Data-Driven Portfolio Optimisation

Transcription

1 Data-Driven Portfolio Optimisation Victor DeMiguel 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 London Optimisation Workshop King s College London, June 2014

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 / 61

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 / 61

5 Implementation: estimation and rebalancing Portfolio for January 2013 Estimation window Next month Jan 03 Dec 12 Time 5 / 61

6 Portfolio for January 2013 Estimation window Next month Jan 03 Dec 12 Time Portfolio for February 2013 Estimation window Next month Feb 03 Jan 13 Time Will the portfolios for January 2013 and February 2013 be similar? 6 / 61

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 / 61

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 / 61

9 See also [Chopra and Ziemba, 1993] and [Broadie, 1993]

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 / 61

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 / 61

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 / 61

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16 Beating 1/N

17 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 17 / 61

18 I Better covariance matrix To estimate covariance matrix better: 1 Use higher-frequency data more accurate estimates of covariance matrix; [Merton, 1980] 18 / 61

19 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] 19 / 61

20 I Better covariance matrix To estimate covariance matrix better: 1 Use higher-frequency data 2 Use factor models 3 Use shrinkage estimators Honey, I have shrunk the sample covariance matrix [Ledoit and Wolf, 2004b, Ledoit and Wolf, 2004a] Σ LW = α ˆΣ + (1 α)i 20 / 61

21 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 21 / 61

22 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, 2003], [Tütüncü and Koenig, 2003], and others min w st max Σ U w C w Σw w ˆµ/γ Worst-case CVaR [Zhu and Fukushima, 2009], [Gotoh et al, 2013] 22 / 61

23 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 23 / 61

24 Portfolio Selection with Robust Estimation [DeMiguel and Nogales, 2009] Variance Deviation Square function amplifies the impact of outliers 24 / 61

25 Portfolio Selection with Robust Estimation Mad Deviation Absolute value function reduces impact of outliers: [Konno and Yamazaki, 1991] 25 / 61

26 Portfolio Selection with Robust Estimation Deviation M-estimator M-estimators use smooth error function 26 / 61

27 Portfolio Selection with Robust Estimation Deviation S-estimator S-estimators: the impact of outliers is bounded 27 / 61

28 II Better mean To obtain better estimates of mean: 1 Ignore means Minimum-variance portfolio usually performs better out of sample than mean-variance portfolios; [Jorion, 1985], [Jorion, 1986], [Jagannathan and Ma, 2003], [DeMiguel et al, 2009b] To obtain more accurate estimators of mean, use longer time series 28 / 61

29 II Better mean To obtain better estimates of mean: 1 Ignore means 2 Use Bayesian estimates 29 / 61

30 Sample mean Historical return data

31 Mean prior distribution Historical return data Sample mean

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

33 II Better mean To obtain better estimates of mean: 1 Ignore means 2 Use Bayesian estimates 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] 33 / 61

34 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 34 / 61

35 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 35 / 61

36 Improving Portfolio Selection Using Option-Implied Volatility and Skewness DeMiguel, Plyakha, Uppal, and Vilkov, JFQA (2013) 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 36 / 61

37 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 37 / 61

38 Stock Return Serial Dependence and Out Of Sample Performance DeMiguel, Nogales, Uppal, RFS (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 38 / 61

39 II Better mean To obtain better estimates of mean: 1 Ignore historical means 2 Use Bayesian estimates 3 Use robust optimization 4 Use option-implied information 5 Use stock return serial dependence 6 Exploit anomalies 39 / 61

40 Exploit anomalies size: small firms outperform large firms, momentum: past winners outperform past losers, book-to-market: high book-to-market ratio firms outperform low book-to-market ratio firms 40 / 61

41 III Better constraints To improve performance impose: 1 Shortsale constraints 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) Weight on risky asset constrained Σ = ˆΣ λe eλ Time 41 / 61

42 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 42 / 61

43 III Better constraints To improve performance impose: 1 Shortsale constraints 2 Parametric portfolios 3 Performance-based regularization Constrain variance of estimators of portfolio mean and CVaR Performance-based regularization in mean-cvar portfolio optimization, [Karoui et al, 2011] 43 / 61

