PPT 6 - 6. PORTFOLIO THEORY Reading: Luenberger Chapter 6...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6. PORTFOLIO THEORY Reading: Luenberger Chapter 6 •  Consider a portfolio of n assets (e.g. stocks). A portfolio is constructed by weighting asset i by weight wi .Stock picking is accomplished by increasing the weight of the stock that you like, and setting the weight to zero for the stock that you do not want to include in the portfolio, or even shorting it (with negative weight). •  We want to maximize the portfolio return while minimize the portfolio risk (many definitions) by varying the weights, wi. 10/17/2011 Many slides are from Prof. Martin’s previous course material 1 Optimization Problems   Difficult to optimize two quantities together, namely, maximize mean portfolio return while minimize portfolio variance. This may even be ill-defined.   Do one optimization at a time.   Maximize portfolio return, without consideration of risk?   Maximize mean portfolio return for a given portfolio risk (variance).   Minimize portfolio variance for a given portfolio return (Markowitz Nobel winning work) 2 Markowitz (1952, 1959)* mean-variance optimal portfolio theory for the idealized case where optimality depends only on asset return s expected values and covariances, with no restriction on short-selling and no other constraints on portfolio weights. In this case analytical formulas for optimal portfolios on the efficient frontier are obtained. Risk was defined to be the portfolio variance. Comment : The Markowitz theory is to this day often called Modern Portfolio Theory (MPT) in spite of being known for more than fifty years. Recent advances in minimizing downside risks not discussed here. * Markowitz, H. M. (1952). “Portfolio Selection”, Journal of Finance, 7, no. 1, 77-91. Markowitz, H. M. (1959). Portfolio Selection: Efficient Diversification of Investments, Wiley. 3 6.1 Mean Vector and Covariance Matrix Let r′ = (r ,..., rn ) be a vector of single period returns. The vector of 1 expected returns µ and the corresponding covariance matrix Σ are: µ ' = E (r!) = ( E ( r1 ), ..., E ( rn )) = (µ1 , ..., µ n ) $ " = cov(r ) = E & r # µ r # µ % ( )( !' ) ( ) ! is symmetric σ ij = cov(ri , rj ) = E ⎡( ri - µi ) ( rj - µ j )⎤ , 1 ≤ i, j ≤ n, ⎣ ⎦ σ ii = Var (ri ) = σ i2 , 1 ≤ i ≤ n, We assume that ! is non-singular and so has an inverse !"1 . 4 Portfolio Returns (random) Portfolio Mean Returns rP = ∑ i =1 wi ri = w ′r n µ P = E ( rP ) = E (w′r ) = w′E (r ) = w′µ = ∑ i =1 wi µi n Portfolio Variance 2 ! P = var( rP ) = var( w!r ) (( = E w! r " µ )) 2 ! r " µ r " µ ! w) = E(w ( )( n ) n = w!#w = $$ wi w j! ij i =1 j =1 Portfolio Volatility 1/ 2 ! P = ( w!!w ) 5 6.2 Global Minimum Variance Portfolio min w!"w w subject to w ′1 = 1, which is just: n ∑w =1 i =1 i 1 L w = w!!w + ! 1 ! w"1 2 () d L w = !w " ! 1 = 0 dw () ( ) w = !!"11 # Apply the constraint w ' 1 = 1 to get: ! 1 ' !"11 = 1 ( -1 ) ! = 1!" 1 #1 6 w MV !"11 = !"#11 1 µP = w ′µ ⇒ 2 P ! = w!"w # Knowledge of mean returns is not required, except as nuisance parameters in Σ ! µ MV 1!"#1µ = 1!"#11 ! MV = 1 1!"#11 7 6.3 Mean-Variance Efficient Frontier Portfolio mean return: Portfolio variance: Minimize variance: subject to µP = w′µ 2 P ! = w!!w min w!"w w w′µ = µ P (specified by investor ) w′1 = 1 (fully invested) This is to find the minimum variance portfolio for a given return unlike the previous global minimum. Difficult! 