Campbell et la(2012) - New York University

An Intertemporal CAPM with Stochastic Volatility
Campbell, Giglio, Polk and Turley (2012)
Sai Ma
New York University
April 2014
Sai Ma (NYU)
Campbell, Giglio, Polk and Turley(2012)
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Motivation – Comparisons
Static Classical CAPM
E [Ri
Rf ] = βi ,M [E (RM )
Rf ]
Campbell and Vuolteenaho (2004)
E [Ri
Rf ] = γσ2M βi ,CFM + σ2M βi ,DR M
ICAPM with Stochastic Volatility
E [Ri
Sai Ma (NYU)
Rf ] = γσ2M βi ,CFM + σ2M βi ,DR M
Campbell, Giglio, Polk and Turley(2012)
1 2
ωσ β
2 M i ,V M
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Model – Preferences
Representative agent with Epstein-Zin preferences
Vt = ( 1
1 γ
θ
δ) Ct
+ δ Et
h
1 γ
Vt + 1
i
1
θ
θ
1 γ
δ : Discount factor (β in Campbell 1996)
γ : Relative risk aversion
ψ : Elasticity of intertemporal substitution (σ in Campbell 1996)
θ (1 γ) / (1 (1/ψ))
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Model – SDF
The Stochastic Discount Factor
Mt +1 =
δ
Ct
Ct + 1
1/ψ
!θ
Wt Ct
Wt +1
1 θ
Wt : market value of consumption stream owned by the agent,
including current consumption Ct
In Campbell (1996), Rm,t +1 = Wt +1 / (Wt Ct ) is used
The log SDF
mt +1 = θ ln δ
rt +1 = ln (Wt +1 / (Wt
Sai Ma (NYU)
θ
∆ct +1 + (θ
ψ
1 ) rt + 1
Ct )) : log return on invested wealth
Campbell, Giglio, Polk and Turley(2012)
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Model – Log Returns
Rm,t +1 =
W t +1
W t Ct
=
Ct
W t Ct
C t +1
Ct
W t +1
Ct
Log returns can be expressed as
rt +1 =
zt + ∆ct +1 + ht +1
zt = ln ((Wt Ct ) /Ct ) : log value of reinvested wealth per unit
consumption
ht +1 = ln (Wt +1 /Ct +1 ) : future value of a consumption claim
Capture the e¤ects of intertemporal hedging on asset prices.
log SDF can be expressed without reference to consumption growth
mt +1 = θ ln δ
Sai Ma (NYU)
θ
θ
zt + ht +1
ψ
ψ
Campbell, Giglio, Polk and Turley(2012)
γrt +1
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Model – ICAPM
Assumption: asset returns are jointly conditionally lognormal
Allow changing conditional volatility (Campbell 1998 assumed
conditional Homoscedasticity)
Take logs of Euler Equation,
1
Et [mt +1 + ri ,t +1 ] + Vart [mt +1 + ri ,t +1 ] = 0
2
Using mt and risk premium on any test asset, the ICAPM pricing
equation is
Et ri ,t +1
1
rf ,t + Vart [ri ,t +1 ] = γCovt [ri ,t +1 , rt +1 ]
2
θ
Covt [ri ,t +1 , ht +1 ]
ψ
Relates the risk premium on any asset to the asset’s covariance with
wealth return and with shocks to future consumption claim values
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Model – Hedging Component
ht +1 = ln (Wt +1 /Ct +1 )
Step 1: Log-linear about z¯
κ + ρzt +1
ht + 1
ρ = exp (z¯ ) / (1 + exp (z¯ ))
1
C
W
Step 2: Using Euler equation applied to wealth portfolio itself,
zt = ψ ln δ + (ψ
1) Et rt +1 + Et ht +1 +
ψ1
Vart [mt +1 + rt +1 ]
θ2
Combine Step 1 and 2, the innovation in ht +1
ht + 1
Sai Ma (NYU)
Et h t + 1 = ( Et + 1
Et ) ρ
(ψ
1) rt +2 + ht +2
[mt +2 + rt +2 ]
+ ψθ 12 Vart +1
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Model – Hedging Component
Solving forward,
ht + 1
= (ψ
1ψ
+
2θ
= (ψ
Et ht +1
(
1)
(
( Et + 1
∞
Et ) ∑ ρj rt +1 +j
j =1
∞
(Et +1
)
Et ) ∑ ρj Vart +j [mt +1 +j + rt +1 +j ]
j =1
)
1ψ
NRISK ,t +1
1) NDR ,t +1 +
2θ
NDR ,t +1 : "News about discount rates" (revisions in expected future
return) from Campbell and Vuolteenaho (2004)
NRISK ,t +1 : Variance of future log-return and log SDF (revisions in
expectation of future risk)
