420 Jou nal o Re iews on Global Economics, 2017, 6, 420-425
E-ISSN: 1929-7092/17 © 2017 Li escience Global
Real Risk-F ee Ra e, he Cen al Bank, and S ock Ma ke Bubbles
Jukka Ilomäki* and Hannu Lau ila
Facul y o Managemen , FI-33014 Uni e si y o Tampe e, Finland
Abs ac : The cen al bank ac s as a social planne , and adjus s he eal isk- ee a e o e u n o co ec any misp icing
in he s ock ma ke so ha he eme gence o posi i e o nega i e bubbles is a oided. The analysis shows ha he cen al
bank mus aise he isk- ee a e in he case o a posi i e bubble, and ice e sa. Mo eo e , he cen al bank should
in e ene in he s ock ma ke e en i i does no ha e pe ec in o ma ion abou he bubble. This is because he
sequen ial di idend yields in he p icing equa ions a e s a iona y. Thus, e en he delayed eac ion o he cen al bank
p e en s he undamen al alue and he equilib ium p ice om d i ing apa o ex ended pe iods.
Keywo ds: Real In e es Ra e, Mone a y Policy, Po olio Choice.
1. INTRODUCTION
Wha should be he ole o he cen al banks in he
con ex o s ock ma ke bubbles? Fo example,
Be nanke and Ge le (2001), G eenspan (2004), and
Posen (2006) a gue ha he cen al bank should
concen a e on in la ion a ge ing and s able g ow h,
while Bo do & Jeanne (2002), Bean (2004), and
Roubini (2006) claim ha he cen al bank should also
ackle s ock ma ke bubbles. Yellen (2010) suppo s
he la e iew by s a ing ha he cen al bank should
in e ene in s ock ma ke p icing, i he bubble ge s
dange ously la ge.
Viewing he ques ion om ano he angle, Maio
(2014) p o ides empi ical suppo o “don’ igh he
Fed” –policy. He shows ha he policy yields
economically and s a is ically signi ican sho - e m
gains compa ing o buy-and-hold s a egy. Obeying he
“don’ igh he Fed” p inciple means ha in es o s
should eac by inc easing/dec easing isky
in es men s in he s ock ma ke s when he cen al bank
educes/ aises he isk- ee a e o e u n.
Fu he mo e, he e a e some p ac ical examples o
cen al banks’ a acks on cu ency ma ke bubbles by
adjus ing he isk- ee a e. Fo example, he cen al
bank o Russia eac ed o i s na ional cu ency ma ke
bubble in 2014 by aising he key in e es a e (Gilenko
2017).
Conce ning he means o in e en ion, Conlon
(2015) sugges ha he cen al bank should use i s
in o ma ional ad an age, and in e ene by wa ning he
*Add ess o co espondence o his au ho a he Facul y o Managemen ,
FI-33014 Uni e si y o Tampe e, Finland; Tel: +358401366117;
E-mail: jukka.ilomaki@u a. i
JEL Classi ica ion: E43, E52, G11
in es o s abou no iceable bubbles. Thus, inac ion o
he cen al bank implici ly means ha he s ock p ices
can ise. Fu he mo e, he in e en ion by he cen al
bank may make hings e en wo se i i does no ha e
p i a e in o ma ion abou he bubble. Fische (2016)
makes a s onge poin by a guing ha , in he case o
economy-wide bubbles, he cen al bank should
in e ene by mone a y policy.
A qui e common pe cep ion abou he p ope cu e
o bubbles is he so called “leaning agains he wind”
ype policy, which means ha he isk- ee a e mus be
li ed in o de o de-in la e a posi i e bubble. Howe e ,
he e a e con on ing iews, oo. Fo example Gali
(2014) claims ha such a policy may ac ually make he
bubble in la e e en u he . This may happen because
he undamen al componen o an asse p ice
co esponds o he discoun ed s eam o payo s, bu
he bubble componen has no payo s o discoun .
