highlevelofinternationalrisksharingwhentheproductivitygrowthcontainslongrunrisk Chang,Yanqin MunichPersonalRePEcArchive

high level of international risk sharing when the productivity growth contains long run risk

Chang, Yanqin

September 2007

Online at https://mpra.ub.uni-muenchen.de/4476/

MPRA Paper No. 4476, posted 15 Aug 2007 UTC

H igh Le ve l of I n t e r n a t ion a l Risk sh a r in g W h e n t h e Pr odu ct ivit y Gr ow t h

Con t a in s Lon g r u n Risk

Yanqin Chang

July 2007

Com m ent s Welcom e

H igh Le ve l of I n t e r n a t ion a l Risk sh a r in g W h e n t h e Pr odu ct ivit y Gr ow t h Con t a in s Lon g r u n Risk

July 2007

Abst r a ct

This t heoret ical paper inv est igat es int ernat ional risk sharing and it s im plicat ions for equit y hom e bias. A general equilibrium m odel, feat uring t wo closed econom ies wit h nont rivial product ion sect ors, is developed. Moreover, product ivit y cont ains a sm all but persist ent highly correlat ed long run risk t hat becom es t he m aj or det erm inant of t he int ert em poral m arginal rat e of subst it ut ion ( I MRS) in a m odel wit h t he recursive preferences. Despit e adopt ing t he m odel of closed econom ies and aut arkic asset holdings—a scenario leading t o t he low est lev el of int ernat ional risk sharing under t he sam e condit ions—our m odel is st ill able t o generat e int ernat ional risk sharing indexes always over 96% for a broad range of param et er v alues, except ing t w o cases: where t he elast icit y of int ert em poral subst it ut ion ( EI S) is t he reciprocal of t he relat ive risk aversion ( RRA) ; and where EI S is around 0.7. I n t hose cases, t he risk sharing index drops shar ply t o about 30% . This result sheds light on w hy t he benchm ark m odel, feat uring a power ut ilit y whereby EI S is t he reciprocal of RRA, generat es int ernat ional risk sharing as low as 30% . However, when EI S t akes t hese values, our m odel’s result s cannot be reconciled wit h asset m arket dat a- m odel yields low volat ilit y of t he logarit hm s of I MRS, even lower t han Hansen- Jagannat han low er bound.

The im plicat ion is t hat t he low proport ion of foreign asset s in a dom est ic agent ’s port folio, a phenom enon observed in t he dat a, m ight not be a puzzle or a depart ure from t he agent 's opt im alit y condit ion. Aft er all, risk has already been well shared int ernat ionally due t o t he high correlat ions across count ries of t he long run product ivit y shocks. Hence, t here is not m uch incent ive left for an agent t o hold foreign asset s in her port folio t o furt her share t he risk int ernat ionally. Therefore, equit y hom e bias m ight not be a puzzle as claim ed by t he benchm ark m odel, in t he sense t hat it can be adequat ely reconciled wit h our t heoret ical result s.

1 . I n t r odu ct ion

This is a t heoret ical paper regarding int ernat ional risk sharing and it s im plicat ions for t he equit y hom e bias puzzle. Our m odel is a general equilibrium m odel feat uring t w o closed econom ies w it h nont rivial product ion sect ors. As a result of our closed econom y set up, asset holding is aut arkic in each count ry. Our m odel also adopt s t he Epst ein- Zin- Weil recur sive ut ilit y funct ion inst ead of t he pow er ut ilit y com m only used in t he benchm ark m odel.

Furt herm ore, t he product ivit y process cont ains a sm all but per sist ent long run risk and a lar ge short run risk. The int ernat ional corr elat ion of long run risk is high - w hile t hat of t he shor t run risk is low in order t o m at ch t he correlat ion across count ries of t he overall product ivit y in t he dat a.

The sm all but persist ent long run risk becom es t he m aj or det erm inant of t he int ert em poral m arginal rat e of subst it ut ion ( I MRS) in a m odel feat uring t he Epst ein- Zin- Weil preferences. Even t hough t he m odel deals w it h closed econom ies and aut ar kic asset holding—a scenario leading t o t he low est level of int ernat ional r isk sharing under t he sam e condit ion—our m odel is st ill able t o generat e int ernat ional risk sharing indexes alw ays over 96% for a br oad range of param et er values, except ing t w o cases: w here t he elast icit y of int ert em poral subst it ut ion ( EI S) is t he recipr ocal of t he r elat ive r isk aversion ( RRA) ; and w here EI S is around 0.7. I n t hose cases, t he risk sharing index drops sharply t o about 30% . This result explains w hy t he benchm ark m odel, w it h a pow er ut ilit y w hereby EI S is t he reciprocal of RRA, generat es an int ernat ional r isk sharing index as low as 30% . Our paper show s t hat t he

int ernat ional risk sharing generat ed from t he benchm ark m odel is not a general result . Rat her, it is a special one arising from t he use of EI S= 1/ RRA. Generally, in cases ot her t han t hat , t he m odel generat es a m uch higher degree of int ernat ional risk sharing.

The im plicat ion of our result s for t he equit y hom e bias puzzle is t hat t he low proport ion of foreign asset s held in a dom est ic agent ’s port folio, a phenom enon observed in t he dat a, m ight not be a puzzle or a depart ure from t he agent ’s opt im alit y condit ion. Aft er all, risk has already been w ell shared int ernat ionally due t o t he high correlat ions acr oss count ries of t he long run product ivit y shocks. Hence, t here is not m uch incent ive left for t he agent t o hold foreign asset s in order t o furt her share her risk w it h foreigner s.

Therefore, t he phenom enon of equit y hom e bias m ight not be a puzzle as claim ed by t he benchm ark m odel, in t he sense t hat it can be w ell reconciled wit h our t heoret ical result s.

The m odel feat uring t he aforem ent ioned recursive preferences and an exogenous st ochast ic process w it h long run r isk has t he pot ent ial t o solve ot her puzzles arising from applying t he benchm ark m odel in int ernat ional econom ics and financial econom ics. For inst ance, t he benchm ark m odel generat es uncovered int erest parit y ( UI P) , w hile dat a exhibit t he forw ard prem ium puzzle [ see Fam a ( 1984) ; Backus, Foresi and Telm er ( 2001) ] . The Backus- Sm it h Puzzle is anot her exam ple; t he volat ilit y of t he exchange rat e produced in t he benchm ark m odel is m uch low er t han t hat of t he dat a [ see Backus, Sm it h ( 1993) ] . Thir d, Real Business Cycle ( RBC) m odels have

difficult y in explaining asset prices, a phenom enon t hat has been dubbed t he equit y prem ium puzzle. Using reasonable param et er values, t he benchm ark m odel generat es an equit y prem ium t hat is m uch sm aller and less volat ile t han t hat found in t he act ual dat a fr om t he asset m arket [ see Mehra, Prescot t ( 1988) ] . Despit e it s success in sim ulat ing quant it y var iables such as out put , consum pt ion, and invest m ent , t he general equilibrium RBC m odel is not orious for it s unsat isfact or y record in explaining asset prices such as equit y ret urn and exchange rat es.

