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University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School1
April 2007
Global Price of Market Risk And Country Inflation
Devraj Basubased on joint work with …
• Daniel Hung Durham Business School
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School2
investigating international asset pricing anomalies …
Introduction overview literature results extensionstheory
• International asset pricing anomalies have been observed-the value effect (Fama-French 1998), the momentum effect (Rouwenhorst 1998)
• The international CAPM is unable to price these anomalies
• The Fama-French factors, global or local arose to explain these anomalies
• However Bansal, Hsieh and Vishwnathan (1993) suggested that nonlinear SDFs work better than linear models
International Anomalies
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School3
investigating international asset pricing anomalies …
Introduction overview literature results extensionstheory
• Higher moments of the market return had been used in the US context starting with Kraus and Litzenberger (1978)
• More recently Harvey and Siddique (2000) and Dittmar (2002) used them to investigate asset pricing anomalies in the US
• Errunza and Sy (2005) used country inflation and its higher powers to explain the international anomalies
• Our goal is to use both global and local factors to try to price size, value and momentum effects in the US, UK and Japan
Nonlinear SDFs …
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School4
investigating international asset pricing anomalies …
Introduction overview literature results extensionstheory
• We first consider a global model with the World factor, its square and cube…
• … and add to it country inflation and its square
• A crucial aspect of our analysis is that the factor risk premiums are all time varying and are functions of lagged global variables (World index and term structure variables)
• We compare performance of our model with the country-specific Fama-French factors
Time-Varying Risk Premiums…
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School5
what is “conditioning information”?
generic model:
Introduction overview literature results extensions
asset returns given by …
with …
… no loss of generality!
theory
ttt zR )( 1
)()(var 11 ttt z
vector of traded asset returns
unpredictable shock
predicted component
set of (lagged) instruments
for example …macro-economic indicatorschanges in the term-structuremarket sentiment indicators
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School6
what can we do with “conditioning information”?
using conditioning info …
Introduction overview literature results extensions
we can model …
• asset return predictability• serial correlation• seasonalities
• time-varying moments• stochastic volatility• time-varying correlations
• non-normality of returns• kurtosis (‘fat tails’)• skewness
• time-varying risk premia
theory
this allows us to study …
• economic value of predictability• market timing• active asset management
• conditional asset pricing• tests of factor models• time-varying ‘betas’
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School7
asset pricing in the presence of conditioning information
pricing equation:
Introduction overview literature results extensions
an SDF must satisfy …
theory
1)(1 ttt RmE
stochastic discount factor
vector of ‘ones’
vector of traded asset returns
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School8
asset pricing in the presence of conditioning information
pricing equation:
Introduction overview literature results extensions
an SDF must satisfy …
theory
vector of ‘weights’ given as …functions of instruments t-1 = f(zt-1)
stochastic discount factor
vector of ‘ones’
1)(1 ttt RmE 1t 1t
vector of traded asset returns
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School9
asset pricing in the presence of conditioning information
pricing equation:
Introduction overview literature results extensions
an SDF must satisfy …
theory
)()( EE
taking unconditional expectations …
1)(1 ttt RmE 1t 1t
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School10
asset pricing in the presence of conditioning information
pricing equation:
Introduction overview literature results extensions
an SDF must satisfy …
theory
)()( EE 1)(1 ttt RmE 1t 1t
suppose we choose weights …so that E(1’t-1) = 1
this can be interpreted as thereturn on a managed portfolio!
