The equi y p emium puzzle and dec easing ela i e isk a e sion
Mau ice J. Roche
The Na ional Uni e si y o I eland, Maynoo h, Co. Kilda e, I eland.
Abs ac
Agen s a e assumed o ha e a powe isk a e sion u ili y unc ion in an o he wise s anda d asse
p icing model. These p e e ences a e shown o be capable o elimina ing one e sion o he
equi y p emium and isk ee a e puzzles when hey display dec easing ela i e isk a e sion.
Keywo ds: asse p icing; equi y p emium; isk a e sion
JEL classi ica ion: G10, G12
1. In oduc ion
I has been wen y yea s since he seminal pape o Meh a and P esco (1985) ha i s
a icula ed he equi y p emium puzzle. In a ecen c i ical e iew o he li e a u e Meh a and
P esco (2003) poin ou ha many o he esolu ions o sol e his puzzle ha e ailed. The
assump ion ha agen s ha e p e e ences ha exhibi dec easing ela i e isk a e sion has
ecen ly been shown o elimina e one e sion o he equi y p emium and isk- ee a e puzzles
(see Meye and Meye (2005)). Howe e hei u ili y unc ion is oo cumbe some o use in many
o he applica ions and lacks in ui ion. In his pape we show ha a powe isk a e sion u ili y
unc ion ha displays dec easing ela i e isk a e sion is also capable o elimina ing he puzzles.
2. A s anda d asse p icing model
Meh a and P esco (1985) de i e wo Eule equa ions om a s anda d asse p icing model
whe e all agen s choose consump ion so as o maximize he p esen discoun ed alue o u u e
expec ed u ili y a ising om andom consump ion s eams. The wo equa ions a e gi en by

111
() 0
()
eb
Uc
ERR
Uc







(1)
and
11
() 1
()
b
Uc
ER
Uc







0

(2)
whe e E is he expec a ions ope a o condi ional on in o ma ion a ime , is he ma ginal
u ili y o eal consump ion pe capi a,
( )
Uc

1
e
R

is he g oss eal e u n on equi y, 1
b
R
 is he g oss eal
e u n on bonds and

is a cons an discoun ac o . Koche lako a (1996) uses he law o i e a ed
1
expec a ions o eplace he condi ional expec a ion in equa ions (1) and (2) wi h an uncondi ional
expec a ion and es ima es he popula ion means o

1
11
()
()
ee
Uc
eR
Uc









1
b
R


(3)
and
1
1
() 1
()
b
Uc
eR
Uc









1
b



(4)
using annual U.S. da a om 1889-1978. Koche lako a (1996) assumes ha agen s ha e u ili y
unc ions ha exhibi cons an ela i e isk a e sion and inds ha one o bo h o indi idual null
hypo heses 10
e
e and 10
b
e a e ejec ed o he same pa ame e o cons an ela i e isk
a e sion which he a ied om 0 o 10. The null hypo hesis 10
e
e

is no ejec ed, a he 95%
signi icance le el, o alues o he pa ame e o cons an ela i e isk a e sion g ea e han
se en, while he null hypo hesis 10
b
e

is no ejec ed, a he 95% signi icance le el, o alues
o he pa ame e o cons an ela i e isk a e sion less han one. This inding cha ac e izes one
e sion o he equi y p emium puzzle.
3. Dec easing ela i e isk a e sion
Meye and Meye (2005) poin ou ha a eason why p e e ences ha allow habi
o ma ion, as in Campbell and Coch ane (1999), can “ educe o elimina e he equi y p emium
puzzle” is because hei u ili y unc ion displays dec easing ela i e isk a e sion. Meye and
Meye (2005) conside a se o u ili y unc ions whe e ma ginal u ili y is gi en by
()
Uc c




(5)
2
o 0

 and 0.

 Rela i e isk a e sion is gi en by / .
c


The pa ame e

go e ns he a e
o dec ease in ela i e isk a e sion. Using he same da a se as Koche lako a (1996) hey
no malize he le el o eal consump ion pe capi a in 1889 o uni y and calcula e ela i e isk
a e sion o a e age consump ion as /2.3.


Meye and Meye (2005) choose alues o

anging om 0.5 o 2 and adjus

so as ela i e isk a e sion o a e age consump ion is in he
0.5-10 ange. A sample o hei esul s is p esen ed in Table 1. They show ha o a la ge ange
o alues o

and

he -s a is ics o es ing whe he he null hypo heses 1
e
e0 and 10
b

e

a e no ejec ed when ela i e isk a e sion o a e age consump ion is in he 6-10 ange and
0.99.