Since t he correlat ions of consum pt ion grow t h across count ries are low , t he benchm ark m odels generat e low levels of int er nat ional r isk sharing aft er applying quant it y dat a. How ever, t he com plet e asset m arket calls for perfect int ernat ional r isk sharing, w hich is achieved w hen agent s across count ries are holding global port folios t hat are ident ical in com posit ion.

We know t hat m odels w ill derive an int ert em poral Euler equat ion aft er solving t he consum er’s int ert em poral choice problem . I n a com plet e m arket , t his first - order condit ion im plies t hat I MRS, also know n as t he st ochast ic discount fact or ( SDF) , should be equalized eit her across agent s in a closed econom y or across count ries in an open econom y. I n t he benchm ark m odel, t he equalit y of I MRS t ranslat es int o having t he sam e consum pt ion gr ow t h rat es acr oss count r ies, which in t urn r equires t hat agent s across count ries are holding global port folios t hat ar e ident ical in com posit ion. I n r ealit y, people hold t he m aj orit y of t heir respect ive port folios in dom est ic asset s and only a sm all port ion in foreign asset s. The problem in reconciling t he above

t heoret ical result s w it h t he real- w orld dat a has been dubbed t he equit y hom e bias puzzle.

Brandt , Cochrane, and Sant a- Clara ( 2006) generat ed a m uch higher degree of int ernat ional risk sharing by using asset price dat a inst ead of quant it y dat a ( such as consum pt ion dat a) . They m aint ained t hat as long as m arket prices reflect t he agent ’s I MRS, t heir result of high int ernat ional risk sharing holds t rue. Our paper is an at t em pt t o r econcile t he result s of int ernat ional risk sharing from bot h price dat a and quant it y dat a; t hat is, w e aim t o generat e high int ernat ional risk sharing aft er applying quant it y dat a.

A close look at t he benchm ar k m odel r eveals t w o feat ures. One is t he pow er ut ilit y, w hich keeps EI S equal t o t he r eciprocal of RRA. Moreover , t he benchm ark m odel assum es t hat t he exogenous st ochast ic process—be it consum pt ion grow t h in an endow m ent econom y or product ivit y grow t h in a product ion econom y—is exposed only t o i.i.d. shocks.

The lim it at ion of benchm ark m odel feat uring t hose t w o propert ies point s t o a pot ent ial unified way of solving t he afor em ent ioned puzzles. The t wo aforem ent ioned feat ures m ight be t he culprit s t hat cause t he benchm ark m odel t o generat e anom alist ic result s. Hence, t he lit erat ure m odifies t he benchm ar k m odel along t wo lines: fir st , replacing t he power ut ilit y w it h t he Epst ein- Zin- Weil recursive pr eferences; and, second, by assum ing an exogenous st ochast ic process cont aining long run risk inst ead of being exposed only t o i.i.d. shocks. So far, t he m odified m odel has had som e

success in solving t he equit y prem ium puzzle [ see Bansal, Yaron ( 2004) ] , t he forw ard prem ium puzzle [ see Bansal, Shaliast ovich ( 2006) Backus, For esi and Telm er ( 2001) ] , and t he Backus- Sm it h puzzle [ see Brandt , Cochrane and Sant a- Clara ( 2006) ; Colacit o, Croce ( 2005) ] .

Are t hose t w o m odificat ions j ust ified and appropriat e, or are t hey arbit rary changes t hat have been m ade only t o fit int o a t heoret ical exercise?

First , w e know t hat preference is unobservable. There is no clear reason for favouring a power ut ilit y against ot her reasonable ut ilit y form s. Second, Bansal and Yaron ( 2004) show ed t hat w hen using consum pt ion dat a only, and w it hout relying on pr ice dat a, it is hard t o different iat e bet w een t he t w o hypot heses—first , t hat consum pt ion is a random w alk process, or, second, t hat it cont ains long run grow t h risk. To put it anot her w ay, t he consum pt ion dat a does not rej ect long run risk hypot hesis in favour of a random w alk hypot hesis.

Based on t he above j ust ificat ion, in t his paper, w e w ill adopt t hose t w o m odificat ions t o t he benchm ark m odel, w it h t he goal being t o shed light on int ernat ional risk sharing and exam ine t he im plicat ions for t he equit y hom e bias puzzle.

What is t he int uit ion behind t he claim t hat t he m odified m odel m ight have t he pot ent ial t o yield high int er nat ional r isk sharing aft er applying quant it y dat a? Sim ilar t o t he benchm ark m odel, t he m odified one derives an int ert em poral Euler equat ion as consum er’s fir st - order condit ion, w hich

im plies, in a com plet e m arket , an ident ical I MRS across count ries. However, in t he m odified m odel t he quant it y im plicat ion of t he first - or der condit ion is quit e different from t hat of t he benchm ark m odel. Thanks t o t he recursive preferences, equal I MRS across count ries does not necessar ily t ranslat e int o t he sam e consum pt ion grow t h rat es. Therefore, t he m odified m odel does not im ply t hat agent s across count ries are holding global port folios of ident ical com posit ion—t he root of t he hom e bias puzzle.

I n a m odel w it h t he recursive preferences, I MRS depends on bot h t he consum pt ion grow t h rat e and t he ret urn on a hypot het ical asset t hat pays a count ry’s aggregat e consum pt ion as it s dividend. Aft er calibrat ing param et ers w it h reasonable values, I MRS in t he m odel can be m ainly det erm ined by it s second it em , t he ret urn on t ot al wealt h.

Given t hat t he product ivit y grow t h process cont ains long run risk—even t hough it is sm all com pared wit h t r ansit ory risk—w hen long r un risk is very persist ent , t he ret ur n on w ealt h is quit e sensit ive t o it . We know t hat asset prices reflect not only present condit ions but also t he expect at ion of fut ure condit ions. An innovat ion in long run risk changes bot h condit ions. Therefore, asset prices could be quit e sensit ive t o long run r isk. As a r esult , a m odel w it h sm all but persist ent long run risk could generat e a high and volat ile equit y prem ium . I n t his sense, long run risk lit erat ure is credit ed w it h having t he pot ent ial t o solve t he equit y prem ium puzzle.

