiversificationbeta coefficientsecurity market line. market premium for riskcapital asset pricing modelcost of capitalmean, variance, standard deviationcorrelation

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MPT Modern Portfolio Theory

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    1. MPT – Modern Portfolio Theory Business 2039

    2. Key Terms Harry Markowitz MPT Expected return required return portfolio systematic risk unsystematic risk diversification beta coefficient security market line market premium for risk capital asset pricing model cost of capital mean, variance, standard deviation correlation capital market line

    3. Harry Markowitz Modern portfolio theory was initiated by University of Chicago graduate student, Harry Markowitz in 1952. Markowitz showed how the risk of a portfolio is NOT just the weighted average sum of the risks of the individual securities…but rather, also a function of the degree of comovement of the returns of those individual assets.

    4. Risk and Return - MPT Prior to the establishment of Modern Portfolio Theory, most people only focused upon investment returns…they ignored risk. With MPT, investors had a tool that they could use to dramatically reduce the risk of the portfolio without a significant reduction in the expected return of the portfolio.

    5. Correlation The degree to which the returns of two stocks co-move is measured by the correlation coefficient. The correlation coefficient between the returns on two securities will lie in the range of +1 through - 1. +1 is perfect positive correlation. -1 is perfect negative correlation.

    6. Perfect Negatively Correlated Returns over Time

    7. Ex Post Portfolio Returns Simply the Weighted Average of Past Returns

    8. Ex Ante Portfolio Returns Simply the Weighted Average of Expected Returns

    9. Grouping Individual Assets into Portfolios The riskiness of a portfolio that is made of different risky assets is a function of three different factors: the riskiness of the individual assets that make up the portfolio the relative weights of the assets in the portfolio the degree of comovement of returns of the assets making up the portfolio The standard deviation of a two-asset portfolio may be measured using the Markowitz model:

    10. Risk of a Three-asset Portfolio

    11. Risk of a Four-asset Portfolio

    12. Diversification Potential The potential of an asset to diversify a portfolio is dependent upon the degree of co-movement of returns of the asset with those other assets that make up the portfolio. In a simple, two-asset case, if the returns of the two assets are perfectly negatively correlated it is possible (depending on the relative weighting) to eliminate all portfolio risk. This is demonstrated through the following chart.

    13. Example of Portfolio Combinations and Correlation

    14. Example of Portfolio Combinations and Correlation

    15. Example of Portfolio Combinations and Correlation

    16. Example of Portfolio Combinations and Correlation

    17. Example of Portfolio Combinations and Correlation

    19. An Exercise using T-bills, Stocks and Bonds

    20. Results Using only Three Asset Classes

    26. CML versus SML Please notice that the CML is used to illustrate all of the efficient portfolio combinations available to investors. It differs significantly from the SML that is used to predict the required return that investors should demand given the riskiness (beta) of the investment.

    27. Data Limitations Because of the need for so much data, MPT was a theoretical idea for many years. Later, a student of Markowitz, named William Sharpe worked out a way around that…creating the Beta Coefficient as a measure of volatility and then later developing the CAPM.

    28. CAPM The Capital Asset Pricing Model was the work of William Sharpe, a student of Harry Markowitz at the University of Chicago. CAPM is an hypothesis …

    29. Capital Asset Pricing Model

    30. CAPM This model is an equilibrium based model. It is called a single-factor model because the slope of the SML is caused by a single measure of risk … the beta. Although this model is a simplification of reality…it is robust (it explains much of what we see happening out there) and it enjoys widespread use in a great variety of applications. Although it is called a ‘pricing model’ there are not prices on that graph….only risk and return. It is called a pricing model because it can be used to help us determine appropriate prices for securities in the market.

    31. Risk Risk is the chance of harm or loss; danger. We know that various asset classes have yielded very different returns in the past:

    32. Historical Returns and Standard Deviations 1948 - 941 Average Return Standard Deviation Canadian common stock 12.73% 16.81% U.S. common stock (Cdn $) 14.09 16.60 Long term bonds 7.01 10.20 Small cap stocks 15.67 24.40 Inflation 4.52 3.54 Treasury bills 6.15 4.17 ___________________ 1The Alexander Group

    33. Risk and Return The foregoing data point out that those asset classes that have offered the highest rates of return, have also offered the highest risk levels as measured by the standard deviation of returns. The CAPM suggests that investors demand compensation for risks that they are exposed to…and these returns are built into the decision-making process to invest or not.

    34. Capital Asset Pricing Model

    35. CAPM The foregoing graph shows that investors: demand compensation for expected inflation demand a real rate of return over and above expected inflation demand compensation over and above the risk-free rate of return for any additional risk undertaken. We will make the case that investors don’t need compensation for all of the risk of an investment because some of that risk can be diversified away. Investors require compensation for risk they can’t diversify away!

    36. Beta Coefficient The beta is a measure of systematic risk of an investment. Systematic risk is the only relevant risk to a diversified investor according to the CAPM since all other risk may be diversified away. Total risk of an investment is measured by the securities’ standard deviation of returns. According to the CAPM total risk may be broken into two parts…systematic (non-diversifiable) and unsystematic (diversifiable) TOTAL RISK = SYSTEMATIC RISK + UNSYSTEMATIC RISK The beta can be determined by regressing the holding period returns (HPRs) of the security over 30 periods against the returns on the overall market.

