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  1. 1. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosQuantile estimationand optimal portfoliosArthur Charpentier & Abder OulidiENSAI-ENSAE-CREST & IMA AngersJournée SFdS, Mai 20071
  2. 2. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosPortfolio management and optimal allocationsIdea: allocating capital among a set of assets to maximize return and minimizerisk.If diversification effects were intuited early, and Markowitz (1952) proposed amathematical model.• return is measured by the expected value of the portfolio return,• risk is quantified by the variance of this return.Agenda1. statistical issue in the mean-variance framework2. portfolio optimization with general risk measures2
  3. 3. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosPortfolio optimization (parametric framework)Consider a risk measure R (variance or Value-at-Risk). Solveω∗=argmin{R(ωtX)},u.c. ωt1 = 1 and E(ωtX) ≥ η,where X ∼ L(θ), θ unknown.θ is unknown but can be estimated using a sample {X1, . . . , Xn}.“The parameters governing the central tendency and dispersion of returns areusually not known, however; and are often estimated or guessed at usingobserved returns and other available data. In empirical applications, theestimated parameters are used as if they were the true value” (Coles& Loewenstein (1988)).If ω∗= ψ(θ) (e.g. mean-variance)ω∗= ψ(θ).3
  4. 4. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosOptimization of standard deviation (or variance)Allocationinthefirst assetAllocation in the second assetStandard deviation of the portfolio−200 −100 0 100 200−200−1000100200 Allocation in the first assetAllocationinthesecondassetFigure 1: Portfolio variance optimization problem.4
  5. 5. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosPortfolio optimization (parametric framework)In the case of no explicit expression of the optimum, solve (numerically)ω∗=argminR(ωtX),u.c.ω ∈ {(ωk)k∈{1,...,m}}where X ∼ L(θ).The idea is to generate samples Xi’s,• either from a parametric distribution L(θ),• or from a nonparametric distribution (bootstrap approach).5
  6. 6. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosOptimization of Value-at-RiskVaR of the portfolio−4 −2 0 2 4 6−2−1012345−4−20246VaR of the portfolio−4 −2 0 2 4 6012345−4−20246Figure 2: Optimization in the mean-VaR framework.6
  7. 7. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosClassical mean-variance allocation problemConsider d risky assets, with weekly returns X = (X1, . . . , Xd). Denoteµ = E(X) and Σ = var(X).Let ω = (ω1, . . . , ωd) ∈ Rddenote the weights in all risky assets.• the expected return of the portfolio is E(ωtX) = ωtµ,• the variance of the portfolio is var(ωtX) = ωtΣω.ω∗∈ argmin{ωtΣω}u.c. ωtµ ≥ η and ωt1 = 1convex⇐⇒ω∗∈ argmax{ωtµ}u.c. ωtΣω ≤ η and ωt1 = 17
  8. 8. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosClassical mean-variance allocation problemThe solution can be given explicitly (see Markowitz (1952)) asω∗= ψ(µ, Σ) = p + ηqwhere µ = E(X), Σ = var(X),p =1dbΣ−11 − aΣ−1µ and q =1dcΣ−1µ − aΣ−11 ,and a = 1tΣ−1µ, b = µtΣ−1µ, c = 1tΣ−11, d = bc − a2.Note that p is an allocation, and q indicates how the original portfolio shouldbe modified.8
  9. 9. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosEfficient frontierfirst asset−0.2 0.0 0.2 0.4 −0.4 0.0 0.4−0.20.00.2−0.20.00.20.4second assetthird asset−0.20.20.6−0.2 0.0 0.2−0.40.00.4−0.2 0.2 0.6fourth assetPortfolio with 4assets0.010 0.015 0.020 0.025 0.0300.0000.0010.0020.0030.0040.0050.006Efficient FrontierStandard deviationExpectedvalueFigure 3: Solving a variance optimization problem.9
  10. 10. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosInference issuesIn practice, µ = [µi] and Σ = [Σi,j] are unknown, and should be estimated.A natural idea is to defineµi =1nnt=1Xi,t et Σi,j =1n − 1nt=1(Xi,t − µi)(Xj,t − µj).Given n observed observed returns,µ|Σ ∼ N µ,Σnand nΣ|Σ ∼ W (n − 1, Σ) .where the two random variables µ and Σ are independent, given Σ.10
  11. 11. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios0.05 0.10 0.15 0.20 0.250.000.050.100.150.200.25Efficient Frontier, with 250 past observationsStandard deviationExpectedvalue0.05 0.10 0.15 0.20 0.250.000.050.100.150.200.25Efficient Frontier, with 1000 past observationsStandard deviationExpectedvalueFigure 4: Efficient frontiers and estimation.11
