Regret-based Selection

Transcription

1 Regret-based Selection David Puelz (UT Austin) Carlos M. Carvalho (UT Austin) P. Richard Hahn (Chicago Booth) May 27, 2017

2 Two problems 1. Asset pricing: What are the fundamental dimensions (risk factors) of the financial market? 1

3 Two problems 1. Asset pricing: What are the fundamental dimensions (risk factors) of the financial market? 2. Investing: Among thousands of choices, which passive funds should I invest in? 1

4 Two problems 1. Asset pricing: What are the fundamental dimensions (risk factors) of the financial market? 2. Investing: Among thousands of choices, which passive funds should I invest in? How are they connected? 1

5 The context for this talk Both problems can be studied using variable selection techniques from statistics. 2

6 Separating priors from utilities Our view: Subset selection is a decision problem. We need a suitable loss function, not a more clever prior. 3

7 Separating priors from utilities Our view: Subset selection is a decision problem. We need a suitable loss function, not a more clever prior. This leads us to think of selection in a post-inference world by comparing models based on regret. 3

8 Where we are headed... Risk factor selection in SUR models κ = 2 % ME4.BM5 ME2.BM4 ME3.BM3 ME2.BM3 ME5.BM4 ME3.BM5 ME2.BM2 BIG.HiBM ME3.BM4 ME5.BM2 ME4.BM3 ME2.BM5 SMALL.HiBMME1.BM4 BIG.LoBM ME3.BM2 SMALL.LoBM SMB ME1.BM3 ME4.BM2 ME4.BM4 ME1.BM2 ME2.BM1 ME4.BM1 Mkt.RF ME3.BM1 ME5.BM3 κ = 4 % ME5.BM3 ME3.BM4 ME5.BM4 ME5.BM2 ME3.BM5 ME4.BM4 ME4.BM1 ME2.BM4 BIG.LoBM ME2.BM1 SMALL.LoBM ME4.BM5 ME2.BM3 Mkt.RF ME4.BM2 SMB ME1.BM4 ME4.BM3 ME3.BM1 ME1.BM3 BIG.HiBM ME3.BM3 ME2.BM2 ME1.BM2 ME2.BM5 ME3.BM2 SMALL.HiBM κ = 12.5 % ME5.BM4 ME3.BM4 ME4.BM4 BIG.HiBM ME4.BM5 ME5.BM2 ME4.BM1 HML ME3.BM5 ME4.BM2 ME2.BM4 ME2.BM1 SMALL.LoBM Mkt.RF ME4.BM3 ME3.BM1 ME2.BM5 BIG.LoBM SMALL.HiBM SMB ME3.BM3 ME5.BM3 ME2.BM2 ME2.BM3 ME1.BM3 ME1.BM2 ME1.BM4 ME3.BM2 κ = 32.5 % Sparse dynamic portfolios QMJ ME4.BM2 BIG.LoBM ME4.BM3 ME5.BM3 SMALL.LoBM ME5.BM2 RMW ME4.BM5 ME4.BM4 CMA BIG.HiBM ME2.BM5 ME3.BM1 HML ME1.BM2 ME2.BM1 Mkt.RF SMALL.HiBM ME3.BM5 SMB ME5.BM4 ME3.BM4 ME2.BM3 ME1.BM3 ME2.BM4 ME1.BM4 ME4.BM1 ME3.BM3 ME2.BM2 ME3.BM2 κ = 47.5 % BIG.LoBM ME1.BM4 ME1.BM3 ME3.BM1ME2.BM1 ME2.BM2 ME2.BM3 ME5.BM4 ME5.BM3 ME3.BM3 QMJ ME4.BM5 ME3.BM5 HML ME2.BM5 Mkt.RF SMB ME2.BM4 ME4.BM1 LTR CMA SMALL.LoBM SMALL.HiBM ME3.BM4 ME4.BM3 ME3.BM2 ME4.BM2 BIG.HiBM ME4.BM4 BAB ME1.BM2 RMWME5.BM2 κ = % ME1.BM2 ME2.BM1 LTR ME5.BM3 ME5.BM2 ME1.BM3 ME4.BM5 ME3.BM1 ME3.BM2 ME2.BM5 RMW BIG.HiBM CMA ME4.BM3 ME4.BM4 SMB Mkt.RF SMALL.HiBM ME3.BM4 BAB HML ME4.BM2 SMALL.LoBM ME3.BM5 ME2.BM3 ME3.BM3 ME1.BM4 ME2.BM4 QMJ STR ME5.BM4 BIG.LoBM ME2.BM2 ME4.BM1 4