price of portfolio
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School11
asset pricing in the presence of conditioning information
pricing equation:
Introduction overview literature results extensions
an SDF must satisfy …
hence …
… for all managed returns rt
theory
)()( EE 1)(1 ttt RmE 1t 1t
1)( ttrmE
implication:
for any candidate SDF mt …… the following are equivalent:
• mt prices all traded assets… conditionally correctly
• mt prices managed portfolios… unconditionally correctly
= 1
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School12
asset pricing with conditioning information
literature:
Introduction overview literature results extensions
• Chamberlain, RothchildE 51 (1983)
• Hansen, RichardE 55 (1987)
• Ferson, SiegelJF 56 (2001)
see also …• Cochrane “Asset Pricing”
theory
construct orthogonal representationof unconditionally efficient frontier
extend CR’s framework to incorporateconditioning information
derive weights of efficient portfoliosas function of conditioning instruments
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School13
tests of conditional asset pricing models
literature:
Introduction overview literature results extensions
• Bansal, Hsieh, Vishwanathan (1993)
• Ferson, Harvey (1993)
• Harvey, Siddique (2000)
• Errunza, Sy (2005)
theory
Nonlinear SDFs for international pricing
examine conditional international asset pricing models
test augmented CAPM on US portfolios
Country specific inflation factors
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School14
conditional factor models
generic factor model:
Introduction overview literature results extensions
asset returns are given by …
theory
vector of traded asset returns
vector of factors
conditional expected return
ttttt FR 11
vector of factor loadings
residual (idiosyncratic risk)
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School15
conditional factor models
generic factor model:
Introduction overview literature results extensions
asset returns are given by …
theory
ttttt FR 11
factors loadings …
can be time-varying function …… of conditioning infocapture asset’s exposure …… to systematic risk factors
• traditional approach:linear function of instruments
• in our framework:solve for optimal loadings
1101 tt z
11 tt z typicallynon-linear!
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School16
conditional factor models
generic factor model:
Introduction overview literature results extensions
asset returns are given by …
theory
ttttt FR 11
the residuals …
capture idiosyncratic risk …… should not be priced!
properties:
• zero mean• orthogonal to factor risk:
can be used to estimate model… e.g. GMM
0),cov( 1 ttt F
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School17
conditional factor models
generic factor model:
Introduction overview literature results extensions
asset returns are given by …
theory
ttttt FR 11
SDF representation:
the following is equivalent:
• there exist at-1 and Bt-1 so that
… is an admissible SDF
that is
… for all managed returns rt
tttt FBam 11
1)( ttrmE
1
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School18
tests of conditional factor models
step 1:
Introduction overview literature results extensions
factor-mimicking portfolios
with …
theory
1)1( tftfi
t rRrf
)( 111
11 ti
ittt q
ity 1:
the managed portfolio that has …maximal correlation with factor
second-moment matrix of returns:
mixed asset-factor moments:
)1)(1(11 ftfttt rRrRE
tfttt FrREQ )1(11
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School19
tests of conditional factor models
step 2:
Introduction overview literature results extensions
managed portfolio of factors
• consider the set of returns …… that can be written as
with
• denote by RF …
… the set of all such returns rt
theory
m
i
itf
itftt rfrr
11
01 )(
1)( 01 tE
managed portfolios made up of …factor-mimicking portfolios
portfolio constraint
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School20
tests of conditional factor models
step 2:
Introduction overview literature results extensions
frontier spanned by factors
• consider the set of returns …… that can be written as
with
• denote by RF …
… the set of all such returns rt
theory
m
i
itf
itftt rfrr
11
01 )(
1)( 01 tE
• maximum Sharpe ratio …… spanned by base assets:
… and by the factors:
)()(
supt
ft
RrF r
rrEF
t
)()(
supt
ft
Rr rrrE
t
managed portfolios of …traded base assets
now define …
managed portfolios of …factor-mimicking portf’s
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School21
tests of conditional factor models
step 2:
Introduction overview literature results extensions
managed portfolios of factors
• consider the set of returns …… that can be written as
with
• denote by RF …
… the set of all such returns rt
theory
m
i
itf
itftt rfrr
11
01 )(
1)( 01 tE
theorem:
we show …
for given set of factors …… the following are equivalent:
• the model admits an SDF… of the form
• there is a factor-portfolio that is… unconditionally efficient
in other words, if
1
22F
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School22
tests of conditional factor models
just for the record …
Introduction overview literature results extensions
the maximum Sharpe ratios are given by …
• the latter is attained by the portfolio
with
theory
)( 21
2 tHE 1
111
21
ttttH
)( 21,
2 tFF HE 1
111
1
11
111
1111
112
1,
ttttttttttttF YYYYH
11
11
1
11
111
1121,
1 )(1
tttttttt
tF
ft YYY
HE
r
10
1 )1( tftftF
t rfrr
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School23
tests of conditional factor models
An intermediate step
Introduction overview literature results extensions
• We also examine unconditional pricing… that is if E(mR)=1 where R are the base assets
• This is via taking unconditional expectations of the original pricing equation
• We test this by comparing
• This is similar to the Gibbons-Ross-Shanken test, except that it is possible for the optimal factor Sharpe ratio to be higher than the fixed-weight asset Sharpe ratio
theory
22FFW
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School24
tests of conditional factor models
An intermediate step
Introduction overview literature results extensions
• We derive a heuristic test for
is asymptotically distributed as a chi-squared variable with (N-K)(1+J) degrees of freedom. The extra J degrees of freedom are to allow it to price the managed strategies
theory
22* F
2
22*
1 F
F
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School25
synopsis:
Introduction overview literature results extensions
in this paper …… we develop the theory:
• explicitly construct …… factor-mimicking portfolio
• develop definition of…… ‘spanning distance’
• show distance proportional …… to Sharpe ratio difference
• explicitly characterize …… optimal factor loadings
theory
objectives:
develop methodology …… and investigate …
• does optimal ‘scaling’ …… improve performance?
