0 Portfolio Management Albert Lee Chun Construction of Portfolios: Markowitz and the Efficient Frontier Session 4 25 Sept 2008

Albert Lee Chun Portfolio Management 1 Plan for Today A Quick Review A Quick Review Optimal Portfolios of N risky securites Optimal Portfolios of N risky securites - Markowitz`s Portfolio Optimization - Two Fund Theorem Optimal Portfolios of N risky securities and a risk-free asset Optimal Portfolios of N risky securities and a risk-free asset - Capital Market Line - Capital Market Line - Market Portfolio -Different Borrowing and Lending rates

Albert Lee Chun Portfolio Management 2 Une petite révision

Albert Lee Chun Portfolio Management 3 We started in a simple universe of 1 risky asset and 1 risk-free asset

Albert Lee Chun Portfolio Management 4 Optimal Weights Depended on Risk Aversion E(r)  Rf Lender Borrower AA 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. The optimal choice is the point of tangency between the capital allocation line and the agent`s utility function.

Albert Lee Chun Portfolio Management 5 Utility maximization Take the derivative and set equal to 0

Albert Lee Chun Portfolio Management 6 We then looked at a universe with 2 risky securities

Albert Lee Chun Portfolio Management 7 Correlation and Risk E(R) ρ DE = 0.00 ρ DE = ρ DE = ρ DE = f g h i j k D E

Albert Lee Chun Portfolio Management 8 Minimum Variance Portfolio 1>  > -1  = -1  = 0  = 1 Asset with the lowest variance, in the absence of short sales.

Albert Lee Chun Portfolio Management 9 Maximize Investor Utility The solution is:

Albert Lee Chun Portfolio Management 10 Then we introduced a risk-free asset

Albert Lee Chun Portfolio Management 11 Optimal Portfolio is the Tangent Portfolio E(r)   CAL 1 CAL 2 CAL 3 Every investor holds exactly the same optimal portfolio of risky assets! Every investor holds exactly the same optimal portfolio of risky assets! Intuition : the optimal solution is the CAL with the maximum slope! E E D D

Albert Lee Chun Portfolio Management 12 Optimal Portfolio Weights The solution is:

Albert Lee Chun Portfolio Management 13 Optimal Borrowing and Lending P E(r) rfrf CAL  The optimal weight on the optimal risky portfolio P depends on the risk-aversion of each investor. Lender Borrower D D E E w*<1 w* >1

Albert Lee Chun Portfolio Management 14 Now imagine a universe with a multitude of risky securities

Albert Lee Chun Portfolio Management 15 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.

Albert Lee Chun Portfolio Management 16 Markowitz Efficient Frontier D E Efficient Frontier σ*σ* µ*µ*

Albert Lee Chun Portfolio Management 17 The Problem of Markowitz I The Problem of Markowitz I Subject to the constraint: Maximize the expected return of the portfolio conditioned on a given level of portfolio variance. Weights sum to 1

Albert Lee Chun Portfolio Management 18 The problem of Markowitz II The problem of Markowitz II Subject to the constraint: Minimize the variance of the portfolio conditioned on a given level of expected return. Weights sum to 1

Albert Lee Chun Portfolio Management 19 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? 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. 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? However, Markowitz’s theory takes asset returns as given. How are these returns determined?

Albert Lee Chun Portfolio Management 20 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. 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.

Albert Lee Chun Portfolio Management 21 Two-Fund Theorem A B Interesting Fact: Any two efficient portfolios will generate the entire efficient frontier! Every point on the efficient frontier is a linear combination of any two efficient portfolios A and B.

Albert Lee Chun Portfolio Management 22 Now imagine a risky universe with a risk-free asset

Albert Lee Chun Portfolio Management 23 Capital Market Line rfrf D E CML maximizes the slope. Tangent Portfolio M

Albert Lee Chun Portfolio Management 24 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.... 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. 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.

Albert Lee Chun Portfolio Management 25 Market Portfolio M D E M Capital Market Line rfrf Market Portfolio w*<1 w* >1

Albert Lee Chun Portfolio Management 26 Separation Theorem M Capital Market Line rfrf Separation of investment decision from the financing decision. Lender Borrower w*<1 w* >1 w* =1

Albert Lee Chun Portfolio Management 27 Who holds only the Market Portfolio? M CML rfrf Lender A>A M Borrower A<A M A=A M w*<1 w* >1 w* =1

Albert Lee Chun Portfolio Management 28 Note that we have reduce the complexity of this universe down to simply 2 points

Albert Lee Chun Portfolio Management 29 Different Borrowing and Lending Rates rLrL rBrB Lender Borrower MLML MBMB

Albert Lee Chun Portfolio Management MBMB 30 Who are the Lenders and Borrowers rLrL rBrB Lender Borrower MLML A>A M L A<A M B

Albert Lee Chun Portfolio Management MBMB 31 Who are the Lenders and Borrowers rLrL rBrB Lender Borrower MLML A>A M L A<A M B

Albert Lee Chun Portfolio Management MBMB 32 Who holds only risky assets? rLrL rBrB Prêteur Emprunteur MLML A>A M L A<A M B A M B <A<A M L

Albert Lee Chun Portfolio Management MDMD 33 Efficient Frontier rLrL rBrB Lender Borrower MLML A>A M L A<A M B A M B <A<A M L

Albert Lee Chun Portfolio Management 34 Where is the market portfolio? rfrf The market portfolio can be anywhere here rBrB

Albert Lee Chun Portfolio Management 35 Only Risk-free Lending rLrL Lender MLML Low risk averse agents cannot borrow, so they hold only risky assets. Least risk-averse lender

Albert Lee Chun Portfolio Management 36 Efficient Frontier rLrL The market portfolio can be anywhere here Lenders All lenders hold this portfolio of risky securities

Albert Lee Chun Portfolio Management For Next Week Next week we will Next week we will - do a few examples, both numerical and in Excel. - discuss Appendix A – diversification. - discuss the article from the course reader. - wrap up Chapter 7 and pave the way for the Capital Asset Pricing Model. 37

Albert Lee Chun Portfolio Management 38 The Power of Diversification Standard Deviation of Return Number of Stocks in the Portfolio Standard Deviation of the Market (systematic risk) Systematic Risk Total Risk Non systematic risk (idiosyncratic, non diversifiable) 90% of the total benefit of diversification is obtained after holding stocks.