Aft er ext ending t he long run risk lit erat ure fr om closed t o open econom ies, and furt her assum ing t hat long run risk has a com m on global or igin, t he m odel has t he abilit y t o yield high correlat ion bet w een I MRS across count ries.

A highly correlat ed I MRS across count ries in t urn could deliver a high level of int ernat ional risk sharing¹.

To rest ore t he RBC m odel’s good record in sim ulat ing quant it y variables, w e can adj ust t he m agnit ude of t ransit ory r isk t o m ake sure t he m odel’s quant it y im plicat ions, such as t he proper t ies of product ion and consum pt ion, are also in line w it h t he dat a. The int ernat ional business cycle dat a show t hat quant it y variables have low volat ilit ies and poor cor relat ions across count ries, which m eans t he t r ansit ory risk in t he m odel needs t o be large and less correlat ed across count ries t o m at ch t he dat a.

I n sum m ary, in a m odel feat uring recursive preferences and a st ochast ic product ivit y process wit h long run r isk, if such risk com es fr om a com m on global origin, t hen I MRS can be highly correlat ed across count ries, which could yield high level of int ernat ional risk sharing. Thus, our m odel serves as m edium of reconciliat ion bet w een t he result s of int ernat ional risk sharing from quant it y dat a and from price dat a. Furt herm ore, hom e bias is no longer a phenom enon t hat cannot be reconciled w it h a general equilibrium m odel.

1 The degree of int ernat ional risk sharing is, how ev er, not t he sam e concept as I MRS correlat ion. See m ore det ail and an index of int ernat ional risk sharing in Brandt , Cochrane, and Sant a- Clara ( 2006) .

An agent can keep m ost of her port folio in dom est ic asset s and only a sm all port ion in foreign asset s, w hich is a scenario in line w it h t he dat a, and t he m odel can st ill achieve high degree of int ernat ional risk sharing. I t t hus appears t hat port folio decisions are irrelevant t o t he first - order condit ion.

Sect ion t w o will review t he r elat ed lit er at ure and it s relevance t o t he problem at hand, w hile sect ion t hree w ill present t he specifics of our m odel and it s log- linear approxim at ing solut ion. Sect ion four t hen serves t o display our m odel’s result s aft er par am et er calibrat ion. Finally, sect ion five will conclude our discussion.

2 . Re vie w of t h e Lit e r a t u r e

This paper cont ribut es t o t he lit erat ure on bot h int er nat ional risk sharing and on long run risk. Our m odel is an ext ension of long run risk m odel from t he endow m ent econom y t o t he product ion econom y. Using a general equilibrium product ion m odel along w it h long run risk, our paper aim s t o shed light on int ernat ional risk sharing. The m ost r elat ed previous w ork in t his area is t hat of Bansal and Yaron ( 2004) ; Brandt , Cochrane, and Sant a- Clara ( 2006) ; Colacit o and Croce ( 2005) ; and Croce ( 2006) .

Bansal and Yaron ( 2004) w as t he pioneering paper in t he grow ing lit erat ure on t he asset pricing field t hat goes under t he nam e of “ risks for t he long run” . Their m odel w as based on an endow m ent econom y feat uring recur sive preferences and st ochast ic consum pt ion grow t h process w it h long

run risk. The m odel has exhibit ed som e success in solving t he equit y prem ium puzzle.

Brandt , Cochrane, and Sant a- Clara ( 2006) w as t he first paper t o point out t hat int er nat ional risk sharing is act ually high aft er applying price dat a inst ead of applying quant it y dat a, w it h t he lat t er being a st andard pract ice in t he lit erat ure. How ever, since t heir paper lacks a t heoret ical m odel, it provides no form al explanat ion as t o w hy t he t w o appr oaches cannot be reconciled, nor does it explore how one m ight reconcile t hem in a single m odel. They concluded t hat a “ surpr isingly high level of r isk sharing” holds t rue as long as t w o condit ions are m et . The first condit ion requires t hat asset prices reflect I MRS, and t he second calls for eit her a com plet e asset m arket or an incom plet e one w it h a reasonably sized uninsurable risk. Alt ernat ively, if risks really are poorly shared, Brandt , Cochrane, and Sant a- Clara ( 2006) m aint ained t hat exchange rat es in t he dat a are m uch t oo sm oot h as com pared t o t he predict ion of t he m odel.

Colacit o and Cr oce ( 2005) at t em pt ed t o pr ovide Brandt , Cochrane, and Sant a- Clar a ( 2006) w it h a r igorous t heor et ical foundat ion. Colacit o and Croce felt t hat t he pioneering w ork r egarding long run r isk in Bansal and Yaron ( 2004) m ight have t he pot ent ial t o fully achieve t heir goal of generat ing high int ernat ional r isk sharing fr om quant it y dat a. Tr ying t o fill a gap left by Brandt , Cochrane, and Sant a- Clara ( 2006) , Colacit o and Croce ( 2005) set out t o reconcile t he result s of int er nat ional risk sharing from quant it y dat a wit h t he result s from asset price dat a. Aft er ext ending Bansal and Yaron’s ( 2004)

closed econom y endow m ent m odel t o a t w o- count ry endow m ent m odel, Colacit o and Croce ( 2005) show ed t hat as long as long run risks are highly correct ed across t w o count ries, t he m odel succeeds in generat ing high int ernat ional risk sharing from quant it y dat a.

Croce ( 2006) furt her cont ribut ed t o t he long run risk lit erat ure by ext ending it from an endow m ent econom y t o a general equilibrium w it h nont rivial product ion sect or. However, he focused on t he issue of welfare cost , rat her t han t hat of int ernat ional risk sharing.

Follow ing Croce ( 2006) , our m odel also feat ures t he general equilibrium product ion econom y wit h long run risk. Our focus, however, is on st udying t he issue of int ernat ional risk sharing. Sim ilar t o Colacit o and Croce ( 2005) , w e at t em pt t o provide a rigorous t heoret ical foundat ion t o Brandt , Cochr ane, and Sant a- Clara ( 2006) . Nevert heless, our m odel is a gener al equilibr ium product ion m odel cont aining long run risk, w hereas Colacit o and Cr oce ( 2005) ’s m odel was based on t he endowm ent econom ies wit h long run risk.

3 . M ode l

We const r uct a general equilibr ium m odel wit h nont rivial pr oduct ion sect ors.

There are t w o count ries in t he m odel, denot ed respect ively as hom e count ry ( h) and foreign count ry ( f) . We st udy t he degree of int ernat ional risk sharing generat ed by t he m odel w hen each count ry runs a closed econom y and

agent s’ asset holdings are aut arkic—a scenario leading t o t he lowest level of int ernat ional risk sharing under t he sam e condit ion.

