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Capital Market Line

Capital Market Line. Line from RF to L is capital market line (CML) x = risk premium = E(R M ) - RF y = risk =  M Slope = x/y = [E(R M ) - RF]/  M y-intercept = RF. L. M. E(R M ). x. RF. y.  M. Risk. Capital Market Line.

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Capital Market Line

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  1. Capital Market Line • Line from RF to L is capital market line (CML) • x = risk premium = E(RM) - RF • y = risk = M • Slope = x/y = [E(RM) - RF]/M • y-intercept = RF L M E(RM) x RF y M Risk

  2. Capital Market Line • Slope of the CML is the market price of risk for efficient portfolios, or the equilibrium price of risk in the market • Relationship between risk and expected return for portfolio P (Equation for CML):

  3. Security Market Line • CML Equation only applies to markets in equilibrium and efficient portfolios • The Security Market Line depicts the tradeoff between risk and expected return for individual securities • Under CAPM, all investors hold the market portfolio • How does an individual security contribute to the risk of the market portfolio?

  4. Security Market Line • Equation for expected return for an individual stock similar to CML Equation

  5. Security Market Line • Beta = 1.0 implies as risky as market • Securities A and B are more risky than the market • Beta > 1.0 • Security C is less risky than the market • Beta < 1.0 SML E(R) A E(RM) B C RF 0 0.5 1.0 1.5 2.0 BetaM

  6. Security Market Line • Beta measures systematic risk • Measures relative risk compared to the market portfolio of all stocks • Volatility different than market • All securities should lie on the SML • The expected return on the security should be only that return needed to compensate for systematic risk

  7. SML and Asset Values Er UnderpricedSML: Er = rf +  (Erm – rf) Overpriced rf β Underpriced  expected return > required return according to CAPM  lie “above” SML Overpriced  expected return < required return according to CAPM  lie “below” SML Correctly priced  expected return = required return according to CAPM  lie along SML

  8. CAPM’s Expected Return-Beta Relationship • Required rate of return on an asset (ki) is composed of • risk-free rate (RF) • risk premium (i [ E(RM) - RF ]) • Market risk premium adjusted for specific security ki = RF +i [ E(RM) - RF ] • The greater the systematic risk, the greater the required return

  9. Estimating the SML • Treasury Bill rate used to estimate RF • Expected market return unobservable • Estimated using past market returns and taking an expected value • Estimating individual security betas difficult • Only company-specific factor in CAPM • Requires asset-specific forecast

  10. Estimating Beta • Market model • Relates the return on each stock to the return on the market, assuming a linear relationship Rit =i +i RMt + eit

  11. Test of CAPM • Empirical SML is “flatter” than predicted SML • Fama and French (1992) • Market • Size • Book-to-market ratio • Roll’s Critique • True market portfolio is unobservable • Tests of CAPM are merely tests of the mean-variance efficiency of the chosen market proxy

  12. Arbitrage Pricing Theory • Based on the Law of One Price • Two otherwise identical assets cannot sell at different prices • Equilibrium prices adjust to eliminate all arbitrage opportunities • Unlike CAPM, APT does not assume • single-period investment horizon, absence of personal taxes, riskless borrowing or lending, mean-variance decisions

  13. Factors • APT assumes returns generated by a factor model • Factor Characteristics • Each risk must have a pervasive influence on stock returns • Risk factors must influence expected return and have nonzero prices • Risk factors must be unpredictable to the market

  14. APT Model • Most important are the deviations of the factors from their expected values • The expected return-risk relationship for the APT can be described as: E(Rit) =a0+bi1 (risk premium for factor 1) +bi2 (risk premium for factor 2) +… +bin (risk premium for factor n)

  15. APT Model • Reduces to CAPM if there is only one factor and that factor is market risk • Roll and Ross (1980) Factors: • Changes in expected inflation • Unanticipated changes in inflation • Unanticipated changes in industrial production • Unanticipated changes in the default risk premium • Unanticipated changes in the term structure of interest rates

  16. Problems with APT • Factors are not well specified ex ante • To implement the APT model, the factors that account for the differences among security returns are required • CAPM identifies market portfolio as single factor • Neither CAPM or APT has been proven superior • Both rely on unobservable expectations

  17. Two-Security Case • For a two-security portfolio containing Stock A and Stock B, the variance is:

  18. Two Security Case (cont’d) Example Assume the following statistics for Stock A and Stock B:

  19. Two Security Case (cont’d) Example (cont’d) What is the expected return and variance of this two-security portfolio?

  20. Two Security Case (cont’d) Example (cont’d) Solution: The expected return of this two-security portfolio is:

  21. Two Security Case (cont’d) Example (cont’d) Solution (cont’d): The variance of this two-security portfolio is:

  22. Minimum Variance Portfolio • The minimum variance portfolio is the particular combination of securities that will result in the least possible variance • Solving for the minimum variance portfolio requires basic calculus

  23. Minimum Variance Portfolio (cont’d) • For a two-security minimum variance portfolio, the proportions invested in stocks A and B are:

  24. Minimum Variance Portfolio (cont’d) Example (cont’d) Assume the same statistics for Stocks A and B as in the previous example. What are the weights of the minimum variance portfolio in this case?

  25. Minimum Variance Portfolio (cont’d) Example (cont’d) Solution: The weights of the minimum variance portfolios in this case are:

  26. Minimum Variance Portfolio (cont’d) Example (cont’d) Weight A Portfolio Variance

  27. Correlation and Risk Reduction • Portfolio risk decreases as the correlation coefficient in the returns of two securities decreases • Risk reduction is greatest when the securities are perfectly negatively correlated • If the securities are perfectly positively correlated, there is no risk reduction

  28. The n-Security Case • For an n-security portfolio, the variance is:

  29. The n-Security Case (cont’d) • A covariance matrix is a tabular presentation of the pairwise combinations of all portfolio components • The required number of covariances to compute a portfolio variance is (n2 – n)/2 • Any portfolio construction technique using the full covariance matrix is called a Markowitz model

  30. Computational Advantages • The single-index model compares all securities to a single benchmark • An alternative to comparing a security to each of the others • By observing how two independent securities behave relative to a third value, we learn something about how the securities are likely to behave relative to each other

  31. Computational Advantages (cont’d) • A single index drastically reduces the number of computations needed to determine portfolio variance • A security’s beta is an example:

  32. Portfolio Statistics With the Single-Index Model • Beta of a portfolio: • Variance of a portfolio:

  33. Portfolio Statistics With the Single-Index Model (cont’d) • Variance of a portfolio component: • Covariance of two portfolio components:

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