8 Solution via Lagrange Multipliers 1 L w = w!!w + !1 µ P ! w"µ + !2 1 ! w"1 2 ! "# #$ ! "# #$ () ( ) Constraint 1 Solving ( ) Constraint 2 d L(w) = 0 dw gives the general form of optimal weight vector: w = !1!"1µ + !2 !"11 Messy algebra left to Appendix A1. Shows hyperbola: 2 a⎞ ⎛ 2 ⎜ µP − ⎟ σP ⎝ c⎠ − =1 2 1c dc a = µ ' !"11 c = 1!!"11 b = µ ' !"1µ d = bc ! a 2 9 Mathematics in Appendix A1 Give the Results Below The upper part is the Efficient Frontier µP = a c + d c σ P µP a c µMV = a c µP = a c − d c σ P µP = ± σ MV = 1 c d2d σP − 2 c c 2 for σ P ≥ 1 c σP A hyperbola called the Portfolio Frontier 10/14/2010 Copyright R. Douglas Martin 10 0.06 EFFICIENT FRONTIER 0.04 GENZ ORCL 0.02 0.03 LLTC MSFT 0.01 MEAN RETURN 0.05 SHARPE RATIO = 0.294 0.0 HON 0.0 0.05 0.10 0.15 0.20 SIGMA 10/14/2010 Copyright R. Douglas Martin 11 $w.mv: GENZ HON LLTC MSFT .308 .375 .083 .146 ORCL .088 $mu.mv: [1] 0.026 $sigma.mv: [1] 0.087 10/14/2010 Copyright R. Douglas Martin 12 6.4 Efficient Frontier with Cash The optimization problem becomes much simpler when we add cash. First consider one stock with expected return µ = E[ r ] and variance σ 2 = var(r ) and some cash earning risk-free return (deterministic) µ f Portfolio return with given fraction α in cash: µ P = E[α rf + (1 − α )r ] = α rf + (1 − α ) µ . Or µ P ,e = (1 − α ) µe , where µe ≡ µ − rf and µ P ,e = µ P − rf σ P = (1 − α )σ . 13 Linear relationship between excess portfolio return and standard deviation µ P ,e = (1 − α ) µe σ P = (1 − α )σ . µP rf σ σP 14 14 What if there are n stocks?   LU 6.9: One-Fund Theorem: Treat this collection of stocks as a fund. The argument in the previous slide still holds with α in cash and (1 − α ) in this fund.   How to allocate (1 − α ) among the n stocks? 15 Efficient Frontier with Cash Let ∑ n i =1 wi be invested in risky assets, and 1 − ∑ i =1 wi n in a risk-free asset with a rate of return portfolio return and expected return are rP = (1 − w ′1) rf + w ′r So we minimize subject to 2 . Then, the µP = (1 − w ′1) rf + w ′µ and 1 rf w!"w (1 # w!1) r f + w!µ = µ P 16 Solution via Lagrange Multipliers (only one constraint, easier) 1 L( w ) = w!"w + ! $ 1 # w!1 rf + w!µ # µ P & % ' 2 d w = ! ! "#1 µ ! rf 1 = ! ! "#1µe . L ( w) = 0 ⇒ dw ( ) ) ( µ P = (1 − w′1) rf + w′µ =γµe ' Σ −1ΣγΣ −1µe ' = rf + w′µe = rf + γ ⋅ µ 'e Σ µe −1 µ P ,e = γµe ' Σ µe −1 σ P 2 = w ' Σ −1w =γ 2 µe ' Σ −1µe σP = γ µe ' Σ −1µe 17 What is γ ?   The fraction allocated to stocks: w ' 1 = γµe ' Σ −11 = (1 − α ) γ = (1 − α )( µe ' Σ −11) −1 So γ ∝ (1 − α ) All cash: α =1, γ =0. 10/17/2011 18 Linear relationship between excess portfolio return and standard deviation µ P ,e σP µP = µe ' Σ −1µe rf σP 10/17/2011 19 19 The straight line looks as follows depending on . µP rf . γ : T1 γ 1 > 0 T2 γ 2 < 0 σP Main Message : γ will be negative if and only if the global minimum variance portfolio has a mean return less than that of the risk free rate. This case can and does sometimes occur in practice, and no risk averse investor will invest in risky assets in this case, and will instead invest in a risk–free rate instrument. 20 The All-Risky-Assets Portfolio T ( w! 1 ! 1 " ! = 1!! µe T In the limit: This portfolio must lie on the hyperbola. Portfolio T weights: Expected excess returns: Risk "1 ) !1 !"1µe wT = "1 1#! µe µ 'e !"1µe µT ,e = µ 'e wT = 1#!"1µe ! T = var( wT ' r ) = | 1!"#1µe |#1 µ 'e "#1µe 21 Tangency portfolio T is the tangency portfolio shown below and the efficient frontier with cash is the straight line tangent to T. µP T µT . rf σT σP Why can t the line cut the hyperbola? 22 The Sharpe Ratio For any portfolio P: µP µP rf T .. P θ σP 10/17/2011 µ P − rf SRP = = Sharpe Ratio σP = tan θ The tangency portfolio has the maximum Sharpe Ratio σP 23 0.06 EFFICIENT FRONTIER 0.04 GENZ ORCL 0.02 0.03 LLTC MSFT 0.01 MEAN RETURN 0.05 SHARPE RATIO = 0.294 0.0 HON 0.0 0.05 0.10 0.15 0.20 SIGMA 10/14/2010 Copyright R. Douglas Martin 24 $w.mv: GENZ HON LLTC MSFT ORCL 0.3081789 0.3753527 0.08328089 0.1456418 0.08754582 $mu.mv: [1] 0.02577599 $sigma.mv: [1] 0.08702582 $w.t: GENZ HON LLTC MSFT ORCL 0.5548272 -0.03873531 0.233836 0.1492129 0.1008592 $mu.t: [1] 0.03648147 $sigma.t: [,1] [1,] 0.107126 10/14/2010 Copyright R. Douglas Martin 25 6.5 Risk Penalized Utility Solution (No constraint; User Input for the Risk Aversion Parameter) Maximize 1 2 U (w) = µ P (w) ! ! ! ! P (w) 2 Risk aversion parameter 1 = w!