44 III Better constraints To improve performance impose: 1 Shortsale constraints 2 Parametric portfolios 3 Performance-based regularization 4 Norm constraints 44 / 61

45 A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms DGNU (MS, 2009) min w st w ˆΣw w δ Diversification: 2-norm, Leverage constraint: 1-norm, Shortsale constraint: tight 1-norm constraint, Sparsity constraint: quasi-norm (l p with p < 1); [Chen et al, 2013, Fengmin et al, 2014b], Cardinality constraint: 0-norm; [Brito and Vicente, 2012, Ruiz-Torrubiano and Suárez, 2009, Fengmin et al, 2014a] 45 / 61

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47 Optimization can help Research opportunities Help wanted 47 / 61

48 Optimization can help Research opportunities Help wanted Integer variables VaR: [Gaivoronski and Pflug, 2005], [Benati and Rizzi, 2007], Vera and Zuluaga (2013), fixed costs and cardinality constraints 48 / 61

49 Optimization can help Research opportunities Help wanted Integer variables Multistage optimization return predictability, transaction costs, Mei, D, Nogales, (2013), Multiperiod Portfolio Optimization with Many Risky Assets and General Transaction Costs 49 / 61

50 Optimization can help Research opportunities Help wanted Integer variables Multistage optimization Calibration, calibration, calibration: 50 / 61

51 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 51 / 61

52 Optimization can help Research opportunities Help wanted Integer variables Multistage optimization Calibration, calibration, calibration Size Matters: Optimal Calibration of Shrinkage Estimators for Portfolio Selection, [Martin-Utrera, D, Nogales, 2013], choose criterion carefully, get nonparametric Martin-Utrera, D, Nogales, JFQA (2014), Parameter Uncertainty in Multiperiod Portfolio Optimization with Transaction Costs 52 / 61

53 Optimization can help Research opportunities Help wanted Integer variables Multistage optimization Calibration, calibration, calibration Statistics/big data/real-time estimation and optimization: stock return prediction: [Welch and Goyal, 2008] show that 15 predictors from the literature fail out of sample, and [Rapach et al, 2010] show that combining forecasts helps hedge fund replication: [Roncalli and Weisang, 2011] show that l 1 regularization helps when replicating hedge fund returns using 12 factors high-frequency trading 53 / 61

54 High frequency trading 54 / 61

55 Thank you Victor DeMiguel 55 / 61

56 Barry, C B (1974) Portfolio analysis under uncertain means, variances, and covariances Journal of Finance, 29(2): Benati, S and Rizzi, R (2007) A mixed integer linear programming formulation of the optimal mean/value-at-risk portfolio problem European Journal of Operational Research, 176(1): Brandt, M W, Santa-Clara, P, and Valkanov, R (2009) Parametric Portfolio Policies: Exploiting Characteristics in the Cross-Section of Equity Returns Review of Financial Studies, 22(9): Brito, R and Vicente, L (2012) Efficient cardinality/mean-variance portfolios Universidade de Coimbra working paper Broadie, M (1993) Computing efficient frontiers using estimated parameters Annals of Operations Research, 45:21 58 Brown, S (1979) the effect of estimation risk on capital market equilibrium Journal of Financial and Quantitative Analysis, 14(2): Chan, L K C, Karceski, J, and Lakonishok, J (1999) On portfolio optimization: Forecasting covariances and choosing the risk model Review of Financial Studies, 12(5): Chen, C, Li, X, Tolman, C, Wang, S, and Ye, Y (2013) 56 / 61

57 Sparse portfolio selection via quasi-norm regularization arxiv preprint arxiv: Chopra, V K and Ziemba, W T (1993) The effect of errors in means, variances, and covariances on optimal portfolio choice Journal of Portfolio Management, 19(2):6 11 DeMiguel, V, Garlappi, L, Nogales, F, and Uppal, R (2009a) A Generalized Approach to Portfolio Optimization: Improving Performance By Constraining Portfolio Norms Management Science, 55(5): DeMiguel, V, Garlappi, L, and Uppal, R (2009b) Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, 22(5): DeMiguel, V, Martín Utrera, A, and Nogales, F J (2013a) Parameter uncertainty in multiperiod portfolio optimization with transaction costs Forthcoming in Journal of Financial and Quantitative Analysis DeMiguel, V, Martin-Utrera, A, and Nogales, F J (2013b) Size matters: Optimal calibration of shrinkage estimators for portfolio selection Journal of Banking & Finance, 37(8): DeMiguel, V, Mei, X, and Nogales, F J (2013c) Multiperiod portfolio optimization with general transaction costs London Business School working paper DeMiguel, V, Nogales, F, and Uppal, R (2014) Stock return serial dependence and out-of-sample portfolio performance 57 / 61