We furt her assum e, in each count ry, hom ogeneit y am ong consum ers and const ant r et urns t o scale in product ion. As a result , t he m odel can be set up w it h a represent at ive consum er and a represent at ive firm in each count ry. To keep t he m odel sim ple and focus on our cent ral issue, w e assum e t hat t he represent at ive agent in each count ry lives infinit ely and her labour supply is fixed. Moreover, w e assum e t hat t here is a single good in t he world econom y.

The good is produced in each count ry by it s firm in a com pet it ive environm ent . Since each count ry runs closed econom y w it hout exchanging good w it h anot her, t he agent in each count r y derives ut ilit y solely from consum ing t he good produced in her count ry.

3.1. Preferences

Alt hough t he use of t he pow er ut ilit y is st andar d pract ice in benchm ark m odels, w e abandon t he use of t he pow er ut ilit y as preference. I nst ead, w e adopt t he recur sive preferences nam ed aft er Epst ein and Zin ( 1989, 1991) and Weil ( 1989) . The m ain feat ure of t hose Epst ein- Zin- Weil preferences is t heir disent angling of t he elast icit y of int ert em poral subst it ut ion ( EI S) , denot ed ψ , from t he coefficient of t he relat ive risk aversion ( RRA) , denot ed γ . How ever, w it h a pow er ut ilit y, EI S alw ays equals t he reciprocal of RRA;

t hat is, 1

ψ = γ. Yet , it is not clear t hat t hese t w o concept s should be linked so t ight ly. As Cam pbell ( 2003) st at ed: “ [ R] isk aversion describes t he

consum er's reluct ance t o subst it ut e consum pt ion across st at es of t he world and is m eaningful even in an at em poral set t ing, w hereas t he elast icit y of int ert em poral subst it ut ion describes t he consum er's w illingness t o subst it ut e consum pt ion over t im e and is m eaningful even in a det erm inist ic set t ing”

( page 828) .

The r epresent at ive agent s m axim ize t he obj ect ive funct ion, w hich t akes t he Epst ein- Zin- Weil preferences as it s ut ilit y form :

(

¹

)

¹

(

¹¹

)

^{1 1}

i i i

t t t t

U C E U

− −

+ −

⎧ ⎫

⎪ ⎪

= ⎨ − + ⎬

⎪ ⎪

⎩ ⎭

γ θγ

θ γ θ

β β ^{( 1)}

{ }

, i h f

∀ ∈

w her e ⁼

(

¹⁻

)

^{/ 1⎛}_⎜ ^{− ⎞}_⎟

⎝ ⎠

θ γ 1

ψ im plicit ly defines EI S (ψ ^{) . When} γ ^=1/ψ , w e have t hat θ

=

1 and Equat ion ( 1) reduces t o a pow er ut ilit y. U_tⁱ denot es t he ut ilit y of t he agent of t he ith count ry at t im e t, C_tⁱ is her consum pt ion at t im e t, and β denot es t he subj ect ive discount fact or, also know n as t he t im e- preference fact or.

Using dynam ic program m ing, Epst ein and Zin ( 1989) show ed t hat aft er solving t he consum er’s opt im al consum pt ion problem , t he opt im alit y condit ion—t he int ert em poral Euler equat ion—t akes t he form

( )

^{( )1}

1 , 1 , 1 1,

i t i

t i c t j t

t

E C R R

C

− −

+ + +

⎡ ⎤

⎛ ⎞

⎢ ⎥

⎜ ⎟ =

⎢ ⎜ ⎝ ⎟ ⎠ ⎥

⎢ ⎥

⎣ ⎦

θ

ψ θ

βθ ( 2)

w here R_{c t}ⁱ_{, 1+} is t he gross ret urn on a hypot het ical asset bet w een t im e t and 1

t+ w hich pays t he ith count ry’s aggregat e consum pt ion as it s dividends in each period. R_{j t, 1+} is t he gross r et urn on t he jth asset bet w een t im e t and

1 t+ .

Equat ion ( 2) cont ains an im port ant concept in asset pr icing lit erat ure—

I MRS, also know n as SDF or t he pricing kernel. We display it separat ely below:

( )

^{( )1}

1

, 1 i

i t i

t i c t

t

M C R

C

− −

+ +

⎛ ⎞

= ⎜ ⎜ ⎝ ⎟ ⎟ ⎠

θ

ψ θ

βθ ( 3)

I n a benchm ark m odel feat uring t he pow er ut ilit y, t he pricing kernel M t akes t he form

1

i ti

t i

t

M C

C

−

⎛ + ⎞

= ⎜⎜⎝ ⎟⎟⎠

γ

β ( 4)

I n a com plet e m arket , t he pricing kernel M is unique across count ries, which im plies t hat t he following equat ion holds in a benchm ark m odel:

1 1 h f

h t t f

t h f t

t t

C C

M M

C C

− −

+

⎛

+

⎞

⎛ ⎞

= ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ = ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ =

γ γ

β β ^{( 5)}

Equat ion ( 5) im plicit ly assum es t hat agent s in hom e and foreign count ry have an ident ical RRA (γ ) and an ident ical t im e preference fact or (β ) . I n t he benchm ark m odel, a unique pricing kernel requires t hat hom e and foreign count r y achieve equal consum pt ion gr ow t h rat es, which in t urn necessit at es t hat agent s across count ries are holding global port folios t hat are ident ical in com posit ion. This result from t he benchm ark t heoret ical m odel is in dr am at ic cont rast t o t he act ual dat a. The dat a show s t hat invest ors hold a m aj orit y of t heir port folios in dom est ic asset s and only a sm all port ion in foreign asset s;

t hus, in pract ice, t he com posit ions of t he global equit y port folios held by dom est ic and foreign invest ors are far from ident ical. As a result , t he equit y hom e bias puzzle arises.

Equat ion ( 3) is t he pr icing kernel M from t he Epst ein- Zin- Weil preferences. I n a com plet e m arket , t he presence of a unique pricing kernel im plies t hat t he follow ing equat ion ( 6) holds t rue. Again, w e assum e t hat agent s across count ries share t he sam e param et ers, γ ^, β ^{, and} ψ ^.

( )

^{( )1}

( )

^{( )1}

1 1

, 1 , 1

h f

f

h t h t f

t h c t f c t t

t t

C C

M R R M

C C

− −

−

+ − +

+ +

⎛ ⎞

⎛ ⎞

= ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ = ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ =

θ θ

ψ θ ψ θ

θ θ

β β ^{( 6)}

Equat ion ( 6) show s t hat w it h t he r ecur sive preferences, a unique pricing kernel does not necessarily im ply equal consum pt ion grow t h rat es across

count ries, and, in t urn, does not necessarily t ranslat e t o ident ically com posed global equit y port folios held by agent s in t he t w o count ries. As a result , it appears t hat equit y hom e bias m ight not necessarily be a phenom enon t hat st ands against t he first - order condit ion derived from t heoret ical m odels.

