RoseHulman Undergraduate Mathematics Journal Volume 8 Issue Article 2 A MultiObjective Aroach to Portfolio Otimization Yaoyao Clare Duan Boston College, sweetclare@gmail.com Follow this and additional works at: htts://scholar.rosehulman.edu/rhumj Recommended Citation Duan, Yaoyao Clare (2007) "A MultiObjective Aroach to Portfolio Otimization," RoseHulman Undergraduate Mathematics Journal: Vol. 8 : Iss., Article 2. Available at: htts://scholar.rosehulman.edu/rhumj/vol8/iss/2
A Multiobjective Aroach to Portfolio Otimization Yaoyao Clare Duan, Boston College, Chestnut Hill, MA Abstract: Otimization models lay a critical role in determining ortfolio strategies for investors. he traditional mean variance otimization aroach has only one objective, which fails to meet the demand of investors who have multile investment objectives. his aer resents a multiobjective aroach to ortfolio otimization roblems. he roosed otimization model simultaneously otimizes ortfolio risk and returns for investors and integrates various ortfolio otimization models. Otimal ortfolio strategy is roduced for investors of various risk tolerance. Detailed analysis based on convex otimization and alication of the model are rovided and comared to the mean variance aroach.. Introduction to Portfolio Otimization Portfolio otimization lays a critical role in determining ortfolio strategies for investors. What investors hoe to achieve from ortfolio otimization is to maximize ortfolio returns and minimize ortfolio risk. Since return is comensated based on risk, investors have to balance the riskreturn tradeoff for their investments. herefore, there is no a single otimized ortfolio that can satisfy all investors. An otimal ortfolio is determined by an investor s riskreturn reference. here are a few key concets in ortfolio otimization. First, reward and risk are measured by exected return and variance of a ortfolio. Exected return is calculated based on historical erformance of an asset, and variance is a measure of the disersion of returns. Second, investors are exosed to two tyes of risk: unsystematic risk and systematic risk. Unsystematic risk is an asset s intrinsic risk which can be diversified away by owning a large number of assets. hese risks do not resent enough information about the overall risk of the entire ortfolio. Systematic risk, or the ortfolio risk, is the risk generally associated with the market which cannot be eliminated. hird, the covariance between different asset returns gives the variability or risk of a ortfolio. herefore, a welldiversified ortfolio contains assets that have little or negative correlations []. he key to achieving investors objectives is to rovide an otimal ortfolio strategy which shows investors how much to invest in each asset in a given ortfolio. herefore, the decision variable of ortfolio otimization roblems is the asset weight vector x = [ xx 2 x] n with xi as the weight of asset i in the ortfolio. he exected return for each asset in the
ortfolio is exressed in the vector form = [ 2 n ] with i as the mean return of asset i. he ortfolio exected return is the weighted average of individual asset n return x x x. Variance and covariance of individual asset are characterized by a = = i= i i σ σn variancecovariance matrixv =, where σ ii, is the variance of asset i and σ i, jis σn σ nn the covariance between asset i and asset j. he ortfolio variance is σ xvx xx j n n 2 = = i jσ i, i= j= [2].. Problem Formulations Modern ortfolio theory assumes that for a given level of risk, a rational investor wants the maximal return, and for a given level of exected return, the investor wants the minimal risk. here are also extreme investors who only care about maximizing return (disregard risk) or minimizing risk (disregard exected return). here are generally five different formulations that serve investors of different investment objectives: Model : Maximize exected return (disregard risk) Maximize: x = x Subject to: x = where = [ ]. he constraint x = requires the sum of all asset weights to be equal to. Model 2: Minimize risk (disregard exected return) 2 Minimize: σ = x Vx Subject to: x = Model 3: Minimize risk for a given level of exected return * 2 Minimize: σ = x Vx Subject to: x = and x = * Model 4: Maximize return for a given level of risk Maximize: x = x Subject to: 2* x = and xvx= σ 2* σ 2
