Portfolio Selection with Estimation Risk: a Test-based Approach - - PowerPoint PPT Presentation

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?
• r 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

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• Main issues:

→ Estimation risk overlooked

ie the fact that true (unknown) characteristics = estimated ones (finite sample size...)

→ "Optimality" of this 2-step approach?

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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

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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

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OUTLINE

I - P-value investment rule II - Related literature III - Theoretical comparisons IV - Practical comparisons V - Conclusion

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I - P-VALUE INVESTMENT RULE

⋆ Perform a one-sided test ensuring the portfolio performance is above a given threshold: H0 : Q > c ⋆ Maximize the associated p-value to get optimal weights: θp = arg max

θ

• P( ˆ

Q(θ) > c)

• Why a test?
• Statistical tool to compare random quantities
• Automatically incorporates estimation risk
• Well-defined objective function for portfolio manager

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• 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
• Mean-variance performance: QMV = θ′˜

µ − η

2θ′Σθ

→ How to choose θ according to QMV ?

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• Classical Mean-Variance problem (Markowitz) - 2-step plug-in method

max

θ [QMV ] ⇒ θMV =

• Σ−1˜

µ

• /η ⇒ ˆ

θMV = [ˆ Σ−1ˆ ˜ µ]/η ⋆ 2-fund rule:

• [ˆ

Σ−1ˆ ˜ 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:

max

θ [P(QMV > c)] ⇒ θp(c) =

• ˆ

Σ−1ˆ ˜ µ

• /˜

ηp

with ˜

ηp = η

• QMV (ˆ

θMV ) c ⋆ 2-fund rule with corrected risk-aversion parameter

Interpretation: Favorable financial environment (i.e. sample with high performance) then (likely) c < QMV (ˆ

θMV ) and ˜ ηp > η

i.e. decrease the share invested in risky assets

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• 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 θp(c) for a benchmark c

c∗ = 1 2η ×

• E
• ˆ

˜ µ′ ˆ Σ−1 ˜ µ0

√

ˆ γ2

2

• E

ˆ

˜ µ′ ˆ Σ−1Σ0 ˆ Σ−1 ˆ ˜ µ ˆ γ2

2

where

ˆ γ2 ≡ ˆ ˜ µ′ ˆ Σ−1ˆ ˜ µ

→ c∗ is unfeasible → Without estimation risk: θp(c∗) = θMV

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II - RELATED LITERATURE

• Dealing with Estimation risk:

◮ Frequentist ◮ Axiomatic ◮ Bayesian

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◮ 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

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◮ 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

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• 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 KZ0
• HRW: Feasible counterpart of HRW 0
• KZ3: Feasible counterpart of KZ30
• GUW: Garlappi, Uppal and Wang

◮ Infeasible rules (only for simulated data)

• MV 0: (infeasible) Mean-variance
• KZ0: Kan and Zhou (infeasible) 2-fund
• HRW 0: ter Horst, de Roon and Werker (infeasible) 2-fund
• KZ30: Kan and Zhou (infeasible) 3-fund

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III - THEORETICAL COMPARISONS

• Theoretical rankings derived by Kan and Zhou (2007):

MV 0 >> KZ30 >> KZ0

• B >> MV

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

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• Reinterpretation of investment rules

⋆ Feasible mean-variance rule: ˆ θMV = [ˆ Σ−1ˆ ˜ µ]/η

. [ˆ

Σ−1ˆ ˜ 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: ˜ ηKZ > ˜ ηHRW > η and ˜ ηB > η This paper: ˜ ηp more flexible depending on realized sample:

• if c > QMV then ˜

ηp < η

• if c < QMV then ˜

ηp > 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

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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):

• SRk =

ˆ ˜ µk ˆ σk

⋆ Out-of-sample certainty-equivalent return (CE):

• CEk = ˆ

˜ µk − η 2 ˆ σ2

k

⋆ Portfolio turnover:

Turnoverk =

1 T − Tw − 1

T −1

• t=Tw+1

(|θk,t+1 − θk,t|)′ ι

where ι is the column vector of ones of size N.

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• 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.

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⋆ Results for 3 P-value rules with targets c1 < c2 < c3

• P1 and P2 defeat easily their targets;
• Associated corrected risk-aversion parameters almost always larger than η.