3.2. Product ion

I n t his sect ion, we work on t he st ochast ic pr oduct ivit y gr ow t h process t o int roduce anot her line of m odificat ion t o t he benchm ark m odels. I n t he benchm ark m odels, t he law of m ot ion of product ivit y gr ow t h is oft en assum ed being a st ochast ic process exposed t o i.i.d. shocks alone. I n t his paper, t he product ivit y grow t h process is exposed t o i.i.d. shocks as w ell as long run shocks. We specify t he law of m ot ion of product ivit y gr ow t h cont aining long run risk as follows:

{ }

1 1

1 ,

, ,

i i i

t t t

i i i

t t x t

z x w

x x

i h f

+ +

−

∆ = µ + +

= +

∀ ∈

ρ ε ( 7)

w here z_tⁱ₊₁

=

logZ_tⁱ₊₁ and Z_tⁱ₊₁ denot e t he ith count r y’s t ot al fact or pr oduct ivit y at t im e t, and

µ

is t he logarit hm of t he st eady st at e product ivit y level. w_t+1, w hich denot es t he short - run com ponent of product ivit y grow t h risk, is an i.i.d.

random variable. x_t is t he long- run com ponent of pr oduct ivit y grow t h r isk, w it h ρ m easuring it s persist ence and ε_{x t,} it s cont em poraneous i.i.d. shock.

We specify t he variance- covariance m at rix of pr oduct ivit y gr ow t h shock s as follows:

( )

1, 1, , 1, , 1, 0;

0 0

1 1

1 ,

1

f f

h h

t t x t x t

w

x

w

x

w₊ w₊

ε

₊

ε

₊ N

⎡ ⎤

⎣ ⎦

⎛ ⎞

⎜ ⎟

=⎜⎝ ⎟⎠

⎛ ⎞

⎜ ⎟

=⎜⎝ ⎟⎠

⎛ ⎞

⎜ ⎟

=⎜ ⎟

⎝ ⎠

∼

2 w

2 x hf w hf w

hf x hf x

Σ Σ σ Γ

σ Γ Γ ρ

ρ Γ ρ

ρ

( 8)

w here σ²x denot es t he variance of t he long run pr oduct ivit y gr ow t h shocks in each count ry while σ²w denot es t he short - run variance. ρ^hfx is t he cor relat ion of t he long run shocks across count ries, w hereas ρ^hfw is t he corresponding correlat ion of t he short run risks. We assum e t he cross correlat ions bet w een short run and long r un risks, eit her w it hin a count r y or across count r ies, are bot h zero.

A high ρ^hfx and a high ρ are t wo crit ical fact or s t hat allow our m odel t o generat e high degree of int ernat ional risk sharing. The reason for t his is t hat , in a m odel incor porat ing t he Epst ein- Zin- Weil preferences, if long run risk is very persist ent , it becom es t he m aj or det erm inant of I MRS. Furt her m ore, if long run risk is highly correlat ed acr oss count ries, I MRS w ill also be highly correlat ed across count ries. The high correlat ion bet w een I MRS across

count ries could furt her t ranslat e t o a claim t hat int ernat ional risk sharing is high.

The rest of our m odel is set up as follow s. I n a general equilibrium product ion m odel, t he firm in each count ry m axim izes it s pr esent value t o ow ners,² subj ect t o t he capit al st ock law of m ot ion and t he st ochast ic product ivit y growt h process. The firm pays it s w orker t he com pet it ive w age rat e, w hich is equal t o t he m arginal product of labour. The firm t hen pays it s shareholder dividends. We assum e a Cobb- Douglas product ion funct ion

( ) ( ) ( ) ( )

^{1 1 ,}

i i i i i i

t t t t t t

Y = Z N ^−α K ^α = Z ^−α K ^α ( 9)

w her e Y_tⁱ denot es t he ith count ry’s out put at t im e t, and N_tⁱ denot es it s firm ’s labor dem and for period t. At equilibrium , t he fir m ’s labour dem and equals t he w orker’s labour supply, w hich w e norm alized t o unit y. K_tⁱ is t he capit al st ock ow ned by t he firm of t he ith count ry at t he beginning of period t, ^α denot es capit al’s share, and ^1−α denot es labour’s share. Capit al st ock is chosen one period before it becom es product ive, and labour can be adj ust ed inst ant aneously.

Adding product ion t o a m odel m akes it s asset price im plicat ions even w orse, since t he agent can now sm oot h her consum pt ion even bet t er w it h

2 The firm ’s present value t o owners is t he sum t ot al of all it s current and fut ure expect ed dividends discount ed by a m arket SDF deem ed valid for every owner.

product ion t echnology t han in an endow m ent econom y. To deal w it h t his problem , a st andard pract ice in t he product ion- based asset pricing lit erat ure is t o im pose t he adj ust m ent cost in fir m ’s invest m ent . Follow ing t he lit er at ur e, we adopt t he adj ust m ent cost in our m odel. Hence, t he firm ’s capit al st ock evolves according t o t he following law of m ot ion:

( )

1 1 ,

i ti i i

t i t t

t

K I K K

K

δ

+

⎛ ⎞

= Ψ ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ + −

( 10)

w her e K_tⁱ₊₁ denot es t he capit al st ock of t he firm of t he ith count ry at t he

beginning of period t+1, I_tⁱ is t he invest m ent m ade by t he firm during period t,

δ

denot es t he depreciat ion rat e, and

Ψ

is a funct ion form of adj ust m ent cost , w hich is increasing and concave in invest m ent I

(

Ψ > 0, Ψ > 0, Ψ < 0^{′ ′′}

)

. This set t ing reflect s t he idea t hat changing t he capit al st ock rapidly is m ore cost ly t han changing it slowly.