58 The Review of Financial Studies, 27(4): DeMiguel, V and Nogales, F J (2009) Portfolio selection with robust estimation Operations Research, 57(3): DeMiguel, V, Plyakha, Y, Uppal, R, and Vilkov, G (2013d) Improving portfolio selection using option-implied volatility and skewness Journal of Financial and Quantitative Analysis, 48(6): Fengmin, X, Lu, Z, and Xu, Z (2014a) An efficient optimization approach for cardinality constrained index tracking problems Fengmin, X, Xu, Z, and Xue, H (2014b) Sparse index tracking: An l1/2 regularization based model and solution Gaivoronski, A A and Pflug, G (2005) Value-at-risk in portfolio optimization: properties and computational approach Journal of Risk, 7(2):1 31 Goldfarb, D and Iyengar, G (2003) Robust portfolio selection problems Mathematics of Operations Research, 28(1):1 38 Gotoh, J-Y, Shinozaki, K, and Takeda, A (2013) Robust portfolio techniques for mitigating the fragility of cvar minimization and generalization to coherent risk measures Quantitative Finance, (ahead-of-print):1 15 Ilmanen, A (2011) 58 / 61

59 Expected Returns: An Investor s Guide to Harvesting Market Rewards Wiley Jagannathan, R and Ma, T (2003) Risk reduction in large portfolios: Why imposing the wrong constraints helps Journal of Finance, 58(4): Jorion, P (1985) International portfolio diversification with estimation risk Journal of Business, 58: Jorion, P (1986) Bayes-stein estimation for portfolio analysis Journal of Financial and Quantitative Analysis, 21(3): Karoui, N E, Lim, A E, and Vahn, G-Y (2011) Performance-based regularization in mean-cvar portfolio optimization London Business School working paper Klein, R W and Bawa, V S (1976) The effect of estimation risk on optimal portfolio choice Journal of Financial Economics, 3(3): Konno, H and Yamazaki, H (1991) Mean-absolute deviation portfolio optimization model and its applications to tokyo stock market Management science, 37(5): Ledoit, O and Wolf, M (2004a) Honey, i shrunk the sample covariance matrix 59 / 61

60 Journal of Portfolio Management, 30(4): Ledoit, O and Wolf, M (2004b) A well-conditioned estimator for large-dimensional covariance matrices Journal of Multivariate Analysis, 88: Merton, R C (1980) On estimating the expected return on the market: An exploratory investigation Journal of Financial Economics, 8: Pástor, Ľ (2000) Portfolio selection and asset pricing models Journal of Finance, 55(1): Pástor, Ľ and Stambaugh, R F (2000) Comparing asset pricing models: An investment perspective Journal of Financial Economics, 56(3): Rapach, D E, Strauss, J K, and Zhou, G (2010) Out-of-sample equity premium prediction: Combination forecasts and links to the real economy Review of Financial Studies, 23(2): Roncalli, T and Weisang, G (2011) Tracking problems, hedge fund replication, and alternative beta Journal of Financial Transformation, 31:19 29 Ruiz-Torrubiano, R and Suárez, A (2009) A hybrid optimization approach to index tracking Annals of Operations Research, 166(1): / 61

61 Tütüncü, R and Koenig, M (2003) Robust asset allocation Working Paper, Carnegie Mellon University Welch, I and Goyal, A (2008) A comprehensive look at the empirical performance of equity premium prediction Review of Financial Studies, 21(4): Zhu, S and Fukushima, M (2009) Worst-case conditional value-at-risk with application to robust portfolio management Operations Research, 57(5): / 61