Maximizing the Sharpe Ratio and Information Ratio in the Barra Optimizer

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1 Maximizing te Sarpe Ratio and Information Ratio in te Barra Optimizer June 5, 2009 Leonid Kopman Scott Liu 2009 MSCI Barra. All rigts reserved. of 4

2 Maximizing te Sarpe Ratio ABLE OF CONENS. INRODUCION OBJECIVE FUNCIONS IN DEAIL MAXIMIZING HE RISK-ADJUSED REURN MAXIMIZING HE SHARPE RAIO MAXIMIZING HE INFORMAION RAIO OBJECIVE FUNCION COMPARISON APPROACHES AND ALGORIHMS HE NONLINEAR PROGRAMMING APPROACH (DIREC SOLUION) LINE SEARCH APPROACH HOMOGENIZAION APPROACH SUMMARY... 8 REFERENCES... 9 APPENDIX MSCI Barra. All rigts reserved. 2 of 4

3 Maximizing te Sarpe Ratio INRODUCION Optimization tecniques can provide significant assistance to an investor in te portfolio construction process. In particular, tey elp accommodate eac investor s specific objective. According to te canonical Markowitz portfolio optimization model, investors sould coose among portfolios tat belong to te efficient frontier. is means tat a portfolio sould ave te minimum risk for a given level of return, or te maximum return for a given level of risk. In practice, investors will ave to deal wit te trade-off between portfolio risk and return. e traditional mean-variance portfolio optimization metodology offers a uniform way of dealing wit tis trade-off. Its goal is to find a portfolio tat satisfies all constraints wile maximizing te following risk-adjusted return function (i.e., utility function): Utility function = portfolio return - risk aversion portfolio variance, were risk aversion is a parameter reflecting te investor s risk tolerance or preference. An alternative approac is to coose a portfolio tat, instead of maximizing te risk-adjusted return, maximizes te ratio of return to risk, or te ratio of active return to te tracking error. In oter words, te objective is to find a portfolio tat maximizes eiter te Sarpe Ratio or Information Ratio. e Barra Optimizer can assist investors in te process of eiter portfolio construction or rebalancing. It can accommodate bot mean-variance and ratio objectives: users can minimize tracking error, maximize risk-adjusted return, or maximize te Sarpe Ratio or Information Ratio. e rest of tis article is structured as follows: first, we will formally describe te problems te Barra Optimizer will solve wen users want to maximize mean-variance and ratio objectives. en we will compare and examine te solutions obtained by optimizing various objective functions in a simple but typical portfolio construction problem. We will also explain te algoritms for portfolio optimization tat acieve Sarpe Ratio and Information Ratio maximization. OBJECIVE FUNCIONS IN DEAIL Barra Optimizer users can coose to maximize eiter a standard mean-variance utility function or te Information Ratio or Sarpe Ratio of teir portfolios. e portfolio optimization problems wit tese objective functions ave te following general form, described below. Maximizing te Risk-Adjusted Return e objective in te standard mean-variance portfolio optimization problem is to maximize riskadjusted return. In te active setting (i.e., in te presence of a bencmark) te problem can be formulated as follows: Maximize r ( B) ( B) V( B) C( ) (MAV ) s.t. la A ua, l u, 2009 MSCI Barra. All rigts reserved. 3 of 4