The firm m axim izes it s value t o shareholder subj ect t o t he product ion funct ion ( 9) , t he law of m ot ion of t he capit al st ock ( 10) , and t he st ochast ic process of product ivit y grow t h w it h long run risk ( 7) . The first - order condit ions for m axim izing t he firm ’s value are:

( )

( )

1

1 1

1 1 1

1 1 1

1 1 1

1

1

1

s

s s

s s s

s

s s s

s s s

s s

s s

q I

K

W Z K

I

K I I

R Z K

K K

I K

α α

α α

α

δ α

−

+ +

− − +

+ + +

+ + +

= ⎛ ⎞

Ψ ⎜′ ⎟

⎝ ⎠

= −

⎡ ⎛ ⎞ ⎤

− + Ψ

⎢ ⎜⎝ ⎟⎠ ⎥ ⎛ ⎞

⎢ ⎥ ′

=⎢⎢ + Ψ′⎛⎜ ⎞⎟ − ⎥⎥Ψ ⎜⎝ ⎟⎠

⎢ ⎝ ⎠ ⎥

⎣ ⎦

( 11)

3.3. Log- linear Approxim at ing Solut ion

Aft er log- linearizing t he syst em around it s st eady st at e, w hich cont ains a const ant grow t h rat e g , w e get t he follow ing approxim at ing solut ion, including a solut ion t o consum pt ion grow t h and I MRS ( See Appendix 1 for det ails) :

( )

( )

1 1 1 1

1

t t kx t kw t

kx kk t

kw kk t

kk kk t

g I Y I g

c c v x v w

g C C C g

Y Y I g I g

v v x

C C C g C g

Y I g I g

v v w

C C g C g

Y I Y I g I g

v v k

C C g C C g C g

v

α

α

α α

+

− = − 1+ + δ

+

+ ⎡ ⎢ ⎣ − α − 1+ + δ ⎤ ⎥ ⎦

+

⎧ ⎡ 2 + − δ 1+ ⎤ ⎫

+ ⎨ ⎩ − α + ⎢ ⎣ + + δ − + δ ⎥ ⎦ ⎬ ⎭

⎡ 2 + − δ 1+ ⎤

+ ⎢ ⎣ + + δ − + δ ⎥ ⎦

⎧ 1− δ ⎡ 2 + − δ 1+ ⎤ ⎫

+ ⎨ ⎩ + + δ + ⎢ ⎣ + + δ − + δ ⎥ ⎦ ⎬ ⎭

= − ( )

( ) ( ) ( )( )

, 1 1 1

1 1

kx x t kw t

kx kk t

kw kk t

kk kk t

g I Y I g

v w

g C C C g

Y Y I g I g

v v x

C C C g C g

Y I g I g

v v w

C C g C g

Y I Y I g I g

v v k

C C g C C g C g

α

α

α α

+

⎡ ⎤

+

1+ 1+

+ ⎢ − α − ⎥

+ δ ⎣ + δ ⎦

⎧ ⎡ 1− δ + 1− ρ + 1+ ⎤ ⎫

⎪ ⎪

+ ⎨ ⎪ ⎩ − α + ⎢ ⎣ + + δ − + δ ⎥ ⎦ ⎬ ⎪ ⎭

⎡ 2 + − δ 1+ ⎤

+ ⎢ ⎣ + + δ − + δ ⎥ ⎦

⎧ 1− δ ⎡ 2 + − δ 1+ ⎤ ⎫

+ ⎨ ⎩ + + δ + ⎢ ⎣ + + δ − + δ ⎥ ⎦ ⎬ ⎭

ε

( 12)

( ) ( )

( )

( )

( ) ( )

( ) ( )( )

1 1 , 1

, , 1 , 1 1

, 1

1

1 1

1

1 1

1

x

t t t a t

m x t m w t t t

kx x t

kx

kk kk

kk

kx kk

m c c r

w E m

v g I

g C

n v

Y Y I Y g I g I

v v

n C n v C g C C g C g C

n Y I g g I

v v

C C g g C

α α

α

+ + +

ε + + +

+

= −θ − + θ −1 ψ

= η + η +

= γ 1+

+ δ

⎡ 1− δ ⎛ 2 + − δ 1+ ⎞⎤

⎛ − γ⎞ − α + ⎢ + + ⎜ + − ⎟⎥

⎜ψ ⎟ 1− ⎣ + δ ⎝ + δ + δ ⎠⎦

⎝ ⎠

+ 1− ρ + ⎡⎢⎣ + 1− δ + 1− ρ+ δ + − 1++ δ ε

ε

( )

( )

( ) ( )

, 1

1

1 1

1 1

1

1

1 1

x t

kw t

kk

kw t

kk kk

kk

kx

g I Y

v w

g C C

Y g I g I

C g C v g C

n v w

n Y I Y g I g I

v v

n v C g C C g C g C

Y Y I

C v C C

α

α α

α

+

+

+

⎧ ⎫

⎪ ⎪

⎪ ⎪ε

⎨ ⎬

⎪ ⎤ ⎪

⎪ ⎥ ⎪

⎩ ⎦ ⎭

⎡ 1+ ⎤

+γ⎢⎣ + δ − − α ⎥⎦

⎧ +2 + − δ − 1+ ⎫

⎪ + δ + δ ⎪

⎛ ⎞ ⎪ ⎪

+ ⎜⎝ψ− γ⎟⎠ ⎨⎪⎪⎩+ 1− ⎡⎢⎣ +1− δ+ δ + ⎛⎜⎝ +2 + − δ+ δ − 1++ δ ⎞⎟⎠⎤⎥⎦⎬⎪⎪⎭

1− δ + 1−

− − α + +

ψ

( )(

¹

)

1

1

kk t

kw kk t

kk kk t

g g I

v x

g g C

Y g I g I

v v w

C g C g C

Y I Y g I g I

v v k

C g C C g C g C

α

α α

⎧ ⎡ ρ + 1+ ⎤⎫

⎪ − ⎪

⎨ ⎢ + δ + δ ⎥⎬

⎪ ⎣ ⎦⎪

⎩ ⎭

⎛ 2 + − δ 1+ ⎞

−ψ ⎜⎝ + + δ − + δ ⎟⎠

⎡ 1− δ ⎛ 2 + − δ 1+ ⎞⎤

−ψ⎢⎣ + + δ + ⎜⎝ + + δ − + δ ⎟⎠⎥⎦ ( 13)

Here _,

mεx

η is t he exposure of I MRS t o innovat ion in t he long run

product ivit y shock, ε_{x t, 1+} ^{, and}

η

m w, is t he exposur e of I MRS t o innovat ion in t he short run shock, w_t+1.

3.4. I ndex of I nt ernat ional Risk sharing

Brandt , Cochrane, and Sant a- Clara ( 2006) const ruct ed t he follow ing index t o gauge t he degree of int ernat ional risk sharing:

( ) ( ) ( )

2 1 1

2 2

1 1

1 ,

f d

t t

f d

t t

m m

IIRS

m m

+ + + +

σ −

= −σ + σ ( 14)

w here I I RS denot es t he index of int ernat ional risk sharing, m is t he logarit hm of I MRS, and

σ

² denot es uncondit ional variances. The use of uncondit ional variances is j ust ified by t w o considerat ions. First , since w e are using a discret e- t im e m odel, st art ing w it h E m_t

(

_t+1R_t+1

)

=1, w e can condit ion dow n t o ^{E mR}

( )

^=1;3 second, aft er calibrat ion, t he result s of our m odel can be com pared wit h t he uncondit ional m om ent s obt ained from t he dat a.