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Model – ICAPM Pricing Equation
With innovation of hedging component, ICAPM becomes
1
rf ,t + Vart ri ,t +1
2
= γCov [ri ,t +1 , rt +1 ] + (γ 1) Covt [ri ,t +1 , NDR ,t +1 ]
1
Covt [ri ,t +1 , NRISK ,t +1 ]
2
Et ri ,t +1
An extension of ICAPM in Campbell (1993)
All else equal, if γ > 1, assets which hedge aggregate discount rates
(Covt [ri ,t +1 , NDR ,t +1 ] < 0) or aggregate risk
(Covt [ri ,t +1 , NRISK ,t +1 ] > 0) have a lower expected returns
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Model – ICAPM Pricing Equation
Following Campbell and Vuolteenaho (2004), decomposing the
market return into cash-‡ow news and discount-rate news
1
Et ri ,t +1 rf ,t + Vart ri ,t +1
2
= γCov [ri ,t +1 , NCF ,t +1 ] + Covt [ri ,t +1 , NDR ,t +1 ]
1
Covt [ri ,t +1 , NRISK ,t +1 ]
2
If γ > 1, price of risk for cash-‡ow news is larger than that for
discount-rate news
Risk averse investors will demand a higher premium to hold assets that
covary with market’s cash-‡ow news
Extra term shows the risk premium associated with exposure to news
about future risks
By-product of relaxing conditional homoscedasticity
An asset providing positive return when risk expectation increase will
o¤er a lower return
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Model – From Risks to Volatility
NRISK ,t +1 = (Et +1
Et ) ∑j∞=1 ρj Vart +j [mt +1 +j + rt +1 +j ]
Suppose the economy is described by …rst-order VAR
xt +1 = x¯ + Γ (xt
x¯ ) + σt ut +1
xt +1 : n 1 vector of state variables that has rt +1 as its …rst element,
σ2t +1 as its second element, and n 2 other variables that help to
predict the …rst and second moments of aggregate returns.
x¯ and Γ: n 1 vector and n n matrix of constant parameters
ut +1 is a vector of shocks to state variables normalized so that its …rst
element has unit variance
Assumption: ut +1 has a constant variance-covariance matrix Σ with
element Σ11 = 1
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Model – From Risks to Volatility
Key assumption: σ2t , equal to the conditional variance of market
returns, governs time-variation in the variance of all shocks to this
system.
Both market returns and state variables, including volatility itself, have
innovations whose variances move in proportion to one another
Makes the stochastic volatility process a¢ ne.
Given this structure, news about discount rate becomes
∞
NDR ,t +1 = (Et +1
=
Sai Ma (NYU)
e10 ρΓ (I
Et ) ∑ ρj rt +1 +j
j =1
1
ρΓ)
Campbell, Giglio, Polk and Turley(2012)
σ t ut + 1
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Model – From Risks to Volatility
Recall:
mt +1 = θ ln δ
θ
θ
zt + ht +1
ψ
ψ
γrt +1
Linear in state variables.
All shocks to mt are proportional to σt
Vart [mt +1 + rt +1 ] ∝ σ2t
The conditional variance Vart (mt +1 + rt +1 ) /σ2t
that independent of state variables
Sai Ma (NYU)
Campbell, Giglio, Polk and Turley(2012)
ω is a constant
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Model – From Risks to Volatility
News of risk is proportional to news of market return variance Nv
∞
Et ) ∑ ρj Vart +j [mt +1 +j + rt +1 +j ]
NRISK ,t +1 = (Et +1
= ω
n
j =1
ρe20
(I
ρΓ)
1
σ t ut + 1
= ωNv ,t +1
o
Empirically testable ICAPM with stochastic volatility
1
rf ,t + Vart ri ,t +1 = γCov [ri ,t +1 , NCF ,t +1 ]
2
1
ωCovt [ri ,t +1 , Nv ,t +1 ]
+Covt [ri ,t +1 , NDR ,t +1 ]
2
Et ri ,t +1
Covariances with news about three key attributes of market portfolio
(cash ‡ows, discount rates and volatility) describe the cross section of
average return.