9 Regret-based selection: Primitives Let d be a decision, λ be a complexity parameter, Θ be a vector of model parameters, and Ỹ be future data. 1. Loss function L(d, Ỹ ) measures utility. 2. Complexity function Φ(λ, d) measures sparsity. 3. Statistical model Π(Θ) characterizes uncertainty. 4. Regret tolerance κ characterizes degree of comfort from deviating from a target decision (in terms of posterior probability). 5

10 Regret-based selection: Procedure Optimize expected loss (1) + complexity (2). The expectation is over p(ỹ, Θ Y) (3). Calculate regrets versus a target d for decisions indexed by λ. ρ(d λ, d, Ỹ ) = L(d λ, Ỹ ) L(d, Ỹ ) Select d λ as the decision satisfying the regret tolerance. π λ = P[ρ(d λ, d, Ỹ ) < 0] Select d λ s.t. π d λ > κ (3,4) 6

11 Which risk factors matter?

12 The Factor Zoo (Cochrane, 2011) Market Size Value Momentum Short and long term reversal Betting against β Direct profitability Dividend initiation Carry trade Liquidity Quality minus junk Investment Leverage... 8

13 The Factor Zoo (Cochrane, 2011) Market Size Value Momentum Short and long term reversal Betting against β Direct profitability Dividend initiation Carry trade Liquidity Quality minus junk Investment Leverage... 9

14 The setup for determining important factors Let the return on test assets be R, and the return on factors be F. R = γf + ɛ, ɛ N(0, Ψ) Primitives: 1. Loss: L(γ, R, F ) = log p( R F ) 2. Complexity: Φ(λ, γ) = λ γ Model: R F with normal errors and conjugate g-priors and F via gaussian linear latent factor model. 4. Regret tolerance: Let s consider several κ s. Assume the target is the λ = 0 model. 10

15 Factor selection graph (κ = 12.5%) R: 25 Fama-French portfolios, F : 10 factors from finance literature Size4 Size1 BM3 Size1 BM4 BM2 Size1 BM3 Size4 Size2 BM2 Size3 BM2 BM2 Size5 BM2 Size2 BM3 SMB Size2 BM1 Size5 BM1 Size3 BM1 Mkt.RF Size2 BM5 Size1 BM5 Size1 BM1 Size5 BM3 Size3 BM5 Size2 BM4 HML Size4 BM5 Size3 BM3 Size4 BM1 Size3 BM4 Size5 BM4 Size4 BM4 Size5 BM5 11

16 Selected graphs under different regret tolerances κ κ = 2 % ME4.BM5 ME2.BM4 ME3.BM3 ME2.BM3 ME5.BM4 ME3.BM5 ME2.BM2 BIG.HiBM ME3.BM4 ME5.BM2 ME4.BM3 ME2.BM5 SMALL.HiBMME1.BM4 BIG.LoBM ME3.BM2 SMALL.LoBM SMB ME1.BM3 ME4.BM2 ME4.BM4 ME1.BM2 ME2.BM1 ME4.BM1 Mkt.RF ME3.BM1 ME5.BM3 κ = 4 % ME5.BM3 ME3.BM4 ME5.BM4 ME5.BM2 ME3.BM5 ME4.BM4 ME4.BM1 ME2.BM4 BIG.LoBM ME2.BM1 SMALL.LoBM ME4.BM5 ME2.BM3 Mkt.RF ME4.BM2 SMB ME1.BM4 ME4.BM3 ME3.BM1 ME1.BM3 BIG.HiBM ME3.BM3 ME2.BM2 ME1.BM2 ME2.BM5 ME3.BM2 SMALL.HiBM κ = 12.5 % ME5.BM4 ME3.BM4 ME4.BM4 BIG.HiBM ME4.BM5 ME5.BM2 ME4.BM1 HML ME3.BM5 ME4.BM2 ME2.BM4 ME2.BM1 SMALL.LoBM Mkt.RF ME4.BM3 ME3.BM1 ME2.BM5 BIG.LoBM SMALL.HiBM SMB ME3.BM3 ME5.BM3 ME2.BM2 ME2.BM3 ME1.BM3 ME1.BM2 ME1.BM4 ME3.BM2 κ = 32.5 % QMJ ME4.BM2 BIG.LoBM ME4.BM3 ME5.BM3 SMALL.LoBM ME5.BM2 RMW ME4.BM5 ME4.BM4 CMA BIG.HiBM ME2.BM5 ME3.BM1 HML ME1.BM2 ME2.BM1 Mkt.RF SMALL.HiBM ME3.BM5 SMB ME5.BM4 ME3.BM4 ME2.BM3 ME1.BM3 ME2.BM4 ME1.BM4 ME4.BM1 ME3.BM3 ME2.BM2 ME3.BM2 κ = 47.5 % BIG.LoBM ME1.BM4 ME1.BM3 ME3.BM1ME2.BM1 ME2.BM2 ME2.BM3 ME5.BM4 ME5.BM3 ME3.BM3 QMJ ME4.BM5 ME3.BM5 HML ME2.BM5 Mkt.RF SMB ME2.BM4 ME4.BM1 LTR CMA SMALL.LoBM SMALL.HiBM ME3.BM4 ME4.BM3 ME3.BM2 ME4.BM2 BIG.HiBM ME4.BM4 BAB ME1.BM2 RMWME5.BM2 κ = % ME1.BM2 ME2.BM1 LTR ME5.BM3 ME5.BM2 ME1.BM3 ME4.BM5 ME3.BM1 ME3.BM2 ME2.BM5 RMW BIG.HiBM CMA ME4.BM3 ME4.BM4 SMB Mkt.RF SMALL.HiBM ME3.BM4 BAB HML ME4.BM2 SMALL.LoBM ME3.BM5 ME2.BM3 ME3.BM3 ME1.BM4 ME2.BM4 QMJ STR ME5.BM4 BIG.LoBM ME2.BM2 ME4.BM1 12