4 Maximizing te Sarpe Ratio were is a olding vector, r is a return vector, B is bencmark, is a risk aversion, V is covariance matrix, la, l, ua, uare bounds; A is te coefficient matrix for linear constraints (including factor constraints and te budget constraint) and C( ) is te transaction cost term. Note tat te active return term r ( B ) in te objective can be replaced by is constant and will not affect te results of optimization. Wen te bencmark is not present, te problem becomes: Maximize r V C( ) (MV ) s.t. la A ua, l u. Maximizing te Sarpe Ratio r, since r B e Sarpe Ratio problem -- (SR) for sort -- is defined as te excess return of a portfolio divided by its total risk. e portfolio optimization problem tat maximizes te Sarpe Ratio as te following form: Maximize r rf C( ) V (SR) s.t. la A ua, l u, were r f is te risk-free interest rate. Maximizing te Information Ratio e Information Ratio problem -- (IR) for sort -- is defined as te active return of a portfolio divided by its tracking error. e portfolio optimization problem tat maximizes te Information Ratio as te following form: Maximize r ( B ) C( ) ( B) V( B) (IR) s.t. la A ua, l u. OBJECIVE FUNCION COMPARISON o construct a portfolio tat suits teir needs, users can maximize eiter te standard meanvariance utility function or te ratio type function. In tis section, we will demonstrate te e IR definition used ere is different from te one given by Grinold and Kan, wo denote by IR te ratio of portfolio residual return to portfolio residual risk MSCI Barra. All rigts reserved. 4 of 4

5 Maximizing te Sarpe Ratio relationsip between tese two types of portfolio optimization problems. We will first discuss te relationsip conceptually, ten use a two-stock case for illustration. Because optimization problems maximizing te Sarpe Ratio and Information Ratio ave similar properties, we will ereafter concentrate on discussing te problem tat maximizes te Sarpe Ratio. e following discussion can be applied to a problem tat maximizes te Information Ratio, except wen mentioned oterwise. Assume tat is te optimal portfolio maximizing te Sarpe Ratio problem. Let ˆ r rf C( ) =. 2 V From eorem in te Appendix, we know tat is also an optimal portfolio for te standard mean-variance portfolio optimization problem -- (MV ) for sort. erefore, te following olds: ere exists a risk aversion ˆ suc tat portfolio optimization problems for (SR) and ave te same optimal portfolio. ˆ (MV ˆ ) In general, we ave te following relationsip between portfolio optimization problems ( MV ) and (SR): (): As intended, solving te Sarpe Ratio maximization problem (SR) will produce a portfolio wit te maximum Sarpe Ratio. Ordinarily, te optimal portfolio in te standard meanvariance portfolio optimization problem ( MV ) is not necessarily te one wit te maximum Sarpe Ratio. If we use te correct risk aversion, te optimal portfolio of ( MV ) will ave te maximum Sarpe Ratio, too. (2): e standard mean-variance portfolio optimization problem ( MV ) is more flexible. By setting proper risk aversion, users will get te portfolio tat suits teir risk tolerance and maximizes te risk-adjusted return. For a plain vanilla mean-variance portfolio optimization problem, follow tis: Maximize r V s.t. e=, were it is possible to obtain a closed-form formula for te optimal portfolio: V r e V r V e ( ) = + ( ). 2 2 ev e As te risk aversion increases, te optimal portfolio ( ) increasingly resembles te minimum risk portfolio, i.e., = V e/ e V e. C As te risk aversion decreases, users will take more risk. At te extreme, wen te risk aversion is zero, one completely disregards te risk. e optimal portfolio in a long-only case will old only te stock tat as te largest alpa. Figure illustrates te relative positions of minimum risk, maximum Sarpe Ratio, and maximum return portfolios on te risk-return frontier MSCI Barra. All rigts reserved. 5 of 4

6 Maximizing te Sarpe Ratio Figure. Maximum Sarpe Ratio Case Study Now let s look at a two-stock portfolio optimization problem and compare te optimal solutions of te two problem types. For simplicity, let s limit te discussion to te long-only case and assume tere are no constraints besides te total investment constraint. We also ignore te transaction costs and assume te risk-free rate is 0. e optimization problems to solve in tis case study are: (MV) (SR) Maximize Maximize r V s.t. e=, 0. r V s.t. e=.0, 0, were: r =, V, e. 0. = = e optimal portfolio for te Sarpe Ratio problem (SR) is: , r , , SR SR = rsr σ SR = = = σ SR MSCI Barra. All rigts reserved. 6 of 4

