Risk Premia and the Conditional Tails of Stock Returns
Bryan Kelly
NYU Stern and Chicago Booth
B. Kelly
Risk Premia and Conditional Tails
Outline
Introduction
An Economic Framework
Econometric Methodology
Empirical Findings
Conclusions
B. Kelly
Risk Premia and Conditional Tails
Tail Risk in the Big Picture
Value of assets depends on the potential for infrequent, extreme
payoff events
1
Peso problems (Krasker, 1980)
2
Potential for rare disasters can explain equity puzzles (Rietz, 1988; Barro,
2006; Weitzman 2007; Gabaix 2009; Wachter 2009)
Plausible mechanism – OR – convenient (though unrealistic)
explanation?
B. Kelly
Risk Premia and Conditional Tails
The Trouble with Tail Risk
Tail risk is difficult to measure, even unconditionally
Few risks are static: Feasibility of conditional tail measures?
My solution: An economically-motivated conditional tail risk
measure extracted from the cross section of asset returns
B. Kelly
Risk Premia and Conditional Tails
Objectives
1
Structural understanding of how tail risk is priced
I derive tractable expressions for expected returns as a function of a
tail risk state variable
I derive the distribution of return tail events implied by the model
2
I econometrically identify the conditional tail distribution of returns
Directly estimable from the cross section of asset prices by exploiting
restriction implied by economic theory
3
I evaluate theories relating tail risk to risk premia using my
estimated series
B. Kelly
Risk Premia and Conditional Tails
Preview of Empirical Results
Tail risk varies substantially over time and is highly persistent
Tail measure predicts market returns over horizons of one month to
five years, outperforms commonly studied predictors
A one standard deviation increase in tail risk increases expected returns
by 4.4% per year
Large explanatory power for cross section of returns
Stocks that covary highly with tail risk earn annual expected returns
2% to 6% lower than stocks that with low tail risk covariation
B. Kelly
Risk Premia and Conditional Tails
Outline
Introduction
An Economic Framework
Econometric Methodology
Empirical Findings
Conclusions
B. Kelly
Risk Premia and Conditional Tails
Structural Models and Tail Risk
Emergence in varied theoretical settings, for example
1
Long run risks + heavy-tailed shocks (similar to Eraker and Shaliastovich
2008, Drechsler and Yaron 2009)
2
3
Time-varying rare disasters (similar to Gabaix 2009, Wachter 2009)
Long run risks + large swings in confidence (similar to Bansal and
Shaliastovich 2009)
B. Kelly
Risk Premia and Conditional Tails
A Tail Risk State Variable in the Long Run Risks
Framework
Epstein-Zin preferences:
mt+1 = θ ln β −
θ
∆ct+1 + (θ − 1)rc,t+1
ψ
Dynamics of the real economy:
p
∆ct+1 = µ + xt + σc σt zc,t+1 + Λt Wc,t+1
xt+1 = ρx xt + σx σt zx,t+1
2
σt+1
= σ̄ 2 (1 − ρσ ) + ρσ σt2 + σσ zσ,t+1
Λt+1 = Λ̄(1 − ρΛ ) + ρΛ Λt + σΛ zΛ,t+1
∆di,t+1 = µi + φi ∆ct+1 + σi σt zi,t+1 +qi
fW (w ) =
B. Kelly
p
Λt Wi,t+1
1
exp(−|w |), w ∈ R
2
Risk Premia and Conditional Tails
What Is Tail Risk?
Gaussian baseline: is variance sufficient to characterize risk of extreme
events?
Illustrative example: Normal-Laplace distribution
Since at least Mandelbrot (1963) and Fama (1963), economists have
argued for power law return tails
P(R > x|R > u) =
x −ζ
u
, u some high threshold
−ζ ≡ tail risk measure
Does ζ change through time?
What kind of world?
B. Kelly
Risk Premia and Conditional Tails
What Is Tail Risk?