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Model – From Risks to Volatility
They showed that ω solves
1
0 = ω 2 xV ΣxV0
4
ω 1
(1
γ) xCF ΣxV0 + (1
0
γ)2 xCF ΣxCF
xCF ut +1 = σ1t NCF ,t +1 and xV ut +1 = σ1t NV ,t +1 : error-to-news
vectors that map VAR innovations to volatility-scaled news terms
The only valid solution is
1
ω=
Sai Ma (NYU)
(1
γ) xCF ΣxV0
s
(1 (1 γ) xCF ΣxV0 )2
0 )
(1 γ)2 (xV ΣxV0 ) (xCF ΣxCF
1
0
2 xV ΣxV
Campbell, Giglio, Polk and Turley(2012)
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Model – Conditions for Existence
ω has a real solution only if
( ρn
1) (1
γ) σcf σv
1
ρn is the correlation between news terms NCF and NV ,
σcf and σv : standard deviation of the scaled news
N CF ,t +1
σt
and
N V ,t +1
σt
They rationalized this condition with related to the existence of a
value function
Given VAR estimates, the real solution to ω for the range of
γ 2 [0, 6.9]
The later empirical analysis will respect this condition
Given high average return in risky assets in the historical data, the
estimate of γ often hits the upper bound of 6.9
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
Sai Ma (NYU)
Campbell, Giglio, Polk and Turley(2012)
\ t +1
RVAR
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
\ t +1
RVAR
RVARt : within-quarter realized variance of daily returns for time t
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
\ t +1
RVAR
RVARt : within-quarter realized variance of daily returns for time t
\ t +1 : Predicted values for RVAR at each time t + 1 via WLS
RVAR
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
\ t +1
RVAR
RVARt : within-quarter realized variance of daily returns for time t
\ t +1 : Predicted values for RVAR at each time t + 1 via WLS
RVAR
3
PE : Price-Earning ratio (S&P 500 index)
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
\ t +1
RVAR
RVARt : within-quarter realized variance of daily returns for time t
\ t +1 : Predicted values for RVAR at each time t + 1 via WLS
RVAR
3
4
PE : Price-Earning ratio (S&P 500 index)
TY : Term yield spread (Global Financial Data)
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
\ t +1
RVAR
RVARt : within-quarter realized variance of daily returns for time t
\ t +1 : Predicted values for RVAR at each time t + 1 via WLS
RVAR
3
4
PE : Price-Earning ratio (S&P 500 index)
TY : Term yield spread (Global Financial Data)
Di¤erence between the log yield on 10 year US constant maturity bond
and the log yield on the 3-Month US treasury bill.
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
\ t +1
RVAR
RVARt : within-quarter realized variance of daily returns for time t
\ t +1 : Predicted values for RVAR at each time t + 1 via WLS
RVAR
3
4
PE : Price-Earning ratio (S&P 500 index)
TY : Term yield spread (Global Financial Data)
Di¤erence between the log yield on 10 year US constant maturity bond
and the log yield on the 3-Month US treasury bill.
5
VS : Small-stock value spread (six benchmark equity portfolios from
French)
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
\ t +1
RVAR
RVARt : within-quarter realized variance of daily returns for time t
\ t +1 : Predicted values for RVAR at each time t + 1 via WLS
RVAR
3
4
PE : Price-Earning ratio (S&P 500 index)
TY : Term yield spread (Global Financial Data)
Di¤erence between the log yield on 10 year US constant maturity bond
and the log yield on the 3-Month US treasury bill.
5
6
VS : Small-stock value spread (six benchmark equity portfolios from
French)
DEF : Default Spread, di¤erence between the log yield on Moody’s
BAA and AAA bonds.
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State Variables
Six state variables x for VAR, the quarterly data from 1926:2 to 2011:4
1
rM , log real return on Market (CRSP VW)
2
EVAR, expected market variance, EVARt
\ t +1
RVAR
RVARt : within-quarter realized variance of daily returns for time t
\ t +1 : Predicted values for RVAR at each time t + 1 via WLS
RVAR
3
4
PE : Price-Earning ratio (S&P 500 index)
TY : Term yield spread (Global Financial Data)
Di¤erence between the log yield on 10 year US constant maturity bond
and the log yield on the 3-Month US treasury bill.
5
6
VS : Small-stock value spread (six benchmark equity portfolios from
French)
DEF : Default Spread, di¤erence between the log yield on Moody’s
BAA and AAA bonds.
Shocks to the DEF should re‡ect some news about aggregate default
probabilities
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State Variables – Short-run Volatility Estimation
Given heteroscedasticity of innovations to their variables, they
estimate the regression using Weighted Least Squares (WLS )
The results indicated Past realized variance strongly predicts future
realized variance.
An increase in either PE or DEF predicts higher future realized
volatility
Higher PE predicts higher RVAR might seem surprising
PE cleans up the information in DEF concerning future volatility
R 2 is heavily in‡uenced by occasional spikes in realized variance
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State Variables – Summary Statistics
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State Variables – Correlation
In Full sample, a high correlation between DEF and both PE and VS.
RVAR, EVAR have high persistence and high correlation with DEF .