Thus, he la e componen g ows a he a e o in e es ,
a leas in expec a ion.
Inspi ed by he abo e a gumen a ions, we p esen a
simple inancial ma ke model, whe e he cen al bank
is in cha ge o he adjus men o s ock ma ke bubbles.
The adjus men is achie ed by con olling he ma ke
eal e u n o isk- ee asse s ia mone a y policy. The
maneu e abili y o bubbles is s udied bo h wi h and
wi hou pe ec in o ma ion abou e icien s ock ma ke
p icing and hus abou he exis ence o bubbles. The
model yields a “leaning agains he wind” ype ule o
moni o ing and co ec ing any misp icing, and
s abilizing he inancial ma ke .
The pape p oceeds as ollows. Sec ion 2 p esen s
ela ed li e a u e. Sec ion 3 de ines he basic model
wi h in o med and unin o med in es o s. Sec ion 4
p esen s he analysis o he equilib ium in he inancial
Real Risk-F ee Ra e, he Cen al Bank, and S ock Ma ke Bubbles Jou nal o Re iews on Global Economics, 2017, Vol. 6 421
ma ke , and desc ibes he in e en ions needed om
he cen al bank o s abilize he inancial ma ke .
Sec ion 5 discusses he indings.
2. LITERATURE REVIEW
In ou model, he key building block is he possibili y
o bubbles. The seminal e iciency a gumen by
Samuelson (1973) is ha , wi h pe ec ly a ional isk-
neu al in es o s, he equilib ium s ock p ice (P ) is
equal o he expec ed discoun ed di idends o he
sha eholde , ha is he undamen al alue (V ). Ti ole
(1982) shows ha
P
=V
holds also in he a ional
expec a ions equilib ium wi h long-li ed isk a e se
in es o s so ha in ini e bubbles a e impossible.
Howe e , Ti ole (1985) a gues ha , in an
o e lapping gene a ions model wi h sho li ed
in es o s and in ini ely li ed asse s bubbles a e
possible so ha
P
!V
may occasionally happen.
San os & Wood o d (1997) indica es ha bubbles a e
impossible in a ional ma ke s only in he long un, bu
ha he e is a possibili y o
P
!V
in he sho un.
The o e lapping gene a ions model is he e
in e p e ed so ha a ional in es o s ha e a sho - e m
in es ing ho izon. Fo example Shlei e and Vishny
(1997) a gue ha he sho -li ed-in es o assump ion
can be mo i a ed so ha he majo i y o in es o s in he
ma ke a e p o essional weal h manage s who handle
o he people’s money wi h pe o mance moni o ing in
sho in e als. Delong e al. (1990), F oo e al. (1992),
Campbell and Kyle (1993), Kyle and Xiong (2001),
Shille (2014), and Ilomäki (2016) a gue ha , i he e
a e in o med and unin o med sho - e m ( ha is sho -
li ed) in es o s in he ma ke ,
P
!V
exis s because o
he asymme y and co ela ed beha io o he
uni o med in es o s.
Fu he mo e, Allen e al. (2006), and Bacche a &
Van Wincoop (2008) show ha
P
!V
occu s in he
case o sho -li ed in es o s, because hei noisy
p i a e in o ma ion inco po a es a ional highe -o de
belie s in o he equilib ium. Cespa & Vi es (2015) also
uses a model o sho -li ed in es o s wi h asymme ic
in o ma ion, and ends o wo ex eme equilib iums: a
high in o ma ion equilib ium
P
=V
wi h low ola ili y
and high liquidi y, and a low in o ma ion equilib ium
P
!V
wi h high ola ili y and low liquidi y.