Normal (µ,σ2)
Laplace (Λ)
Var(XN−L)=σ2+2Λ2
Normal−Laplace (µ,σ2,Λ)
P(XN−L>x|XN−L>u)=exp(−[x−u]/Λ)
u
−10
−8
B. Kelly
−6
−4
−2
0
2
4
6
8
10
Risk Premia and Conditional Tails
What Is Tail Risk?
Gaussian baseline: is variance sufficient to characterize risk of extreme
events?
Illustrative example: Normal-Laplace distribution
Since at least Mandelbrot (1963) and Fama (1963), economists have
argued for power law return tails
P(R > x|R > u) =
x −ζ
u
, u some high threshold
−ζ ≡ tail risk measure
Does ζ change through time?
What kind of world?
Return
B. Kelly
Risk Premia and Conditional Tails
A Tail Risk State Variable in the Long Run Risks
Framework
Epstein-Zin preferences:
mt+1 = θ ln β −
θ
∆ct+1 + (θ − 1)rc,t+1
ψ
Dynamics of the real economy:
p
∆ct+1 = µ + xt + σc σt zc,t+1 + Λt Wc,t+1
xt+1 = ρx xt + σx σt zx,t+1
2
σt+1
= σ̄ 2 (1 − ρσ ) + ρσ σt2 + σσ zσ,t+1
Λt+1 = Λ̄(1 − ρΛ ) + ρΛ Λt + σΛ zΛ,t+1
∆di,t+1 = µi + φi ∆ct+1 + σi σt zi,t+1 +qi
fW (w ) =
B. Kelly
p
Λt Wi,t+1
1
exp(−|w |), w ∈ R
2
Risk Premia and Conditional Tails
Prices and Excess Returns
Proposition
The log wealth-consumption ratio and log price-dividend ratio for asset (i)
are linear in state variables,
2
wct+1 = A0 + Ax xt+1 + Aσ σt+1
+ AΛ Λt+1
2
pdi,t+1 = Ai,0 + Ai,x xt+1 + Ai,σ σt+1
+ Ai,Λ Λt+1 .
Proposition
The expected return on asset (i) in excess of the risk free rate is
1
Et [ri,t+1 − rf ,t ] = βi,c λc (σc2 σt2 + 2Λt ) + βi,x λx σx2 σt2 + βi,σ λσ σσ2 + βi,Λ λΛ σΛ2 − Var (ri,t+1 ).
2
Proof
B. Kelly
Risk Premia and Conditional Tails
Key Implications
1
Tail risk forecasts excess stock returns
High tail risk ⇒ high future returns
2
Covariance with tail risk impacts cross section of expected returns
High return tail risk beta ⇒ low expected returns
B. Kelly
Risk Premia and Conditional Tails
Implied Distribution of Returns
Proposition
The lower and upper tail distributions of arithmetic returns are
asymptotically equivalent to a power law,
Pt (Ri,t+1 < r Ri,t+1 < u) ∼
Pt (Ri,t+1
> r Ri,t+1 > u) ∼
ai ζt
r
u
−ai ζt
r
u
√
where ai = max(φi , qi )−1 and ζt = 1/ Λt .
B. Kelly
Risk Premia and Conditional Tails
Key Implications
Tail risk state variable drives risk premia and tail exponent
1
Tail risk (and thus tail exponent) forecasts excess stock returns
High tail risk ⇒ high future returns
2
Covariance with tail risk (and thus tail exponent) impacts cross
section of expected returns
High return tail risk beta ⇒ low expected returns
Other Structural Models
B. Kelly
Risk Premia and Conditional Tails
Outline
Introduction
An Economic Framework
Econometric Methodology
Empirical Findings
Conclusions
B. Kelly
Risk Premia and Conditional Tails
Econometric Intuition
˛
Pt (Ri,t+1 < r ˛ Ri,t+1 < u) ∼
„ «ai ζt
r
u
Single process drives tail dynamics of entire panel of returns
B. Kelly
Risk Premia and Conditional Tails
Definition: Dynamic Power Law Model
Individual returns on asset (i), conditional upon exceeding threshold u and
given Ft , obey
Fu,i,t (r ) = P(Ri,t+1 > r |Ri,t+1 > u, Ft ) =
−ai ζt
r
u
with exponent
1
= π0 + π1
ζt+1
1
ζtupd
+ π2
1
ζt
and observable update
1
ζtupd
=
B. Kelly
Kt
Rk,t
1 X
ln
Kt
u
k=1
Risk Premia and Conditional Tails
Definition: Dynamic Power Law Model
Individual returns on asset (i), conditional upon exceeding threshold u and
given Ft , obey
Fu,i,t (r ) = P(Ri,t+1 > r |Ri,t+1 > u, Ft ) =
−ai ζt
r
u
with exponent
1
= π0 + π1
ζt+1
1
ζtupd
∞
+ π2
X j 1
1
π0
=
+ π1
π2 upd
ζt 1 − π2
ζ
j=0
t−j
and observable update
1
ζtupd
Kt
Rk,t
1 X
=
ln
Kt
u
B. Kelly
k=1
(
Hill (1975) Estimator
applied to cross section
Risk Premia and Conditional Tails
Quasi-Likelihood Estimator for Dynamic Power Law Model
Proposal: Assume tail observations in time t cross section are identical
and independent
But theory (and years of empirical work) suggests...