→ P1 and P2 investors reinterpreted as more conservative MV-investors

(b/c corrected risk-aversion parameter larger than the actual risk-aversion parameter)

• P3 defeats c3 most of the time, but not as easily:

→ despite a highly aggressive investment strategy → corrected risk-aversion parameter smaller than η with out-of-sample probability 1/3.

• Compare investors P2 and P3:

→ CE performances really close to each other → Riskier strategy for P3 (larger share in the risky assets) → Setting a target too high can be detrimental

• Strategy P1 with lowest target = most stable and affordable

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Out-of-sample CE for 3 P-value rules with 5% deviation D3 (as a function of the size of the rolling window) Target c1 < Target c2 < Target c3

100 150 200 250 300 350 400 450 1 2 3 4 5 6 7 8 9 T CE Optimum θ(c1) Target 1 θ(c2) Target 2 θ(c3) Target 3

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⋆ Results for Infeasible rules

• Feasibility cost decreases with the sample size

→ All feasible rules perform similarly when the size of the rolling window is larger than 360

(30 years of monthly data).

• In addition: comparing rules HRW and KZ confirms previous findings that the variance is

easier to accurately estimate than the mean (fixed time span)

→ even for small sample sizes (still with N/T small), the cost of ignoring the estimation

risk of the variance is low.

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⋆ Results for Feasible rules

• Observed ranking (in terms of SR or CE) irrespective of the deviation:

P2, KZ3 >> B, KZ, HRW >> MV >> GUW >> EQ

→ quite consistent with the partial theoretical ranking → except for KZ: KZ0 expected to outperform B, HRW 0 and any P-value rule

(Theoretical ranking: MV 0 >> KZ30 >> KZ0 >> HRW 0 >> P )

Group 1 (D1, D4) Group 2 (no dev., D3) Small rolling window sizes CE

P2 >> KZ3 >> KZ KZ3 >> P2 >> KZ

SR

P2 >> KZ3, KZ KZ3, P2 >> KZ

Large rolling window sizes CE

KZ3, KZ >> P2 KZ3 >> KZ >> P2

SR

P2 >> KZ3, KZ KZ3, P2 >> KZ

• Very good performance of P2, especially with smaller rolling window sizes

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Out-of-sample SR for feasible rules with 5% deviation D1 (as a function of the size of the rolling window)

100 150 200 250 300 350 400 450 0.35 0.4 0.45 0.5 0.55 T SR Optimum P−value(c2) MV Eq B KZ2 HRW KZ3 GUW

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• Empirical study:

⋆ Empirical dataset:

Monthly excess returns on 10 Industry portfolios in the United States from January 1981 to July 2008.

⋆ Results:

P1 is the only one who beats his target; most stable and reliable performance; cheapest

→ despite aggressive investment strategy for P2 and P3

ie larger share in risky assets than MV with out-of-sample probability 0.87

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Out-of-sample CE for above 3 three P-value rules with dataset Sector (as a function of the size of the rolling window) Target c1 < Target c2 < Target c3

40 60 80 100 120 140 160 180 −10 −5 5 T CE Optimum θ(c1) Target 1 θ(c2) Target 2 θ(c3) Target 3

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⋆ Industry portfolios

SR ranking: KZ3, P1, P2, P3 >> B, MV >> HRW, KZ >> GUW >> EQ CE ranking:

P1 >> KZ3, GUW >> KZ >> P2 >> B >> HRW >> MV >> EQ >> P 3

→ Very good performance of P1; also very cheap.

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Out-of-sample SR for feasible rules with dataset Industry (as a function of the size of the rolling window)

40 60 80 100 120 140 160 180 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 T SR Optimum P−value(c2) MV B KZ2 HRW KZ3 GUW

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CONCLUSION

⋆ New portfolio selection method to account for estimation risk: → No maximization of a mean-variance performance or a related loss function → More conservative definition of optimality: guaranteeing some minimal performance ⋆ Performance of p-value investment rule depends on the benchmark: → Simulation study considered a wide range of benchmarks → Performs relatively well especially for small sample T → Stable rule ⋆ Simplified framework: → Closed-form investment rule → Interpretation as a mean-variance behavior with a corrected, sample-dependent

risk-aversion parameter

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