1. CHAPTER THREE: Portfolio Theory, Fund Separation and CAPM

3. The Trade-Off Between Expected Return and Risk Portfolio of two assets Markowitz’s contribution 1: The measurement of return and risk Expected Return Risk Weight Asset 1 Asset 2 is correlation coefficient :

4. Mini Case 1: Portfolio of the Riskless Asset and a Single Risky Asset Is the portfolio efficient ? Suppose , how to achieve a target expected return ?

5. The Diversification Principle Mini Case 2: Portfolio of Two Risky Assets The Diversification Principle — The standard deviation of the combination is less than the combination of the standard deviations. Asset 1 Asset 2 Expected Return 0.14 0.08 Standard Deviation 0.20 0.15 Correlation Coefficient 0.6

6. Hyperbola Frontier of Two Risky Assets Combination Minimum Variance Portfolio The Optimal Combination of Two Risky Assets R 0 100% 8% 0.15 C 10% 90% 8.6% 0.1479 Minimum Variance Portfolio 17% 83% 9.02% 0.1474 D 50% 50% 11% 0.1569 Symbol Proportion in Asset 1 Proportion in Asset 2 Portfolio Expected Return Portfolio Standard Deviation S 100% 0 14% 0.20 .2000 C 0 .1569 .1500 .1479 .0860 .0902 .1100 .1400 S D R .0800

7. — Diversification 0 Systematic Exposure Markowitz’s contribution 2: Diversification. Suppose , Then Let , Let ,

8. Mini Case 3: Portfolio of Many Risky Assets ? Resolving the quadratic programming, get the minimum variance frontier Expected return : : Covariance : :

9. Efficient Frontier of Risky Assets The Mean-Variance Frontier 0 Indifference Curve of Utility Optimal Portfolio of Risky Assets

10. Proposition! The variance of a diversified portfolio is irrelevant to the variance of individual assets. It is relevant to the covariance between them and equals the average of all the covariance.

12. Proposition! Only unsystematic risks can be diversified. Systematic risks cannot be diversified. They can be hedged and transferred only. Markowitz’s contribution 3: Distinguishing systematic and unsystematic risks.

13. Proposition! There is systematic risk premium contained in the expected return. Unsystematic risk premium cannot be got through transaction in competitive markets. Only systematic risk premium contained, no unsystematic risk premium contained. Both systematic and unsystematic volatilities contained

14. Two Fund Separation The portfolio frontier can be generated by any two distinct frontier portfolios. Theorem: Practice: If individuals prefer frontier portfolios, they can simply hold a linear combination of two frontier portfolios or mutual funds. 0

15. Orthogonal Characterization of the Mean-Variance Frontier

16. Orthogonal Characterization of the Mean-Variance Frontier

17. P(x)=0 P(x)=1 R* 1 E=0 E=1 Re* Proposition: Every return r i can be represented as 0

18. Efficient Frontier of Risky Assets The Portfolio Frontier: where is R*? 0 R* w 1 w 2 w 3

19. Some Properties of the Orthogonal Characterization

20. Capital Market Line (CML) CML CAL — Capital Allocation Line 0 Indifference Curve 2 Indifference Curve 1 CAL 1 CAL 2 P can be the linear combination of M and

21. Combination of M and Risk-free Security — The weight invested in portfolio M — The weight invested in risk-free security

25. 0 1.0 SML Derivation of CAPM: Security Market Line E(r M ) -r F

28. 1.0 SML The market becomes more aggressive The market becomes more conservative Risk neutral 0