1 University of Zurich Finance Seminar April 2007 Devraj Basu Cass Business School April 2007 Global...

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



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





• 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 …





• 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…



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



what can we do with “conditioning information”?

using conditioning info …


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’



asset pricing in the presence of conditioning information

pricing equation:


an SDF must satisfy …

theory

1)(1 ttt RmE

stochastic discount factor

vector of ‘ones’





pricing equation:



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





pricing equation:



theory

)()( EE

taking unconditional expectations …

1)(1 ttt RmE 1t 1t




pricing equation:



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




pricing equation:



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



asset pricing with conditioning information

literature:


• 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



tests of conditional asset pricing models

literature:


• 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



conditional factor models

generic factor model:


asset returns are given by …

theory


vector of factors

conditional expected return

ttttt FR 11

vector of factor loadings

residual (idiosyncratic risk)







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!







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







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



tests of conditional factor models

step 1:


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




step 2:


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




step 2:


frontier spanned by factors


with



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




step 2:


managed portfolios of factors


with



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




just for the record …


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




An intermediate step


• 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




An intermediate step


• 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



synopsis:


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?



results: Global versus Fama-French

test results: US size portfolios


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



results: Adding Country Inflation Factors

test results



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



results: US momentum portfolios

test results: US Momentum Decile Portfolio



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**


… … + kurtosis WLD SMB HML INF INF2 1.05 1.51

test assets: US momentum decile portfolios; instruments: WORLD,TB1M, TSPR, CONV



results: Japanese Value Portfolios

test results: Japan Book-to-Market Portfolios



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**


… … + kurtosis WLD SMB HML INF INF2 0.74 1.26

test assets: Japanese Value Portfolios; instruments: WORLD,TB1M, TSPR, CONV



frontiers: US portfolios




frontiers: UK portfolios




frontiers: Japanese portfolios




Premiums


• 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



Time Variation in Factor Risk Premiums


• 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

1 University of Zurich Finance Seminar April 2007 Devraj Basu Cass Business School April 2007 Global...

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