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Portfolio Managment 3-228-07 Albert Lee Chun

Portfolio Managment 3-228-07 Albert Lee Chun. Proof of the Capital Asset Pricing Model. Lecture 6. Course Outline. Sessions 1 and 2 : The Institutional Environment Sessions 3, 4 and 5: Construction of Portfolios Sessions 6 and 7: Capital Asset Pricing Model

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Portfolio Managment 3-228-07 Albert Lee Chun

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  1. Portfolio Managment3-228-07Albert Lee Chun Proof of the Capital Asset Pricing Model Lecture 6

  2. Course Outline Sessions 1 and 2 : The Institutional Environment Sessions 3, 4 and 5: Construction of Portfolios Sessions 6 and 7: Capital Asset Pricing Model Session 8: Market Efficiency Session 9: Active Portfolio Management Session 10: Management of Bond Portfolios Session 11: Performance Measurement of Managed Portfolios

  3. Plan for Today • Fun Proof of the CAPM • Zero-Beta CAPM (not on the syllabus) • A few examples • Revision for the mid-term

  4. A Fun Proof of the CAPM

  5. CAPM Says that for any security i that we pick, the expected return of that security is given by Capital Market Line M security i

  6. Why does CAPM work? Green line traces out the set of possible portfolios P using security i and M by varying w, Capital Market Line where w is the weight on securityi in portfolio P M P security i

  7. Why does CAPM work? Note that w=1corresponds to security i and w=0gives us the market portfolio M, Capital Market Line where w is the weight on securityi in portfolio P w = 0 M P security i w = 1

  8. Why does CAPM work? For any weight w, we can easily compute the expected return and the variance of portfolio P, Capital Market Line where w is the weight on securityi in portfolio P w = 0 M P security i w = 1

  9. Why does CAPM work? Note that the CML (orange line) is tangent to both the risky efficient frontier (blue line) and the green line at M. Capital Market Line w = 0 M P Intuition: The orange line, the blue line and the green line all touch at only 1 point M. Why? security i w = 1

  10. Why does CAPM work? Slope of the green line at M, is equal to the slope of the blue line at M which is equal to the slope of the CML(orange line)! Capital Market Line w = 0 M Intuition: The orange line, the blue line and the green line all touch at only 1 point M. Why? security i

  11. Why does CAPM work? Slope of the green line at M, is equal to the slope of the blue line at M which is equal to the slope of the CML(orange line)! Capital Market Line w = 0 M The slope of the CML security i

  12. Why does CAPM work? (slope = slope = slope) Capital Market Line w = 0 M Therefore, the slope of all 3 lines at Mis security i

  13. Why does CAPM work? Mathematically the slope of the green line at M is: Capital Market Line w = 0 M The slope of all 3 lines at Mis security i

  14. Why does CAPM work? Note that we can also express the slope of the green line asas: This slope has to equal the slope of the CML at M! w = 0 M = security i

  15. Proof of CAPM We want to find the slope of the green line by differentiating these at w = 0 and using this relation to set the slope at (w = 0) equal to the slope of the CML =

  16. Proof of CAPM = w = 0 M security i To prove CAPM we use the fact that the green slope has to equal the slope of the CML at M.

  17. Let’s Take a Few Derivatives Derivative of expected return w.r.t w. 16

  18. Let’s Take a Few Derivatives Derivative of standard deviation w.r.t. w Evaluate the derivative at w = 0, which is at the market portfolio! 17

  19. Equate the Slopes = =

  20. Equating the Slopes Capital Market Line w = 0 M security i

  21. Now Solve for E(Ri) Voila! We just proved the CAPM!!

  22. We just showed that for any security i that we pick, the expected return of that security is given by M security i So we just won the Nobel Prize!

  23. Zero-Beta Capital Asset Pricing Model(Not on the Syllabus: However, understanding this might be useful for solving other problems on the exam.)

  24. Efficient frontier s Suppose There is No Risk Free Asset Can we say something about the expected return of a particular asset in this economy?

  25. Zero Beta CAPM Fisher Black (1972) There exists an efficient portfolio that is uncorrelated with the market portfolio, hence it has zero beta.

  26. Efficient frontier s Zero-Beta CAPM World Zero-Beta Portfolio

  27. Zero-Beta SML SML

  28. Example CAPM Suppose there are 2 efficient risky securities: Security E(r) Beta Egg 0.07 0.50 Bert 0.10 0.80 You do not know E(Rm) or Rf. Suppose that Karina is thinking about buying the following: Security E(r) Beta Karina 0.16 1.30 Should she buy the security? 27

  29. Under Valued or Overvalued Undervalued Buy! SML Market Bert Egg Overvalued Don`t Buy! 28

  30. Example CAPM We know that for the two efficient securities: E(REgg) = rf + BEgg(E(Rm)- Rf) E(RBert) = rf + BBert(E(Rm)- Rf) And if Karina is an efficient security we would have: E(RKarina) = rf + BKarina(E(Rm) - Rf) 29

  31. Example CAPM First find the expected return on the market and the risk-free retrun by solving 2 equations in 2 unknowns: E(REgg) = (1- BEgg)Rf + BEgg E(Rm) E(RBert) = (1- BBert)Rf + BBert E(Rm) Some algebra: (E(REgg) - (1- BEgg) Rf )/ BEgg = (E(RBert) - (1- BBert) Rf )/ BBert Rf = [BBertE(REgg) - BEgg E(RBert)]/ [BEgg(1-BBert ) + BBert (1- BEgg) ] E(Rm)= (E(REgg) - (1- BEgg) Rf )/ BEgg 30

  32. Example CAPM Security E(r) Beta Egg .07 .5 Bert .1 .8 Karina .16 1.3 Rf = [BBertE(REgg) - BEgg E(RBert)]/ [-BEgg(1-BBert ) + BBert (1- BEgg) ] = .02 E(Rm)= (E(REgg) - (1- BEgg) Rf )/ BEgg = .12 E(RKarina) = rf + BKarina(E(Rm) - Rf) =.02 + 1.3*(.12 - .02) = .15 < .16 31

  33. Stock is Under Valued Undervalued Buy! Karina 16% SML Market 15% Bert Egg 32

  34. Another Example

  35. Example The expected return on the market portfolio is 9%. A) Determine the covariance between the return on Dingo and the return on the market portfolio. B) Determine the rate of return on Dingo using CAPM. Would you recommend that investors buy shares of Dingo? (Justify your answer)

  36. Solution : E(re) = 13,00% E(rd) = 12,55% Var(re) = 0,004860 Var(rd) = 0,002365 STD(re) = 0,069714 STD(rd) = 0,048629 STD Market= 0,030220 Var Market = 0,000913 Covariance of Dingo with the market = 0,001237 Beta of Dingo = 1,35 Expected Reeturn of the Market = 9% Expect Return of Dingo according to CAPM : E(rd) = Rf + BetaDingo (E(Rm) - Rf) = 11,13% 12,55% > 11,13% - Buy! Lies above the SML.

  37. Midterm • Focus on solving examples that I gave you to do at home and what we did in class. • Do the math as well as know the intuition. • The lecture notes are more important than the book, although the book is important too. • Focus on Lectures 3 – 6

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