Capital markets and portfolio theory

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Capital Markets and Portfolio Theory Roland Portait From the class notes taken by Peng Cheng Novembre 2000 2 Table of Contents Table of Contents PART I Standard (One Period) Portfolio Theory . . . . . . . . . . . . . . . . . . . . . 1 1 Portfolio Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.A Framework and notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.A.i No Risk-free Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.A.ii With Risk-free Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.B Efficient portfolio in absence of a risk-free asset . . . . . . . . . . . . . . . . . . . . . . 6 1.B.i Efficiency criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.B.ii Efficient portfolio and risk averse investors . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.B.iii Efficient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.B.iv Two funds separation (Black) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.C Efficient portfolio with a risk-free asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.D HARA preferences and Cass-Stiglitz 2 fund separation . . . . . . . . . . . . . . 14 1.D.i HARA (Hyperbolic Absolute Risk Aversion) . . . . . . . . . . . . . . . . . . . . . . . . 14 1.D.ii Cass and Stiglitz separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Capital Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.A CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.A.i The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.A.ii Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.A.iii CAPM as a Pricing and Equilibrium Model . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.A.iv Testing the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.B Factor Models and APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.B.i K-factor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.B.ii APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.B.iii Arbitrage and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.B.iv References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 PART II Multiperiod Capital Market Theory : the Probabilistic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.A Probability Space and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.B Asset Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.B.i DeÞnitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.C Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.C.i Notation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.C.ii Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.C.iii Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 i Table of Contents 4 AoA, Attainability and Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.A DeÞnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.B Propositions on AoA and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.B.i Correspondance between Q and Π : Main Results . . . . . . . . . . . . . . . . . . . 35 4.B.ii Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Alternative SpeciÞcations of Asset Prices . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.A Ito Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.B Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.C Diffusion state variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.D Theory in the Ito-Diffusion Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.D.i Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.D.ii Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.D.iii Redundancy and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.D.iv Criteria for Recognizing a Complete Market . . . . . . . . . . . . . . . . . . . . . . . . 44 PART III State Variables Models: the PDE Approach. . . . . . . . . . . . . . . . 45 6 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7 Discounting Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.A Ito’s lemma and the Dynkin Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.B The Feynman-Kac Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8 The PDE Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 8.A Continuous Time APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 8.A.i Alternative decompositions of a return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 8.A.ii The APT Model (continuous time version) . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.B One Factor Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.C Discounting Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 9 Links Between Probabilistic and PDE Approaches . . . . . . . . . . . . . . . 55 9.A Probability Changes and the Radon-Nikodym Derivative . . . . . . . . . . . 55 9.B Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 9.C Risk Adjusted Drifts: Application of Girsanov Theorem . . . . . . . . . . . . 56 PART IV The Numeraire Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 10 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 11 Numeraire and Probability Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 11.A Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 11.A.i Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ii Table of Contents 11.A.ii Numeraires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.B Correspondence Between Numeraires and Martingale Probabilities . 11.B.i Numeraire → Martingale Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.B.ii Probability → Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.C Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 62 62 63 63 12 The Numeraire (Growth Optimal) Portfolio . . . . . . . . . . . . . . . . . . . . . . . 65 12.A DeÞnition and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12.A.i DeÞnition of the Numeraire (h, H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12.A.ii Characterization and Composition of (h, H) . . . . . . . . . . . . . . . . . . . . . . . . 65 12.A.iii The Numeraire Portfolio and Radon-Nikodym Derivatives . . . . . . . . . . . . 69 12.B First Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 12.B.i CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 12.B.ii Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 PART V Continuous Time Portfolio Optimization. . . . . . . . . . . . . . . . . . . . 72 13 Dynamic Consumption and Portfolio Choices (The Merton Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.A Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.A.i The Capital Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.A.ii The Investors (Consumers)’ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.B The Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.B.i Sketch of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.B.ii Optimal portfolios and L + 2 funds separation . . . . . . . . . . . . . . . . . . . . . . 77 13.B.iii Intertemporal CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 14 THE ”EQUIVALENT” STATIC PROBLEM (Cox-Huang, Karatzas approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 14.A Transforming the dynamic into a static problem . . . . . . . . . . . . . . . . . . . . 80 14.A.i The pure portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 14.A.ii The consumption-portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 14.B The solution in the case of complete markets . . . . . . . . . . . . . . . . . . . . . . . . 83 14.B.i Solution of the pure portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 14.B.ii Examples of speciÞc utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 14.B.iii Solution of the consumption-portfolio problem . . . . . . . . . . . . . . . . . . . . . . 86 14.B.iv General method for obtaining the optimal strategy x∗∗ . . . . . . . . . . . . . . . 87 14.C Equilibrium: the consumption based CAPM . . . . . . . . . . . . . . . . . . . . . . . . 88 PART VI STRATEGIC ASSET ALLOCATION . . . . . . . . . . . . . . . . . . . . . . . 90 15 The problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 16 The optimal terminal wealth in the CRRA, mean-variance iii Table of Contents and HARA cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 16.A Optimal wealth and strong 2 fund separation. . . . . . . . . . . . . . . . . . . . . . . 92 16.B The minimum norm return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 17 Optimal dynamic strategies for HARA utilities in two cases . . . . 93 17.A The GBM case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 17.B Vasicek stochastic rates with stock trading . . . . . . . . . . . . . . . . . . . . . . . . . 