Risk-BudgetingAllocation versus …

1 portfolio optimizationRisk-budgeting techniquesSome illustrationsPortfolio OptimizationversusRisk-Budgeting Allocation1 Thierry Roncalli??Lyxor Asset Management, France & vry University, FranceWG RISK ESSEC, January 18, 20121 The opinions expressed in this presentation are those of the author and are notmeant to represent the opinions or official positions of Lyxor Asset RoncalliPortfolio optimization vs Risk-Budgeting Allocation1 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsOutline1 portfolio optimizationSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio Theory2 Risk-budgeting techniquesRisk-budgeting principlesThe ERC portfolio3 Some

2 IllustrationsRisk-based indexationRisk parity fundsBond portfolios managementThierry RoncalliPortfolio optimization vs Risk-Budgeting Allocation2 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheorySome modelsMost Popular Models in Asset AllocationMean-variance portfolio selection (Markowitz, 1952)(minimum-variance strategy, tangency portfolio , strategic assetallocation, market-cap indexation, etc.)

3 Dynamic optimization (Merton, 1971)(constant-mix strategy, liability-driven investment, lifecycle funds,target date funds, etc.)Tactical asset allocation (Black-Litterman, 1992)(equilibrium portfolios, flexible views, market timing, etc.) These 3 models are based on optimization RoncalliPortfolio optimization vs Risk-Budgeting Allocation3 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheoryThe Markowitz frameworkIn the portfolio theory of Markowitz, we maximize the expected return fora given level of volatility:max (w) = > (w) = w> w= ?

4 Optimized portfolios with respect tovolatility and expected returnThe optimal portfolio is the tangencyportfolioConfusion between volatility anddiversification conceptsThe solution is not robust; it is highlysensitive to expected return inputsHigh turnover of the portfolioThierry RoncalliPortfolio optimization vs Risk-Budgeting Allocation4 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheoryStabilityWe consider the minimum-variance portfolio (because it does not dependon expected returns).

5 2 assets with 1(t) = 2(t) =20%and 1,2(t) =100%:w?1(t) =w?2(t) =50%Int+1, if the volatility of the first 1(t+1) =19,9%, we obtain:w?1(t+1) =100%andw?2(t+1) =0%Thierry RoncalliPortfolio optimization vs Risk-Budgeting Allocation5 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheoryAn example3 assets with 1(t) =20%, 2(t) =22%and 3(t) =23%and auniform correlation . We assume that the true correlation is 90%.

6 Table: Optimal correlationWith short selling90%85%95%85%95%AssetEW90%No short sellingMichaud (1989),FAJ:The Markowitz optimization Enigma: Is Optimized Optimal?Thierry RoncalliPortfolio optimization vs Risk-Budgeting Allocation6 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheoryOn the importance of the information matrixLet and be the vector of expected returns and the covariance solutions are of the following form:w?

7 1 In the case of the minimum-variance portfolio , the form is:w? 11 The important quantity isI= 1, which is called the RoncalliPortfolio optimization vs Risk-Budgeting Allocation7 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheoryWhich factors are important?Eigendecomposition of the information matrixThe eigendecomposition ofI= 1is the same as the one of , butwith reverse order of eigenvectors and inverse eigenvalues:Vi(I)=Vn i( ) i(I)=1 n i( )Table.

8 Example with the previous covariance matrix (with correlation 90%)Asset / Factor matrixInformation matrixThierry RoncalliPortfolio optimization vs Risk-Budgeting Allocation8 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheorySolutionsBecause the optimal solution depends principally on the last factors of thecovariance matrix, we have to introduce some regularization techniques:regularization of the objective function by using resamplingtechniquesregularization of the covariance matrix:Factor analysisShrinkage methodsRandom matrix of the program specification by introducing someconstraintsThierry RoncalliPortfolio optimization vs Risk-Budgeting Allocation9 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheoryMain resultWe consider a universe ofnassets.

9 We denote by the vector of theirexpected returns and by the corresponding covariance matrix. Wespecify the optimization problem as follows:min12w> 1>w=1 >w ?w Rn Cwherewis the vector of weights in the portfolio andCis the set ofweights constraints. We define:the unconstrained portfoliow?orw?( , ):C=Rnthe constrained portfolio w:C(w ,w+)={w Rn:w i wi w+i}Thierry RoncalliPortfolio optimization vs Risk-Budgeting Allocation10 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheoryTheoremJagannathan and Ma (2003) show that the constrained portfolio is thesolution of the unconstrained problem: w=w?

10 ( , )with:{ = = + ( + )1>+1( + )>where and +are the Lagrange coefficients vectors associated to thelower and upper bounds. Introducing weights constraints is equivalent to introduce somerelative views (similar to theBlack-Littermanapproach).Thierry RoncalliPortfolio optimization vs Risk-Budgeting Allocation11 / 54 portfolio optimizationRisk-budgeting techniquesSome illustrationsSome ModelsRobustness of the Markowitz frameworkWeights constraints and portfolio TheoryProof for the global minimum-variance portfolioWe define the Lagrange function asf(w; 0) =12w> w 0(1>w 1)with 0 0.}