The 11th Annual Financial Econometrics Conference "Econometric Modeling in Risk Management" Portfolio Selection with Estimation Risk: a Test-based Approach Bertille Antoine Simon Fraser University, Vancouver March 27th, 2009
INTRODUCTION • How to choose the optimal portfolio? or the best allocation between available risky assets and the riskless asset? → depends on the performance measure Q examples: mean-variance, Sharpe-ratio... • Q depends on characteristics of returns distributions → true characteristics are unknown • 2-step procedure: (1) maximize Q � you get an infeasible optimal portfolio (2) plug-in � use estimates to make it feasible –2–
• Main issues: → Estimation risk overlooked ie the fact that true (unknown) characteristics � = estimated ones (finite sample size...) → "Optimality" of this 2-step approach? –3–
THIS PAPER : ⋆ Integrated portfolio selection method that accounts for estimation risk → 1-step procedure → Feasible investment rule (no plug-in required) ⋆ Allocations compared through their probability of beating a benchmark ⋆ For reasonable benchmarks, the definition of optimality is more conservative ⋆ Of interest for several industries, e.g. pension funds, mutual funds → Very general procedure –4–
Main results: - when performance measure Q is mean-variance - when estimation risk of variance is ignored ⋆ Theoretical results : - Closed-form optimal allocation - No nuisance parameter - No suboptimal plug-in step - Reinterpretation of our investor as a mean-variance investor → with a corrected, sample-dependent risk-aversion parameter ⋆ Practical results : (for reasonable benchmarks): - Simulation & Empirical study: very good performance → especially small sample sizes → stable investment rule –5–
OUTLINE I - P-value investment rule II - Related literature III - Theoretical comparisons IV - Practical comparisons V - Conclusion –6–
I - P-VALUE INVESTMENT RULE ⋆ Perform a one-sided test ensuring the portfolio performance is above a given threshold: H 0 : Q > c ⋆ Maximize the associated p-value to get optimal weights: � � P ( ˆ θ p = arg max Q ( θ ) > c ) θ • Why a test? - Statistical tool to compare random quantities - Automatically incorporates estimation risk - Well-defined objective function for portfolio manager –7–
• Very general procedure - Any performance measure Q - Any (consistent) estimate ˆ Q • Simplifications and Assumptions - Portfolio performance evaluated by mean-variance criterion - Estimation risk of the variance neglected (number of assets over sample size kept small) - Asymptotic test (for tractability) • Formally: - N risky assets iid (mean µ and variance Σ ) and the riskfree asset - θ vector of weights invested in the risky assets µ − η - Mean-variance performance: Q MV = θ ′ ˜ 2 θ ′ Σ θ → How to choose θ according to Q MV ? –8–
• Classical Mean-Variance problem (Markowitz) - 2-step plug-in method � � /η ⇒ ˆ θ MV = [ˆ Σ − 1 ˆ Σ − 1 ˜ max θ [ Q MV ] ⇒ θ MV = µ µ ] /η ˜ ⋆ 2-fund rule: - [ˆ Σ − 1 ˆ µ ] = sample tangency portfolio allocates wealth among risky assets ˜ - η = risk-aversion parameter weights the share of wealth assigned to risky assets • P-value rule for a benchmark c : � � � Q MV (ˆ θ MV ) ˆ Σ − 1 ˆ max θ [ P ( Q MV > c )] ⇒ θ p ( c ) = µ ˜ / ˜ η p with ˜ η p = η c ⋆ 2-fund rule with corrected risk-aversion parameter Interpretation: Favorable financial environment (i.e. sample with high performance) then (likely) c < Q MV (ˆ θ MV ) and ˜ η p > η i.e. decrease the share invested in risky assets –9–
• How to choose the benchmark? - Not really a choice variable → determined heterogeneously → historic performance - Still we can define the (mathematical) optimal benchmark Definition: maximize the expected portfolio performance associated with the optimal p-value rule θ p ( c ) for a benchmark c � � �� 2 µ ′ ˆ Σ − 1 ˜ ˆ ˜ µ 0 √ E c ∗ = 1 γ 2 γ 2 ≡ ˆ µ ′ ˆ ˆ Σ − 1 ˆ 2 η × ˆ ˜ ˜ � � ˆ �� 2 µ where µ ′ ˆ Σ − 1 ˆ Σ − 1 Σ 0 ˆ ˜ µ ˜ E γ 2 ˆ → c ∗ is unfeasible → Without estimation risk: θ p ( c ∗ ) = θ MV –10–
II - RELATED LITERATURE • Dealing with Estimation risk: ◮ Frequentist ◮ Axiomatic ◮ Bayesian –11–