7 Maximizing te Sarpe Ratio e following table presents optimal solutions and oter pertinent information about te (MV) problem wit different levels of risk aversion Risk Aversion (input) Asset Weigt (output) 2 Asset 2 Weigt (output) r(%) Return V (%) Risk r V Sarpe Ratio ( ˆ ) , In te first row labeled Risk Aversion, we ave applied varying levels of, creating eigt (8) different portfolios. e subsequent rows display corresponding results for olding weigts (Assets and 2), portfolio return, risk, and te Sarpe Ratio. is table illustrates tat wen risk aversion is 0., te optimal portfolio will old 00% of stock 2 and te portfolio return is 0% -- te largest possible portfolio return. As te risk aversion increases, te optimal portfolio will increasingly old more of stock, wile bot portfolio return and risk will decrease. However, te Sarpe Ratio will increase at first, ten steadily decrease. As illustrated above, te maximum Sarpe Ratio is reaced wen risk aversion is (see under 4). Wen risk aversion is equal to or larger tan 50, an optimal portfolio will be same as a minimum risk portfolio. As designed, solving a Sarpe Ratio (SR) problem will provide a portfolio wit te maximum Sarpe Ratio. Solving a standard mean-variance optimization problem (MV ) does not necessary get te maximum Sarpe Ratio portfolio. However, (MV ) will give users more control on te risk level of te optimal portfolio MSCI Barra. All rigts reserved. 7 of 4

8 Maximizing te Sarpe Ratio APPROACHES AND ALGORIHMS ere are several algoritms for solving te maximum Sarpe Ratio or Information Ratio problem. For example, te nonlinear programming approac (direct solution), te line searc approac (including Newtons metod and Golden Section searc), and te omogenization approac. 4.. e Nonlinear Programming Approac (Direct Solution) is approac utilizes a nonlinear programming solver to optimize te Sarpe Ratio or Information Ratio directly. Because te nonlinear terms appear only in te objective function and all constraints are linear or piecewise-linear, we can use, for example, a reduced-gradient algoritm in conjunction wit a quasi-newton algoritm for solving tis kind of problem. e nonlinear programming approac as, essentially, no limitations. However, in cases were te objective can never assume positive values, te problem migt become non-convex, wic removes an optimality guarantee and migt result in numerical difficulties Line Searc Approac is approac solves a sequence of standard portfolio optimization (MV ) problems. As illustrated by eorem in te Appendix, under certain conditions we get te optimal portfolio for a Sarpe Ratio problem (SR) by solving a standard portfolio optimization problem wit te correct risk aversion. It is also illustrated by eorem 2 in te Appendix tat te Sarpe Ratio for portfolios on te efficient frontier is first increasing te risk aversion, until it reaces te maximum, and ten it starts decreasing. erefore, te Golden Section searc algoritm can be utilized to find te optimal risk aversion Homogenization Approac A Sarpe Ratio maximization problem (SR) witout transaction costs and penalties can be solved quickly (solving one linear programming problem and one convex quadratic programming problem) by utilizing te omogenization metod described by eorem 3 in te Appendix. However, tis metod as certain drawbacks: piecewise linear transaction costs, presence of a bencmark, and even asset-variable bounds tat need special andling. For instance, accommodating piecewise linear transaction costs will require splitting asset-olding variables, wic will result in increased problem size. SUMMARY Besides te standard mean-variance portfolio optimization tat maximizes te risk-adjusted return, te Barra Optimizer offers users an option to maximize te Sarpe Ratio or Information Ratio. However, te standard mean-variance portfolio optimization is more flexible. It gives users a control on te risk level of te optimal portfolio. Also, users will be able to use more functions and features in te Barra Optimizer wen maximizing risk-adjusted return MSCI Barra. All rigts reserved. 8 of 4

