Modeling Portfolios that Contain Risky Assets Risk and Return III: - PowerPoint PPT Presentation

Modeling Portfolios that Contain Risky Assets Risk and Return III: Basic Markowitz Portfolio Theory C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling January 26, 2012 version � 2011 Charles David Levermore c

Modeling Portfolios that Contain Risky Assets Risk and Return I: Introduction II: Markowitz Portfolios III: Basic Markowitz Portfolio Theory Portfolio Models I: Portfolios with Risk-Free Assets II: Long Portfolios III: Long Portfolios with a Safe Investment Stochastic Models I: One Risky Asset II: Portfolios with Risky Assets Optimization I: Model-Based Objective Functions II: Model-Based Portfolio Management III: Conclusion

Risk and Return III: Basic Markowitz Portfolio Theory 1 . How to Build a Portfolio Theroy 2 . Constrained Minimization Problem 3 . Frontier Portfolios 4 . General Portfolio with Two Risky Assets 5 . Simple Portfolio with Three Risky Assets 6 . Application: Efficient-Market Hypothesis

Risk and Return III: Basic Markowitz Portfolio Theory The 1952 Markowitz paper initiated what subsequently became known as modern portfolio theory (MPT). Because 1952 was long ago, this name has begun to look silly and some have taken to calling it Markowitz Portfolio Theory (still MPT), to distinguish it from more modern theories. (Markowitz simply called it portfolio theory, and often made fun of the name it aquired.) How to Build a Portfolio Theroy. Portfolio theories strive to maximize return for a given risk — or what is related, mimimize risk for a given return. They do this by quantifying the notions of return and risk, and identifying a class of idealized portfolios for which an analysis is tractable. Here we present MPT, the first such theory.

Markowitz chose to use the return rate mean µ as the proxy for the return of a portfolio, and the volatility σ = √ v as the proxy for its risk. He also chose to analyze the class that we have dubbed Markowitz portfolios. Then for a portfolio of N risky assets characterized by m and V the problem of minimizing risk for a given return becomes the problem of minimizing � f T Vf σ = over f ∈ R N subject to the constraints T f = 1 , T f = µ , 1 m where µ is given. Here 1 is the N -vector that has every entry equal to 1 . Additional constraints can be imposed. For example, if only long positions are to be considered then we must also impose the entrywise constraints f ≥ 0 , where 0 denotes the N -vector that has every entry equal to 0 .

Constrained Minimization Problem. Because σ > 0 , minimizing σ is equivalent to minimizing σ 2 . Because σ 2 is a quadratic function of f , it is easier to minimize than σ . We therefore choose to solve the constrained minimization problem � 1 2 f T Vf : 1 T f = 1 , m T f = µ � min . f ∈ R N Because there are two equality constraints, we introduce the Lagrange multipliers α and β , and define Φ( f , α, β ) = 1 2 f T Vf − α � T f − 1 � � T f − µ � 1 − β m . By setting the partial derivatives of Φ( f , α, β ) equal to zero we obtain 0 = ∇ f Φ( f , α, β ) = Vf − α 1 − β m , T f + 1 , 0 = ∂ α Φ( f , α, β ) = − 1 T f + µ . 0 = ∂ β Φ( f , α, β ) = − m

Because V is positive definite we may solve the first equation for f as f = α V − 1 1 + β V − 1 m . By setting this into the second and third equations we obtain the system T V − 1 1 + β 1 T V − 1 m = 1 , α 1 T V − 1 1 + β m T V − 1 m = µ . α m If we introduce a , b , and c by T V − 1 1 , T V − 1 m , T V − 1 m , a = 1 b = 1 c = m then the above system can be expressed as � � � � � � a b α 1 = . b c β µ Because V − 1 is positive definite, the above 2 × 2 matrix is positive definite if and only if the vectors 1 and m are not co-linear ( m � = µ 1 for every µ ).

We now assume that 1 and m are not co-linear, which is usually the case in practice. We can then solve the system for α and β to obtain � � � � � � � � 1 1 α c − b 1 c − bµ = = . β ac − b 2 − b a µ ac − b 2 aµ − b Hence, for each µ there is a unique minimizer given by f ( µ ) = c − bµ ac − b 2 V − 1 1 + aµ − b ac − b 2 V − 1 m . The associated minimum value of σ 2 is T V σ 2 = f ( µ ) T Vf ( µ ) = � α V − 1 1 + β V − 1 m � � α V − 1 1 + β V − 1 m � � � � � � � � � � � 1 a b α c − b 1 � � = α β = 1 µ ac − b 2 b c β − b a µ � 2 � = 1 a µ − b a + . ac − b 2 a

Remark. In the case where 1 and m are co-linear we have m = µ 1 for some µ . If we minimize 1 2 f T Vf subject to the constraint 1 T f = 1 then, by Lagrange multipliers, we find that Vf = α 1 , for some α ∈ R . Because V is invertible, for every µ the unique minimizer is given by 1 T V − 1 1 V − 1 1 . f ( µ ) = 1 The associated minimum value of σ 2 is T V − 1 1 = 1 1 σ 2 = f ( µ ) T Vf ( µ ) = a . 1 Remark. The mathematics behind MPT is fairly elementary. Markowitz had cleverly indentified a class of portfolios which is analytically tractable by elemetary methods, and which provides a framework for useful models.