◮ Frequentist: Minimize the loss function of replacing the true (unknown) mean of the portfolio by its sample estimate → very complicated problem → restriction to the class of 2-fund rules - ter Horst, de Roon and Werker (2006): ignore estimation risk of variance - Kan and Zhou (2007): incorporate both estimation risks also they consider the class of 3-fund rules (add sample global mean-variance portfolio) → All rules are unfeasible → Additional (non-optimal) plug-in step required –12–
◮ Axiomatic (Garlappi, Uppal and Wang (2007)): - Sequential min-max approach i.e. max. the worst performance - Standard MV framework modified by adding a preliminary minimization step: . Restrict expected returns to fall into CI around estimated value . Minimize over the possible expected returns subject to this constraint - Solid axiomatic foundation: → model allows multi priors and the investor is averse to ambiguity → but sequentiality cannot be directly liked to an optimality criterion ◮ Bayesian (Bawa, Brown and Klein (1979)): - Maximize the expected portfolio performance: expectation computed according to the predictive distribution (= combination of historical observations and the prior) - Unknown parameters = random variables → Uninformative priors –13–
• Investment rules: ◮ Feasible rules (both for simulated and empirical data) - P1: P-value with target 10% Optimum or MV-CE - P2: P-value with target 50% Optimum or MV-CE - P3: P-value with target 90% Optimum or MV-CE - EQ: Equi-weighted portfolio - MV: Feasible counterpart of MV 0 - B: Bayesian with diffuse priors - KZ: Feasible counterpart of KZ 0 - HRW: Feasible counterpart of HRW 0 - KZ3: Feasible counterpart of KZ 3 0 - GUW: Garlappi, Uppal and Wang ◮ Infeasible rules (only for simulated data) - MV 0 : (infeasible) Mean-variance - KZ 0 : Kan and Zhou (infeasible) 2-fund - HRW 0 : ter Horst, de Roon and Werker (infeasible) 2-fund - KZ 3 0 : Kan and Zhou (infeasible) 3-fund –14–
III - THEORETICAL COMPARISONS • Theoretical rankings derived by Kan and Zhou (2007): � � B >> MV � � MV 0 >> KZ 3 0 >> KZ 0 � HRW 0 � � � GUW >> means "outperforms in terms of expected mean-variance performance" → In practice: unfeasible rules need to be estimated → Theoretical ranking not guaranteed anymore –15–
• Reinterpretation of investment rules ⋆ Feasible mean-variance rule: ˆ θ MV = [ˆ Σ − 1 ˆ µ ] /η ˜ . [ˆ Σ − 1 ˆ µ ] sample tangency portfolio allocates wealth among risky assets ˜ . η = risk-aversion parameter weights the share of wealth assigned to risky assets ⋆ Any 2-fund rule = mean-variance rule with corrected risk-aversion � Literature: ˜ η KZ > ˜ η HRW > η and ˜ η B > η � This paper : ˜ η p more flexible depending on realized sample: - if c > Q MV then ˜ η p < η - if c < Q MV then ˜ η p > η Interpretation: moderate benchmark & profitable financial environment (ie a sample associated to a relatively high performance) → smaller part invested in the risky assets → avoid extreme positions –16–
IV - COMPARATIVE STUDY • Design: - Rolling window approach: window of size 120 (ie 10 years of monthly data) → Series of (monthly) out-of-sample returns for each strategy k - Compare strategies according to: ⋆ Out-of-sample Sharpe-ratio (SR): ˆ ˜ µ k � SR k = ˆ σ k ⋆ Out-of-sample certainty-equivalent return (CE): µ k − η � CE k = ˆ σ 2 ˜ 2 ˆ k ⋆ Portfolio turnover: T − 1 � 1 ( | θ k,t +1 − θ k,t | ) ′ ι Turnover k = T − T w − 1 t = T w +1 where ι is the column vector of ones of size N . –17–
• Monte-Carlo study: ⋆ DGP: - Asset returns generated according to a distribution F that deviates slightly from the normal distribution F = (1 − h ) N + hD - Generate the joint normal distribution N through factor model with 4 risky assets including 1 factor, and 1 riskless asset - 3 different proportions h : no deviation; 2.5%; 5% - 3 different deviation distributions D (i) deterministic: expected return of the asset plus 5 standard deviations; (ii) binomial: expected return of the asset plus 5 standard deviations with prob. 0.5 and to the expected return of the asset minus 5 standard deviations with prob. 0.5; (iii) normal distribution with mean equal to the expected return of the asset plus 5 standard deviations and same covariance matrix as N . –18–
Recommend
More recommend
