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MBA 643 Managerial Finance Lecture 8: Modern Portfolio Theory, Part II

MBA 643 Managerial Finance Lecture 8: Modern Portfolio Theory, Part II. Spring 2006 Jim Hsieh. What do we know about portfolio risk? -- Recap. Most stocks are positively correlated. Average correlation between two stocks is 0.65.

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MBA 643 Managerial Finance Lecture 8: Modern Portfolio Theory, Part II

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  1. MBA 643Managerial FinanceLecture 8: Modern Portfolio Theory, Part II Spring 2006 Jim Hsieh

  2. What do we know about portfolio risk? -- Recap • Most stocks are positively correlated. • Average correlation between two stocks is 0.65. • So long as the stocks in the portfolio are not perfectly correlated (i.e., <1), the standard deviation of the portfolio will be less than the weighted average of the standard deviation of the stocks in the portfolio. • When we add more securities in the portfolio, we can lower the risk of the portfolio even further. • This is because the added securities would not be perfectly correlated with existing securities in the portfolio. • Q: Can we eliminate the portfolio risk completely by adding more assets/securities into our portfolio?

  3. Stand-alone Risk in Individual Securities • Stand-alone risk consists of: • Diversifiable risk • Company or industry specific • Also called unsystematic, unique, or idiosyncratic risk • Non-diversifiable risk • Related to market as a whole • Also called systematic, portfolio, or market risk • It shows the degree to which a stock moves systematically with other stocks. • Total risk of security = unsystematic risk + portfolio risk • We can eliminate unsystematic risk by adding more securities into the portfolio.

  4. Diversification reduces risk • In this example, the table shows portfolio risk (std dev) by adding one more stock.

  5. Diversification Unsystematic Risk Market Risk

  6. Question • If Stephen holds a one-stock portfolio and Jennifer holds a multiple-stock portfolio. Thus Stephen is exposed to more risk than Jennifer. Do you think Stephen should be compensated for all the risk he bears?

  7. Answer • No! • Even though Stephen holds only one stock, he will not be compensated for the additional risk he bears. • Stand-alone risk as a whole is not as important to a well-diversified investor, and most of it can be eliminated at virtually no cost through diversification. • Thus, bearing the diversifiable risk should not be rewarded. • Rational risk averse investors are concerned with P, which is based on market risk.

  8. The Systematic Risk Principal • The reward for bearing risk depends only on the systematic risk of an investment since unsystematic risk can be diversified away. • This implies that the expected return on any asset depends only on that asset’s systematic risk. • If investors are well diversified, they only care how a stock correlates with the rest of their portfolio (the “market portfolio”). The variance of that stock (i2) is, essentially, irrelevant. • Important Implication: If two assets have the same correlation with the market portfolio, they must have the same expected return.

  9. Measuring Systematic Risk • Beta (β) measures a stock’s market (or systematic) risk. It shows the relative volatility of a given stock compared to the average stock. An average stock (or the market portfolio) has a beta = 1.0. • Beta shows how risky a stock is if the stock is held in a well-diversified portfolio. • β=1 → stock has average risk. • β>1 → stock is riskier than average. • β<1 → stock is less risky than average. • β=0 → risk free assets (e.g., Treasury bills) • Q: Can β be negative?

  10. More on Beta • Definition of Beta: • Note that stock i’s beta has two components: • Covariance of returns between stock i and market portfolio. • Variance of return on market portfolio • NO variance of return on stock i  NO i2

  11. Beta Coefficients for Selected Companies (From Table 10.7)

  12. Portfolio Betas • The beta of a portfolio (βP) is the weighted average of the betas from its constituent securities. • βP = w1β1 + w2β2 + … + wNβN for N securities • Example 1: You have $6,000 invested in IBM, $4,000 in GM. You estimate that IBM has a beta of 0.95 and GM has a beta of 1.15. What is the beta of your portfolio? • βP = 0.6*0.95 + 0.4*1.15 = 1.03

  13. Beta and Risk Premium • Consider a portfolio which consists of stock A with a beta of 1.2 and expected return of 18%, and a Treasury bill with a 7% return. • E(RP)=wARA+wFRF=(wA)(18%)+(wF)(7%) • P=wAA+wFF=(wA)(1.2) 0*18% + 1*7% = 7% 0*1.2 = 0 .25*18% + .75*7% = 9.75 0.25*1.2 = 0.3 .50*18% + .50*7% = 12.5 0.50*1.2 = 0.6 .75*18% + .25*7% = 15.25 0.75*1.2 = 0.9 1*18% + 0*7% = 18% 1*1.2 = 1.2 1.5*18% + (-.5)*7% = 23.5% 1.5*1.2 = 1.8

  14. The Relationship Between E(R) and Beta 18% 7%=Rf

  15. Reward to Risk Ratio • We can vary the amount invested in each type of asset and get an idea of the relation between portfolio expected return and beta: • It estimates the expected risk premium per unit of risk. • We can also calculate the reward to risk ratio for all individual securities.

  16. What happens if two securities have different reward-to-risk ratios? • Investors would only buy the securities (portfolios) with a higher reward-to-risk ratio. Here, it would be A. • Eventually, all securities will have the same reward-to-risk ratio. • Because the reward-to-risk ratio is the same for all securities, it must hold for the market portfolio too. • Result:

  17. The Capital Asset Pricing Model (CAPM) • Since we know: • Then: E(RA) = Rf + A[E(RM) – Rf] • The CAPM describes the relationship between the expected risk premium on a security, E(Ri)-Rf, and the risk, . • What determines a security’s expected return? • The risk-free rate • The market risk premium • The beta coefficient • The CAPM holds for individual assets as well as portfolios of those assets. This is CAPM!!

  18. An Example of CAPM • Suppose the risk-free rate is 4%, the market risk premium is 8.6%, and a stock has a beta of 1.3. Based on the CAPM, what is the expected return on this stock? What would the expected return be if the beta were to double? • E(R) = Rf + [E(RM) – Rf] = 4% + 1.3*8.6% = 15.18% • E(R) = Rf + [E(RM) – Rf] = 4% + 2.6*8.6% = 26.36%

  19. The Security Market Line (SML)CAPM: E(Ri) = Rf + i[E(RM) – Rf] • The security market line (SML) is a line that shows the relationship between risk () and the required rate of return on individual securities. Required Rate of Return SML: E(Ri) = Rf + i*[E(RM)-Rf] = E (RM) – Rf E (RM) Rf Risk, i M= 1.0

  20. SML • All of the “fair-valued” securities should fall on the SML. Required Rate of Return Over-valued/Under-priced SML Under-valued/Over-priced Rf Risk, i

  21. Summary of Risk and Return • Investors like high E(R) and low standard deviation. Portfolios that offer the highest E(R) for a given standard deviation are called efficient. • If investors are well diversified, they only care how a stock correlates with the rest of their portfolio (the “market portfolio”). They only care about the covariance of that stock with other stocks in the portfolio. • A stock’s sensitivity to changes in the value of the market portfolio is its beta. • The CAPM is a simple, linear model that links risk premia on assets to risk premium on the market portfolio, using beta as a measure of risk.

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