• do World skewness and kurtosis and country inflation… really matter?
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results: Global versus Fama-French
test results: US size portfolios
Introduction overview literature results extensionstheory
model factor(s) Sharpe ratio
fixed scaled
asset frontier
1.27 2.21
factor frontier
CAPM WLD 0.56 0.64
+skew+ kurtosis WLD SKEW KURT 1.12 1.67
FAMA-FRENCH WLD SMB HML 0.73 1.34
test assets: US Size decile portfolios; instruments: WORLD,TB1M, TSPR, CONV
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School27
results: Adding Country Inflation Factors
test results
Introduction overview literature results extensionstheory
model factor(s) Sharpe ratio
fixed scaled
asset frontier
1.27 2.21
factor frontier
… … + inf MMRF
SKEW KURT INF INF2 1.13 2.08**
FAMA-FRENCH
… … + inf MMRF
SMB HML INF INF2 1.14 1.82
test assets: US Size decile portfolios; instruments: WORLD,TB1M, TSPR, CONV
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School28
results: US momentum portfolios
test results: US Momentum Decile Portfolio
Introduction overview literature results extensionstheory
model factor(s) Sharpe ratio
fixed scaled
asset frontier
1.29 1.87
factor frontier
CAPM WLD 0.83 1.01
WLD SKEW KURT 0.98 1.30
… … + inf WLD SKEW KURT INF INF2 1.01 1.75**
FAMA-FRENCH WLD SMB HML 0.85 1.29
… … + kurtosis WLD SMB HML INF INF2 1.05 1.51
test assets: US momentum decile portfolios; instruments: WORLD,TB1M, TSPR, CONV
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School29
results: Japanese Value Portfolios
test results: Japan Book-to-Market Portfolios
Introduction overview literature results extensionstheory
model factor(s) Sharpe ratio
fixed scaled
asset frontier
0.83 1.51
factor frontier
CAPM WLD 0.03 0.55
WLD SKEW KURT 0.43 1.04
… … + inf WLD SKEW KURT INF INF2 0.47 1.53**
FAMA-FRENCH WLD SMB HML 0.74 1.00
… … + kurtosis WLD SMB HML INF INF2 0.74 1.26
test assets: Japanese Value Portfolios; instruments: WORLD,TB1M, TSPR, CONV
University of Zurich Finance Seminar April 2007
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frontiers: US portfolios
Introduction overview literature results extensionstheory
University of Zurich Finance Seminar April 2007
Devraj BasuCass Business School31
frontiers: UK portfolios
Introduction overview literature results extensionstheory
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frontiers: Japanese portfolios
Introduction overview literature results extensionstheory
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Devraj BasuCass Business School33
Premiums
Introduction overview literature results extensionstheory
• The US size, value and momentum premiums are 10.45%, 10.57% and 9.98% and the scaled five factor model captures over 90% of these premiums
• The UK premiums are 8.27%, 12.59% and 7.84% and performance is similar although not quite as good
• The Japanese premiums are lower and our model captures over 90% of these.
• Scaled five factor augmented Fama-French under-performs except for US momentum and UK value premium. Unscaled model works best for US and UK value premiums
Model Implied Size, Value and Momentum Premiums
University of Zurich Finance Seminar April 2007
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Time Variation in Factor Risk Premiums
Introduction overview literature results extensionstheory
• The predictive variables are market (World) and business cycle (term structure--How are the factor risk premiums correlated with these ?
• The skewness and kurtosis factor risk premiums appear to be correlated with the market variable
• The inflation risk factors appear to be functions of the term structure variables
• Surprisingly, the Fama-French factor risk premiums appear to be functions of the market variables rather than term structure
Factor Risk Premiums