Portfolio Management 3-228-07 Albert Lee Chun

1. Portfolio Management3-228-07Albert Lee Chun Construction of Portfolios: Markowitz and the Efficient Frontier Session 4 25 Sept 2008

2. Plan for Today • A Quick Review • Optimal Portfolios of N risky securites - Markowitz`s Portfolio Optimization - Two Fund Theorem • Optimal Portfolios of N risky securities and a risk-free asset - Capital Market Line - Market Portfolio -Different Borrowing and Lending rates

3. Une petite révision

4. We started in a simple universe of1 risky asset and 1 risk-free asset

5. Optimal Weights Depended on Risk Aversion Each investor chooses an optimal weight on the risky asset, where w*> 1 corresponds to borrowing at the risk-free rate, and investing in the risky asset. E(r) Borrower Rf Lender  A The optimal choice is the point of tangency between the capital allocation line and the agent`s utility function.

6. Utility maximization Take the derivative and set equal to 0

7. We then looked at a universe with 2 risky securities

8. Correlation and Risk E(R) f E g ρDE = -1.00 h i j ρDE = +1.00 k ρDE = +0.50 D ρDE = 0.00

9. Minimum Variance Portfolio 1> > -1  = -1  = 0 Asset with the lowest variance, in the absence of short sales.  = 1

10. Maximize Investor Utility The solution is:

11. Then we introduced a risk-free asset

12. CAL 3 Everyinvestorholdsexactly the same optimal portfolio of riskyassets! E(r) CAL 2 CAL 1 E Intuition : the optimal solution is the CAL with the maximum slope! s Optimal Portfolio is the Tangent Portfolio D

13. Optimal Portfolio Weights The solution is:

14. CAL E(r) P rf s Optimal Borrowing and Lending Borrower w* >1 The optimal weight on the optimal risky portfolio P depends on the risk-aversion of eachinvestor. E Lender w*<1 D

15. Now imagine a universe with a multitude of risky securities

16. Harry Markowitz 1990 Nobel Prize in Economics for having developed the theory of portfolio choice. The multidimensional problem of investing under conditions of uncertainty in a large number of assets, each with different characteristics, may be reduced to the issue of a trade-off between only two dimensions, namely the expected return and the variance of the return of the portfolio.

17. Markowitz Efficient Frontier Efficient Frontier E µ* D σ*

18. The Problem of Markowitz I Subject to the constraint: Weightssum to 1 Maximize the expected return of the portfolio conditioned on a givenlevel of portfolio variance.

19. The problem of Markowitz II Subject to the constraint: Weightssum to 1 Minimize the variance of the portfolio conditioned on a givenlevel of expected return.

20. Does the Risk of an Individual Asset Matter? • Does an asset which is characterized by relatively large risk, i.e., great variability of the return, require a high risk premium? • Markowitz’s theory of portfolio choice clarified that the crucial aspect of the risk of an asset is not its risk in isolation, but the contribution of each asset to the risk of an entire portfolio. • However, Markowitz’s theory takes asset returns as given. How are these returns determined?

21. Citation de Markowitz So about five minutes into my defense, Friedman says, well Harry I’ve read this. I don’t find any mistakes in the math, but this is not a dissertation in economics, and we cannot give you a PhD in economics for a dissertation that is not in economics. He kept repeating that for the next hour and a half. My palms began to sweat. At one point he says, you have a problem. It’s not economics, it’s not mathematics, it’s not business administration, and Professor Marschak said, “It’s not literature”. So after about an hour and a half of that, they send me out to the hall, and about five minutes later Marschak came out and said congratulations Dr. Markowitz.

22. Two-Fund Theorem Interesting Fact: Any two efficient portfolios will generate the entire efficient frontier! B Every point on the efficient frontier is a linear combination of any two efficient portfolios A and B. A

23. Now imagine a risky universe with a risk-free asset

24. Capital Market Line CML maximizes the slope. Capital Market Line Tangent Portfolio E M D rf

25. Tobin’s Separation Theorm • James Tobin ... in a 1958 paper said if you hold risky securities and are able to borrow - buying stocks on margin - or lend - buying risk-free assets - and you do so at the same rate, then the efficient frontier is a single portfolio of risky securities plus borrowing and lending.... • Tobin's Separation Theorem says you can separate the problem into first finding that optimal combination of risky securities and then deciding whether to lend or borrow, depending on your attitude toward risk. He then showed that if there's only one portfolio plus borrowing and lending, it's got to be the market.

26. Market Portfolio Capital Market Line w* >1 Market Portfolio E w*<1 M M D rf

27. Separation Theorem Borrower Capital Market Line w* >1 w* =1 Lender M w*<1 Separation of investment decision from the financing decision. rf

28. Who holds only the Market Portfolio? w* >1 Borrower A<AM CML A=AM w* =1 w*<1 Lender A>AM M rf

29. Note that we have reduce the complexity of this universe down to simply 2 points

30. Different Borrowing and Lending Rates Borrower MB Lender rB ML rL

31. Who are the Lenders and Borrowers Borrower A<AMB MB Lender rB ML A>AML rL

32. Who are the Lenders and Borrowers Borrower A<AMB MB Lender rB ML A>AML rL

33. Who holds only risky assets? Emprunteur A<AMB AMB<A<AML MB Prêteur rB ML A>AML rL

34. Efficient Frontier Borrower A<AMB AMB<A<AML MD Lender rB ML A>AML rL

35. Where is the market portfolio? The market portfolio can be anywhere here rB rf

36. Only Risk-free Lending Lowrisk averse agents cannotborrow, sotheyholdonlyriskyassets. Least risk-averse lender Lender ML rL

37. Efficient Frontier The market portfolio can be anywhere here All lendersholdthis portfolio of riskysecurities Lenders rL

38. For NextWeek • Nextweekwewill • do a few examples, bothnumerical and in Excel. - discussAppendix A – diversification. • discuss the article from the course reader. • wrap up Chapter 7 and pave the way for the Capital AssetPricing Model.

39. The Power of Diversification 90% of the total benefit of diversification is obtained after holding 12-18 stocks. Standard Deviation of Return Non systematic risk (idiosyncratic, non diversifiable) Total Risk Standard Deviation of the Market (systematic risk) Systematic Risk Number of Stocks in the Portfolio