In Early sample (1926-1963), PE is negatively correlated with RVAR,
VS and EVAR
Re‡ects the high volatility that occurred during Great Depression when
prices were relatively low
In modern sample (1963-2011), PE is uncorrelated with RVAR but
positively correlated with VS and thus positively correlated with
EVAR
Re‡ects the episodes with high volatility and high stock prices,
Technology boom of the later 1990s were more prevalent in the modern
than episodes with high volatility and low stock prices as in early 1980s
recession
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VAR Estimation
xt +1 = x¯ + Γ (xt
x¯ ) + σt ut +1
State Variables
xt +1 = [rM ,t +1 , EVARt +1 , PEt +1 , TYt +1 , DEFt +1 , VSt +1 ]
x¯ : vector of the means of variables
Γ:6
σ2t ,
6 matrix of constant parameters
proxied for by EVAR, scales ut +1 ’s variance covariance matrix Σ
Estimate the second-stage VAR using WLS, with weight of each
observation pair (xt +1 , xt ) initially based on (EVARt ) 1 .
Still constrain both the weights across observations and the …tted
values of regression forecasting EVAR.
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VAR Estimation
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VAR Estimation – Full Sample
PE , DEF negatively predict future returns, though marginal
signi…cance.
Higher conditional variance EVAR is associated with higher future
returns
Both high PE and DEF predict higher future conditional variance of
returns
High past market returns forecast a lower EVAR, higher PE and lower
DEF
High correlation among PE , DEF , VS and EVAR complicates the
interpretation of individual e¤ect.
They also documented sample correlation and autocorrelation
matrices of both the unscaled residual σt ut +1 and the scaled residuals
ut + 1
Much of the sample autocorrelation in the unscaled residuals ut +1 is
eliminated by WLS approach.
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VAR Estimation – VAR Speci…cation Test
EVAR signi…cantly predicts with a positive sign on all the squared
errors of the VAR supports the underlying assumption that σ2t drives
the volatility of all innovations.
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News Terms – Covariance Between News
News about future variance has signi…cant volatility, with nearly a
third of the variability of discount-rate news
Variance news is negatively correlated with cash-‡ow news
"Leverage E¤ect", news about low cash ‡ows is associated with news
about high future volatility.
Variance news correlates negatively with discount-rate news
News of high volatility coincides with news of low future real return.
Slightly negative correlation of 0.02 between volatility news and
contemporaneous market return
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News Terms – Corr. Between Shocks and Innovations
Decompose innovations σ2t ut +1 into NCF ,t +1 , NDR ,t +1 and NV ,t +1
and unpack EVAR to express the news terms as a function of
rM , PE , TY , VS, DEF , and RVAR.
Innovations to RVAR are mapped more than one-to-one to news
about future volatility.
Innovations in PE , DEF and VS are associated with news of higher
future volatility.
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News Terms – Smoothed Series for News
There is considerable time variation in NV
The episodes of news of high future volatility during Great Depression
and just before the beginning of WWII
Spikes in news about future volatility are found in early and late 70s,
and the 1987 crash of the stock market
The recession of the late 2000s is characterized by strongly negative
cash-‡ow news
The recovery from …nancial crisis has brought positive cash-‡ow news
together with news about lower future volatility
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Long-run Volatility
Volatility is strongly predictable by realizations of volatility itself, PE
and DEF .
In order to capture the long-horizon component of volatility, they
regress realized discounted long-run variance up to period
h, LHRVARh on di¤erent forecasting models of long-run variance
LHRVARh =
4
∑hj=1 ρj
1 RVAR
t +j
∑hj=1 ρj
1
They also estimated two standard GARCH-type models, designed to
capture the long-run dynamics of volatility process
EGARCH and FIGARCH
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Long-run Volatility
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Long-run Volatility
Both EGARCH and FIGARCH forecasts by themselves capture a
signi…cant portion of the variation in long-run realized volatility
VAR variables provide as good or between explanatory power
RVAR, PE and DEF are strongly statistically signi…cant
VAR-implied forecast produces good forecasts of volatility in both
direction and magnitude .
PE has no information about low-frequency variation in volatility but
DEF forecasts nearly 22% of the variation in LHRVAR40
With DEFO, the R 2 increase signi…cantly, where as not the case for
PEO, indicating its strong predictive power.
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Long-run Volatility
DEF and PE contain information about future volatility not captured
by simple univariate models
Even those like FIGARCH or EGARCH that are designed to …t
long-run movements in volatility, and the VAR method for calculating
long-horizon forecasts, preserves this information.
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Test Assets
Constructed three sets of portfolios to use as test assets.
1
The primary cross section consists of the excess returns on 25 ME and
BE /ME -sorted portfolios (Characteristics-sorted test assets)
In the empirical analysis, they consider two main sub-samples: early
(1931:3 - 1963:3) and modern (1963:4-2011:4)
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Test Assets
Constructed three sets of portfolios to use as test assets.