93 18 Assessing the theoretical grounds of the popular advice . . . . . . . . . 94 18.A The bond/stock allocation puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 18.B The conventional wisdom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 REFERENCES 95 iv PART I Standard (One Period) Portfolio Theory Chapter 1 Portfolio Choices Chapter 1 Portfolio Choices 1.A Framework and notations In all the following we consider a single period or time interval (0 1), hence two instants t = 0 and t = 1 Consider an asset whose price is S(t) (no dividends or dividends reinvested). The return of this asset between two points in time (t = 0, 1) is: R= S (1) − S (0) S (0) We now consider the case of a portfolio. and distinguish the case where a riskless asset does not exist from the case where a risk free asset is traded. 1.A.i No Risk-free Asset There are N tradable risky assets noted i = 1, ..., N : • The price of asset i is Si (t), t = 0, 1. • The return of asset i is Ri = Si (1) − Si (0) Si (0) 2 Chapter 1 Portfolio Choices • The number of units of asset i in the portfolio is ni . The portfolio is described by the vector n(t); ni can be >0 (long position) or <0 (short position). • Then the value of the portfolio, denoted by X (t), is X (t) = n0 · S (t) with n (0) = n (1) = n (no revision between 0 and 1), the prime denotes a transpose. S (t) stands for the column vector (S1 (t), ..., SN (t))0 • The return of the portfolio is: RX = X (1) − X (0) X (0) • Portfolio X can also be deÞned by weights, i.e. xi (0) = xi = ni S (0) X (0) (Note that xi (1) 6= xi ). Besides the weights sum up to one: x0 · 1=1 where x= (x1 , x2 , ..., xN )0 and 1 is the unit vector. • The return of the portfolio is the weighted average of the returns of its components: RX = x0 R 3 Chapter 1 Portfolio Choices Proof 1 + RX = = = = X (1) X (0) n0 S (1) X (0) N X ni Si (1) Si (0) · X (0) Si (0) i=1 N X xi · i=1 = N X Si (1) Si (0) xi · (1 + Ri ) i=1 = 1+ N X xi Ri i=1 Q.E.D. 0 • DeÞne µi = E [Ri ] and µ= (µ1 , µ2 , ..., µN ) , then: µX = E (RX ) = x0 µ • Denote the variance-covariance matrix of returns ΓN×N = (σ ij ), where σ ij = cov (Ri , Rj ), then: var (RX ) = var (x0 R) = x0 Γx N N X X = xi xj σ ij i=1 j=1 1.A.ii With Risk-free Asset We now have N +1 assets, with asset 0 being the risk-free asset, and the remaining N assets being the risky assets. 4 Chapter 1 Portfolio Choices • S0 (1) = S0 (0) · (1 + r) with r a deterministic interest rate. • Again we can deÞne the portfolio in units, with n= (n0 , n1 , n2 , ..., nN )0 • The portfolio can be similarly deÞned in weights: xi = ni S (0) X (0) for the N risky assets (i = 1, 2, ..., N ), and x0 = 1 − N X xi i=1 Note that now x0 · 1 6= 1 where x= (x1 , x2 , ..., xN )0 denotes the weights in the N risky assets. • The return of the portfolio is: RX = x0 r + N X xi Ri = r + i=1 N X i=1 xi (Ri − r) The term (Ri − r) is the excess return of asset i over r. Moreover: µX = E (RX ) = r + x0 π where π is the risk premium vector of the E (Ri − r) • Also denote ΓN×N as the variance-covariance matrix of the risky assets, then: var (RX ) = x0 Γx Γ is always positive semi-deÞnite (meaning that ∀x, x0 Γx ≥ 0). In some cases it is positive deÞnite (∀x 6= 0, x0 Γx > 0). DeÞnition 1 Assets i = 1, 2, ..., N are redundant if there exist N scalars λ1 , λ2 , ..., λN such P that N i=1 λi Ri = k, where k is a constant. Then the portfolio λ is risk-free. Proposition 1 The N assets i = 1, 2, ..., N are not redundant iff singular or invertible). 5 Γ is positive deÞnite (i.e. non- Chapter 1 Portfolio Choices Proof PNAssume that the assets are redundant, then there exist N scalars λ1 , λ2 , ..., λN such that i=1 λi Ri = k. Consider the portfolio deÞned by the weights λ. The variance of its return = var (k) = 0 = λ0 Γλ, i.e. Γ is singular and not positive deÞnite. Conversely if Γ is singular and not positive deÞnite there existP a non 0 vector λ such that λ0 Γλ = 0; Then the return of N portfolio λ has zero variance and i=1 λi Ri = k Q.E.D. Remark 1 In the following sections we will assume that the assets are non-redundant (it is always possible to “drop” redundant assets if any). 