Brandt , Cochrane, and Sant a- Clara ( 2006) em phasized t he difference bet w een t he index and t he correlat ion bet w een m across count ries. They argued t hat “ [ r] isk sharing requires t hat dom est ic and foreign m arginal ut ilit y grow t h are equal, not j ust perfect ly correlat ed, and our index det ect s violat ions of scale as w ell as of correlat ion” ( page 672) . They also provided an exam ple t o show t he difference: “ [ F] or exam ple, if lnm^f

= ×

2 lnm^d, r isks are not perfect ly shared despit e perfect correlat ion. I n t his case, our index is 0.8” ( page 672) .

4 . Th e M ode l’s Re su lt s Aft e r Pa r a m e t e r iz a t ion

4.1. Param et erizat ion

3 See Brandt , Cochrane, and Sant a- Clara ( 2006) foot not e 5 for m ore det ail.

I n t he lit er at ure on calibrat ion, param et er values are chosen “ [ F] or purposes of ‘calibrat ion’ in a quart erly m odel” ( Cam pbell ( 1994) p467) . Follow ing t his pract ice, w e choose t he param et er values as follow s. g, t he logarit hm of grow t h rat e at st eady st at e, is chosen t o be 0.005 ( 2% at an annual rat e) . r, logarit hm of st eady st at e r et ur n on r isky asset , is set at 0.015 ( 6% at an annual rat e) . β , subj ect ive discount fact or, is set at 0.99923 ( 0.997 at an annual rat e) ; depreciat e rat e δ is set at 0.025 ( 10% at an annual rat e) ;⁴ capit al’s share, denot ed as α, is set t o be 0.36, a st andard value in lit er at ure.

ζ

, a param et er r elat ed t o invest m ent adj ust m ent cost , w it h 1

ζ

being t he elast icit y of invest m ent - capit al rat io wit h respect t o m ar ginal q, is set at 0.8306.⁵

ρ

, t he persist ence of long run product ivit y risk, is set at 0.987.⁶ n₁, t he param et er of linearizat ion of t he ret urn t o consum pt ion equit y defined by

1

n P

P C

= + ^, is set at 0.9965, an average num ber bet w een 0.996 from

Cam pbell ( 2003) and 0.997 from Bansal and Yaron ( 2004) . K

C , st eady st at e level of capit al- consum pt ion rat io, is chosen t o be 12.5544, an average level bet ween 11.3755 in Cam pbell ( 1994) and 13.7333 in Uhlig ( 1999) ;

4 These four param et er values are in line wit h Kydland and Prescot t ( 1982) .

5 Eberly ( 1997) est im at ed t he elast icit y for t he U.S., denot ed ¹_ζ , at a 95% confidence

int erval of [ 1.08, 1.36] .

6 Croce ( 2006) set t his num ber at 0.98.

Analogously, Y

C , st eady st at e level of out put - consum pt ion rat io, is set at 1.3387, an average bet ween 1.3423 and 1.3350, fr om Cam pbell ( 1994) and Uhlig ( 1999) r espect ively. To cover t he w hole range of possibilit y, w e set EI S,

{ }

ψ ∈ 20,10,5, 2,1.7,1.5,1.1,1,0.9,0.71,0.5,0.2,0.1,1/15 and RRA, γ ∈ 15,10,5

{ }

. This range covers t he cases of bot h

ψ ≠ 1

γ

^and

ψ = 1

γ

. This range also includes

ψ =1

. Table 1 sum m arizes t he key param et er values chosen for our m odel.

Table 1

Param et er Values

Param et er Quart erly Value Equivalent Annual

Rat e

1 St eady st at e growt h rat e g 0.005 2%

2 St eady st at e ret urn on risky asset r 0.015 6%

3 Subj ect ive discount fact or β 0.99923 0.997

4 Depreciat ion δ 0.025 10%

5 Capit al’s share α 0.36

6 Param et er relat ed t o Adj ust m ent cost 0.8306

7 Persist ence of long run product ivit y shock 0.987

8 ⁿ¹ 0.9965

9 St eady st at e capit al- consum pt ion rat io 12.5544

10 St eady st at e out put - consum pt ion rat io

1.3387

11 ^{EI S}

⎧ 20,10,5, 2, ⎫

⎪ 1.7,1.5,1.1,1, ⎪

⎪ ⎪

⎨ 0.9,0.71,0.5, ⎬

⎪ ⎪

⎪ 0.2,0.1,1/15 ⎪

⎩ ⎭

12 ^RRA

{

^15,10,5

}

We calibr at e t he var iance- covar iance m at r ix of product ivit y gr ow t h shocks t o m at ch t he dat a. Boldrin, Chr ist iano, and Fisher ( 2001) par am et er ized t he quart er ly volat ilit y of t he log product ivit y grow t h, σ_∆z , t o be 0.018. Since Equat ion ( 7) show s t hat a product ivit y process is const it ut ed by an AR ( 1) long run risk and an i.i.d. short run risk, t he overall product ivit y’s uncondit ional variance is:

2 2 2 2 2

2

1 0.018

z x w 1 w

∆ 2ε

σ = σ + σ = σ + σ =

− ρ ( 15)

Backus, Kehoe, and Kydland ( 1995) sum m arized product ivit y’s int ernat ional cor relat ions based on int er nat ional business cycles dat a. For exam ple, corr z( _eu,z_us)=0.56 and corr z( _japan,z_us)=0.58. We t ake t he average of t he t w o, 0.57, as our param et er value for t he int ernat ional correlat ion of t he log product ivit y growt h. Then, w e get t he follow ing equat ion:

( ) ( )

( )

2

2

2

cov( , ) cov( , )

( , )

1 , ,

1

1 1

1 1 ,

1 0.57,

1 1

h f h f

h f

z

h f h f

w

w

h f

x x w w

corr z z

corr corr w w

corr w w n n

∆ 2ε

ε2

∆ ∆ = +

σ

ε ε σ + σ

= − ρ

σ + σ

− ρ

− ρ× +

= =

− ρ+

( 16)

w here

2

n w₂ ε

= σ

σ , t he rat io bet w een t he variance of t he short run shock t o t he variance of t he long r un shock. Recalling t hat our m odel assum es a sm all but persist ent long run shock, along w it h a large short run shock, n needs t o be a large num ber. We set n=64 . Equat ions ( 15) and ( 16) im ply t hat

1.42%

σ =w and ,corr w w( ^h, ^f)=0.31. Consequent ly, t he variance of t he long run shock,

σ

²_ε , explains about 0.97% of t he variance of t he product ivit y grow t h. We assum e t hat t he sm all but per sist ent long r un product ivit y grow t h shocks have a com m on global origin, and t hat , as a result , t he correlat ion acr oss t he t wo count ries of long run product ivit y gr ow t h shocks is set t o be 1, i.e., corr(ε ε =^h, ^f) 1. Table 2 displays a sum m ary of our choices of param et ers for t he product ivit y process.

