Modeling Portfolios that Contain Risky Assets Optimization II: - PowerPoint PPT Presentation

Modeling Portfolios that Contain Risky Assets Optimization II: Model-Based Portfolio Management 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

Optimization II: Model-Based Portfolio Management 1 . Reduced Maximization Problem 2 . One Risk-Free Rate Model 3 . Two Risk-Free Rates Model 4 . Long Portfolio Model

Optimization II: Model-Based Portfolio Management We now address the problem of how to manage a portfolio that contains N risky assets along with a risk-free safe investment and possibly a risk-free credit line. Given the mean vector m , the covariance matrix V , and the risk-free rates µ si and µ cl , the idea is to select the portfolio distribution f that maximizes an objective function of the form σ 2 − χ ˆ µ − 1 ˆ Γ( f ) = ˆ 2 ˆ σ , where � T f � T f ,  µ = µ rf ˆ 1 − 1 + m T f < 1 , µ si for 1  µ rf = T f > 1 . � µ cl for 1 f T Vf , σ = ˆ  √ Here χ = ζ/ T where ζ ≥ 0 is the risk aversion coefficient and T > 0 is a time horizon that is usually the time to the next portfolio rebalancing . Both ζ and T are chosen by the investor.

Reduced Maximization Problem. Because frontier portfolios minimize ˆ σ for a given value of ˆ µ , the optimal f clearly must be a frontier portfolio. Because the optimal portfolio must also be more efficient than every other portfolio with the same volatility, it must lie on the efficient frontier. Recall that the efficient frontier is a curve µ = µ ef ( σ ) in the σµ -plane given by an increasing, concave, continuously differentiable function µ ef ( σ ) that is defined over [0 , ∞ ) for the unconstrained One Risk-Free Rate and Two Risk-Free Rates models, and over [0 , σ mx ] for the long portfolio model. The problem thereby reduces to finding σ that maximizes 2 σ 2 − χ σ . ef ( σ ) = µ ef ( σ ) − 1 Γ This function has the continuous derivative Γ ′ ef ( σ ) = µ ′ ef ( σ ) − σ − χ . Because µ ef ( σ ) is concave, Γ ′ ef ( σ ) is strictly decreasing.

Because Γ ′ ef ( σ ) is strictly decreasing, there are three possibilities. • Γ ef ( σ ) takes its maximum at σ = 0 , the left endpoint of its interval of definition. This case arises whenever Γ ′ ef (0) ≤ 0 . • Γ ef ( σ ) takes its maximum in the interior of its interval of definition at the unique point σ = σ opt that solves the equation Γ ′ ef ( σ ) = µ ′ ef ( σ ) − σ − χ = 0 . This case arises for the unconstrained models whenever Γ ′ ef (0) > 0 , and for the long portfolio model whenever Γ ′ ef ( σ mx ) < 0 < Γ ′ ef (0) . • Γ ef ( σ ) takes its maximum at σ = σ mx , the right endpoint of its in- terval of definition. This case arises only for the long portfolio model whenever Γ ′ ef ( σ mx ) ≥ 0 .

This reduced maximization problem can be visualized by considering the family of parabolas parameterized by Γ as 2 σ 2 . µ = Γ + χσ + 1 As Γ varies the graph of this parabola shifts up and down in the σµ -plane. For some values of Γ the corresponding parabola will intersect the efficient frontier, which is given by µ = µ ef ( σ ) . There is clearly a maximum such Γ . As the parabola is strictly convex while the efficient frontier is concave, for this maximum Γ the intersection will consist of a single point ( σ opt , µ opt ) . Then σ = σ opt is the maximizer of Γ ef ( σ ) . This reduction is appealing because the efficient frontier only depends on general information about an investor, like whether he or she will take short positions. Once it is computed, the problem of maximizing any given ˆ Γ( f ) over all admissible portfolios f reduces to the problem of maximizing the associated Γ ef ( σ ) over all admissible σ — a problem over one variable.

In summary, our approach to portfolio selection has three steps: 1. Choose a return rate history over a given period (say the past year) and calibrate the mean vector m and the covariance matrix V with it. 2. Given m , V , µ si , µ cl , and any portfolio constraints, compute µ ef ( σ ) . √ 3. Finally, choose χ = ζ/ T and maximize the associated Γ ef ( σ ) ; the maximizer σ opt corresponds to a unique efficient frontier portfolio. Below we will illustrate the last step on some models we have developed.