  12. 12. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosParametric bootstrapAssume that X ∼ L(θ). estimate θ by θn. The procedure is the following1. generate n returns X1, . . . , Xn from L(θn);2. estimate µ and Σ, i.e. µn and Σn,3. solve the minimization problem, i.e.ω∗=1dbΣ−11 − aΣ−1µ + η1dcΣ−1µ − aΣ−11 ,Using several simulations, the distribution of the ω∗k and vark(ω∗ktX) can beobtained.12
  13. 13. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosNonparametric bootstrapA nonparametric procedure can also be considered. Consider a n sample{X1, . . . , Xn}1. generate a bootstrap sample from {X1, . . . , Xn}2. estimate µ and Σ, i.e. µn and Σn,3. solve the minimization problem, i.e.ω∗=1dbΣ−11 − aΣ−1µ + η1dcΣ−1µ − aΣ−11 ,Using several simulations, the distribution of the ω∗k and vark(ω∗ktX) can beobtained.13
  14. 14. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios0.4 0.6 0.8 1.0012345Allocation in the first assetAllocation weightDensity0.0 0.2 0.4012345Allocation in the second assetAllocation weightDensity−0.3 −0.1 0.0 0.102468Allocation in the third assetAllocation weightDensity0.05 0.15 0.250246810Allocation in the fourth assetAllocation weightDensityFigure 5: Distributions of optimal allocations ω∗k’s.14
  15. 15. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinsecondassetJoint distribution of optimal allocations (1−2)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinthirdassetJoint distribution of optimal allocations (1−3)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinfourthassetJoint distribution of optimal allocations (1−4)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the second assetAllocationinfourthassetJoint distribution of optimal allocations (2−4)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the second assetAllocationinthirdassetJoint distribution of optimal allocations (2−3)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the third assetAllocationinfourthassetJoint distribution of optimal allocations (3−4)Figure 6: Joint distributions of optimal allocations ω∗k’s.15
  16. 16. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios0.05 0.06 0.07 0.08 0.09 0.10 0.11020406080100Density of estimated optimal standard deviationOptimal standard deviationDensityFigure 7: Distribution of vark(ω∗ktX)16
  17. 17. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosValue-at-Risk minimizationWith V aR(X, p) = F−1(p) = sup{x, F(x) < p}, the program isω∗∈ argmin{VaR(ωtX, α)}u.c. E(ωtX) ≥ η,ωt1 = 1nonconvexω∗∈ argmax{E(ωtX)}u.c. {VaR(ωtX, α)} ≤ η ,ωt1 = 1In the previous framework (mean-variance), it could be done easily since• there are only a few estimates of the variance• there exists an analytical expression of the optimal allocation,In the case of Value-at- Risk minimization,• there are several estimators of quantiles (see Charpentier & Oulidi(2007)),• numerical optimization should be considered.17
  18. 18. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosQuantile estimation• raw estimator of the quantile, X[pn]:n = Xi:n = F−1n (i/n) such thati ≤ pn < i + 1.• weighted average of F−1n (p), e.g. αXi:n + (1 − α)Xi+1:n,• weighted average of F−1n (p), e.g.ni=1αiXi:n =10αuF−1n (u)du,• smoothed version of the cdf, F−1K (p) where FK(x) =1nhni=1Kx − Xih• semiparametric approach, based on Hill’s estimator, Xn−k:nnk(1 − p)−ξk,where ξk =1kki=1log Xn+1−i:n − log Xn−k:n (if ξ > 0),• fully parametric approach, Xn + u1−pvar(X) (if X ∼ N(µ, σ2))18
  19. 19. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 Empirical quantile estimationValueProbability0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0Empirical quantile estimationValueProbabilityFigure 8: Classical estimation of the quantile, based on F−1(·).19
  20. 20. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios0.0 0.5 1.0 1.5 2.00.00.51.01.5Smoothed empirical quantile estimationValueSmootheddensity0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0Smoothed empirical quantile estimationValueProbabilityFigure 9: Smoothed estimation of the quantile, based on F−1K (·).20
  21. 21. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosA short extention to general risk measuresIn a much more general setting, spectral risk measures can be considered, i.e.R(X) =10φ(p)F−1X (p)dp,for some distortion function φ : [0, 1] → [0, 1].21