Frontier Portfolios. We have seen that for every µ there exists a unique Markowitz portfolio with mean µ that minimizes σ 2 . This minimum value is � 2 � σ 2 = 1 a µ − b a + . ac − b 2 a This is the equation of a hyperbola in the σµ -plane. Because volatility is nonnegative, we only consider the right half-plane σ ≥ 0 . The volatility σ and mean µ of any Markowitz portfolio will be a point ( σ, µ ) in this half- plane that lies either on or to the right of this hyperbola. Every point ( σ, µ ) on this hyperbola in this half-plane represents a unique Markowitz portfolio. These portfolios are called frontier portfolios . We now replace a , b , and c with the more meaningful frontier parameters � ac − b 2 mv = 1 µ mv = b σ √ a , a , ν as = . a

The volatility σ for the frontier portfolio with mean µ is then given by � � 2 � � µ − µ mv � σ 2 � σ = σ f ( µ ) ≡ mv + . ν as It is clear that no portfolio has a volatility σ that is less than σ mv . In other words, σ mv is the minimum volatility attainable by diversification. Markowitz interpreted σ 2 mv to be the contribution to the volatility due to the systemic risk of the market, and interpreted ( µ − µ mv ) 2 /ν 2 as to be the contribution to the volatility due to the specific risk of the portfolio. The frontier portfolio corresponding to ( σ mv , µ mv ) is called the minimum volatility portfolio . Its associated distribution f mv is given by = 1 � b � a V − 1 1 = σ 2 mv V − 1 1 . f mv = f ( µ mv ) = f a This distribution depends only upon V , and is therefore known with greater confidence than any distribution that also depends upon m .

The distribution of the frontier portfolio with mean µ can be expressed as f f ( µ ) ≡ f mv + µ − µ mv V − 1 � � m − µ mv 1 . ν 2 as T V − 1 � � Because 1 m − µ mv 1 = b − µ mv a = 0 , we see that � = σ 2 µ − µ mv f T T V − 1 � � � f f ( µ ) − f mv m − µ mv 1 = 0 . mv V 1 mv ν 2 as The vectors f mv and f f ( µ ) − f mv are thereby orthogonal with respect to the V -inner product, which is given by ( f 1 | f 2 ) V = f T 1 Vf 2 . In the associated V -norm, which is given by � f � 2 V = ( f | f ) V , we find that � 2 � µ − µ mv � f mv � 2 V = σ 2 � f f ( µ ) − f mv � 2 mv , V = . ν as Hence, f f ( µ ) = f mv + ( f f ( µ ) − f mv ) is the orthogonal decomposition of f f ( µ ) into components that account for the contributions to the volatility due to systemic risk and specific risk respectively.

Remark. Markowitz attributed σ mv to systemic risk of the market because he was considering the case when N was large enough that σ mv would not be significantly reduced by introducing additional assets into the portfolio. More generally, one should attribute σ mv to those risks that are common to all of the N assets being considered for the portfolio. Definition. When two portfolios have the same volatility but different means then the one with the greater mean is said to be more efficient because it promises greater return for the same risk. Definition. Every frontier portfilio with mean µ > µ mv is more efficient than every other portfolio with the same volatility. This segment of the frontier is called the efficient frontier . Definition. Every frontier portfilio with mean µ < µ mv is less efficient than every other portfolio with the same volatility. This segment of the frontier is called the inefficient frontier .

The efficient frontier is represented in the right-half σµ -plane by the upper branch of the frontier hyperbola. It is given as a function of σ by � σ 2 − σ 2 µ = µ mv + ν as mv , for σ > σ mv . This curve is increasing and concave and emerges vertically upward from the point ( σ mv , µ mv ) . As σ → ∞ it becomes asymptotic to the line µ = µ mv + ν as σ . The inefficient frontier is represented in the right-half σµ -plane by the lower branch of the frontier hyperbola. It is given as a function of σ by � σ 2 − σ 2 µ = µ mv − ν as mv , for σ > σ mv . This curve is decreasing and convex and emerges vertically downward from the point ( σ mv , µ mv ) . As σ → ∞ it becomes asymptotic to the line µ = µ mv − ν as σ .