    37. Measuring Risk of the Individual Security Risk is the possibility that the actual return that will be realized, will turn out to be different than what we expect (or have forecast). This can be measured using standard statistical measures of dispersion for probability distributions. They include: variance standard deviation coefficient of variation

    38. Standard Deviation The formula for the standard deviation when analyzing population data (realized returns) is:

    39. Standard Deviation The formula for the standard deviation when analyzing forecast data (ex ante returns) is: it is the square root of the sum of the squared deviations away from the expected value.

    40. Using Forecasts to Estimate Beta The formula for the beta coefficient for a stock ‘s’ is: Obviously, the calculate a beta for a stock, you must first calculate the variance of the returns on the market portfolio as well as the covariance of the returns on the stock with the returns on the market.

    41. Systematic Risk The returns on most assets in our economy are influenced by the health of the ‘system’ Some companies are more sensitive to systematic changes in the economy. For example durable goods manufacturers. Some companies do better when the economy is doing poorly (bill collection agencies). The beta coefficient measures the systematic risk that the security possesses. Since non-systematic risk can be diversified away, it is irrelevant to the diversified investor.

    42. Systematic Risk We know that the economy goes through economic cycles of expansion and contraction as indicated in the following:

    44. Companies and Industries Some industries (and by implication the companies that make up the industry) move in concert with the expansion and contraction of the economy. Some lead the overall economy. (stock market) Some lag the overall economy. (ie. automotive industry)

    45. Amount of Systematic Risk Some industries may find that their fortunes are positively correlated with the ebb and flow of the overall economy…but that this relationship is very insignificant. An example might be Imperial Tobacco. This firm does have a positive beta coefficient, but very little of the returns of this company can be explained by the beta. Instead, most of the variability of returns on this stock is from diversifiable sources. A Characteristic line for Imperial Tobacco would show a very wide dispersion of points around the line. The R2 would be very low (.05 = 5% or lower).

    46. Characteristic Line for Imperial Tobacco

    47. High R2 An R2 that approaches 1.00 (or 100%) indicates that the characteristic (regression) line explains virtually all of the variability in the dependent variable. This means that virtually of the risk of the security is ‘systematic’. This also means that the regression model has a strong predictive ability. … if you can predict what the market will do…then you can predict the returns on the stock itself with a great deal of accuracy.

    48. Characteristic Line General Motors

    49. Diversifiable Risk (non-systematic risk) Examples of this type of risk include: a single company strike a spectacular innovation discovered through the company’s R&D program equipment failure for that one company management competence or management incompetence for that particular firm a jet carrying the senior management team of the firm crashes the patented formula for a new drug discovered by the firm. Obviously, diversifiable risk is that unique factor that influences only the one firm.

    50. Partitioning Risk under the CAPM Remember that the CAPM assumes that total risk (variability of a security’s returns) can be separated into two distinct components: Total risk = systematic risk + unsystematic risk 100% = 40% + 60% (GM) or 100% = 5% + 95% (Imperial Tobacco) Obviously, if you were to add Imperial Tobacco to your portfolio, you could diversify away much of the risk of your portfolio. (Not to mention the fact that Imperial has realized some very high rates of return in addition to possessing little systematic risk!)

    51. Using the CAPM to Price Stock The CAPM is a ‘fundamental’ analyst’s tool to estimate the ‘intrinsic’ value of a stock. The analyst needs to measure the beta risk of the firm by using either historical or forecast risk and returns. The analyst will then need a forecast for the risk-free rate as well as the expected return on the market. These three estimates will allow the analyst to calculate the required return that ‘rational’ investors should expect on such an investment given the other benchmark returns available in the economy.

    52. Required Return The return that a rational investor should demand is therefore based on market rates and the beta risk of the investment. To find this, you solve for the required return in the CAPM: This is a formula for the straight line that is the SML.

    53. Security Market Line This line can easily be plotted. Draw Cartesian coordinates. Plot the yield on 91-day Government of Canada Treasury Bills as the risk-free rate of return on the vertical axis. On the horizontal axis set a scale that includes Beta=1 (this is the beta of the market) Plot the point in risk-return space that represents your expected return on the market portfolio at beta =1 Draw a straight line to connect the two points. Plot the required and expected returns for the stock at it’s beta.

    54. Plot the Risk-Free Rate

    55. Plot Expected Return on the Market Portfolio

    56. Draw the Security Market Line

    57. Plot Required Return (Determined by the formula = Rf + bs[kM - Rf]

    58. Plot Expected Return E(k) = weighted average of possible returns

    59. If Expected = Required Return The stock is properly (fairly) priced in the market. It is in EQUILIBRIUM.

    60. If E(k) < R(k) The stock is over-priced. The analyst would issue a sell recommendation in anticipation of the market becoming ‘efficient’ to this fact. Investors may ‘short’ the stock to take advantage of the anticipated price decline.

    61. Let’s Look at the Pricing Implications In this example: E(k) = 9% R(k) = 13.6% If the market expects the company to pay a dividend of $1.00 next year, and the stock is currently offering an expected return of 9%, then it should be priced at: But, given the other rates in the economy and our judgement about the riskiness of this investment we think that this stock should be worth:

    62. Practical Use of the CAPM Regulated utilities justify rate increases using the model to demonstrate that their shareholders require an appropriate return on their investment. Used to price initial public offerings (IPOs) Used to identify over and under value securities Used to measure the riskiness of securities/companies Used to measure the company’s cost of capital. (The cost of capital is then used to evaluate capital expansion proposals). The model helps us understand the variables that can affect stock prices…and this guides managerial decisions.

    63. Rf rises

    64. The Slope of The SML rises (indicates growing pessimism about the future of the economy)

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