The e o e, ou model builds on he assump ion o
asymme ic in o ma ion among he in es o s, and on
he a gumen o Loewens ein and Willa d (2006) ha
he a ia ion o he isk- ee a e o e u n assu es ha
P
=V
. O iginally, he la e poin de i es om he
p ope y ha he limi ed supply o isk- ee asse s pulls
oge he he equilib ium p ice and he undamen al
alue, implying ha
P
!V
is impossible ega dless o
he in es o s’ beha io . In ou model, he cen al bank
in e enes he ma ke by con olling he supply o isk-
ee asse s hus se ing he isk- ee a e.
3. THE MODEL
The model ollows Ilomäki (2016) wi h he ex ension
o a ime- a ying eal isk- ee a e o e u n. The eal
a e is co ec ed o in la ion so ha whe e he g oss
in la ion is de e mined as
1+
!
"p
p #1
, (1)
whe e
p
is he gene al p ice le el in he economy. The
economy consis s o an in ini e se o a ional cons an
absolu e isk-a e se (CARA) in es o s, who ha e
asymme ic in o ma ion so ha 0<
µ
<1 o hem a e
in o med, and
1!
µ
o hem a e unin o med in e e y
pe iod. The a omis ic in es o s li e o wo pe iods,
in es ing in he i s pe iod, and consuming in he
second pe iod.
The e is an in ini ely li ed isky asse (sha e o i m
F), and a isk- ee al e na i e wi h ime- a ying eal
isk- ee a e o e u n
. The in es o s alloca e hei
in es men s be ween isk- ee and isky asse s. The
po olio choice is simpli ied, because he assump ion
o wo-pe iod li ed CARA in es o s omi s he possibili y
o hedging agains changes in expec ed e u ns, and
because he assump ion o an in ini ely li ed isky asse
cons i u es limi s o a bi age in he o e lapping
gene a ions model (Shlei e and Vishny, 1997). In
addi ion, assuming no mally dis ibu ed excess
consump ion, CARA in es o s assu e ha he
condi ional a iance in excess e u ns is cons an .
The e a e ou ypes o a ional in es o s in e e y
pe iod : young in o med and unin o med in es o s who
open hei posi ions ( he demand side), and old
in o med and unin o med in es o s who close hei
posi ions ( he supply side). Fo simplici y, excess
e u ns a e no mally dis ibu ed, he ime- a ying isk
p emium is common o all in es o s, and he e a e
nei he ansac ion cos s no axes. The budge
cons ain comes om he assump ion ha all young
in es o s a ime ha e he same indi idual endowmen
w
y.
The na u al loga i hm o he di idend
D
on i m
F’s s ock ollows andom walk so ha
ln D =ln D !1+e
d
, (2)
422 Jou nal o Re iews on Global Economics, 2017, Vol. 6 Ilomäki and Lau ila
whe e
e
d~WN (0,
!
d
2)
. This means ha he change in
he di idend a is pe manen . In pe iod , he di idends
a e paid o old in es o s.
In pe iod , in o ma ion common o all in o med and
unin o med in es o s consis s o he his o y o
equilib ium eal p ices
(...P
!3,P
!2,P
!1)
, and he
cu en alue o he ime a ying eal isk- ee a e
(
). Mo eo e , all in es o s obse e cu en eal
di idends (
D
), bu he young in o med in es o s ha e
also p i a e in o ma ion on D +1.
Assume ha , in any pe iod , he cen al bank ac s
as a social planne wi h he aim o s abilize he inancial
ma ke s by adjus ing he eal isk- ee a e o e u n.
The cen al bank may ha e pe ec o impe ec
in o ma ion on he inancial ma ke . In any case, he
cu en alue o he eal isk- ee a e o e u n
is
de e mined by he adjus men o possible misp icing so
ha i is cons an wi h ze o a iance om pe iod o
pe iod +1. Supposing ha he supply o he isk- ee
asse is in sole con ol o he cen al bank, he ma ke
clea ing condi ion o he isk- ee asse eads
ay
y
!"bcb
cb
!=0
, (3)
whe e
ay
is he o al demand by he young in es o s,
and bcb is he o al supply by he cen al bank in e e y
pe iod.
