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Betting Against Beta: A State-Space ApproachAn Alternative to Frazzini and Pederson (2014)
David Puelz and Long Zhao
UT McCombs
October 14, 2015
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Overview
Background
Frazzini and Pederson (2014)
A State-Space Model
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Background
I Investors care about portfolio Return and Risk1
I Objective: Maximize Sharpe Ratio = ReturnRisk
I Maximum Sharpe Ratio portfolio called Tangency Portfolio
1standard deviation of portfolio return 2
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Key Question
How can I price an asset’s expected return?
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The Capital Asset Pricing Model (Sharpe, 1964) (Lintner,1965)
I r∗m = Market Portfolio
I rf = risk-free rate
I For asset i :
E[ri ] = rf + βi [E[r∗m]− rf ] (1)
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Let’s derive the CAPM!
I Portfolio of N assets defined by weights: {xim}Ni=1
I Covariance between returns i and j : σij = cov(ri , rj)
I Standard deviation of portfolio return:
σ(rm) =N∑i=1
ximcov(ri , rm)
σ(rm)(2)
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Maximizing Portfolio Return
I Choosing efficient portfolio =⇒ maximizes expected returnfor a given risk: σ(rp)
I Choose {xim}Ni=1 to maximize:
E[rm] =N∑i=1
ximE[ri ] (3)
with constraints: σ(rm) = σ(rp) and∑N
i=1 xim = 1
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What does this imply? (I)
The Lagrangian:
L(xim, λ, µ) =N∑i=1
ximE[ri ] + λ (σ(rp)− σ(rm)) + µ
(N∑i=1
xim − 1
)(4)
Taking derivatives, setting equal to zero:
E[ri ]− λcov(ri , r
∗m)
σ(r∗m)+ µ = 0 ∀i (5)
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What does this imply? (II)
From 5, we have:
E[ri ]− λcov(ri , r
∗m)
σ(r∗m)= E[rj ]− λ
cov(rj , r∗m)
σ(r∗m)∀i , j (6)
Assume ∃ r0 that is uncorrelated with portfolio r∗m. From 6, we
have:
E[r∗m]− E[r0]
σ(r∗m)= λ (7)
E[ri ]− E[r∗m] = −λσ(r∗m) + λcov(ri , r
∗m)
σ(r∗m)(8)
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Bringing it all together
7 and 8 =⇒
E[ri ] = E[r0] + [E[r∗m]− E[r0]]βi (9)
where
βi =cov(ri , r
∗m)
σ2(r∗m)(10)
Linear relationship between expected returns of asset and r∗m!
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Capital Asset Pricing Model (CAPM)
I r∗m = Market Portfolio
I r0 = rf
I For asset i :
E[ri ] = rf + βi [E[r∗m]− rf ] (11)
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Capital Asset Pricing Model (CAPM)
I For portfolio of assets:
E[r ] = rf + βP [E[r∗m]− rf ] (12)
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Background
”Lever up” to increase return ...
E[r ] = rf + βP [E[r∗m]− rf ]
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Background
I Investors constrained on amount of leverage they can take
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Background
Due to leverage constraints, overweight high-β assets instead
E[r ] = rf + βP [E[r∗m]− rf ]
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Background
Market demand for high-β
=⇒
high-β assets require a lower expected return than low-β assets
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Can we bet against β ?
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Monthly Data
I 4,950 CRSP US Stock Returns
I Fama-French Factors
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Frazzini and Pederson (2014)
1. For each time t and each stock i , estimate βit
2. Sort βit from smallest to largest
3. Buy low-β stocks and Sell high-β stocks
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F&P (2014) BAB Factor
Buy top half of sort (low-β stocks) and Sell bottom half of sort(high-β stocks) ∀t
rBABt+1 =1
βLt(rLt+1 − rf )− 1
βHt(rHt+1 − rf ) (13)
βLt = ~βTt ~wL
βHt = ~βTt ~wH
~wH = κ(z − z̄)+
~wL = κ(z − z̄)−
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F&P (2014) BAB Factor
βit estimated as:
β̂it = ρ̂σ̂iσ̂m
(14)
I ρ̂ from rolling 5-year window
I σ̂’s from rolling 1-year window
I β̂it ’s shrunk towards cross-sectional mean
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Decile Portfolio α’s
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Low, High-β and BAB α’s
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Sharpe Ratios
Decile Portfolios (low to high β):
P1 P2 P3 P4 P5 P6 P7 P8 P9 P100.74 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 Market0.71 0.48 0.76 0.41
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Motivation
0 50 100 150 200 250
01
23
4
beta
Beta Plot of 200th Stock
Cor 5, SD 5Cor 5, SD 1
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Motivation
0 50 100 150 200 250
01
23
4
beta
Beta Plot of 200th Stock
Cor 5, SD 5Cor 5, SD 1Cor 1, SD 1
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Our Model
Reit = βitR
emt + exp
(λt2
)εt (15)
βit = a + bβit−1 + wt (16)
λit = c + dλit−1 + ut (17)
εt ∼ N[0, 1]
wt ∼ N[0, σ2β]
ut ∼ N[0, σ2λ]
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Our Model
Reit = βitR
emt + exp
(λt2
)εt (18)
βit = a + bβit−1 + wt (19)
λit = c + dλit−1 + ut (20)
εt ∼ N[0, 1]
wt ∼ N[0, σ2β]
ut ∼ N[0, σ2λ]
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The Algorithm
1. P(β1:T |Θ, λ1:T ,DT ) (FFBS)
2. P(λ1:T |Θ, β1:T ,DT ) (Mixed Normal FFBS)
3. P(Θ|β1:T , λ1:T ,DT ) (AR(1))
I βt |Θ, λ1:T ,Dt
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Comparison: Decile Portfolio α’s
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Comparison: With β Shrinkage
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Comparison: Without β Shrinkage
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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
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High Frequency Estimation
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High Frequency Estimation
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High Frequency Estimation
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