1
2
3
Dependent observations (factor structure)
Heterogeneous volatility
Heterogeneous tail exponent
Result: Despite mis-specification, estimator consistent and asymptotically
normal
B. Kelly
Risk Premia and Conditional Tails
Quasi-Likelihood Estimator for Dynamic Power Law Model
1
Assume (provisionally) tail observations are cross-sectionally
independent and each obey
−ζ̃t
x
F̃u,i,t (Xi,t+1 ; π) =
u
ζt
f˜u,i,t (Xi,t+1 ; π) =
u
2
−(1+ζ̃t )
x
u
Construct log quasi-likelihood using only u-exceedences
T
T Kt+1 Xk,t+1
1 X ˜
1 XX 1
L(X ; π) =
ln fu,t (Xt+1 ; π) =
− ln
T
T
u
ζ̃t
t=0
3
t=0 k=1
Maximize
QML Estimator: π̂QL ≡ arg max L(X ; π)
π∈Π
B. Kelly
Risk Premia and Conditional Tails
Asymptotic Properties of QML Estimator
Proposition
Let the true DGP of {Rt }T
t=1 be given by the Dynamic Power Law model
with parameter values π ∗ . Under standard GMM regularity conditions,
p
π̂QL → π ∗
and
√
d
T (π̂QL − π ∗ ) → N(0, Ψ)
where
Ψ = S −1 GS −1 , S = E [∇π s(Xt ; π ∗ )], and G = E [s(Xt ; π ∗ )s(Xt ; π ∗ )0 ].
B. Kelly
Risk Premia and Conditional Tails
Proof Sketch
First order condition of quasi-likelihood maximization
Kt+1
s(Xt+1 ; π) ≡ ∇π ln f˜t (Xt+1 ; π) =
Kt+1 X Xk,t+1
−
ln
=0
u
ζ̃t
k=1
MLE identification condition: expected value of FOC equals zero
Mis-specified MLE is GMM – FOC moment condition holds
Lemma
E [s(Xt+1 ; π)] = 0 when the true model is the Dynamic Power Law.
Proof: E [s(Xt+1 ; π)]
=
=
=
B. Kelly
Kt+1
X Xk,t+1 ˜
ˆ Kt+1
E Et [
−
ln
]
u
ζ̃t
k=1
P
Kt+1 n1 i ai
Kt+1
−
E[
]
ζt
ζ̃t
P
1
ai
1
0 when
= n i . ζt
ζ̃t
Risk Premia and Conditional Tails
Volatility and Other Considerations
By varying threshold each period, accommodate time-varying
volatility
Cross sectional differences in volatility?
Explicitly modeling dependence?