One Risk-Free Rate Model. This is the easiest model to analyze. You first compute σ mv , µ mv , and ν as from the return rate history. The model assumes that µ si = µ cl < µ mv . Then its tangency parameters are � � � 2 � 2 � � µ mv − µ rf � � ν as σ mv � � ν tg = ν as � 1 + , σ tg = σ mv � 1 + , ν as σ mv µ mv − µ rf where µ rf = µ si = µ cl , while its efficient frontier is µ ef ( σ ) = µ rf + ν tg σ for σ ∈ [0 , ∞ ) . ef ( σ ) = µ ef ( σ ) − 1 2 σ 2 − χσ , we have Because Γ Γ ′ ef ( σ ) = ν tg − σ − χ . When χ ≥ ν tg we see that Γ ′ ef (0) = ν tg − χ ≤ 0 , whereby σ opt = 0 , while when χ < ν tg there is a positive solution of Γ ′ ef ( σ ) = 0 . We obtain  0 if ν tg ≤ χ ,  σ opt = ν tg − χ if χ < ν tg . 

The optimal return rate µ opt = µ ef ( σ opt ) is expressed in terms of the return rate µ tg of the tangency portfolio and the risk-free rate µ rf as   σ σ opt opt  µ rf + µ opt =  1 − µ tg , σ tg σ tg where ν 2 as σ 2 mv µ tg = µ mv + . µ mv − µ rf The optimal efficient frontier portfolio has the distribution f opt = f ef ( σ opt ) which is expressed in terms of the tangency portfolio f tg as σ 2 σ opt mv V − 1 � � f opt = f tg , where f tg = m − µ rf 1 . σ tg µ mv − µ rf

It follows from the distribution f opt that the optimal efficient frontier portfolio can be built from the tangency portfolio f tg and the risk-free assets as follows. There are four possibilities: 1. If σ opt = 0 then the investor will hold only the safe investment. opt ∈ (0 , σ tg ) then the investor will place 2. If σ σ tg − σ opt of the portfolio value in the safe investment , σ tg σ opt of the portfolio value in the tangency portfolio f tg . σ tg 3. If σ opt = σ tg then the investor will hold only the tangency portfolio f tg .

4. If σ opt ∈ ( σ tg , ∞ ) then the investor will place σ opt of the portfolio value in the tangency portfolio f tg , σ tg σ opt − σ tg by borrowing of this value from the credit line . σ tg In order to see which of these four cases arises as a function of µ rf , we must compare ν tg with χ and σ opt = ν tg − χ with σ tg . Because ν tg > ν as , the condition χ ≥ ν tg cannot be met unless χ > ν as , in which case it can be expressed as µ rf ≥ η ex ( χ ) where  µ mv if χ ≤ ν as ,  η ex ( χ ) = � χ 2 − ν 2 µ mv − σ mv if χ > ν as . as  This is called the exit rate for the investor because when µ rf ≥ η ex ( χ ) the investor is likely to sell all of his or her risky assets.

In order to compare ν tg − χ with σ tg , notice that ν tg − χ = σ tg whenever µ = µ rf solves the equation � � � 2 � 2 � � � � µ mv − µ ν as σ mv � � ν as � 1 + − σ mv � 1 + = χ . ν as σ mv µ mv − µ The left-hand side of this equation is a strictly decreasing function of µ over the interval ( −∞ , µ mv ) that maps onto R . Let µ = η tg ( χ ) be the unique solution of this equation in ( −∞ , µ mv ) . Then because σ opt = ν tg − χ , we see that the four cases arise as follows σ opt = 0 η ex ( χ ) ≤ µ rf , if and only if σ opt ∈ (0 , σ tg ) η tg ( χ ) < µ rf < η ex ( χ ) , if and only if σ opt = σ tg if and only if µ rf = η tg ( χ ) , σ opt ∈ ( σ tg , ∞ ) if and only if µ rf < η tg ( χ ) . Notice that if χ ≤ ν as then η ex ( χ ) = µ mv , so the first case does not arise.

In particular, when χ = 0 the first case does not arise, while µ = η tg (0) is the solution of � � � 2 � 2 � � � � µ mv − µ ν as σ mv � � ν as � 1 + − σ mv � 1 + = 0 . ν as σ mv µ mv − µ This can be solved explicitly to find that η tg (0) = µ mv − σ 2 mv . Therefore the three remaining cases arise as follows: µ mv − σ 2 σ opt ∈ (0 , σ tg ) if and only if mv < µ rf , µ rf = µ mv − σ 2 σ opt = σ tg if and only if mv , µ rf < µ mv − σ 2 σ opt ∈ ( σ tg , ∞ ) if and only if mv . Specifically, when χ = 0 we have µ opt = µ rf + ν 2 γ opt = µ rf + 1 2 ν 2 opt = ν tg , σ tg , tg .