Capital markets and portfolio theory (2000)

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 3Framework 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 :MainResults . 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 6Framework 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 Numerai re Approach .59 10 Introduction 60 11 Numeraire and Probability Changes 61 11.AFramework 61 11.A.i Assets . 61 ii Table of Contents 11.A.ii Numeraires . 61 11.B Correspondence Between Numeraires and Martingale Probabilities . 62 11.B.i Numeraire → Martingale Probabilities . 62 11.B.ii Probability → Numeraire . 63 11.CSummary 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.BFirst 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.AFramework 73 13.A.i The Capital Market . 73 13.A.ii The Investors (Consumers)’ Problem . 74 13.BThe 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.ATransforming the dynamic into a static problem 80 14.A.i The pure portfolio problem . 80 14.A.ii The consumption-portfolio problem 82 14.BThe 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.AThe bond/stock allocation puzzle . 94 18.BThe 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),hencetwo instants t =0and 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 S i (t),t=0, 1. • The return of asset i is R i = S i (1)− S i (0) S i (0) 2 Chapter 1 Portfolio Choices • The number of units of asset i in the portfolio is n i . The portfolio is described by the vector n(t); n i can be >0(longposition)or<0 (short position). • Then the value of the portfolio, denoted by X (t),is X (t)=n 0 · 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 (S 1 (t), ., S N (t)) 0 • The return of the portfolio is: R X = X (1) − X (0) X (0) • Portfolio X can also be deÞned by weights, i.e. x i (0) = x i = n i S (0) X (0) (Note that x i (1) 6= x i ). Besides the weights sum up to one: x 0 · 1=1 where x=(x 1 ,x 2 , ., x N ) 0 and 1 is the unit vector. • The return of the portfolio is the weighted average of the returns of its components: R X = x 0 R 3 Chapter 1 Portfolio Choices Proof 1 + R X = X (1) X (0) = n 0 S (1) X (0) = N X i=1 n i S i (1) X (0) · S i (0) S i (0) = N X i=1 x i · S i (1) S i (0) = N X i=1 x i · (1 + R i ) = 1 + N X i=1 x i R i Q.E.D. • DeÞne µ i = E [R i ] and µ=(µ 1 ,µ 2 , ., µ N ) 0 , then: µ X = E (R X )=x 0 µ • Denote the variance-covariance matrix of returns Γ N×N =(σ ij ),where σ ij = cov (R i ,R j ),then: var (R X )=var (x 0 R) = x 0 Γx = N X i=1 N X j=1 x i x j σ ij 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 [...]... Consider any two ecient portfolio x and y: 1 Any convex combination of x and y is ecient, i.e. u [0, 1] , ux+ (1 u) y ES 2 Any ecient portfolio is a combination of x and y (not necessarily a convex combination) 3 The whole parabola (ecient and inecient 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 bX and b Y , we have:... Chapter 1 Portfolio Choices We recognize in the ịrst term the minimum variance portfolio (k1 ) and we call k2 the second term: k1 k2 1 1 = 0 1 1 1 ã á 10 1 à 1 à 0 1 ã1 = 1 1 Then the solution of (P ) writes: x = k1 + b 2 k Note that k01 1 = 1 and x0 1 = 1, therefore k02 1 = 0 Any ecient portfolio is thus the sum of k1 (the minimum variance portfolio) and k2 which is a zero weight (zero investment) portfolio. .. combination of 0 and X yielding R = uRX + (1 u) r, lies on the straight line connecting 0 and X in the (, 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 ecient frontier and is called the Capital Market Line 4 (Tobins Two-fund Separation) Any ecient portfolio is a combination of any two ecient portfolios,... portfolios, for instance 0 and M 5 Any ecient portfolio writes: 6 Ă Â x = b 1 àr1 The tangent portfolio (m,M ) is: Ă Â m = bM 1 àr1 1 bM = Ă Â 0 1 1 àr1 Proof 1, 2, 3, 4 are standard and easy to prove Let us proove 5 and 6: x ES solves: The ịrst order condition is: Ă Â max 1 r + x0 à r1 x0 x 2 à r1 = x Then: x  1 1 Ă à r1 Ă Â = b 1 à r1 = Ă Â The tangent portfolio is an ecient portfolio, therefore,... then: bM = 10 1 Q.E.D 13 1 Ă Â à r1 Chapter 1 Portfolio Choices : Remark 5 Given a risk tolerance b b < bM , the portfolio is long in 0 and m b > bM , the portfolio shorts 0 Remark 6 We deịne 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 ecient iif its representative point... necessary and sucient condition for the market portfolio to be ecient is that it coincides with the tangent portfolio m (which is the only ecient portfolio of EFR, in presence of a risk free asset) Would all investors face the same ecient 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. .. 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 Efficient frontier in presence of a riskless asset Figure 1.1 12 Chapter 1 1 Portfolio Choices Proposition 2 Asset 0 is ecient 2 Consider any portfolio. .. W 16 Chapter 2 Capital Market Equilibrium Chapter 2 Capital Market Equilibrium 2.A 2.A.i CAPM The Model Consider again N risky assets (a risk free asset may exist or not) The market value of asset i is Vi , then (by deịnition of the market portfolio) its weight in the market portfolio is: Vi mi = PN i=1 Vi The return of the market portfolio is: RM = m0 R Hypothesis 1 (H) : The market portfolio M is... Remark 7 The market portfolio would be ecient if all investors would hold ecient portfolios (since a combination of ecient portfolios is ecient) Theorem 3 (General CAPM ) 1 If (H) is true, then there exist and such that, for i = 1, , N : ài = E [Ri ] = + cov (RM , Ri ) 2 Conversely, if there exist and such that, for i = 1, , N : ài = + cov (RM , Ri ), then (H) is true 17 Chapter 2 Capital Market Equilibrium... U (W ) belongs to HARA class if it writes: ã á1 W b+ U (W ) = 1 Some restrictions are imposed on the coecients and b and the domain of deịnition The absolute risk tolerance (ART) and absolute risk aversion (ARA) are: 1 ARA U0 = 00 U W = b+ ART = 14 Chapter 1 Portfolio Choices and the relative risk tolerance (RRT) is: RRT = b 1 + W In particular: 1 b=0 U (W ) = W 1 1 We obtain CRRA, i.e constant