Model 5: Maximize return and Minimize risk Maximize: x Subject to: x = = x and Minimize: σ = x Vx 2 he five models above include both rational and extreme investors with different investment objectives. Model 3 and 4 are extensions of Model and 2 with fixed constraints. he classic solution to ortfolio otimization is the mean variance otimization roosed by Nobel Prize winner Harry Markowitz in 990 [2]. he mean variance method aims at minimizing variance of a ortfolio for any given level of exected return, which shares the same formulation of model 3. Since the mean variance method assumes all investors objectives are to minimize risk, it may not be the best model for those who are extremely risk seeking. Also, the formulation does not allow investors to simultaneously minimize risk and maximize exected return..2 Introduction to Multiobjective otimization Multiobjective otimization, develoed by FrenchItalian economist V. Pareto, is an alternative aroach to the ortfolio otimization roblem [3]. he multiobjective aroach combines multile objectives f( x), f2( x),, fn ( x) into one objective function by assigning a weighting coefficient to each objective. he standard solution technique is to minimize a ositively weighted convex sum of the objectives using singleobjective method, that is, n Minimize F x = a f x, a > 0, i =, 2,, n ( ) ( ) i= i i i he concet of otimality in multiobjective otimization is characterized by Pareto otimality. Essentially, a vector x * is said to be Pareto otimal if and only if there is no x such * * f x f x for all i =, 2,, n. In other words, x is the Pareto oint if that i( ) i( ) F( x * )achieves its minimal value [4]. Since investors are interested in minimizing risk and maximizing exected return at the same time, the ortfolio otimization roblem can be treated as a multiobjective otimization roblem (Model 5). One can attain Pareto otimality in this case because the formulation of Model 5 belongs to the category of convex vector otimization, which guarantees that any local otimum is a global otimum [5]. his aer focuses on the analysis and alication of the multiobjective aroach to ortfolio otimization based on convex vector otimization. 3
2. Methodology Before going into details about multiobjective otimization, it is essential to introduce the concet of convex vector otimization. 2. Convex Vector Otimization As shown in Figure, a set any air of oints in S, that is, S R n is a convex set if it contains all line segments joining ( ) x, y S, θ > 0 θx+ θ y S Convex Convex NonConvex Figure (a) Figure (b) Figure (c) n A function f :R R is convex if its domain dom f is convex and for all x, y dom f, θ [0,] ( θ + ( θ) ) θ ( ) + ( θ) ( y ) f x y f x f A function f is concave if f is convex. Geometrically, one can think of the curve of a convex function as always lying below the line segment of any two oints. Here is an examle of a convex function: Figure 2 Convex Function A vector convex otimization has standard form: Minimize (w.r.t. x ) f0 ( ) f ( x ) 0, Subject to i i =,, m h x = j =,, j x ( ) 0, 4
Here ( x ) is the convex o f ( ) f bjective function, x are the convex inequality constraint 0 functions, and hj ( x ) are the equality constraint functions which can be exressed in linear form Ax + B. 2.2 Multiobjective Formulation i he secific formulation for the ortfolio otimization roblem in Model 5 can be 2 determined by recognizing that the two objectives minimizing ortfolio risk σ = x Vx and maximizing ortfolio exected return exected return Model 5: x x = x and ortfolio risk = x are equivalent to minimizing negative ortfolio σ = x Vx. his gives the new formulation of 2 x f x f x x x Vx Minimize w.r.t. ( ) ( ) Subject to: x = (, 2 ) ( =, ) (Model 5) his multiobjective otimization can be solved using scalarization, a standard technique for finding Pareto otimal oints for any vector otimization roblem by solving the ordinary scalar otimization [4]. Assign two weighting coefficients, 2 0 λ λ > for objective functions f ( ) x and f2 ( x ) resectively. By varying λ and λ 2, one can obtain different Pareto otimal solutions of the vector otimization roblem. Without loss of generality, one can take λ = and λ 2 = µ > 0 : Minimize: (Modified Model 5) x+ µ x Vx Subject to: x = he weighting coefficient µ reresents how much an investor weights risk over exected return. One can consider µ as a risk aversion index that measures the risk tolerance of an investor. A smaller value of µ indicates that the investor is more riskseeking, and a larger value of µ indicates that the investor is more riskaverse. All Pareto otimal ortfolios can be obtained by he objective function in Modified Model 5 is convex because V is ositive semidefinite. A twice differentiable function f is convex if and only if the second derivative of f is ositive semidefinite for all x dom f [5]. 5
