Practical Portfolio Optimization

Transcription

1 Practical Portfolio Optimization Victor DeMiguel Professor of Management Science and Operations London Business School Based on joint research with Lorenzo Garlappi Alberto Martin-Utrera Xiaoling Mei U of British Columbia U Carlos III de Madrid U Carlos III de Madrid Francisco J Nogales Yuliya Plyakha Raman Uppal U Carlos III de Madrid Goethe University Frankfurt EDHEC Business School Grigory Vilkov Goethe University Frankfurt International Conference on Continuous Optimization July 30, 2013

2 The motivation behind my dissertation was to apply mathematics to the stock market Harry Markowitz This is not a dissertation in economics, and we cannot give you a PhD in economics for this Milton Friedman

3 Mean-variance efficient frontier Efficient frontier: Investor concerned only about mean and variance of returns chooses portfolio on efficient frontier Expected Return Variance 3 / 70

4 Mean Variance portfolio optimization min w st w Σw }{{} variance w C }{{} constraints }{{} w µ /γ mean w portfolio weight vector Σ covariance matrix of asset returns µ mean asset returns γ risk aversion parameter 4 / 70

5 Implementation: estimation and rebalancing Portfolio for January 2009 Estimation window Next month Jan 98 Dec 08 Time 5 / 70

6 Portfolio for January 2009 Estimation window Next month Jan 98 Dec 08 Time Portfolio for February 2009 Estimation window Next month Feb 98 Jan 09 Time Will the portfolios for January 2009 and February 2009 be similar? 6 / 70

7 Problem: unstable portfolios The portfolios for consecutive months usually differ greatly: Unstable portfolios: Extreme weights that fluctuate a lot as we rebalance the portfolio Why? Weight on risky asset Time 7 / 70

8 Problem: unstable portfolios The portfolios for consecutive months usually differ greatly: Unstable portfolios: Extreme weights that fluctuate a lot as we rebalance the portfolio Why? Estimation error!!! Weight on risky asset Time 8 / 70

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10 The 1/N paper

11 The 1/N paper DeMiguel, Garlappi, Uppal (RFS, 2009) Objective: : Quantify impact of estimation error by comparing mean-variance and equal-weighted portfolio (1/N) Why use the 1/N portfolio as a benchmark? Estimation-error free, Simple but not simplistic: Does have some diversification, though not optimally diversified; With rebalancing = contrarian; Without rebalancing = momentum Ancient wisdom Rabbi Issac bar Aha (Talmud, 4th Century): Equal allocation A third in land, a third in merchandise, a third in cash Investors use it, even nowadays 11 / 70

12 Thomson Reuters equal weighted commodity index

13 What we do Empirically: Compare fourteen portfolio rules to 1/N across seven datasets Sharpe ratio SR = mean stddev Analytically: Derive critical estimation window length for mean-variance strategy to outperform 1/N Simulations: Extend analytical results to models designed to handle estimation error 13 / 70

14 What we find Empirically: None of the fourteen portfolio models consistently dominates 1/N across seven separate datasets (SR and turnover) Analytically: Based on US stock market data, critical estimation window for sample-based mean variance (MV) to outperform 1/N is Approximately 3,000 months for 25-asset portfolio Approximately 6,000 months for 50-asset portfolio Simulations: Even models designed to handle estimation error need unreasonably large estimation windows to outperform 1/N 14 / 70

15 Bayesian portfolios Portfolios tested 15 / 70

16 Sample mean Historical return data

17 Mean prior distribution Historical return data Sample mean

18 Mean prior distribution Bayes rule Mean posterior distribution Historical return data Sample mean

19 Bayesian portfolios Portfolios tested Diffuse prior [Barry, 1974], [Klein and Bawa, 1976], [Brown, 1979], Informative empirical prior [Jorion, 1986] Prior belief on asset pricing model [Pástor, 2000] and [Pástor and Stambaugh, 2000] 19 / 70

20 Bayesian portfolios Portfolios tested Portfolios with moment restrictions Minimum-variance portfolio often outperforms mean-variance portfolios; [Jagannathan and Ma, 2003], Value-weighted market portfolio optimal in a CAPM world, Missing-factor portfolio [MacKinlay and Pástor, 2000] impose Σ = νµµ + σ 2 I N 20 / 70

21 Bayesian portfolios Portfolios tested Portfolios with moment restrictions Portfolios subject to shortsale constraints [Jagannathan and Ma, 2003] show they perform well in practice 21 / 70

22 Bayesian portfolios Portfolios tested Portfolios with moment restrictions Portfolios subject to shortsale constraints Optimal combinations of portfolios [Kan and Zhou, 2007] w KZ = a w mean-variance + b w minimum-variance, where a and b minimize portfolio loss 22 / 70

23 Empirical results: Out-of-sample Sharpe ratios - I Sector Indust Inter l Mkt/ FF FF Strategy Portf Portf Portf SB/HL 1-fact 4-fact 1/N mean var (in sample) Classical approach that ignores estimation error mean variance Bayesian approach to estimation error Bayes Stein data and model Moment restrictions minimun variance market portfolio missing factor / 70

