2. 0 Portfolio Managment 3-228-07 Albert Lee Chun Proof of the Capital Asset Pricing Model Lecture 6

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

4. Albert Lee Chun Portfolio Management 2 Plan for Today Plan for Today Fun Proof of the CAPM Fun Proof of the CAPM Zero-Beta CAPM (not on the syllabus) Zero-Beta CAPM (not on the syllabus) A few examples A few examples Revision for the mid-term Revision for the mid-term

5. Albert Lee Chun Portfolio Management 3 A Fun Proof of the CAPM

6. Albert Lee Chun Portfolio Management 4 CAPM Says that security i Capital Market Line for any security i that we pick, the expected return of that security is given by M

7. Albert Lee Chun Portfolio Management 5 Why does CAPM work? security i P Capital Market Line Green line traces out the set of possible portfolios P using security i and M by varying w, M where w is the weight on security i in portfolio P

8. Albert Lee Chun Portfolio Management 6 Why does CAPM work? security i Capital Market Line w = 1 P M w = 0 where w is the weight on security i in portfolio P Note that w=1 corresponds to security i and w=0 gives us the market portfolio M,

9. Albert Lee Chun Portfolio Management 7 Why does CAPM work? security i Capital Market Line For any weight w, we can easily compute the expected return and the variance of portfolio P, w = 1 P M where w is the weight on security i in portfolio P w = 0

10. Albert Lee Chun Portfolio Management 8 Why does CAPM work? security i Capital Market Line Intuition: The orange line, the blue line and the green line all touch at only 1 point M. Why? w = 1 P Note that the CML (orange line) is tangent to both the risky efficient frontier (blue line) and the green line at M. M w = 0

11. Albert Lee Chun Portfolio Management 9 Why does CAPM work? security i Capital Market Line 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)! Intuition: The orange line, the blue line and the green line all touch at only 1 point M. Why? M w = 0

12. Albert Lee Chun Portfolio Management 10 Why does CAPM work? security i Capital Market Line 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)! The slope of the CML M w = 0

13. Albert Lee Chun Portfolio Management 11 Why does CAPM work? security i Capital Market Line Therefore, the slope of all 3 lines at M is M w = 0 (slope = slope = slope)

14. Albert Lee Chun Portfolio Management 12 Why does CAPM work? security i Capital Market Line Mathematically the slope of the green line at M is: M w = 0 The slope of all 3 lines at M is

15. Albert Lee Chun Portfolio Management 13 Why does CAPM work? security i Note that we can also express the slope of the green line as as: = This slope has to equal the slope of the CML at M! M w = 0

16. Albert Lee Chun Portfolio Management 14 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

17. Albert Lee Chun Portfolio Management 15 Proof of CAPM security i = To prove CAPM we use the fact that the green slope has to equal the slope of the CML at M. M w = 0

18. Albert Lee Chun Portfolio Management 16 Let’s Take a Few Derivatives Derivative of expected return w.r.t w.

19. Albert Lee Chun Portfolio Management 17 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!

20. Albert Lee Chun Portfolio Management 18 Equate the Slopes = =

21. Albert Lee Chun Portfolio Management 19 Equating the Slopes security i Capital Market Line M w = 0

22. Albert Lee Chun Portfolio Management 20 Now Solve for E(R i ) Voila! We just proved the CAPM!!

23. Albert Lee Chun Portfolio Management 21 We just showed that security i for any security i that we pick, the expected return of that security is given by M So we just won the Nobel Prize!

24. Albert Lee Chun Portfolio Management 22 Zero-Beta Capital Asset Pricing Model (Not on the Syllabus: However, understanding this might be useful for solving other problems on the exam.)

25. Albert Lee Chun Portfolio Management 23 Suppose There is No Risk Free Asset Can we say something about the expected return of a particular asset in this economy? Efficient frontier 

26. Albert Lee Chun Portfolio Management 24 Zero Beta CAPM Fisher Black (1972) There exists an efficient portfolio that is uncorrelated with the market portfolio, hence it has zero beta.

27. Albert Lee Chun Portfolio Management 25 Zero-Beta CAPM World Efficient frontier  Zero-Beta Portfolio

28. Albert Lee Chun Portfolio Management 26 Zero-Beta SML SML

29. Albert Lee Chun Portfolio Management Example CAPM Suppose there are 2 efficient risky securities: SecurityE(r) Beta Egg 0.070.50 Bert0.100.80 You do not know E(Rm) or Rf. Suppose that Karina is thinking about buying the following: SecurityE(r) Beta Karina0.161.30 Should she buy the security? Should she buy the security? 27

30. Albert Lee Chun Portfolio Management 28 Under Valued or Overvalued Undervalued Buy! Overvalued Don`t Buy! SML Egg Bert Market

31. Albert Lee Chun Portfolio Management Example CAPM We know that for the two efficient securities: E(R Egg ) = r f + B Egg (E(R m) - R f ) E(R Bert ) = rf + B Bert (E(R m) - R f ) And if Karina is an efficient security we would have: E(R Karina ) = rf + B Karina (E(R m) - R f ) E(R Karina ) = rf + B Karina (E(R m) - R f ) 29

32. Albert Lee Chun Portfolio Management Example CAPM First find the expected return on the market and the risk-free retrun by solving 2 equations in 2 unknowns: E(R Egg ) = (1- B Egg ) R f + B Egg E(R m) E(R Egg ) = (1- B Egg ) R f + B Egg E(R m) E(R Bert ) = (1- B Bert ) R f + B Bert E(R m) Some algebra: Some algebra: (E(R Egg ) - (1- B Egg ) Rf )/ B Egg = (E(R Bert ) - (1- B Bert ) Rf )/ B Bert (E(R Egg ) - (1- B Egg ) Rf )/ B Egg = (E(R Bert ) - (1- B Bert ) Rf )/ B Bert R f = [ B Bert E(R Egg ) - B Egg E(R Bert )]/ [B Egg (1-B Bert ) + B Bert (1- B Egg ) ] E(R m) = (E(R Egg ) - (1- B Egg ) Rf )/ B Egg 30

33. Albert Lee Chun Portfolio Management Example CAPM 31 SecurityE(r) Beta Egg.07.5 Bert.1.8 Karina.161.3 Rf = [B Bert E(R Egg ) - B Egg E(R Bert )]/ [-B Egg (1-B Bert ) + B Bert (1- B Egg ) ] =.02 E(Rm)= (E(R Egg ) - (1- B Egg ) Rf )/ B Egg =.12 E(R Karina ) = rf + B Karina (E(Rm) - Rf) =.02 + 1.3*(.12 -.02) =.15 <.16

34. Albert Lee Chun Portfolio Management 32 Stock is Under Valued Undervalued Buy! SML Egg Bert Market Karina 16% 15%

35. Albert Lee Chun Portfolio Management Another Example State of the Economy ProbabilityReturn Eggbert Rerurn Dingo Risk-Free Rate Bad0.200.040.070.03 Good0.450.10 0.03 Great0.350.220.190.03 Expected Return ?? Variance?? Coefficient of Correlation with the market 0.7120.842 Covariance with the Market 0.0015?

36. Albert Lee Chun Portfolio Management 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)

37. Albert Lee Chun Portfolio Management Solution : 35 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.

38. Albert Lee Chun Portfolio Management 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