varying µ excet for two extreme cases where µ 0 and µ. As µ 0, the variance term µ xvx 0 and the objective function is dominate d by the execte d return term x. his relicates Model where investors only want to maximize return and disregard risk. In this case, the investor is being extremely risk seeking. he otimal strategy for this extreme case is to concentrate the ortfolio entirely on the asset that gives the highest exected return. As µ, µ xvx. he objective function is dominated by the variance term µ x Vx. his relicates Model 2 wher e the investor only wants to minimize risk without regard to exected return. In this case, the investor is being extremely risk averse. he otimal strategy for such tye of investor is to invest all resources on the asset that has the minimal variance. By varying µ, one can generate various otimization models that serve investors of any risk tolerance. 2.3 Solving Multiobjective otimization he multiobjective otimization can be solved using Lagrangian multilier: Lx ( ) = x+ µ xvx+ λ( x ) δ L Set = 0, it follows that δ x ( x = V )( λ ) 2µ (.2) o solve the Lagrangian multilier λ, substitute equation.2 to the constraint x = : V V (.3) V 2µ λ = Let a = V and a2 = V, both of which are scalars, equation.3 can be written as: a a 2 λ = 2µ a he otimized solution for the ortfolio weight vector x is * V a2 2µ x = V ( ) 2µ 2µ a a Detailed derivation of the otimal solution is rovided in Aendix A. (.4) 6
3 Alications of Multiobjective Portfolio Otimization he mathematical results from the multiobjective ortfolio otimization (.4) can be alied to ortfolios consisting of any number of assets. As a secific examle, assume that an investor is interested in owning a ortfolio that contains five of his favorite stocks: IBM (IBM), Microsoft (MSF), Ale (AAPL), Quest Diagnostics (DGX), and Bank of America (BAC). Assuming that the investor is not sohisticated in finance, cases involving of short selling are excluded in this examle. he exected return and variance of each stock in the ortfolio is calculated based on historical stock rice and dividend ayment from February, 2002 to February, 2007. (Aendix C) Stock Ex. Return Variance IBM 0.400% 0.00646 MSF 0.53% 0.0039 AAPL 4.085% 0.02678 DGX.006% 0.00559836 BAC.236% 0.00622897 able Exected Return and Variances of Selected Stocks Using Matlab to imlement the multiobjective otimization on this ortfolio, the investor can see the otimal asset allocation strategy for any value of risk aversion index µ. Figure shows how much the investor should invest in each stock given different values of µ. Asset Weight 00% 90% 80% 70% 60% 50% 40% 30% 20% 0% 0% 0.0 2 0 50 00 000 Risk Aversion Index BAC DGX AAPL MSF IBM Figure 3 Risk Aversion Index vs. Otimal Asset Allocations 7
he otimized results of this simle examle agree with intuition. An investo r with µ = 0.0 is highly risk seeking, and the otimal ortfolio for such an investor is to concentrate 00% on the highest exected return stock AAPL. As µ increases, the investor is becoming more sensitive to risk, and the comosition of ortfolio starts to show a mix of other lower return (lower variance) stocks. When µ equals to 50, the otimal ortfolio strategy shows that the investor should invest in a mix of assets; for this examle the investor should invest 2.05% of total resources in AAPL stock, 6.22% in BAC stock, 9.77% in MSF stock, and 6.96% in DGX stock. he allocation on AAPL stock has significantly decreased from 00% to 2.05% as µ increases from 0.0 to 50 because AAPL has the highest return variance. Note that none of the otimal ortfolio strategies indicate any asset allocation in IBM stock. hat is because IBM gives the lowest return but somewhat high variance comared to other four stocks in the ortfolio. Another imortant observation from Figure 3 is that there is no significant difference in asset allocation strategy as µ increases from 00 to 000. he actual data suggests that µ 00 is the meaningful range of risk index µ. Figure 4 illustrates that and 00 are two thresholds o f µ that are determinate to the investor s ortfolio strategy. When µ <, the otimal solutions indicate that the investor should invest all his resources on the highest return stock AAPL. For µ 00, the otimal solutions indicate a variety of asset allocation strategies. As µ > 00, the asset allocation strategy has little change. Figure 4 Risk Aversion Index vs. Asset Weight 8
