Betting Against Beta: A State-Space Approach

Transcription

1 Betting Against Beta: A State-Space Approach An Alternative to Frazzini and Pederson (2014) David Puelz and Long Zhao UT McCombs April 20, 2015

2 Overview Background Frazzini and Pederson (2014) A State-Space Model 1

3 Background Investors care about portfolio Return and Risk Objective: Maximize Sharpe Ratio = Excess Return Risk Maximum Sharpe Ratio portfolio called Tangency Portfolio 2

4 Let s derive the CAPM! Portfolio of N assets defined by weights: {x im } N i=1 Covariance between returns i and j: σ ij = cov(r i, r j ) Standard deviation of portfolio return: σ(r m ) = N i=1 x im cov(r i, r m ) σ(r m ) (1) 3

5 Maximizing Portfolio Return Choosing efficient portfolio = maximizes expected return for a given risk: σ(r p ) Choose {x im } N i=1 to maximize: E[r m ] = N x im E[r i ] (2) i=1 with constraints: σ(r m ) = σ(r p ) and N i=1 x im = 1 4

6 What does this imply? (I) The Lagrangian: L(x im, λ, µ) = ( N N ) x im E[r i ] + λ (σ(r p ) σ(r m )) + µ x im 1 i=1 i=1 (3) Taking derivatives, setting equal to zero: E[r i ] λ cov(r i, r m) σ(r m) + µ = 0 i (4) 5

7 What does this imply? (II) From 4, we have: E[r i ] λ cov(r i, r m) σ(r m) = E[r j ] λ cov(r j, r m) σ(r m) i, j (5) Assume r 0 that is uncorrelated with portfolio r m. From 5, we have: E[r m] E[r 0 ] σ(r m) = λ (6) E[r i ] E[r m] = λσ(r m) + λ cov(r i, r m) σ(r m) (7) 6

8 Bringing it all together 6 and 7 = where E[r i ] = E[r 0 ] + [E[r m] E[r 0 ]] β i (8) β i = cov(r i, r m) σ 2 (r m) (9) Linear relationship between expected returns of asset and r m! 7

9 Capital Asset Pricing Model (CAPM) r m = Market Portfolio For asset i: E[r i ] = r f + β i [E[r m] r f ] (10) 8

10 Capital Asset Pricing Model (CAPM) For portfolio of assets: E[r] = r f + β P [E[r m] r f ] (11) 9

11 Background Lever up to increase return... E[r] = r f + β P [E[r m] r f ] 10

12 Risk / Return Space 11

13 Background Investors constrained on amount of leverage they can take 12

14 Background Due to leverage constraints, overweight high-β assets instead E[r] = r f + β P [E[r m] r f ] 13

15 Background Market demand for high-β = high-β assets require a lower expected return than low-β assets 14

16 Can we bet against β? 15

17 Monthly Data 4,950 CRSP US Stock Returns from Fama-French Factors from

18 Frazzini and Pederson (2014) 1. For each time t and each stock i, estimate β it 2. Sort β it from smallest to largest 3. Buy low-β stocks and Sell high-β stocks 17

19 F&P (2014) BAB Factor Buy top half of sort (low-β stocks) and Sell bottom half of sort (high-β stocks) t rt+1 BAB = 1 βt L (rt+1 L r f ) 1 βt H (r H t+1 r f ) (12) βt L = β t T w L βt H = β t T w H w H = κ(z z) + w L = κ(z z) 18

20 F&P (2014) BAB Factor β it estimated as: ˆβ it = ˆρ ˆσ i ˆσ m (13) ˆρ from rolling 5-year window ˆσ s from rolling 1-year window ˆβ it s shrunk towards cross-sectional mean 19

21 Decile Portfolio α s 20

22 Low, High-β and BAB α s 21

23 Sharpe Ratios Decile Portfolios (low to high β): P1 P2 P3 P4 P5 P6 P7 P8 P9 P Low, High-β and BAB Portfolios: Low-β High-β BAB Market

24 Motivation Beta Plot of 200th Stock beta Cor 5, SD 5 Cor 5, SD

25 Motivation Beta Plot of 200th Stock beta Cor 5, SD 5 Cor 5, SD 1 Cor 1, SD

26 Our Model R e it = β it R e mt + exp ( ) λt ɛ t (14) 2 β it = a + bβ it 1 + w t (15) λ it = c + dλ it 1 + u t (16) ɛ t N[0, 1] w t N[0, σ 2 β ] u t N[0, σ 2 λ ] 24

27 Our Model R e it = β it R e mt + exp ( ) λt ɛ t (17) 2 β it = a + bβ it 1 + w t (18) λ it = c + dλ it 1 + u t (19) ɛ t N[0, 1] w t N[0, σ 2 β ] u t N[0, σ 2 λ ] 25

28 The Algorithm 1. P(β 1:T Θ, λ 1:T, D T ) (FFBS) 2. P(λ 1:T Θ, β 1:T, D T ) (Mixed Normal FFBS) 3. P(Θ β 1:T, λ 1:T, D T ) (AR(1)) β t Θ, λ 1:T, D t 26

29 Comparison: Decile Portfolio α s 27

30 Comparison: With β Shrinkage 28

31 Comparison: Without β Shrinkage 29

32 Comparison: Sharpe Ratios and α s Shrinkage? Method BAB Sharpe BAB α Yes BAB Paper SS Approach No BAB Paper SS Approach

33 High Frequency Estimation 31

34 High Frequency Estimation 32

35 High Frequency Estimation 33