B. Kelly
Risk Premia and Conditional Tails
Outline
Introduction
An Economic Framework
Econometric Methodology
Empirical Findings
Conclusions
B. Kelly
Risk Premia and Conditional Tails
Data
Primary sample: Daily NYSE/AMEX/NASDAQ stock returns from
CRSP
Fama-French factors; Ken French’s data library
Federal Reserve macro data
Goyal and Welch (2008) data
OptionMetrics
Other (VIX, Hao Zhou’s variance risk premium)
Count
B. Kelly
Risk Premia and Conditional Tails
Dynamic Power Law Estimates
1
ζt+1
= π0 + π1
1
upd
ζt
+ π2 ζ1
t
Table: 1963-2008
B. Kelly
Both Tails
Lower Tail
Upper Tail
ζ̄
2.110
(0.021)
2.201
(0.044)
1.872
(0.018)
π1
0.188
(0.014)
0.072
(0.010)
0.239
(0.058)
π2
0.798
(0.015)
0.923
(0.011)
0.683
(0.092)
Risk Premia and Conditional Tails
Dynamic Power Law Estimates: Exponent Series
6
−1.5
5
−2.0
4
−2.5
3
−3.0
2
1963
Tail Risk (− ζt)
Aggregate Price−Dividend Ratio (Log)
Exponent (both)
1967
1971
1975
1979
1983
1988
1992
1996
2000
2004
−3.5
2008
ρ(Exponent, log P/D) = −14%
B. Kelly
Risk Premia and Conditional Tails
Dynamic Power Law Estimates: Exponent Series
6
−1.5
5
−2.0
Tail Risk (− ζt)
Aggregate Price−Dividend Ratio (Log)
Lower
Upper
−2.5
4
−3.0
3
−3.5
2
1963
1967
1971
1975
1979
1984
ρ(Lower Exponent, log P/D) = −15%,
B. Kelly
1988
1992
1996
2000
2004
−4.0
2008
ρ(Upper Exponent, log P/D) = −14%
Risk Premia and Conditional Tails
Dynamic Power Law Estimates: Threshold
0.8
Volatility
Threshold (both)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1963
1967
1971
1975
1979
1984
1988
1992
1996
2000
2004
2008
ρ(Volatility , Threshold) = 60%
B. Kelly
Risk Premia and Conditional Tails
Testing Model Implications: Predicting Stock Returns
Theory suggests increases in tail risk forecast increases in excess
returns
Predictive regressions of excess returns on aggregate market over
short (one month) and long (up to five year) horizons
Compare against common alternatives (dividend-price ratio, term
spread, etc.)
Robustness
B. Kelly
Risk Premia and Conditional Tails
Testing Model Implications: Predicting Stock Returns
Univariate Prediction
Tail Risk (-ζ lower)
Book-to-market
Cross section premium
Default return spread
Default yield spread
Dividend payout ratio
Dividend price ratio
Dividend yield
Earnings price ratio
Inflation
Long term return
Long term yield
Net equity expansion
Stock volatility
Term Spread
Treasury bill rate
Variance risk premium
One month horizon
Coef.
t-stat
R2
One year horizon
Coef.
t-stat
R2
Five year horizon
Coef.
t-stat
R2
6.70
0.81
5.53
1.73
4.65
-0.27
2.55
2.65
2.80
-6.72
5.35
-0.70
-4.34
0.15
4.49
-3.08
9.73
4.44
1.76
-4.19
-0.08
2.26
0.88
3.19
3.18
2.86
-1.99
2.21
1.50
-2.11
0.44
3.95
-0.81
4.34
5.02
1.15
-4.28
-0.12
2.45
1.90
3.15
3.10
2.17
-0.21
1.18
3.93
-1.04
-0.03
4.00
1.26
-4.64
B. Kelly
2.9
0.3
2.4
0.7
2.0
0.1
1.0
1.1
1.1
2.6
2.3
0.3
2.1
0.1
2.0
1.3
3.3
0.016
0.000
0.010
0.001
0.008
0.000
0.002
0.003
0.003
0.016
0.010
0.000
0.007
0.000
0.007
0.003
0.035
2.1
0.7
1.8
0.1
0.9
0.4
1.2
1.2
1.1
1.1
3.0
0.6
0.9
0.2
1.9
0.3
3.1
0.069
0.011
0.061
0.000
0.018
0.003
0.036
0.036
0.029
0.014
0.017
0.008
0.016
0.001
0.056
0.002
0.038
2.3
0.4
1.9
0.6
1.3
0.7
1.4
1.4
0.9
0.1
2.5
2.0
0.5
0.0
1.8
0.6
2.9
Risk Premia and Conditional Tails
0.272
0.013
0.179
0.000
0.063
0.028
0.092
0.089
0.049
0.000
0.014
0.151
0.010
0.000
0.159
0.016
0.090
Testing Model Implications: Predicting Stock Returns
Bivariate Prediction
Coef.