4 Multiobjective otimization vs. Mean variance otimization he revious sections have demonstrated the alication of multiobjective aroach on the ortfolio otimization. his section comares the multiobjective aroach with the traditional mean variance method. Alying both multiobjective otimization and mean variance otimization to the same ortfolio, the numerical exeriments generate efficient frontiers that show the set of all ossible otimal ortfolio oints on a riskreturn tradeoff curve. Figure 5 shows that efficient frontiers generated by both methods coincide, which indicates that both methods roduce exactly the same set of otimal solutions. Figure 5 Efficient Frontiers of Multiobjective and Mean variance otimization From the analytical oint of view, one can rove that both methods roduce the same otimal solution by rearranging the terms in their corresonding Lagrange multiliers. Examining the Lagrangian multiliers from both formulations, it is not hard to notice that the two Lagrangian δ multiliers share equivalent form. his is because when taking the first derivative L δ x, the constant terms vanish and the remaining arts can all be rearranged to have equivalent form. Details of roof are rovided in Aendix B. he main difference between the mean variance and multiobjective aroach is their roblem formulations. he multiobjective aroach uts two otimization objectives 9
(minimizing risk and maximizing exected return) into one objective function where as the mean variance aroach has only one objective of minimizing risk. he mean variance method laces the exected value as a constraint in the formulation, which forces the otimization model to rovide the minimal risk for each secified level of exected return. here are two comarative advantages for the multiobjective formulation over the mean variance formulation. First, since the mean variance aroach assumes that the investor s sole objective is to minimize risk, it may not be a good fit for investors who are extremely risk seeking. he multiobjective formulation is alicable for investors of any risk tolerance. Second, the mean variance method requires investors to lace an exected value constraint, but there are times when investors do not want to lace any constraints on their investment or do not know what kind of return to exect from his investment. he multiobjective otimization rovides the entire icture of otimal riskreturn trade off. Another key difference between these two methods lies in their aroach to roducing efficient frontiers. he efficient frontier of the multiobjective otimization is determined by the risk aversion index µ because different values of µ determine different values of risk and exected return. he efficient frontier of the mean variance method is generated by varying the roortion of two otimal ortfolios because the wo Fund Searation heorem guarantees that any otimized ortfolio can be dulicated by a combination of two otimal ortfolios [6]. herefore, in the rocess of generating the efficient frontier for the mean variance otimization, one needs to use the minimum variance ortfolio to relicate a secondary ortfolio with the given exected return vector. As a result, using the mean variance method to generate the efficient frontier can be numerically more cumbersome than the multiobjective aroach. 5 Concluding Remarks he traditional singleobjective aroach, such as the mean variance method, solves the roblem by having one of the otimization objectives in the objective function and fixes the other objective as a constraint. Consequently, investors have to choose the otimal solution based on given exected return or risk. he multiobjective otimization rovides an alternative solution to the ortfolio otimization roblem, generating the same otimal solution as the mean variance method. It can be alied to investors of any risk tolerance, including those who are extremely riskseeking and riskaverse. he riskaversion index measures how much an investor weights risk over exected return. Given any secified value of riskaversion index, the multiobjective 0
otimization rovides investors with otimal asset allocation strategy that can simultaneously maximize exected return and minimize risk.
