Portfolio Theory Gorazd Brumen Morgan Stanley September 11-12, - PowerPoint PPT Presentation

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Portfolio Theory Gorazd Brumen Morgan Stanley September 11-12, 2009 Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Topics I will cover Part I: Portfolio Selection in One Period 1 Part II: Portfolio Selection in Continuous Time 2 Part III: Advanced Topics in Portfolio Theory 3 Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Requirements, prior knowledge Basic linear algebra, optimization techniques. Basic probability theory. Stochastic integration, SDE. Basic microeconomics. Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Part I Part I: Portfolio Selection in One Period Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Historical perspective on portfolio selection Even though the concept of diversification is firmly grounded in today’s economic thinking, this was not always the case. Before Markowitz’s seminal contribution investors did not consider portfolio diversification but rather stock picking: Of all stocks in a market pick the one which brings you highest combination of dividends (and capital gains). Portfolio theory answers the question which risks are priced and in what extent. Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Framework and Notations One period model. Vectors will be underlined, such as x , matrices are boldface, e.g. Γ . Begining of period at time 0, end of period at 1. Return on an asset i = 1 , . . . , N in terms of its price S i in this period is R i = S i (1) − S i (0) S i (0) ( R i is a random variable) Let the number of units of asset i be n i . The value of the portfolio X ( t ) at time t holding n i units is X ( t ) = n ′ S ( t ). The return on such a portfolio is R X = X (1) − X (0) . X (0) If x i (0) = x i = n i S i (0) X (0) then x ′ 1 = 1 and R X = x ′ R . (proof left as an exercise.) Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Analysis of mean and variance We focus on the first two moments of R . Let µ = ( µ i ) = E ( R i ) and Γ = ( σ ij ) = ( cov ( R i , R j )). Then E ( R X ) = x ′ µ µ X = � var ( R X ) = x i x j σ ij = x Γ x i , j Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Portfolio with risk-less asset We introduce the riskless asset by S 0 (1) = S 0 (0)(1 + r ) and additionally N � x 0 = 1 − x i i =1 Portfolio returns in this case are N N � � R X = x 0 r + x i R i = r + x i ( R i − r ) i =1 i =1 x ′ ( µ − r 1) , R X − r = i.e. portfolio excess return is a linear combination of excess returns of individual stocks. Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Simple relations Definition Assets i = 1 , . . . , N are redundant if there exists N scalars λ 1 , . . . , λ N such that � N i =1 λ i R i = k for some constant k . The portfolio λ is risk-free. Proposition The assets i = 1 , . . . , N are not redundant if and only if Γ is positive definite. (exercise.) Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Efficiency criteria and optimization program Definition Portfolio ( x ∗ , X ∗ ) is efficient if for every other portfolio y we have that if σ Y < σ X ∗ then µ Y < µ X ∗ and σ Y = σ X ∗ implies µ Y ≤ µ X ∗ . Portfolio optimization problem: s.t. x ′ Γ x = k x ′ 1 = 1 . max E ( R X ) x The Lagrangian of this problem is L ( x , θ 2 , λ ) = x ′ µ − θ 2 x ′ Γ x − λ x ′ 1 First order condition gives us µ − θ Γ x ∗ − λ 1 = 0 (1) Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Optimization program (contd.) Equivalently N � x ∗ µ i = λ + θ j σ ij . j =1 FOC are neccessary and sufficient, since the second derivative is strictly concave (Γ is positive definite). Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Connection to utility theory Criterium of portfolio efficiency is consistent with the economic agents with the following utility E ( R X ) − θ u ( x ) = 2 var ( R X ) , x ′ µ − θ 2 x ′ Γ x . = where the Lagrange parameter θ now represents some degree of risk aversion, i.e. the higher θ is, the more averse the agent is wrt. (variance) risk. Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Competitive economic equilibirum A set of agents i = 1 , . . . , I . A set of assets S j , j = 1 , . . . , N in net supply y . Definition (Competitive equilibrium) Portfolio x ∗ and price system S is a competitive equilibrium if x ∗ i is the solution to the optimization problem s.t. x ′ max u i ( x ) i S = W i x i Markets clear: I � x ∗ i = y i =1 Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Two funds separation (Black) Consider any two efficient portfolios x and y . Then Theorem Any convex comination of x and y, i.e. ux + (1 − u ) y is efficient. Any efficient portfolio is a combination of x and y (not necessarily convex). The efficient frontier is a parabola in the expected return-variance space ( µ, σ 2 ) and a hyperbola in the expected return-standard deviation space ( µ, σ ) . Due to the first bullet point above, any efficient portfolio can be described as a convex combination of just 2 portfolios. Proof of the first two bullet points left as an exercise. Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Proof of last bullet of two fund separation We have shown in (1) that x ∗ = θ Γ − 1 ( µ − λ 1). From 1 ′ x ∗ = 1 we − 1 µ − θ 1 ′ Γ . Therefore x ∗ = k 1 + θ k 2 for appropriate get that λ = 1 ′ Γ 1 k 1 and k 2 . The efficiency set is given by ES = { x ∗ : x ∗ = k 1 + θ k 2 , θ > 0 } from where it follows that portfolio return µ ′ x ∗ is linear and the variance x ∗ ′ Γ x ∗ is quadratic in θ . The efficiency frontier is a parabola in this space. Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Efficiency set with riskless asset Theorem Asset 0 is efficient. Any combination uR X + (1 − u ) r of asset 0 and a portfolio X lies on the straight line between 0 and X in the ( µ, σ ) space. The straight line between asset 0 and asset M is the efficient frontier called the Capital Market Line. (Tobin’s two fund separation) Any efficient portfolio is a combination of only 2 portfolios (e.g. 0 and M). Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Efficiency set with riskless asset (contd.) Theorem (contd.) Any efficient portfolio satisfies x ∗ = ˆ θ Γ − 1 ( µ − r 1) Tangent (market) portfolio ( m , M ) is θ M Γ − 1 ( µ − r 1) ˆ m = 1 ˆ θ M = 1 ′ Γ − 1 ( µ − r 1) (Proof left as an exercise.) Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory Capital Market Equilibirium Since the market portfolio is efficient there exists scalars λ and θ such that µ i = λ + θ cov ( R M , R i ) , It follows that for any portfolio we have N � E ( R X ) = x i µ i i =1 N � = x i ( λ + θ cov ( R M , R i )) i =1 = λ + θ cov ( R M , R X ) Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory CAPM equilibrium In particular for market portfolio it holds that µ M = λ + θσ 2 M from where it follows that θ = µ M − λ and therefore σ 2 M µ i = λ + θ cov ( R M , R i ) = λ + ( µ M − λ ) β i where β i = cov ( R M , R i ) . If we set R i = r the risk-less asset we get σ 2 M E ( R i ) = r + β i ( µ M − r ) . Gorazd Brumen Portfolio Theory