1.B 1.B.i Efficient portfolio in absence of a risk-free asset Efficiency criteria DeÞnition 2 Portfolio (x∗ , X ∗ ) is efficient if ∀y, σY < σX ∗ ⇒ µY < µX ∗ and σ Y = σX ∗ ⇒ µY ≤ µX ∗ Consider any efficient portfolio (x∗ , X ∗ ) and let variance(RX ) = k x∗ solves the optimization program (P ) : max E [RX ] x s.t. x0 Γx = k ; x0 1 = 1 The Lagrangian is: µ ¶ θ θ L x, , λ = x0 µ − x0 Γx − λx0 1 2 2 ¡ ¢ The Þrst order condition ∂L = 0 writes: ∂x µ − θΓx∗ −λ1 = 0 or equivalently, for i = 1, .., N : µi = λ + θ N X j=1 6 x∗j σ ij Chapter 1 Portfolio Choices Remark that these Þrst order conditions are necessary and also sufficient for the solution being a maximum since the second order conditions hold (L(x) is strictly concave -Γ positive deÞnite). Theorem 1 A portfolio (x, X) is efficient iff there exist two scalars λ and θ such that for all i = 1, 2, ..., N : µi = λ + θ · cov (Rx , Ri ) Proof The necessary and sufficientP condition for x to be efficient is that it satisÞes the Þrst order ∗ condition: for all i: µi = λ + θ N j=1 xj σ ij . We then have: µi = λ+θ N X x∗j cov (Ri , Rj ) j=1  = λ + θ · cov Ri, N X j=1 = λ + θcov (Ri , RX )  x∗j Rj  Q.E.D. Remark 2 The second term can be considered as the additional required rate of return (risk premium), proportional to cov (Ri , RX ). Remark 3 If cov (Ri , RX ) = 0, then µi = λ. Remark 4 Also note: var (RX ) = N X N X i=1 j=1 = N X i=1 = N X i=1 xi xj σij  xi · cov Ri , N X j=1  xj Rj  xi · cov (Ri , RX ) The covariance term cov (Ri , RX ) indicates the contribution of asset i to the total risk of the portfolio. Therefore, additional required rate of return should be proportional to this induced risk which is what is stated in the theorem. Moreover cov (Ri , RX ) appears to be the relevant measure of risk for any asset i embedded in the portfolio X. 7 Chapter 1 1.B.ii Portfolio Choices Efficient portfolio and risk averse investors DeÞne another optimization program (P 0 ), equivalent to (P ) : ¶ µ θ 0 0 0 (P ) max x µ− x Γx s.t. : x0 1 = 1 2 ( (P ) and (P 0 ) yield the same solutions since they have the same Lagrangian) (P 0 ) writes, equivalently: θ M ax E [RX ] − var (RX ) , s.t. : x0 1 = 1 2 In (P 0 ) θ is interpreted as a given risk-aversion while in (P ) it is an unknown lagrangian multiplier. In (P ) we are given σ 2X and we solve for θ and λ as functions of σ 2X . In (P 0 ) we are given the risk-aversion parameter θ and solve for σ 2X as function of θ. The Þrst order conditions of (P’) write as for (P): µ − θΓx∗ −λ1 = 0 (with only one multiplier for (P 0 )) • Consider the case of minimum variance portfolio where θ = ∞, i.e. 1 min x0 Γx s.t. : x0 1 = 1 2 The Lagrangian is then: 1 L (x,λ) = x0 Γx − λx0 1 2 Call k1 the solution. The Þrst order condition gives: Γk1 − λ1 = 0 Together with the constraint k01 1 = 1 gives: λ= 1 10 Γ−1 1 Thus: k1 = λΓ−1 1 1 = 0 −1 · Γ−1 1 1Γ 1 8 Chapter 1 1.B.iii Portfolio Choices Efficient set DeÞnition 3 The Efficient Set is the set of all x∗ that obey the Þrst order condition. Equivalently, it is the set of all x∗ that solve the optimization program (P 0 ) ∀θ ≥ 0. Recall that the Þrst order condition for (P 0 ) is: µ − θΓx∗ −λ1 = 0 DeÞne risk tolerance b θ as the inverse of risk aversion, i.e. Then x∗ can be solved as: 1 b θ= θ ¡ ¢ x∗ = b θΓ−1 µ − λ1 To Þnd λ, use the constraint 10 x∗ = 1, i.e. Then: or: This solves for λ: 1 = 10 x∗ ¡ ¢ = 10 · b θΓ−1 µ − λ1 b b 0 Γ−1 1 = 1 θ10 Γ−1 µ − θλ1 b 0 Γ−1 1 = θθ b b θ10 Γ−1 µ − θλ1 10 Γ−1 µ−θ λ= 10 Γ−1 1 Then: ¡ ¢ x∗ = b θΓ−1 µ − λ1 µ ¶ 10 Γ−1 µ − θ −1 = b θΓ µ− ·1 10 Γ−1 1 µ ¶ 10 Γ−1 µ Γ−1 1 −1 b = 0 −1 + θΓ µ − 0 −1 ·1 1Γ 1 1Γ 1 9 Chapter 1 Portfolio Choices We recognize in the Þrst term the minimum variance portfolio (k1 ) and we call k2 the second term: k1 k2 Γ−1 1 = 0 −1 1Γ 1 · ¸ 10 Γ−1 µ −1 µ − 0 −1 ·1 = Γ 1Γ 1 Then the solution of (P ) writes: x∗ = k1 + b θk2 Note that k01 1 = 1 and x∗0 1 = 1, therefore k02 1 = 0. Any efficient portfolio is thus the sum of k1 (the minimum variance portfolio) and k2 which is a zero weight (zero investment) portfolio. As it could be expected, an investor with a zero risk tolerance will hold only k1 ; If he has a positive risk tolerance b θ he will add a risk b taking the form θk2 in order to increase the expected return. The efficient set can now be caracterized as: o n ∗ ∗ b b ES = x |x = k1 + θk2 ∀θ > 0 Since the expected return x∗0 µ is linear in b θ and the variance is quadratic in b θ, in 2 the (σ , R) space the efficient portfolios are represented by the efficient frontier, which is a parabola. Each point on the efficient frontier corresponds to a given θ, the slope of the parabola at this point being equal to 2θ (the shadow price in (P ) of the constraint on variance). In the (σ, R) space the efficient frontier is an hyperbola. 