1
2
The primary cross section consists of the excess returns on 25 ME and
BE /ME -sorted portfolios (Characteristics-sorted test assets)
Second set of six portfolios double-sorted on past risk loadings to
market and variance risk (Risk-sorted test assets)
In the empirical analysis, they consider two main sub-samples: early
(1931:3 - 1963:3) and modern (1963:4-2011:4)
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Test Assets
Constructed three sets of portfolios to use as test assets.
1
2
The primary cross section consists of the excess returns on 25 ME and
BE /ME -sorted portfolios (Characteristics-sorted test assets)
Second set of six portfolios double-sorted on past risk loadings to
market and variance risk (Risk-sorted test assets)
Intended to alleviate Daniel and Titman (1997, 2012) critique on test
asset pricing model using only portfolios sorted by characteristics
known to be related to average return, such as size and value
In the empirical analysis, they consider two main sub-samples: early
(1931:3 - 1963:3) and modern (1963:4-2011:4)
Sai Ma (NYU)
Campbell, Giglio, Polk and Turley(2012)
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Test Assets
Constructed three sets of portfolios to use as test assets.
1
2
The primary cross section consists of the excess returns on 25 ME and
BE /ME -sorted portfolios (Characteristics-sorted test assets)
Second set of six portfolios double-sorted on past risk loadings to
market and variance risk (Risk-sorted test assets)
Intended to alleviate Daniel and Titman (1997, 2012) critique on test
asset pricing model using only portfolios sorted by characteristics
known to be related to average return, such as size and value
Sorted by Market CAPM betas and volatility ∆VAR beta
In the empirical analysis, they consider two main sub-samples: early
(1931:3 - 1963:3) and modern (1963:4-2011:4)
Sai Ma (NYU)
Campbell, Giglio, Polk and Turley(2012)
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Test Assets
Constructed three sets of portfolios to use as test assets.
1
2
The primary cross section consists of the excess returns on 25 ME and
BE /ME -sorted portfolios (Characteristics-sorted test assets)
Second set of six portfolios double-sorted on past risk loadings to
market and variance risk (Risk-sorted test assets)
Intended to alleviate Daniel and Titman (1997, 2012) critique on test
asset pricing model using only portfolios sorted by characteristics
known to be related to average return, such as size and value
Sorted by Market CAPM betas and volatility ∆VAR beta
3
Cross section of option, bond and equity returns for 1986:1-2011:4
(Non-equity test assets)
In the empirical analysis, they consider two main sub-samples: early
(1931:3 - 1963:3) and modern (1963:4-2011:4)
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Test Assets
Constructed three sets of portfolios to use as test assets.
1
2
The primary cross section consists of the excess returns on 25 ME and
BE /ME -sorted portfolios (Characteristics-sorted test assets)
Second set of six portfolios double-sorted on past risk loadings to
market and variance risk (Risk-sorted test assets)
Intended to alleviate Daniel and Titman (1997, 2012) critique on test
asset pricing model using only portfolios sorted by characteristics
known to be related to average return, such as size and value
Sorted by Market CAPM betas and volatility ∆VAR beta
3
Cross section of option, bond and equity returns for 1986:1-2011:4
(Non-equity test assets)
S&P 100 index straddle returns, return on Barclays Capital High Yield
Bond Index (HYRET ) and on Barclays Capital Investment Grade Bond
Index (IGRET ) (Risky bonds)
In the empirical analysis, they consider two main sub-samples: early
(1931:3 - 1963:3) and modern (1963:4-2011:4)
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Test Assets
Constructed three sets of portfolios to use as test assets.
1
2
The primary cross section consists of the excess returns on 25 ME and
BE /ME -sorted portfolios (Characteristics-sorted test assets)
Second set of six portfolios double-sorted on past risk loadings to
market and variance risk (Risk-sorted test assets)
Intended to alleviate Daniel and Titman (1997, 2012) critique on test
asset pricing model using only portfolios sorted by characteristics
known to be related to average return, such as size and value
Sorted by Market CAPM betas and volatility ∆VAR beta
3
Cross section of option, bond and equity returns for 1986:1-2011:4
(Non-equity test assets)
S&P 100 index straddle returns, return on Barclays Capital High Yield
Bond Index (HYRET ) and on Barclays Capital Investment Grade Bond
Index (IGRET ) (Risky bonds)
Also include the return on Market (RMRF ), size (SMB ), and value
(HML) equity factor and cross section of currency portfolios
In the empirical analysis, they consider two main sub-samples: early
(1931:3 - 1963:3) and modern (1963:4-2011:4)
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Beta Measurement
1
rf ,t + Vart ri ,t +1 = γCov [ri ,t +1 , NCF ,t +1 ]
2
1
ωCovt [ri ,t +1 , Nv ,t +1 ]
+Covt [ri ,t +1 , NDR ,t +1 ]
2
Et ri ,t +1
Simpli…cation: Unconditional version
Expressing the ICAPM Pricing Equation in terms of betas
E [Ri
Rf ] = γσ2M βi ,CFM + σ2M βi ,DR M
Cov (r ,N
)
1 2
ωσ β
2 M i ,V M
Cov (r , N
)
βi ,CF M = Var (r i ,t E CF r,t ) , βi ,DR M = Var (r i ,t E DRr ,t ) ,
t 1 M ,t
t 1 M ,t
M ,t
M ,t
Cov (r ,N )
βi ,V M = Var (r i ,tE V ,tr )
M ,t
t 1 M ,t
As comparison, In Campbell and Vuolteenaho (2004)
E [Ri
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Rf ] = γσ2M βi ,CFM + σ2M βi ,DR M
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Beta Measurement - Characteristic-sorted Test Assets
Early Sample
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Beta Measurement - Characteristic-sorted Test Assets
Early Sample
In the pre-1963 sample period, value stocks have both higher
cash-‡ow and higher discount-rate betas growth stocks.