24 Why does minimum-variance portfolio outperform mean-variance portfolio? In sample, min-var portfolio has smallest expected return Expected Return in-sample frontier Variance 24 / 70

25 Why does minimum-variance portfolio outperform mean-variance portfolio? In sample, min-var portfolio smallest expected return Out of sample, minimum-variance portfolio has typically highest Sharpe ratio; Jorion (1985, 1986, 1991) Estimation error in mean larger than in variance; Merton (1980) Jagannathan and Ma (2003): estimation error in the sample mean is so large that nothing much is lost in ignoring the mean altogether Expected Return in-sample frontier out-of-sample frontier Variance 25 / 70

26 Mean variance portfolio returns are extreme 15 1 Mean Variance Equally Weighted Portfolio Returns Jul 84 Jul 88 Sep 92 Nov 96 Jan 01 Apr 05 Jun 09 Dates 26 / 70

27 Empirical results: Out-of-sample Sharpe ratios - II Sector Indust Inter l Mkt/ FF FF Strategy Portf Portf Portf SB/HL 1-fact 4-fact 1/N mean var (in sample) Shortsale constraints mean variance Bayes Stein minimum variance Optimal combinations of portfolios mean & min-var /N & min-var / 70

28 Why do shortselling constraints help? Intuitively, they prevent extreme (large) positive and negative weights in the portfolio Theoretically, imposing shortselling constraints is like reducing the covariances assets that we would short; Jagannathan and Ma (2003) Σ = ˆΣ λe eλ Weight on risky asset constrained Time 28 / 70

29 Results: Turnover relative to 1/N Sector Indust Inter l Mkt/ FF FF Strategy Portf Portf Portf SB/HL 1-fact 4-fact 1/N 305% 216% 293% 237% 162% 198% Mean variance and Bayesian portfolios mean variance Bayes Stein data and model Moment restrictions minimum variance market portfolio missing factor Shortsale constraints mean variance Bayes Stein minimum variance Optimal combinations of portfolios mean & min-var /N & min-var / 70

30 Empirical results: Summary In-sample, mean-var strategy has highest Sharpe ratio Out-of-sample 1/N does quite well in terms of Sharpe ratio and turnover Sample-based mean-var strategy has worst Sharpe ratio Bayesian policies typically do not out-perform 1/N Constrained policies do better than unconstrained, but constraints alone do not help (need extra moment restrictions) Min-var-constrained does well, but: Sharpe Ratios and CEQ statistically indistinguishable from 1/N Better than 1/N in only one dataset (20-size/bm portfolios) Turnover is 2-3 times higher than 1/N 30 / 70

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32 Beating 1/N

33 Beating 1/N min w st w ˆΣw }{{} variance w C }{{} constraints }{{} w ˆµ /γ mean Improve portfolio performance using I better covariance matrix, II better mean, III better constraints 33 / 70

34 I Better covariance matrix To estimate covariance matrix better: 1 Use higher-frequency data more accurate estimates of covariance matrix; [Merton, 1980] 34 / 70

35 I Better covariance matrix To estimate covariance matrix better: 1 Use higher-frequency data 2 Use factor models r }{{} returns = a }{{} intercept + B }{{} slopes f }{{} factors +ɛ more parsimonious estimates; [Chan et al, 1999] 35 / 70

36 I Better covariance matrix To estimate covariance matrix better: 1 Use higher-frequency data 2 Use factor models 3 Use shrinkage estimators [Ledoit and Wolf, 2004] combine sample covariance and identity Σ LW = α ˆΣ + (1 α)i 36 / 70

37 Shrinkage estimators Distribution of deterministic covariance matrix estimator Bias Distribution of shrunk covariance matrix estimator Distribution of sample covariance matrix estimator I Σ LW Σ Variance Σ Variance Σ 37 / 70

38 I Better covariance matrix To obtain better estimates of covariance matrix: 1 Use higher-frequency data 2 Use factor models 3 Use shrinkage estimators 4 Use robust optimization Goldfarb and Iyengar (MOR, 2003) and many others min w st max Σ U w C w Σw w ˆµ/γ 38 / 70

39 I Better covariance matrix To obtain better estimates of covariance matrix: 1 Use higher-frequency data 2 Use factor models 3 Use shrinkage estimators 4 Use robust optimization 5 Use robust estimation 39 / 70

40 Portfolio Selection with Robust Estimation DeMiguel and Nogales (Operations Research, 2009) Variance Deviation Square function amplifies the impact of outliers 40 / 70

41 Portfolio Selection with Robust Estimation Deviation Mad Absolute value function reduces impact of outliers 41 / 70

42 Portfolio Selection with Robust Estimation Deviation M-estimator M-estimators use smooth error function 42 / 70

43 Portfolio Selection with Robust Estimation Deviation S-estimator S-estimators: the impact of outliers is bounded 43 / 70

44 II Better mean To obtain better estimates of mean: 1 Ignore means 44 / 70

45 II Better mean To obtain better estimates of mean: 1 Ignore means 2 Use Bayesian estimates 45 / 70

46 II Better mean To obtain better estimates of mean: 1 Ignore means 2 Use Bayesian estimates 3 Use robust optimization min w st w Σw min µ U w C w µ/γ 46 / 70

