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					                             Fin 2802: Investments
                                  Spring, 2010
                                  Dragon Tang

                                      Lecture 18
                             Optimal Investment Portfolio
                                   March 30, 2010

                              Readings: Chapter 7
                      Practice Problem Sets: 1-15, 17-21

                                  Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                  1
                             Optimal Portfolio Choice
             Objectives:
             • Show how covariance and correlation affect the
               power of diversification
             • Construct efficient portfolio
             • Calculate the composition of the optimal risky
               portfolio
             • Take risk wisely!




                                 Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                 2
                  Diversification and Portfolio Risk

              • Market risk or beta risk
                 – Systematic or Nondiversifiable
              • Firm-specific risk
                 – Diversifiable or nonsystematic




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             3
                                    Portfolio Risk as a
                             Function of the Number of Stocks




                                   Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                   4
                                    Portfolio Risk as a
                             Function of Number of Securities




                                   Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                   5
               Two Asset Portfolio Return – Stock and Bond

                             r w r w r
                              p                B      B            S        S


                             r  Portfolio Return
                              p


                             w  Bond Weight
                                   B

                             r  Bond Return
                              B

                             w  Stock Weig ht
                                   S

                             r  Stock Return
                              S


                                  Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                      6
                               Covariance
              Cov(r1r2) = r1,2s1s2
               r1,2 = Correlation coefficient of
                       returns
                s1 = Standard deviation of
                      returns for Security 1
                s2 = Standard deviation of
                      returns for Security 2
                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             7
                 Correlation Coefficients: Possible Values

             Range of values for r 1,2
                             -1.0 < r < 1.0
            If r = 1.0, the securities would be
            perfectly positively correlated
            If r = - 1.0, the securities would be
            perfectly negatively correlated

                                   Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                   8
                Two Asset Portfolio St Dev – Stock and Bond


              s  w s  w s  2w w s s r
                    2        2   2            2        2
                    p        B   B            S        S               B       S   S   B   B,S


              s  Portfolio Variance
                    2
                    p


               s  Portfolio Standard Deviation
                        2
                        p




                                     Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                                       9
                        In General, For an n-Security Portfolio:


           rp = Weighted average of the
              n securities
           sp2 = (Consider all pair-wise
                  covariance measures)



                                 Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                 10
              Numerical Example: Bond and Stock
                 Returns
                   Bond = 6%        Stock = 10%
                 Standard Deviation
                   Bond = 12%       Stock = 25%
                 Weights
                   Bond = .5 Stock = .5
                 Correlation Coefficient
                   (Bonds and Stock) = 0




                              Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              11
                       Return and Risk for Example

              Return = 8%
                .5(6) + .5 (10)

              Standard Deviation = 13.87%
                [(.5)2 (12)2 + (.5)2 (25)2 + …
                  2 (.5) (.5) (12) (25) (0)] ½
                [192.25] ½ = 13.87




                              Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              12
               Investment Opportunity Set for Stock and Bonds




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             13
                        Investment Opportunity Set for Stock
                         and Bonds with Various Correlations




                                Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                14
     Table 7.1 Descriptive Statistics for Two Mutual Funds




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             15
     Table 7.3 Expected Return and Standard Deviation
            with Various Correlation Coefficients




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             16
                     Figure 7.3 Portfolio Expected Return
                    as a Function of Investment Proportions




                              Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              17
                     Figure 7.4 Portfolio Standard Deviation
                     as a Function of Investment Proportions




                              Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              18
                  Figure 7.5 Portfolio Expected Return
                   as a function of Standard Deviation




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             19
          Table 7.4 Risk Reduction of Equally Weighted Portfolios
                 in Correlated and Uncorrelated Universes




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             20
                             Portfolio Selection

                       • Asset allocation
                       • Security selection
                       • These two are separable!




                               Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                               21
                             Asset Allocation
             • John Bogle: “Asset allocation accounts for 94% of
               the differences in pension fund performance”
             • Identify investment opportunities (risk-return
               combinations)
             • Choose the optimal combination according to
               investor’s risk attitude




                              Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              22
                     Optimal Portfolio Construction

            Step 1: Using available risky securities (stocks) to
              construct efficient frontier.
            Step 2: Find the optimal risky portfolio using risk-
              free asset
            Step 3: Now We have a risk-return tradeoff, choose
              your most favorable asset allocation
            Step 4: Calculate optimal portfolio weights




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             23
          Portfolios Constructed from Three Stocks A, B and C




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             24
                                 The Efficient Frontier of
                             Risky Assets and Individual Assets




                                  Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                  25
                             Optimal Capital Allocation Line for
                                 Bonds, Stocks and T-Bills




                                    Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                    26
                             The Complete Portfolio




                                Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                27
                             The Complete Portfolio – Solution
                              to the Asset Allocation Problem




                                  Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                  28
                         Discussion: Practical Portfolio Rules

            • Rule #1: do not be a amateur stock trader (don’t do it or do it
              full time!), choose to be a trader or investor first!
            • Investment philosophy: define value! Be cost cautious!
            • Investment psychology: do not chicken out!
                  – Don’t get sentimental, history doesn’t matter
                  – Stop loss and let your winner run
                  –…
            •   Research, research, research!
            •   Sector rotation, familiarity, estimation risk
            •   Offense wins game, defense wins championship
            •   Amateurs practice until they get it right, pros practice until
                they can’t get it wrong.


                                   Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                       29
                             Investor Personalities

           •   Measured investors: Rich and greedy
           •   Reluctant investors: Rich and humble
           •   Competitive investors: Like to trade, which is hazardous
           •   Unprepared investors: Poor, greedy, and ignorant




                                Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                30
                             Mistakes Investors Make
           •   Overconfident, underestimate market force
           •   Short-sighted, resulting in unnecessary transactions
           •   Mental accounting, do not see the big picture
           •   Can’t see “everyone is unique, just like everyone else”
           •   Disposition: holding on losers too long and selling winner
               too fast
           •   Averaging down in price rather than up in buying
           •   Buying on tips and rumors
           •   Speculating too heavily in options or futures wanting to get
               rich quick
           •   No investment strategy, or having one without persistence

                                 Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                    31
                                     Summary

            •   Diversification
            •   Optimal risky portfolio and efficient frontier
            •   Allocation among risky and risk-free assets
            •   Next Class: Practical Portfolio Management




                             Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             32

				
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