Book-to-market
Cross section premium
Default return spread
Default yield spread
Dividend payout ratio
Dividend price ratio
Dividend yield
Earnings price ratio
Inflation
Long term return
Long term yield
Net equity expansion
Stock volatility
Term Spread
Treasury bill rate
Variance risk premium
0.73
16.9
1.45
3.36
0.49
1.80
1.78
1.66
-6.31
4.56
-3.45
-2.34
1.02
2.46
-3.92
9.19
One month horizon
Tail Tail
t Coef.
t
0.3
4.7
0.6
1.5
0.2
0.7
0.7
0.6
2.5
2.0
1.5
1.1
0.4
1.0
1.7
2.9
B. Kelly
6.47
16.7
6.42
5.72
6.54
6.27
6.23
6.18
6.05
5.86
7.71
5.66
6.62
5.59
6.96
6.73
2.8
4.6
2.8
2.5
2.8
2.7
2.7
2.6
2.6
2.5
3.2
2.3
2.9
2.3
3.0
2.0
R2
Coef.
One year horizon
Tail Tail
t Coef.
t
0.015
0.070
0.016
0.019
0.015
0.016
0.016
0.016
0.030
0.023
0.019
0.017
0.016
0.017
0.021
0.052
1.70
-2.17
-0.28
1.33
1.42
2.70
2.61
2.11
-1.70
1.64
-0.10
-0.63
1.04
2.70
-1.37
4.63
0.7
0.9
0.5
0.5
0.7
1.1
1.0
0.8
1.0
2.5
0.0
0.3
0.6
1.3
0.5
3.3
4.41
2.97
4.44
4.13
4.59
4.11
4.06
4.04
4.31
4.21
4.46
4.21
4.56
3.45
4.59
7.42
2.1
1.3
2.1
2.0
2.2
2.0
1.9
1.9
2.0
2.0
2.0
2.0
2.1
1.6
2.2
2.6
R2
Coef.
Five year horizon
Tail Tail
t Coef.
t
0.080
0.081
0.070
0.076
0.077
0.096
0.094
0.085
0.080
0.079
0.070
0.071
0.074
0.092
0.077
0.228
1.05
-1.45
-0.32
1.35
2.67
2.44
2.29
1.25
0.17
0.44
2.24
1.08
0.62
2.36
0.57
-3.90
0.4
0.9
1.4
0.7
1.2
1.2
1.2
0.5
0.1
1.4
1.3
0.6
0.6
1.3
0.3
2.9
5.04
4.15
5.07
4.75
5.34
4.74
4.72
4.83
5.07
5.00
4.24
5.44
5.14
4.20
4.99
5.45
Risk Premia and Conditional Tails
2.3
2.0
2.3
2.0
2.3
2.2
2.1
2.1
2.3
2.3
2.0
2.2
2.4
2.1
2.3
1.9
R2
0.283
0.286
0.273
0.290
0.325
0.325
0.319
0.287
0.272
0.273
0.313
0.281
0.275
0.319
0.275
0.313
Testing Model Implications: Predicting Stock Returns
Out-of-Sample Prediction (Lower Tail)
30
25
20
15
10
5
0
−5
1965
1969
1973 1977
1981 1985 1989
1993
1997 2001 2005
2008
Monthly Out-of-Sample R 2 = 1.3%
B. Kelly
Risk Premia and Conditional Tails
Testing Model Implications: The Cross Section of Returns
Theory predicts
1
2
Differential exposure to tail risk state variable implies cross-sectional
difference in expected returns
Negative price of tail risk: assets with high beta on tail risk have hedge
value
Test for cross-sectional relation between individual asset/portfolio
return tails and returns
1
2
3
Returns on tail risk beta-sorted portfolios
Fama-MacBeth tests
Robustness to alternative characteristics
B. Kelly
Risk Premia and Conditional Tails
Testing Model Implications: The Cross Section of Returns
Tail Beta-Sorted Portfolios: NYSE/AMEX/NASDAQ Stocks
Tail Risk Beta
Low
1
Panel A: Tail Risk Beta Only
All
6.40
2
3
4
High
5
Diff.