1.B.iv Two funds separation (Black) Theorem 2 Consider any two efficient portfolio x and y: 1. Any convex combination of x and y is efficient, i.e.∀ u ∈ [0, 1] , ux+ (1 − u) y ∈ ES 2. Any efficient portfolio is a combination of x and y (not necessarily a convex combination) 3. The whole parabola (efficient and inefficient frontier) is generated by (all) combinations of x and y 10 Chapter 1 Portfolio Choices Proof • Since x∈ ES and y∈ ES, for some positive b θX and bθ Y , we have: θ X k2 x = k1 + b y = k +b θY k 1 2 Let z = ux + (1 − u)y, then: h i z = [uk1 + (1 − u) k1 ] + ub θX + (1 − u) b θ Y k2 = k1 + b θZ k2 With b θZ > 0, we can conclude that z∈ ES. • Let z∈ ES, then z = k1 + b θZ k2 for some b θZ > 0. For any x∈ ES and y∈ ES: h i ux + (1 − u) y = k1 + ub θX + (1 − u) b θY k2 By equating b θZ to ub θX + (1 − u) b θY we get: u∗ = b θZ − b θY b θY θX − b Then the combination u∗ x + (1 − u∗ ) y = z Q.E.D. 1.C Efficient portfolio with a risk-free asset Consider Þgure 1 where the upper branch of the hyperbola EFR represents, in the (σ, E) space, the efficient portfolios in absence of a riskless asset. Assume now that exists a risk free asset 0 yielding the certain return r. M stands for the tangency point of the hyperbola EFR with a straight line drown from r representing asset 0. Point M represents a portfolio composed only of risky assets, called the tangent portfolio. 11 Chapter 1 Portfolio Choices E EFR M X r σ • Efficient frontier in presence of a riskless asset Figure 1.1. 12 Chapter 1 1. Portfolio Choices Proposition 2 Asset 0 is efficient 2. Consider any portfolio X. Any combination of 0 and X yielding R = uRX + (1 − u) r, lies on the straight line connecting 0 and X in the (σ, E) space 3. Any feasible portfolio which representative point is not on r − M (such as X) is dominated by portfolios in r − M. The straight line r − M is the efficient frontier and is called the Capital Market Line 4. (Tobin’s Two-fund Separation) Any efficient portfolio is a combination of any two efficient portfolios, for instance 0 and M 5. Any efficient portfolio writes: 6. ¡ ¢ θΓ−1 µ−r1 x∗ = b The tangent portfolio (m,M ) is: ¡ ¢ m = b θM Γ−1 µ−r1 1 b ¢ θM = 0 −1 ¡ 1Γ µ−r1 Proof 1, 2, 3, 4 are standard and easy to prove. Let us proove 5 and 6: x∗ ∈ ES solves: The Þrst order condition is: ¡ ¢ θ max 1 r + x∗0 µ − r1 − x∗0 Γx∗ 2 µ − r1 = θΓx∗ Then: x∗ ¢ 1 −1 ¡ µ − r1 Γ θ ¡ ¢ = b θΓ−1 µ − r1 = ¡ ¢ θ M Γ−1 µ − r1 . Also: m0 1 = 1, The tangent portfolio is an efficient portfolio, therefore, m= b then: b θM = 10 Γ−1 Q.E.D. 13 1 ¡ ¢ µ − r1 Chapter 1 Portfolio Choices Remark 5 Given a risk tolerance bθ: • b θb θ M , the portfolio shorts 0 Remark 6 We deÞne later the market portfolio as a portfolio containing all the risky assets present in the market (and only risky assets). In absence of riskless asset the market portfolio is efficient iif its representative point belongs to the hyperbola EFR. In presence of a risk free asset the necessary and sufficient condition for the market portfolio to be efficient is that it coincides with the tangent portfolio m (which is the only efficient portfolio of EFR, in presence of a risk free asset). Would all investors face the same efficient frontier (it would be the case under homogeneous expectations and horizon) and would they all follow the mean-variance criteria, they would all hold combinations of 0 and M and the tangent portfolio M would necessarily coincide with the market portfolio. 1.D HARA preferences and Cass-Stiglitz 2 fund separation A rational agent (in the sense of Von Neumann-Morgenstern) should maximize the expected utility of wealth E [U (W )]. 1.D.i HARA (Hyperbolic Absolute Risk Aversion) A utility function U (W ) belongs to HARA class if it writes: · ¸1−γ γ W b U (W ) = θ+ 1−γ γ Some restrictions are imposed on the coefficients γ and b θ and the domain of deÞnition. The absolute risk tolerance (ART) and absolute risk aversion (ARA) are: 1 ARA U0 = − 00 U W = b θ+ γ ART = 14
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