Fu he mo e, assume exogenous noise ade s wi h
he dis ibu ion
e
n ~N(0,
!
n
2)
in he s ock ma ke .
Thus, he ma ke clea ing condi ion o he isky asse
eads
xy!so
o
"
y
"+e
n =0
, (4)
whe e xy e e s o o al demand o he s ock by he
young in es o s, and so is he o al supply o he s ock
by he old in es o s. The op imal demand decisions
p oduce he equilib ium p ice in pe iod hus ul illing
he ma ke clea ing condi ion. This happens because
he old in es o s ha e o close hei posi ion o
consume in he second pe iod.
Ra ional in es o s ca e also abou he isk o he
in es men . LeRoy (1973) shows ha i he isk- ee
a e o e u n is ime- a ying, and i all in es o s a e isk
a e se, he p ope discoun a e includes he isk- ee
a e and a isk p emium. S a ing om Ma kowi z
(1952) and Sha pe (1964), isk is de ined as he
a iance o e u ns.
A a ional young in es o maximizes indi idual u ili y
by alloca ing he ini ial endowmen be ween isky and
isk- ee asse s. The maximiza ion p oblem eads
Max[E(!e!
"
c +1|
#
y,w
y)]
s. .
c +1=x (1+
)+x E (R +1)
w
y=x +x ,
(5)
whe e
!
y
is he in o ma ion se , > 0 is he coe icien
o isk a e sion,
c +1
is consump ion when old,
w
y
is
he endowmen , and x and x deno e he amoun o
money in es ed in isk- ee and isky asse s,
espec i ely. The ne excess e u n on a isky sha e is
R +1!P
+1"P
+D +1
P
" .
(6)
Assuming no mally dis ibu ed ex a consump ion,
and plugging he consump ion cons ain in o he u ili y
unc ion yields
E U(c +1)
[ ]
=!e
!
"
x E (R +1)+
"
2
2
x 2
#
2
, (7)
whe e
!
2
is he cons an a iance o excess e u ns.
No e ha , since he in es o s obse e
,
i s a iance
is ze o. Maximize (7) wi h espec o x , and use
equa ion (6) o w i e he i s o de p icing condi ion o
he isky asse ,
P
=P
+1+D +1
1+
+x
!"
2
. (8)
Manipula ion o (8) esul s in he op imal in es men
decision, saying ha he s ock demand o a a ional
young in es o eads
x
=(P
+1+D +1) / P
!(1+
)
"#
2
. (9)
In equa ion (8), he equi ed ne eal a e o e u n is
n=
+x
!
2,
whe e
x
!
2=P
+1+D +1
P
"(1+
)=
#
(10)
deno es he ime- a ying isk p emium. In any pe iod ,
he in es o s know i upon he choice o
x
.
4. THE FINANCIAL MARKET EQUILIBRIUM
In he model, all in o med in es o s ha e he bes
possible in o ma ion, and old/young in es o s
Real Risk-F ee Ra e, he Cen al Bank, and S ock Ma ke Bubbles Jou nal o Re iews on Global Economics, 2017, Vol. 6 423
ecognize ha young/old in es o s obse e his.
Recalling ha he p ope ies o andom walk say ha
he change in he di idend a ime
is pe manen , he
a ional choice ep esen ed by equa ion (8) esul s o e
ime in he pe pe ui y model,
P
=D +1
n=V
. (11)
whe e
n=
+x
i
!
2,
in which xi is he s ock demand
o an in o med in es o . The e o e, i can be concluded
ha equa ion (11) e lec s he undamen al alue o he
isky asse , and ha he in o med in es o s ollow he
Samuelsonian p icing pa e n,
P
i=V
.
The unin o med in es o s obse e he cu en eal
di idend D . Hence, he a ional p icing ule o he
unin o med in es o s is
P
=D
n
, (12)
whe e
n=
+x
u
!