(5-1)
Diff.
t-stat
7.13
6.23
4.44
0.36
-6.03
2.55
Panel B: Market Beta / Tail Risk Beta
Low βMKT
1
6.71
7.41
7.11
2
5.91
6.19
5.97
3
4.32
5.12
4.34
4
2.54
3.36
2.53
High βMKT
5
-0.02
1.78
-0.35
6.40
4.50
3.25
1.06
-1.57
3.65
2.27
0.44
-1.01
-4.45
-3.06
-3.64
-3.88
-3.55
-4.43
1.76
2.01
2.08
1.87
2.20
Panel C: Market Equity / Tail Risk Beta
Small
1
10.16
9.27
10.15
2
2.76
4.48
3.46
3
4.81
5.81
4.99
4
6.71
7.72
6.45
Big
5
6.76
6.82
6.80
10.67
0.51
0.86
4.79
5.68
13.98
-4.33
-5.32
-2.18
1.43
3.82
-7.10
-10.13
-8.89
-5.33
1.72
2.81
4.18
3.69
2.31
Panel D: Book-to-Market / Tail Risk Beta
Growth
1
5.75
6.16
5.23
2
7.50
7.34
7.07
3
9.14
8.78
8.11
4
10.35
9.93
9.00
Value
5
11.09
10.66
11.64
3.05
5.23
7.22
8.98
12.01
-1.88
1.94
4.71
7.11
14.06
-7.63
-5.56
-4.43
-3.24
2.96
3.07
2.37
1.97
1.52
1.42
B. Kelly
Risk Premia and Conditional Tails
Testing Model Implications: The Cross Section of Returns
Stage 2 Fama-MacBeth Results: NYSE/AMEX/NASDAQ Stocks
-ζ (both)
-5.303
2.5
-ζ (lower)
-5.582
2.3
-6.197
2.9
-ζ (upper)
-4.399
2.4
-5.782
2.7
-0.112
0.1
R. Vol.
-5.586
2.9
-0.005
0.0
5.209
2.8
4.663
2.6
4.978
3.0
e
RMKT
-1.249
0.7
-1.150
0.7
-1.087
0.6
SMB
-3.787
2.4
-3.596
2.3
-3.583
2.3
HML
-0.940
0.6
-0.930
0.6
-0.588
0.4
5.264
6.0
0.143
4.883
5.6
0.151
4.647
5.3
0.135
Intercept
R
2
6.001
2.9
2.420
1.1
0.030
B. Kelly
2.268
1.0
0.029
-0.645
-0.2
0.020
4.082
2.2
0.053
6.309
3.1
4.023
2.2
0.056
6.863
3.6
-1.617
1.2
4.561
2.3
0.031
Risk Premia and Conditional Tails
Hedging Tail Risk
Tail Risk Betas: 25 Size/BM Ptfs and VIX
B. Kelly
Risk Premia and Conditional Tails
Outline
Introduction
An Economic Framework
Econometric Methodology
Empirical Findings
Conclusions
B. Kelly
Risk Premia and Conditional Tails
Conclusions
Derive link between return tails and risk premia in an affine pricing
framework with tail risk
Present new methodology for capturing dynamic extreme risk in the
economy
Identify substantial time variation in tails
Empirics consistent with predictions of structural model
1
2
3
Large variation in tail risk over time
Tail exponent forecasts excess market returns
Associated with large cross-sectional differences in average returns
?
What next?
1
Unified pricing with other asset classes (options and credit)
B. Kelly
Risk Premia and Conditional Tails