Betting Against Beta: A State-Space Approach An Alternative to Frazzini and Pederson (2014) David Puelz and Long Zhao UT McCombs April 20, 2015
Overview Background Frazzini and Pederson (2014) A State-Space Model 1
Background Investors care about portfolio Return and Risk Objective: Maximize Sharpe Ratio = Excess Return Risk Maximum Sharpe Ratio portfolio called Tangency Portfolio 2
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
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
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
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
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
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
Capital Asset Pricing Model (CAPM) For portfolio of assets: E[r] = r f + β P [E[r m] r f ] (11) 9
Background Lever up to increase return... E[r] = r f + β P [E[r m] r f ] 10
Risk / Return Space 11
Background Investors constrained on amount of leverage they can take 12
Background Due to leverage constraints, overweight high-β assets instead E[r] = r f + β P [E[r m] r f ] 13
Background Market demand for high-β = high-β assets require a lower expected return than low-β assets 14
Can we bet against β? 15
Monthly Data 4,950 CRSP US Stock Returns from 1926-2013 Fama-French Factors from 1926-2013 16
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
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
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
Decile Portfolio α s 20
Low, High-β and BAB α s 21
Sharpe Ratios Decile Portfolios (low to high β): P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 0.74 0.67 0.63 0.63 0.59 0.58 0.52 0.5 0.47 0.44 Low, High-β and BAB Portfolios: Low-β High-β BAB Market 0.71 0.48 0.76 0.41 22
Motivation Beta Plot of 200th Stock beta 0 1 2 3 4 Cor 5, SD 5 Cor 5, SD 1 0 50 100 150 200 250 23
Motivation Beta Plot of 200th Stock beta 0 1 2 3 4 Cor 5, SD 5 Cor 5, SD 1 Cor 1, SD 1 0 50 100 150 200 250 23
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
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
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
Comparison: Decile Portfolio α s 27
Comparison: With β Shrinkage 28
Comparison: Without β Shrinkage 29
Comparison: Sharpe Ratios and α s Shrinkage? Method BAB Sharpe BAB α Yes BAB Paper 0.76 0.75 SS Approach 0.42 0.58 No BAB Paper 0.04 0.75 SS Approach 0.43 1.73 30
High Frequency Estimation 31
High Frequency Estimation 32
High Frequency Estimation 33