2,
in which xu is he s ock
demand o an unin o med in es o . Using equa ions
(11) and (12), and ecalling ha µ is he sha e o he
in o med in es o s, and 1-µ is he sha e o he
unin o med in es o s, and i deno es in o med and u
deno es unin o med in es o s, he agg ega e p icing
ule in he inancial ma ke eads
P
=
µ
D +1
+
!
i+(1"
µ
)D
+
!
u
. (13)
The ma ke p ice gi en by equa ion (13) esul s
om he asymme y o in o ma ion among he
in es o s.
P oposi ion 1: I he isk- ee a e o e u n
ises/ alls, he in es o s’ a ac ion o isky in es men s
diminishes/inc eases.
P oo : Di e en ia ion o equa ions (11) and (12)
agains
x
i
(x
u )
and
yields
!x
i
!
=!x
u
!
="1
#
2<0,
(14)
showing ha an inc ease in he isk- ee a e educes
isky in es men s o bo h in o med and unin o med
in es o s. Q.E.D.
Co olla y 1: I he isk- ee a e ises/ alls, he
ma ke p ice alls/ ises, ce e is pa ibus.
P oo : Di e en ia e Equa ion (13) agains
P
and
, manipula e, and ge
!P
!
="
µ
D +1
(
+
#
i)2"(1"
µ
)D
(
+
#
u)2<0
, (15)
which shows ha a ise/ all o he isk- ee a e makes
he ma ke p ice o he s ock all/ ise. Q.E.D.
Conside an al e na i e case, whe e he unin o med
in es o s ope a e as echnical ade s saying ha hey
use pas p ices o es ima e he ai p ice o oday.
Manipula ing Equa ion (8), and aking one s ep
backwa ds yields
P
u=(1+ !1
+
"
!1
u)P
!1!D
, which
educes o
P
u=(1+
+
!
u)P
"1"D
(16)
since
!1
=
because he eal isk- ee a e emains
unal e ed unless he cen al bank in e e es, and
!
"1
u=x "1
u
#
2=x
u $
2=
!
u
because he unin o med
in es o makes he in es men decision upon pas
p ices. Di e en ia ion o Equa ion (16) agains
x
u
and
yields again
!x
u
!
="1
#
2<0,
showing ha a ise/ all in he isk- ee a e makes
in es men s in he isky asse all/ ise also o echnical
ade s.
By Equa ions (11) and (16), he agg ega e p icing
ule in he inancial ma ke eads
P
=
µ
V +(1!
µ
) (1+
+
"
)P
!1!D
#
$%
&
. (17)
Co olla y 2: I he isk- ee a e ises/ alls, he
ma ke p ice alls/ ises, ce e is pa ibus.
P oo : Recall Equa ion (11), di e en ia e Equa ion
(16) agains
P
and
, and ge
!P
!
="
µ
D +1
(
+
#
)2+(1"
µ
)P
"1
. (18)
The e ec is nega i e i
µ
D +1
(1!
µ
)P
!1
>
+
"
( )
2
. Tha
is he equilib ium p ice alls, i he weigh ed di idend
e u n is la ge han squa ed ne equi ed e u n, which
is a easonable assump ion. Q.E.D.
5. THE CENTRAL BANK
The ma ke p ice equa ions (13) and (17) imply a
possibili y o a bubble, and he ask o he cen al bank
is o a oid such bubbles by manipula ing he eal isk-
424 Jou nal o Re iews on Global Economics, 2017, Vol. 6 Ilomäki and Lau ila
ee a e o e u n in o de o cause desi ed ma ke
eac ions desc ibed by equa ions (15) and (18). I he
cen al bank possesses pe ec in o ma ion abou he
s ock ma ke and hus on bubbles, i is able o calcula e
he ma ke s abilizing eal isk- ee a e om Equa ion
(13) as ollows:
=
µ
D +1
P
+(1!
µ
)D
P
!
"
.