µ ! ! ! w"#w 2 With no weight constraints, i.e., no full investment constraint, the optimal weights are w = ! !1 " #!1µ The choice ! = 1!"#1µ yields full-investment, with possible short positions. But if this is not the case then 1′w ≠ 1 and if 1′w < 1 the portfolio contains a positive cash position. Q. What if 1′w > 1 ? 26 An interesting solution is obtained by optimizing with the full-investment constraint, i.e., maximize subject to Lagrangian: 1 U ( w ) = w!µe ! ! ! w"#w 2 1′ w = 1 (full investment) 1 L( w ) = w!µe ! ! ! w"!w ! ! ! (1"w ! 1) 2 dL( w ) = µe ! ! ! "w # ! ! 1 = 0 dw ! w = ! !1 ! "#1 ! µe ! ! ! 1 ( ) 27 Use the constraint to solve for the Lagrange multiplier and obtain the following result. Two-Fund Separation. The optimal weight vector has the following representation: #!1µe #!11 w = ! !11"#!1µe + 1 ! ! !11"#!1µe $ !1 1"#!1µe 1"# 1 ( = (! !1 ) ( µ ) $ w + (1 ! ! 1"#!1 e T Tangency portfolio "1 It is achieved at ! = 1!! µ e . 10/14/2010 ) !1 ) 1"#!1µe $ w MV Global minimum variance portfolio It is achieved when λ → ∞ . Copyright R. Douglas Martin 28 A1: Frontier Mathematics Now apply the two constraints w ′µ = µ P and w ′1 = 1 to the optimal weight vector w = !1!"1µ + !2 !"11 and solve for the two Lagrange multipliers: c ⋅ µP − a λ1 = d where b − a ⋅ µP λ2 = d a = µ ' !"11 b = µ ' !"1µ c = 1!!"11 10/14/2010 d = bc ! a 2 Copyright R. Douglas Martin 29 Proposition A1: The optimal weight vector may written as w = g1 + g 2 µ P where g1 and g 2 are fixed vectors determined by µ and ! : g1 = 1 b ! "#11 ! a ! "#1µ d ( ) g2 = 1 c ! "#1µ ! a ! "#11 d ( ) Proof: Exercise. Note that as you vary µ P arbitrarily you will typically get some wi < 0 that correspond to short selling. 10/14/2010 Copyright R. Douglas Martin 30 Proposition A2: Let P , P2 be any two efficient frontier 1 portfolios. Then ( ) cov rP , rP = w!P "w P 1 2 1 2 1 c$ a '$ a' = + & µ P # )& µ P # ) c d % 1 c (% 2 c ( Proof: Exercise (Hint: use Proposition 1) The special case of the above with P = P2 gives the 1 Portfolio Frontier formula: 1 c⎛ a⎞ σ = cov ( rP , rP ) = + ⎜ µP − ⎟ c d⎝ c⎠ 2 2 P 10/14/2010 Copyright R. Douglas Martin 31 The Portfolio Frontier is a Hyperbola Re-arranging gives 2 a⎞ ⎛ 2 ⎜ µP − ⎟ σP ⎝ c⎠ − =1 2 1c dc Recall that the equation of a hyperbola centered at (0, 0) is x 2 y 2 ⎛ x y ⎞⎛ x y ⎞ B − 2 = ⎜ − ⎟⎜ + ⎟ = 1, with asymptotes y = ± x 2 A B ⎝ A B ⎠⎝ A B ⎠ A Thus µ P versus with asymptotes 10/14/2010 σ P is a hyperbola centered at (0, a / c), a d µP = ± σ P c c Copyright R. Douglas Martin 32 A2. Positive Definite Covariance Matrices Variances are non-negative, which gives: 2 ! P = w!!w ! 0 Thus ! is always non-negative definite. A covariance matrix ! is said to be positive definite if 2 ! P = w!"w > 0 !w " 0 It can be shown that if w!"w = 0 for some w ≠ 0 then there is a linear dependency among the returns r of the following form: there exist constants a1 , a2 ,L , an not all of which are zero and a constant c such that Prob( a1r1 + a2 r2 + L + an rn = c ) = 1 10/14/2010 Copyright R. Douglas Martin 33 Some Matrix Algebra Results See for example the Appendix of Matrix Analysis (1979) by Mardia, Kent and Bibby, Academic Press 1)  Definition: A square matrix A is non-singular if and only if its determinant A is non-zero. 2)  The inverse A −1 of a square matrix A exists if and only if A is non-singular. 3)  A positive definite matrix is non-singular and hence its inverse exists 4)  The inverse of a positive definite matrix is positive definite. 10/14/2010 Copyright R. Douglas Martin 34 ...
View Full Document

This note was uploaded on 03/23/2012 for the course AMATH 541 taught by Professor Kk.t during the Winter '11 term at University of Washington.

Ask a homework question - tutors are online