Table 2

Param et er Values for t he Product ivit y Process

Param et er Quart erly Value

1 Uncondit ional st andard deviat ion of product ivit y grow t h

0.018

2 I nt ernat ional correlat ion of product ivit y grow t h 0.57

3 N, t he rat io bet ween t he variance of t he short run

shock t o t he variance of t he long run shock

64

4 I nt ernat ional correlat ion of long run product ivit y grow t h shocks

1

Param et er Value Generat ed in

t he Model

5 St andard deviat ion of short run product ivit y grow t h shocks

1.42%

6 I nt ernat ional correlat ion of short run product ivit y grow t h shocks

0.31

4.2. Result s

Table 3 report s our m odel’s result s for _,

mεx

η and

η

_{m w,} aft er param et erizat ion, w here _,

mεx

η capt ures I MRS’s exposure t o innovat ion in t he long- run

com ponent , ε_{x t, 1+} ^{, and}

η

m w, is I MRS’s exposure t o innovat ion in t he short - run com ponent , w_t+1. Put it anot her w ay, _,

mεx

η is t he innovat ion in m_t+1 driven

by t he innovat ion in ε _{x t+, 1}^{, and}

η

_{m w,} is t he innovat ion in m_t+1 driven by t he innovat ion in w_t+1.

Table 3

Model result s for ηm,ε_x and

η

m w,

EI S RRA _,

mεx

η

η

_{m w,}

20 15 - 612.07 - 12.85

20 10 - 407.37 - 8.57

20 5 - 202.66 - 4.28

10 15 - 550.07 - 12.85

10 10 - 365.49 - 8.57

10 5 - 180.90 - 4.28

5 15 - 466.87 - 12.85

5 10 - 309.15 - 8.57

5 5 - 151.43 - 4.28

2 15 - 306.84 - 12.85

2 10 - 201.06 - 8.57

2 5 - 95.28 - 4.28

1.7 15 - 269.74 - 12.85

1.7 10 - 176.19 - 8.57

1.7 5 - 82.64 - 4.28

1.5 15 - 238.79 - 12.85

1.5 10 - 155.54 - 8.57

1.5 5 - 72.28 - 4.28

1.1 15 - 151.77 - 12.85

1.1 10 - 97.99 - 8.57

1.1 5 - 44.22 - 4.28

1 15 - 121.66 - 12.85

1 10 - 78.30 - 8.57

1 5 - 34.95 - 4.28

0.9 15 - 86.30 - 12.85

0.9 10 - 55.35 - 8.57

0.9 5 - 24.40 - 4.28

0.71 15 1.97 - 12.85

0.71 10 1.05 - 8.57

0.71 5 0.14 - 4.28

0.5 15 157.27 - 12.85

0.5 10 96.40 - 8.57

0.5 5 35.53 - 4.28

0.2 15 681.82 - 12.85

0.2 10 338.05 - 8.57

0.203 5 - 0.59 - 4.28

0.1 15 778.05 - 12.85

0.10113 10 - 3.88 - 8.57

0.1 5 - 822.08 - 4.28

0.0675 15 - 1.78 - 12.85

1/ 15 10 - 1301.78 - 8.57

1/ 15 5 - 2554.82 - 4.28

Table 3 shows t hat _,

mεx

η is m uch larger t han

η

_{m w,} . Accor ding t o our result s, on average, _,

mεx

η is m ore t han 33 t im es gr eat er t han

η

_{m w,} . This is pivot al for our m odel t o be able t o gener at e high int ernat ional r isk sharing.

I n a m odel feat uring recursive preferences and a st ochast ic product ivit y process w it h long run risk, if _,

mεx

η is m uch larger t han

η

_{m w,} , a sm all but persist ent long run risk dom inat es a large short run risk t o becom e a m aj or det erm inant of I MRS m_t+1. Furt herm ore, if long run risk is highly corr elat ed across count ries, so is I MRS. As a result , t he m odel generat es a high degree

of int ernat ional risk sharing, even t hough t he t w o count ries run closed econom ies and agent s’ asset holding are aut arkic.

Finally, Table 4 present s our m odel’s result s for I I RS, index of int ernat ional risk sharing.

Table 4

Model result s for I ndex of I nt ernat ional Risk sharing ( I I RS)

EI S RRA I I RS

20 15 99.9497%

20 10 99.9496%

20 5 99.9491%

10 15 99.9378%

10 10 99.9374%

10 5 99.9361%

5 15 99.9137%

5 10 99.9125%

5 5 99.9088%

2 15 99.8005%

2 10 99.7935%

2 5 99.7701%

1.7 15 99.7420%

1.7 10 99.7313%

1.7 5 99.6948%

1.5 15 99.6711%

1.5 10 99.6556%

1.5 5 99.6016%

1.1 15 99.1915%

1.1 10 99.1388%

1.1 5 98.9457%

1 15 98.7499%

1 10 98.6607%

1 5 98.3277%

0.9 15 97.5597%

0.9 10 97.3709%

0.9 5 96.6545%

0.71 15 31.9534%

0.71 10 31.6154%

0.71 5 31.0348%

0.5 15 99.2465%

0.5 10 99.1104%

0.5 5 98.3801%

0.2 15 99.9595%

0.2 10 99.9268%

0.203 5 31.7722%

0.1 15 99.9689%

0.10113 10 38.6128%

0.1 5 99.9969%

0.0675 15 31.7849%

1/ 15 10 99.9951%

1/ 15 5 99.9997%

Our m odel produces int ernat ional risk sharing levels above 96% for m ost param et er values of EI S and RRA. Only w hen EI S is around eit her 1/ RRA or 0.7 does I I RS drop sharply, falling t o levels as low as approxim at ely 30% .

Using exchange rat e and equit y dat a, Brandt , Cochrane, and Sant a- Clar a ( 2006) report ed int er nat ional r isk shar ing levels above 98% : 0.986 ( US vs.

UK) ; 0.985 ( US vs. Germ any) ; 0.980 ( US vs. Japan) .