(19)
Conside also he case ha he cen al bank does
no possess pe ec in o ma ion abou he ma ke and
bubbles. Assume ha he cen al bank has he same
impe ec in o ma ion as he uni o med in es o s,
knowing only D , and an icipa ing ha P = P -1 and
D +1=D .
Thus, Equa ion (19) eads
!=
µ
D
P
"1
+(1"
µ
)D
P
"1
"
#
.
(20)
P oposi ion 2 The cen al bank should in e ene in
he s ock ma ke e en i i does no ha e p i a e
in o ma ion abou i .
P oo : Sub ac Equa ion (20) om Equa ion (19),
and w i e
!
"=
µ
D +1
P
!D
P
!1
#
$
%&
'
(
. (21)
By Equa ion (21), he di e ence o he eal isk- ee
a es ha a e se wi h pe ec and impe ec in o ma ion
equals he weigh ed di e ence be ween he eal
di idend yield a pe iod
+1
and he eal di idend yield
a pe iod
. The le -hand side o Equa ion (21) is
s a iona y, i he igh -hand side is s a iona y. Recall
Equa ion (2),
ln D =ln D !1+e
d
. Hence, bo h
componen s in he pa en hesis on he igh -hand side
a e s a iona y in Equa ion (21). The s a iona i y o he
di e ence be ween he wo eal isk- ee a es implies
ha P oposi ion 1 and Co olla ies 1-2 apply also i he
cen al bank possesses impe ec in o ma ion on he
s ock ma ke . Q.E.D.
6. CONCLUSIONS
The pape p esen s an o e lapping gene a ions
model o a ional in es o s wi h asymme ic
in o ma ion. The o e lapping gene a ions model is
in e p e ed so ha he weal h manage s con ol
people’s money, moni o ed in sho - e m in e als. The
cen al bank, as a social planne , aims o dampen
possible misp icing in he s ock ma ke by adjus ing he
eal isk- ee a e o e u n. The ac i e ole o he
cen al bank is jus i ied because unin o med in es o s
make mis akes on p icing. By p e en ing bubbles o
in la e he cen al bank a oids social losses caused by
he de elopmen and e en ual bu s ing o a supe
bubble.
The analysis shows ha he cen al bank esponds
o a ma ginal eme gence o a posi i e bubble by using
i s a ailable means o manipula e he eal isk- ee a e
upwa ds hus ealloca ing he in es men s o in es o s
om isky o isk- ee asse s. Likewise, a nega i e
bubble is dampened by making he eal isk- ee a e
all hus di ec ing in es men s om isk- ee o isky
asse s. Thus, he policy adjus men o he eal isk- ee
a e o e u n le els ou any bubbles. Mos , impo an ly,
he me hod is e ec i e e en i he cen al bank has no
p i a e in o ma ion abou bubbles.
The esul s o he pape a e wo h highligh ing,
because he e does no exis a ull consensus nei he
among academics no among p ac i ione s abou he
p ope way o in e ene in ma ke bubbles. One key
issue conce ns he alidi y o he “leaning agains he
wind” policy. Ou s esul s clea ly jus i y he policy. To
be mo e speci ic, a posi i e bubble mus be ackled by
aising he isk- ee a e, and ice e sa.
Ano he impo an ques ion is whe he he cen al
bank should ha e pe ec in o ma ion abou he bubble.
Ou esul s simply show ha , i he loga i hm o
di idends ollows andom walk, hen e en an
in e e ence based on delayed in o ma ion is su icien
o p e en he de eloping o he wedge be ween he
undamen al alue and he equilib ium p ice in he
s ock ma ke s. This happens because he di e ence o
sequen ial di idend yields is s a iona y so ha he
equilib ium p ice ollows a s ochas ic end in he
undamen al alue.
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Recei ed on 04-05-2017 Accep ed on 10-07-2017 Published on 24-08-2017
DOI: h ps://doi.o g/10.6000/1929-7092.2017.06.43
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