17 Passive Investing

18 thousands of investment opportunities 14

19 The setup for sparse passive investing Let R t be a vector of N future asset returns. Let w t be the portfolio weight vector (decision) at time t. We use the log cumulative growth rate for our utility! Primitives: 1. Loss: log ( 1 + N k=1 w t k R ) t k 2. Complexity: λ t w t 1 3. Model: DLM for R t parameterized by (µ t, Σ t ) 4. Regret tolerance: κ = 55%. Assume the target is fully invested (dense) portfolio. 15

20 Static regret tolerance dynamic portfolio decisions Data: Returns on 25 ETFs from κ = 55% decision. Date DIA IWD IWB IWN IWM IYR Dow Jones Value Large Small Small value Real estate

21 Ex ante SR target SR decision evolution Data: Returns on 25 ETFs from κ = 55% decision. Difference in Sharpe ratio dense portfolio as target SPY as target IWB as target

22 Ex post performance of the κ = 55% decision cumulative return sparse dense SPY IWB

23 Last slide Passive investing and factor selection for asset pricing models approached using new variable selection technique. Utility functions can enforce inferential preferences that are not prior beliefs. Variable selection in SUR models with random predictors. Bayesian Analysis (2017). Sparse dynamic portfolios with regret-based selection. Submitted (2017). Thanks! 19

24 Extra slides

25 A complicated posterior! R i t = (β i t) T RF t + ɛ i t, ɛ i t N(0, 1/φ i t), β i t = β i t 1 + w i t, β i 0 D 0 T n i 0 (m i 0, C i 0), φ i 0 D 0 Ga(n i 0/2, d i 0/2), w i t T n i t 1 (0, W i t ), β i t D t 1 T n i t 1 (m i t 1, R i t), R i t = C i t 1/δ β, φ i t D t 1 Ga(δ ɛ n i t 1/2, δ ɛ d i t 1/2), R F t = µ F t + ν t ν t N(0, Σ F t ), µ F t = µ F t 1 + Ω t Ω t N(0, W t, Σ F t ), (µ F 0, Σ F 0 D 0 ) NW 1 (m n0 0, C 0, S 0 ), (µ F t, Σ F t D t 1 ) NW 1 δ F n t 1 (m t 1, R t, S t 1 ), R t = C t 1 /δ c, 21

26 Dynamic regret-based selection Assume N asset returns follow the model: R t Π(µ t, Σ t ) Specifically, let the covariates be the five Fama-French factors, Rt F N(µ F t, Σ F t ), so that: µ t = β T t µ F t Σ t = β t Σ F t β T t + Ψ t Given µ t and Σ t, make portfolio decision for time t

27 Seemingly unrelated regressions Replace R with generic response vector Y and F with generic covariate vector X : R Y and F X Y j = β j1 X β jp X p + ɛ j, ɛ N(0, Ψ), j = 1,, q The proposed framework permits variable selection in SUR models with random predictors! 23

28 Posterior summary plot λ L(γ λ, Θ, R, F ) L(γ 0, Θ, R, F ), π λ P( λ < 0) utility E[ λ ] π λ selected model probability models ordered by decreasing λ π λ = probability that λ-model is no worse than the dense model. 24

29 Regret-based selection: Illustration d λ : sparse decisions, d : target decision. π λ = P[ρ(d λ, d, Ỹ ) < 0]: probability of not regretting λ-decision. sparse decisions target decision 2 π decision 2 decision 2 Density decision 1 decision Loss Regret (difference in loss) 25

30 Ex ante regret evolution Data: Returns on 25 ETFs from κ = 55% decision Regret (difference in loss)