On average 0.12 higher βCF and 0.20 higher in βDR
Small stocks have higher cash-‡ow betas and discount-rate betas
than large stock
On average 0.14 higher βCF and 0.34 higher in βDR
Value stocks and small stocks are also riskier in terms of volatility
betas
Value asset has an on average 0.05 lower βV
Small asset has an on average 0.04 lower βV
In sum, value and small stocks were unambiguously riskier than
growth and large stocks in the early sample
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Beta Measurement - Characteristic-sorted Test Assets
Modern Sample
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Beta Measurement - Characteristic-sorted Test Assets
Modern Sample
In modern sample, Value stock still have higher βCF , but much lower
βDR than growth stocks
Value stock continue to have lower βV and is even greater than that
in the early period
Asset has an on average 0.13 lower βV
42% higher than the corresponding di¤erence in the early period (0.05)
Growth stocks are relative hedges for two key aspects of the
investment opportunity set in the post-1963 sample
Hedge news about future real stock returns (Campbell and
Vuolteenaho (2004))
Novel …nding: Growth stocks hedge news about the variance of market
return
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Beta Measurement - The Changing Volatility Beta
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Beta Measurement - The Changing Volatility Beta
The average βV of the 25 Size- and B/M portfolios change sign from
the early to the modern sub-period
-0.06 for pre-1963 to 0.09 for post-1964
Clear Distinction between single-period realized variance RVAR and
long run volatility news
Related to the change in sign over time in correlation between PE
and market returns with PE -adjusted DEF , proxy for news about
long-horizon variance.
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Beta Measurement - Risk-sorted Test Assets Early Sample
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Beta Measurement - Risk-sorted Test Assets Early Sample
In pre-1963, high CAPM beta stocks have both higher cash ‡ow and
higher discount-rate betas than low CAPM beta stocks
On average 0.19 higher in βCF and 0.44 higher in βDR
Low volatility stocks have higher cash ‡ow and discount-rate betas
than high volatility beta stocks in this sub-sample
On average 0.06 higher in βCF and 0.11 higher in βDR
High CAPM beta and low volatility stocks are also riskier in terms of
volatility beta
High CAPM beta stocks have 0.04 lower in (negative) βV , on average
Low volatility stocks have 0.023 lower in (negative) βV , on average
In summary, high CAPM beta and low volatility beta stocks were
unambiguously riskier than low CAPM beta and high volatility beta
stocks in the pre-1963 periods.
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Beta Measurement - Risk-sorted Test Assets Modern
Sample
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Beta Measurement - Risk-sorted Test Assets Modern
Sample
In the modern period, high CAPM beta stocks again have higher βCF
and βDR
However, high CAPM beta stocks are no longer riskier in terms of
volatility beta βV
0.07 higher (positive) βV than low CAPM, on average.
Three-beta model potentially explains why stocks with high past CAPM
betas have o¤ered relatively little extra return in the modern period.
In the post-1963 period, high volatility stock is still riskier with 0.06
lower (positive) βV than low volatility stock
Sorts on volatility stock also generated spread in βDR , but no spread
in βCF
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Beta Measurement - Non-equity Test Assets
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Beta Measurement - Non-equity Test Assets
Recall: Consist of S&P 100 index straddle position and three equity
factors, and the default bond factor over the period 1986-2011
Consistent with the nature of a straddle bet, (betting for volatility)
Straddle has a very large volatility beta of 0.38. Large negative
discount-rate beta of -1.71, and a large negative cash-‡ow beta of -0.39
With higher βCF , βDR and lower βV , high interest rate countries are
unambiguously riskier than low interest rate countries over 1984-2010
period.