47 II Better mean To obtain better estimates of mean: 1 Ignore means 2 Use Bayesian estimates 3 Use robust optimization 4 Use option-implied information 47 / 70

48 Improving Portfolio Selection Using Option-Implied Volatility and Skewness DeMiguel, Plyakha, Uppal, and Vilkov, forthcoming in JFQA Implied volatilities improve volatility by 10-20% Implied correlations do not improve performance Implied skewness and volatility risk premium proxy mean returns Improve Sharpe ratio even with moderate transactions costs for weekly and monthly rebalancing 48 / 70

49 II Better mean To obtain better estimates of mean: 1 Ignore means 2 Use Bayesian estimates 3 Use robust optimization 4 Use option-implied information 5 Use stock return serial dependence 49 / 70

50 Stock Return Serial Dependence and Out Of Sample Performance, DeMiguel, Nogales, Uppal (2013) Stock return serial dependence Contrarian: buy losers and sell winners Momentum: buy winners and sell losers Can this be exploited systematically with many assets? Yes, for transaction costs below 10 basis points 50 / 70

51 II Better mean To obtain better estimates of mean: 1 Ignore means 2 Use Bayesian estimates 3 Use robust optimization 4 Use option-implied information 5 Use stock return serial dependence 6 Exploit anomalies: size, momentum, value Expected Returns by Antti Ilmanen 51 / 70

52 III Better constraints To improve performance impose: 1 Shortsale constraints 52 / 70

53 III Better constraints To improve performance impose: 1 Shortsale constraints 2 Parametric portfolios [Brandt et al, 2009] impose constraint w 1 w M 1 me 1 btm 1 mom 1 w 2 w = M 2 me + 2 btm 2 mom 2 θme θ btm, θ mom w N w M me N N btm N mom N me = market equity, btm = book-to-market ratio, mom = momentum or average return over past 12 months 53 / 70

54 III Better constraints To improve performance impose: 1 Shortsale constraints 2 Parametric portfolios 3 Norm constraints 54 / 70

55 A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms DGNU (MS, 2009) min w w ˆΣw st w e = 1 w δ Nests 1/N, shrinkage covariance matrix of [Ledoit and Wolf, 2004], and shortsale constraints, Diversification: 2-norm, Leverage constraint: 1-norm 55 / 70

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57 Help wanted Optimization can make a difference in portfolio selection 57 / 70

58 Help wanted Optimization can make a difference in portfolio selection Research opportunities 58 / 70

59 Help wanted Optimization can make a difference in portfolio selection Research opportunities Integer programming: fixed costs and cardinality constraints, value at risk 59 / 70

60 Help wanted Optimization can make a difference in portfolio selection Research opportunities Integer programming: fixed costs and cardinality constraints, value at risk Approximate dynamic programming: dynamic portfolio choice, transaction costs 60 / 70

61 Help wanted Optimization can make a difference in portfolio selection Research opportunities Integer programming: fixed costs and cardinality constraints, value at risk Approximate dynamic programming: dynamic portfolio choice, transaction costs Calibration, calibration, calibration: 61 / 70

62 Same dog, different collar? Devil Is in One Detail: Calibration Constraints Gotoh & Takeda CMS 2011 Robust optimization DGNU MS 2009 Bayesian portfolios Jaganathan Ma JoF, 2003 DGNU MS 2009 Jorion JFQA 1986 Goldfarb Iyengar MOR 2003 Caramanis Mannor Xu (2011) Shrinkage estimators 62 / 70

63 Help wanted Optimization can make a difference in portfolio selection Research opportunities Integer programming: fixed costs and cardinality constraints, value at Risk Approximate dynamic programming: dynamic portfolio choice, transaction costs Calibration, calibration, calibration: Size Matters: Optimal Calibration of Shrinkage Estimators for Portfolio Selection, D, Martin-Utrera, Nogales, (JB&F, 2012): Choose criterion carefully, get nonparametric Parameter Uncertainty in Multiperiod Portfolio Optimization with Transaction Costs, D, Martin-Utrera, Nogales Shrink both target portfolio and trading rate 63 / 70

64 Help wanted Optimization can make a difference in portfolio selection Research opportunities Integer programming: fixed costs and cardinality constraints, value at Risk Approximate dynamic programming: dynamic portfolio choice, transaction costs Calibration, calibration, calibration: Size Matters: Optimal Calibration of Shrinkage Estimators for Portfolio Selection, D, Martin-Utrera, Nogales, (JB&F, 2012): Choose criterion carefully, get nonparametric Parameter Uncertainty in Multiperiod Portfolio Optimization with Transaction Costs, D, Martin-Utrera, Nogales Shrink both target portfolio and trading rate Statistics/big data/real-time estimation and optimization: high-frequency trading 64 / 70

65 High frequency trading 65 / 70

66 High frequency trading 66 / 70

67 Thank you Victor DeMiguel 67 / 70