  22. 22. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosParametric bootstrapAssume that X ∼ L(θ). The procedure is the following1. generate n returns X1, . . . , Xn from L(θ);2. estimate for all ω on a finite grid, estimate VaR(ωtX),3. solve the minimization problem on the grid to get numerically ω∗n.Using several simulations, the distribution of ω∗n and var(ω∗nX) can beobtained.22
  23. 23. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinsecondassetJoint distribution of optimal allocations (1−2)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinthirdassetJoint distribution of optimal allocations (1−3)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinfourthassetJoint distribution of optimal allocations (1−4)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the second assetAllocationinfourthassetJoint distribution of optimal allocations (2−4)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the third assetAllocationinfourthassetJoint distribution of optimal allocations (3−4)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the second assetAllocationinthirdassetJoint distribution of optimal allocations (2−3)Figure 10: Joint distributions of optimal allocations ω∗k’s, smoothed quantileestimator.23
  24. 24. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios0.16 0.18 0.20 0.22 0.2405101520Density of estimated optimal 99% quantileOptimal Value−at−RiskDensityFigure 11: Distribution of VaRk(ω∗ktX, 95%).24
  25. 25. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinsecondassetJoint distribution of optimal allocations (1−2)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinthirdassetJoint distribution of optimal allocations (1−3)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the first assetAllocationinfourthassetJoint distribution of optimal allocations (1−4)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the second assetAllocationinfourthassetJoint distribution of optimal allocations (2−4)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the third assetAllocationinfourthassetJoint distribution of optimal allocations (3−4)−0.2 0.0 0.2 0.4 0.6 0.8 1.0−0.20.00.20.40.60.81.0Allocation in the second assetAllocationinthirdassetJoint distribution of optimal allocations (2−3)Figure 12: Joint distributions of optimal allocations ω∗k’s, raw quantile estima-tor.25
  26. 26. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios0.16 0.18 0.20 0.22 0.24010203040Density of estimated optimal 95% quantileOptimal Value−at−RiskDensityFigure 13: Distribution of VaRk(ω∗ktX, 95%).26
  27. 27. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosConclusionDealing with only 4 assets, it is difficult to get robust optimal allocation, onlybecause of statistical uncertainty of classical estimators. Remark: this wasmentioned in Liu (2003) on high frequency data (every 5 minutes, i.e.n = 10, 000) with 100 assets.27
  28. 28. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfoliosSome referencesColes, J.L. & Loewenstein, U. (1988). Equilibrium pricing and portfolio composition in the presence of uncertainparameters. Journal of Financial Economics, 22, 279-303.Dowd, K. & Blake, D.. (2006). After VaR: the theory, estimation, and insurance applications of quantile-based riskmeasures. Journal of Risk & Insurance, 73, 193-229.Duarte, A. (1999). Fast computation of efficient portfolios. Journal of Risk, 1, 71-94.Duffie, D. & Pan, J. (1997). An overview of Value at Risk. Journal of Derivatives, 4, 7-49.Gaivoronski, A.A. & Pflug, G. (2000). Value-at-Risk in portfolio optimization: properties and computationalapproach. Working Paper 00-2, Norwegian University of Sciences & Technology.Jorion, P. (1997). Value at Risk: the new benchmark for controlling market risk. McGraw-Hill.Kast, R., Luciano, E. & Peccati, L. (1998). VaR and optimization: 2nd international workshop on preferences anddecisions. Trento, July 1998.Klein, R.W. & Bawa, V.S. (1976). The effect of estimation risk on optimal portfolio choice. Journal of FinancialEconomics, 3, 215-231.Larsen, N., Mausser, H. & Uryasev, S. (2002). Algorithms for optimization of Value at Risk. in Financialengineering, e-commerce and supply-chain, Pardalos and Tsitsiringos eds., Kluwer Academic Publichers, 129-157.Litterman, R. (1997). Hot spots and edges II. Risk, 10, 38-42.Rockafellar, R.T. & Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk, 2, 21-41.28

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