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Beta Pricing
Evaluate the performance of …ve asset pricing models
1
Traditional CAPM that restrict βCF and βDR to have the same price of
risks and sets the price of variance risk to zero
Each model is estimated in two di¤erent forms:
1
2
Restricted zero-beta rate equal to T-bill beta
Unrestricted zero-beta rate (Black version)
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Beta Pricing
Evaluate the performance of …ve asset pricing models
1
2
Traditional CAPM that restrict βCF and βDR to have the same price of
risks and sets the price of variance risk to zero
Two beta intertemporal asset pricing model of CV (2004) that restricts
the price of discount-rate risk to σM
Each model is estimated in two di¤erent forms:
1
2
Restricted zero-beta rate equal to T-bill beta
Unrestricted zero-beta rate (Black version)
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Beta Pricing
Evaluate the performance of …ve asset pricing models
1
2
3
Traditional CAPM that restrict βCF and βDR to have the same price of
risks and sets the price of variance risk to zero
Two beta intertemporal asset pricing model of CV (2004) that restricts
the price of discount-rate risk to σM
Three-beta interpemporal asset pricing model that restricts the price of
discount-rate risk to σM , and puts restriction on NCF and NV by
(ρn 1) (1 γ) σcf σv 1
Each model is estimated in two di¤erent forms:
1
2
Restricted zero-beta rate equal to T-bill beta
Unrestricted zero-beta rate (Black version)
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Beta Pricing
Evaluate the performance of …ve asset pricing models
1
2
3
4
Traditional CAPM that restrict βCF and βDR to have the same price of
risks and sets the price of variance risk to zero
Two beta intertemporal asset pricing model of CV (2004) that restricts
the price of discount-rate risk to σM
Three-beta interpemporal asset pricing model that restricts the price of
discount-rate risk to σM , and puts restriction on NCF and NV by
(ρn 1) (1 γ) σcf σv 1
Partially-constrained three-beta model that restricts the restricts the
price of discount-rate risk to σM , but no restriction on NCF and NV
Each model is estimated in two di¤erent forms:
1
2
Restricted zero-beta rate equal to T-bill beta
Unrestricted zero-beta rate (Black version)
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Beta Pricing
Evaluate the performance of …ve asset pricing models
1
2
3
4
5
Traditional CAPM that restrict βCF and βDR to have the same price of
risks and sets the price of variance risk to zero
Two beta intertemporal asset pricing model of CV (2004) that restricts
the price of discount-rate risk to σM
Three-beta interpemporal asset pricing model that restricts the price of
discount-rate risk to σM , and puts restriction on NCF and NV by
(ρn 1) (1 γ) σcf σv 1
Partially-constrained three-beta model that restricts the restricts the
price of discount-rate risk to σM , but no restriction on NCF and NV
Unrestricted three-beta model that freely estimates risk prices for
cash-‡ow, discount rate and volatility betas
Each model is estimated in two di¤erent forms:
1
2
Restricted zero-beta rate equal to T-bill beta
Unrestricted zero-beta rate (Black version)
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Beta Pricing
Parameters are estimated from a cross-sectional regression
R¯ ie = g0 + g1 βˆ i ,CFM + g2 βˆ i ,DR M + g3 βˆ i ,V M + ei
R¯ ie = R¯ i
R¯ rf : sample average simple excess return on asset i
They report composite pricing error, computed as a quadratic form of
the pricing errors.
Standard errors are produced with a bootstrap from 10,000 simulated
realizations
A conditional on estimated news terms but B incorporate full
estimation uncertainty of news term
Implied risk-aversion coe¢ cient, γ =
g1
g2
and implied sensitivity of
news about risk to news about market variance ω =
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g2
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Beta Pricing – Characteristic-sorted Test Assets: Early
Sample
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Beta Pricing – Characteristic-sorted Test Assets: Early
Sample
In pre-1963 period, all models explain the cross-section of stock
returns reasonably well
For the Black version of the three-beta ICAPM, the spread in
volatility betas generates an annualized spread in average returns of
1.6% compared to a comparable spread of 7.3% and 3.2 % for
cash-‡ow and discount-rate betas
Variation in volatility betas accounts for 2% of the variation in
explained returns compared to 38% and 7% for cash ‡ow and
discount-rate betas respectively
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Beta Pricing – Characteristic-sorted Test Assets: Modern
Sample
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Beta Pricing – Characteristic-sorted Test Assets: Modern
Sample
In modern sample, restricted zero-beta CAPM and three-beta model
do a very poor job of explaining cross-sectional variation
Unconstrained zero-beta rate version of the two-beta model does a
better job than CAPM but with implied γ of 20.7
Restrict the risk price for discount-rate and variance news and allow
an unrestricted zero-beta, huge improvement for …t.