9 Maximizing te Sarpe Ratio REFERENCES Grinold, R. and Kan, Ronald, Active Management, McGraw-Hill, Sarpe, W. F.,"e Sarpe Ratio," Journal of Management 2 (): Serali, H., M. Bazaraa, and C. Setty, Nonlinear Programming eory and Algoritms. New York: Wiley, 993. Stefek, Dan, Suceng Scott Liu and Rong Xu, Barra Optimizer.: Features and Functions Guide, MSCI Barra. All rigts reserved. 9 of 4

10 Maximizing te Sarpe Ratio APPENDIX eorem : Assume tat we want to solve a problem: Ax ( ) max Bx ( ) g ( x) 0, i =... N i () were Ax, ( ) Bx ( ), and Ax ( ) Bx ( ) are convex functions. en tere exists a ˆt for wic te optimal solution of problem () is te same as te optimal portfolio of problem (2): max A( x) tb ˆ ( x) g ( x) 0, i =... N i (2) Proof: Ax ( ) Ax ( ) x = ( A( x) -0.5 B( x) ) ; let Bx ( ) Bx ( ) Bx ( ) Ax ( ) b=0.5a, Bx ( ) x be optimal in (), π be te optimal dual vector corresponding to en te part of te KK system involving objective for () is. a A( x ) - b B ( x ) + π g ( x) =0 i i i x. a = Bx ( ), By takingπ 2 = π, ˆ b t =, we get te same part of te KK system for (2). a a Scalingπ by a positive number keeps complementary slackness and sign restrictions on duals satisfied. QED MSCI Barra. All rigts reserved. 0 of 4

11 Maximizing te Sarpe Ratio eorem 2: Let S be a convex set defined by a finite number of inequalities, and let ( ) be defined as follows: r V en te function ( ) = arg max S. SR( ) r ( ) = is unimodal, i.e., monotonically non-decreasing to ( ) V( ) te left of te optimal region, and monotonically non-increasing to te rigt of it. Proof: From eorem it follows tat tere exists = arg max SR( ). Let between and. en tere exists 0 γ for wic r( 2) = γr( ) + ( γ) r( ). Define = γ( ) + ( γ) ( ). Notice tat ( 2 ) C r r = C, C, 2 lie strictly S by convexity of S, and ence ( 2) V( 2) CVC, by te argmax property of ( 2 ). By concavity, V γ ( ) V( ) + ( γ) ( ) V( ). erefore: C C r( ) r γr( ) + ( γ) r( ) SR( 2 ) =, ( ) V( ) V ( ) V( ) ( ) ( ) V( ) 2 C 2 2 C C γ + γ Since = arg max SR( ), r( ) r( ) V V ( ), and ence ( ) ( ) ( ) γr( ) + ( γ) r( ) r( ) V + V V = SR( ) γ ( ) ( ) ( γ) ( ) ( ) ( ) ( ) and SR( ) SR( ) SR( ). QED MSCI Barra. All rigts reserved. of 4

12 Maximizing te Sarpe Ratio eorem 3: Suppose we need to solve te following problem: Maximize µ V s. t. A b (3) e = We can get te optimal solution of (3) by te following procedure. First, solve te following linear programming problem: Maximize µ s. t. A b (3) e =. If te optimum solution in (3) as a positive objective function value, ten solve convex quadratic programming problem (4): Minimize gvg st.. g, Ag ζ b 0, eg µ = ζ = 0, ζ 0 (4) Proof (equivalence of (3) and (4)): A: Let be optimal in (3), and µ =/ζ >0; V =a>0. en, its objective value is f = /( ζ a). Let g=ζ. en (g, ζ ) is feasible in (4), and its objective value is 2 2 f = ζ a = / f. 2 B: Let (g, ζ ) be optimal in (4), wit objective f 2 =a>0. en, = g / ζ is feasible in (3) and its objective value is f = / f2. QED MSCI Barra. All rigts reserved. 2 of 4

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