Aendix A: Derivation of Analytic Solution to Multiobjective otimization Objective function: Minimize (w.r.t. x ) x + µ x Vx Subject to: x = Solution: Using Lagrangian Multilier to solve the multiobjective otimization roblem: ( ) Lx x µ xvx = + + λ( x ) δ L Set = 0. hen we have δ x 2µ Vx = λ thus x ( )( 2 V = λ µ ) () o solve the Lagrangian multilier λ, substitute equation () to the constraint x = : V λ = 2µ ( )( ) thus λ V V = 2µ 2µ V 2µ thus λ = V V (2) Set a = V and a2 = V. Both a and a2 are scalars. Substitute a and a2 into equation (2): a a 2 λ = 2µ a he otimized solution for the asset weight vector: * V a2 2µ x = V ( 2µ 2µ a a ) 2
Aendix B: Proof of Equivalent Analytic Solutions for Multiobjective and Mean variance otimization Lagriangian Equation for Multiobjective otimization: ( ) Multi Objective Lx x µ xvx = + + λ( x ) Since µ can be assigned to either x or x Vx, he Lagrangian Equation for Multiobjective otimization can be rewritten as: ( ) Multi Objective Lx µ x xvx = + + λ( x ) he Lagriangian Equation for Mean variance otimization: Lx ( ) = xvx + λ ( x ) + λ ( x ) Mean Variance * 2 () (2) Now comare equation () and (2), let µ = λ and λ = λ2, δ Lx ( ) Solving for =0 for both the Mean variance and Multiobjective Lagrangian Equations: δ x (MultiObjective) 2Vx λ + λ2 = 0 2 ( ) * Multi Objective x = V λ λ2 (Mean variance) 2 herefore, x 2Vx λ + λ = 0 * Mean Variance x = V λ λ2 = x * Multi Objective * Mean Variance 2 ( ) 3
Aendix C: Exected Return of Five Selected Assets Date IBM MSF AAPL DGX BAC 2//2007 0.52% 0.972%.55%.05% 0.494% /3/2007 2.055% 3.349%.049% 0.794%.57% 2//2006 5.696%.703% 7.44% 0.320% 0.854% //2006 0.20% 2.62% 3.049% 6.90%.03% 0/2/2006 2.673% 4.952% 5.326% 8.543% 0.547% 9//2006.93% 6.443% 3.456% 4.856% 4.083% 8//2006 5.025% 7.200% 0.62% 6.928% 0.97% 7/3/2006 0.763% 3.24% 8.666% 0.486% 7.35% 6//2006 3.856% 2.890% 4.83% 7.502% 0.633% 5//2006 2.6% 5.860% 5.087% 0.08% 2.026% 4/3/2006 0.60%.223% 2.229% 8.855% 9.608% 3//2006 2.780%.242% 8.425% 2.972% 0.40% 2//2006.050% 4.26% 9.297% 6.948% 3.680% /3/2006.0% 7.642% 5.035% 3.802% 4.60% 2//2005 7.535% 5.533% 6.00% 2.780% 0.568% //2005 8.835% 8.036% 7.764% 7.237% 6.027% 0/3/2005 2.070% 0.9% 7.424% 7.420% 3.908% 9//2005 0.493% 6.020% 4.33%.2% 2.57% 8//2005 3.70% 7.2% 9.94% 2.638% 0.47% 7//2005 2.47% 3.22% 5.865% 3.459% 4.42% 6//2005.785% 3.78% 7.420%.466% 0.558% 5/2/2005 0.88% 2.307% 0.26% 0.766% 2.847% 4//2005 6.48% 4.659% 3.463% 0.80% 2.26% 3//2005.294% 3.946% 7.