With implied γ of 6.9, neither version of ICAPM with stochastic
volatility is rejected at the 5% level
For the Black version of the three-beta ICAPM, the spread in
volatility betas generates an annualized spread in average returns of
5.2% compared to a comparable spread of 2.8% and 2.2 % for
cash-‡ow and discount-rate betas
Variation in volatility betas accounts for 104% of the variation in
explained returns compared to 19% and 13% for cash ‡ow and
discount-rate betas respectively
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Beta Pricing – Characteristic-sorted Test Assets: Modern
Sample
The relatively poor performance of the risk-free rate version of the
three beta ICAPM is related to the derived link between γ and ω.
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Beta Pricing – Inclusion of Risk-sorted Test Assets:
Modern Sample
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Beta Pricing – Inclusion of Risk-sorted Test Assets:
Modern Sample
Modern Sample provides the stronger challenge to the asset-pricing
models
With inclusion of risk-sorted assets, Zero-beta rate three-beta ICAPM
is not rejected by the data, while both versions of the CAPM are
rejected
Relatively high R 2 for unrestricted zero-beta rate version of the
volatility ICAPM (68%) is not disproportionately due to
characteristics-sorted portfolios
R 2 for the risk-sorted subset (80%) is not only comparable to but also
larger than R 2 for the characteristics-sorted portfolios (68%)
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Beta Pricing – Non-equity Test Assets
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Beta Pricing – Non-equity Test Assets
Equity-based estimate of the three-beta model explains roughly 31%
of the realized straddle premium.
CAPM and ICAPM models are rejected at 5-percent level
Two-beta ICAPM is not rejected,but with implied γ = 53
Due to extremely low realized returns on the straddle portfolio.
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Beta Pricing – Non-equity Test Assets: Currency
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Beta Pricing – Non-equity Test Assets: Currency
Restricted zero-beta CAPM does a poor job of explaining
cross-sectional variation in currency returns
Unconstrained zero-beta rate version of CV model produces a 60% R 2
but with implied γ of 14.4
Unrestricted zero-beta Three-beta model explained variation in
currency by 81.5%, with a implied γ = 6.9
Not rejected at 5-percent by either set of critical values
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Robustness
Successful zero-beta-rate volatility ICAPM pricing in the modern
period requires PE , DEF and VS in the VAR.
Negative βV for HML and successful zero-beta rate ICAPM pricing in
both time periods are very robust to using di¤erent VAR estimation
methods and di¤erent realized variance estimation
Adding two additional state variables CAY , FIGARCH does not alter
the success of zero-beta rate ICAPM pricing in both time periods
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Concluding Remark – US Financial History
Plot of smoothed shocks for CAPM (NCF NDR ), two-beta ICAPM
(γNCF NDR ) and three-beta ICAPM γNCF NDR 21 ωNV
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Concluding Remark
Extend Campbell (1993) and CV (2004) by allowing for stochastic
volatility
Investment opportunity may deteriorate because of increased volatility
Stocks’risks now determined by news about cash ‡ows, discount rates
and future market volatility
Model has 3-D risk, but prices of all risks determined by risk aversion
Low-frequency movements in market volatility tied to the default
spread
Inclusion of volatility risk help explain the average returns on
non-equity test assets
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Concluding Remark
Extend Campbell (1993) and CV (2004) by allowing for stochastic
volatility
Investment opportunity may deteriorate because of increased volatility
Stocks’risks now determined by news about cash ‡ows, discount rates
and future market volatility
Model has 3-D risk, but prices of all risks determined by risk aversion
Low-frequency movements in market volatility tied to the default
spread
Inclusion of volatility risk help explain the average returns on
non-equity test assets
Future research direction
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Concluding Remark
Extend Campbell (1993) and CV (2004) by allowing for stochastic
volatility
Investment opportunity may deteriorate because of increased volatility
Stocks’risks now determined by news about cash ‡ows, discount rates
and future market volatility
Model has 3-D risk, but prices of all risks determined by risk aversion
Low-frequency movements in market volatility tied to the default
spread
Inclusion of volatility risk help explain the average returns on
non-equity test assets
Future research direction
1
Assumed that wealth portfolio of a representative investor can be
adequately proxied by a diversi…ed equity portfolio (invest 100% in
equity), can we explore stochastic volatility in alternative proxies
(corporate bonds, human capital)?
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Concluding Remark
Extend Campbell (1993) and CV (2004) by allowing for stochastic
volatility
Investment opportunity may deteriorate because of increased volatility
Stocks’risks now determined by news about cash ‡ows, discount rates
and future market volatility
Model has 3-D risk, but prices of all risks determined by risk aversion
Low-frequency movements in market volatility tied to the default
spread
Inclusion of volatility risk help explain the average returns on
non-equity test assets
Future research direction
1
2
Assumed that wealth portfolio of a representative investor can be
adequately proxied by a diversi…ed equity portfolio (invest 100% in
equity), can we explore stochastic volatility in alternative proxies
(corporate bonds, human capital)?
Test only the unconditional implications, how about a full conditional
test?
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