Capital M arkets and Portfolio Theory Roland P ortait From the class notes taken by Pen g C he n g No vembre 2000 2 Table of Contents Table of Contents PART I Stand ard (One Period) P o rtfo lio The o ry 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 A sset 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 E fficient 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 Ca pit al Ma rket Equ ilib rium 17 2.A CAPM 17 2.A.i The Model 17 2.A.ii Geometry 19 2.A.iii CAPM as a P ricing 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 M ult iperiod Capital Market The ory : the Probabilistic Approach 26 3Framework 27 3.A Proba b ility Space an d Information . . . . . . . . . . . . . . 27 3.B Asset Prices 28 3.B.i DeÞnitions and N otations 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 Ao A, Attainability and Com pleteness 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 Framew ork 41 5.D.ii Martingales 42 5.D.iii Redundancy and Completeness 42 5.D.iv Criteria for Recognizing a Complete Mark et 44 PART III State Variables Models: the PDE Approach 45 6Framework 46 7 Discoun ting 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 Link s Be tween Proba bilistic a nd PDE Appro a che s 55 9.A Probab ility Chang es and the Ra do n -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 P ro b a bility Chan g e s 61 11.AFramework 61 11.A.i Assets 61 ii Table of Contents 11.A.ii Numeraires 61 11.B Correspondence B et w een Numeraires and Martingale Probabilities . 62 11.B.i Numeraire → Martingale Probabilities 62 11.B.ii Probabilit y → Numeraire 63 11.CSummary 63 12 The N u m e ra ire (G rowth Optima l) 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 P or tfolio and Radon-Nikodym Derivatives 69 12.BFirst Applications 69 12.B.i CAPM 70 12.B.ii Valuation 70 PART V Contin uous Time Portfolio Optimization 72 13 Dynamic Consump tion and P ortfolio 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 PR OBLEM (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 con sumption-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 G eneral method fo r obtaining the optimal s trategy x ∗∗ 87 14.C Eq uilibriu m : the consu m p tion based CAPM . . . . . . . . . . . . . . . . . . . . . . . . 88 PA RT VI STRATEGIC ASSET ALLOCATION 90 15 The problems 91 16 The optimal termina l w e alt h in the CR RA, me an -varia n ce 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 Op timal dynamic stra te gie s for HARA utilitie s in t wo 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 Po r tfolio Ch oice s 1.A Framework and notations In all the following we co nsider a single period or time i nter val (0 1),hencetwo instants t =0and t =1 Consider an asset whose price is S(t) (no dividends or dividends reinvested). The r et urn of this asset between two points in time (t =0, 1) is: R = S (1) − S (0) S (0) We now consider t he case o f a portfolio. and di stinguish the case where a riskless asset does not ex ist from t he case where a risk free asset is traded. 1.A.i N o R isk-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 o f asset i is R i = S i (1) − S i (0) S i (0) 2 Chapter 1 Portfolio Choices • The n umber of units of asset i in the portfolio is n i . The portfolio is d e scr ibed 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 bet ween 0 and 1), the prime denotes a transpose. S (t) stands for t he column vector (S 1 (t), , S N (t)) 0 • The retur n of the portfolio is: R X = X (1) −X (0) X (0) • Portfolio X can also be deÞned by we ights, i.e. x i (0) = x i = n i S (0) X (0) (Note that x i (1) 6= x i ). Besides the weigh ts sum up to one: x 0 · 1=1 where x=(x 1 ,x 2 , , x N ) 0 and 1 is the unit vect o r. • Th e retur n of the portfolio is the weigh te d av e r age 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 m atrix 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 A s set We now have N +1 assets, with asset 0 being the risk-free asset, and the remaining N assets being the risky assets. 4 [...]... given the portfolio is the same for all 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... u) bY k2 By equating bZ to ubX + (1 u) bY we get: u = bZ bY bX bY Then the combination u x + (1 u ) y = z Q.E.D 1.C Ecient portfolio with a risk-free asset Consider ịgure 1 where the upper branch of the hyperbola EFR represents, in the (, E) space, the ecient 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. .. 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 belongs to the hyperbola EFR... risk free asset the 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... 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, for instance 0 and M 5 Any ecient portfolio. .. to increase the expected return The ecient set can taking the form k2 now be caracterized as: o n b b > 0 ES = x |x = k1 + k2 Since the expected return x0 à is linear in b and the variance is quadratic in b in , 2 the ( , R) space the ecient portfolios are represented by the ecient frontier, which is a parabola Each point on the ecient frontier corresponds to a given , the slope of the parabola... Remark 10 The proof follows the same lines when the portfolio contains a risk free asset with weight x0 Remark 11 and are the same for all assets or portfolios Remark 12 For the market portfolio: àM = + cov (RM , RM ) = + 2 M Therefore: = àM 2 M Then: ài = + cov (RM , Ri ) ã á à = + M2 cov (RM , Ri ) M 18 Chapter 2 Capital Market Equilibrium Deịne: i = cov (RM , Ri ) 2 M Then we may write the CAPM... discounted at the risk-free rate However this asset may be an element of the market portfolio M (unless this claim is in zero net supply ) and therefore the previous pricing formula is not a closed form general equilibrium relation In fact CAPM is an equilibrium condition stemming from the demand side; The equilibrium price can only be otained by specifying the supply side (in the previous example the supply... 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 Ecient portfolio. .. (H) : The market portfolio M is ecient 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 . Capital M arkets and Portfolio Theory Roland P ortait From the class notes taken by Pen g C he n g No vembre 2000 2 Table of Contents Table of Contents PART I Stand ard (One Period). problem 80 14.A.i The pure portfolio problem 80 14.A.ii The con sumption -portfolio problem 82 14.BThe solution in the case of complete markets 83 14.B.i Solution of the pure portfolio problem . 1 Portfolio Choices Proof Assume that the assets are redundant, then there exist N scalars λ 1 , λ 2 , , λ N such that P N i=1 λ i R i = k. Consider the portfolio deÞned by the weights λ . The