% 5.776% 4.548% 2//2005 0.74% 3.947% 6.67% 4.300% 0.586% /3/2005 5.235%.653% 9.40% 0.06%.39% 2//2004 4.605% 0.345% 3.967%.929% 2.564% //2004 5.22% 6.833% 27.977% 7.078% 3.3% 0//2004 4.676%.59% 35.9% 0.600% 3.372% 9//2004.238%.300% 2.348% 3.067% 2.73% 8/2/2004 2.532% 3.908% 6.679% 4.290% 5.82% 7//2004.227% 0.24% 0.65% 3.26% 0.472% 6//2004 0.488% 8.884% 5.966%.396% 2.776% 5/3/2004 0.679% 0.395% 8.844% 2.5% 3.285% 4//2004 4.00% 4.788% 4.660% 2.022% 0.609% 3//2004 4.825% 6.05% 3.043% 0.049% 0.66% 4
Date IBM MSF AAPL DGX BAC /2/2004 7.062%.049% 5.59% 6.57%.294% 2//2003 2.364% 6.429% 2.297% 0.96% 7.730% /3/2003.378%.625% 8.654% 7.867% 0.393% 0//2003.302% 5.440% 0.425%.542% 2.959% 9/2/2003 7.705% 4.787% 8.400%.057% 0.525% 8//2003.37% 0.437% 7.306% 0.4% 4.028% 7//2003.522% 3.07% 0.598% 6.320% 4.502% 6/2/2003 6.284% 4.74% 6.25% 0.678% 7.40% 5//2003 3.882% 3.747% 26.30% 6.064% 0.89% 4//2003 8.246% 5.627% 0.566% 0.03% 0.805% 3/3/2003 0.64% 2.43% 5.859% 3.% 2.548% 2/3/2003 0.20% 0.95% 4.596%.903%.75% /2/2003 0.90% 8.99% 0.279% 5.468% 0.70% 2/2/2002 0.835% 0.397% 7.63%.98% 0.203% //2002 0.33% 7.882% 3.487% 2.600% 0.374% 0//2002 35.368% 22.234% 0.759% 3.759% 9.42% 9/3/2002 22.639% 0.854%.762% 9.748% 8.34% 8//2002 7.3% 2.268% 3.277% 7.84% 5.366% 7//2002 2.223% 2.278% 3.883% 29.805% 5.480% 6/3/2002 0.506% 7.46% 23.948%.569% 6.433% 5//2002 3.754% 2.57% 4.036% 4.878% 4.597% 4//2002 9.470% 3.364% 2.534% 0.99% 6.567% 3//2002 5.992% 3.374% 9.24% 6.825% 6.342% Ex. Returns 0.400% 0.53% 4.085%.006%.236% 5
Aendix D: VarianceCovariance Matrix of Five Selected Assets IBM MSF AAPL DGX BAC IBM 0.00646 0.002983 0.00235487 0.00235487 0.00096889 MSF 0.002983 0.0039 0.00095937 0.000987 0.00063459 AAPL 0.002355 0.000959 0.0267778 0.003572 0.003448 DGX 0.002355 0.0002 0.003572 0.00559836 0.0004942 BAC 0.000969 0.000635 0.003448 0.0004942 0.006229 6
Acknowledgement his work is comleted as an undergraduate indeendent research roject in the Boston College Mathematics Deartment. I am deely grateful to my suervisor Professor Nancy Rallis (Boston College) for her guidance throughout the roject. Meanwhile, I would like to give my secial thanks to Kyle Guan (MI) for his continuous suort and suggestions. his roject is selected for resentation in 2007 Hudson River Undergraduate Mathematics Conference and Pacific Coast Undergraduate Mathematics Conference. 7
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