 Assuming ha 0.99

 we es ima ed he pa ame e s

and

using equa ions (1)
and (2) by gene alized me hod o momen s1 using he da a se om Koche lako a (1996). We
es ima ed

o be 2.18 wi h a s anda d e o o 1.23 and es ima ed

o be 36.32 wi h a s anda d
e o o 17.02. The p obabili y alue o he J- es o o e iden i ying es ic ions is 0.45. The
pa ame e es ima es o

and

a e signi ican a he 10% and 5% le els espec i ely. Mos
alues o

and

used by Meye and Meye (2005) which we e p esen ed in Table 1 a e wi hin
he 95% con idence in e al o ou es ima es o

and .

One p oblem wi h Meye and Meye (2005) p e e ence speci ica ion is ha he
unde lying u ili y unc ion is a he cumbe some. They do no ac ually p esen he u ili y
unc ion in hei pape and s a e “ he exac o m o he u ili y unc ion is unknown”. Using a
ma hema ics so wa e p og am such as Maple one can in eg a e equa ion (5) wi h espec o
consump ion and his gi es he ollowing u ili y unc ion
111
() ,
c
Uc cc



  

 

      

 


 







(6)
3
whe e 1







is a gamma unc ion and 1,
c




 



is an incomple e gamma unc ion. We
belie e ha using his unc ion in o he applica ions would be di icul and lack in ui ion.
In his pape we o e an al e na i e unc ional o m o u ili y, he powe isk a e sion
u ili y unc ion (see Xie (2000)). This unc ion has he p ope y o dec easing ela i e isk
a e sion unde ce ain pa ame e alues. The unc ion is gi en by
1
1
() 1exp 1
1
c
Uc 



 















(7)
No e when 0

and 0

hen equa ion (7) is he commonly used cons an ela i e isk
a e sion u ili y unc ion. Ma ginal u ili y is gi en by
1
() exp 1
1
c
Uc c 














(8)
and ela i e isk a e sion is 1.
c



Thus equa ion (7) exhibi s dec easing ela i e isk
a e sion when 1

 and 0.

 We calcula e he -s a is ics o es ing he asse p icing Eule
equa ions (1) and (2). Simila o Meye and Meye (2005) we show ha o a la ge ange o
alues o

and

he -s a is ics o es ing whe he he null hypo heses 10
e
eand 10
b
e

a e
no ejec ed when ela i e isk a e sion o a e age consump ion is in he 6-10 ange. We
p esen ou esul s in Table 2. Since

go e ns he a e o dec ease in ela i e isk a e sion we
choose alues o

anging om 2 o 5. Then is

calcula ed so ha ela i e isk a e sion o
a e age consump ion akes on alues 6, 8 o 10.
Assuming ha 0.99

 we es ima ed he pa ame e s

and

using equa ions (1) and
(2) by gene alized me hod o momen s.2 We es ima ed

o be 3.95 wi h a s anda d e o o
2.98 and es ima ed

o be 30.93 wi h a s anda d e o o 16.14. The p obabili y alue o he J-
4

es o o e iden i ying es ic ions is 0.39. The pa ame e es ima es o

and

a e signi ican
a he 20% and 10% le els espec i ely. Mos alues used in Table 2 a e wi hin he 95%
con idence in e al o ou es ima es o

and .

4. Conclusions
In a ecen pape Meye and Meye (2005) show ha p e e ences ha display dec easing
ela i e isk a e sion a e capable o elimina ing one e sion o he equi y p emium and isk ee
a e puzzles. We sugges ha hei u ili y unc ion is oo cumbe some o use in many o he
applica ions and lacks in ui ion. We show ha a powe isk a e sion u ili y unc ion ha
displays dec easing ela i e isk a e sion is also capable o elimina ing he puzzles. This
unc ion is ela i ely s aigh o wa d o use and has been employed in o he applica ions (see Xie
(2000).
5
Re e ences
Meye , D. and J. Meye , 2005. Risk p e e ences in mul i-pe iod consump ion models, he equi y
p emium puzzle, and habi o ma ion u ili y. Jou nal o Mone a y Economics, o hcoming.
Koche lako a, N., 1996. The equi y p emium: I ’s s ill a puzzle. Jou nal o Economic Li e a u e
34:42-71.
Meh a, R. and P esco , E., 1985. The equi y p emium: A puzzle. Jou nal o Mone a y
Economics 15:145-61.
Meh a, R. and P esco , E., 2003. The equi y p emium in e ospec . The Handbook o he
Economics o Finance, Eds. Cons an inides, G., Ha is, M., and S ulz, R., No h-Holland:
Ams e dam, 888-936.
Xie, D., (2000). Powe isk a e sion u ili y unc ions. Annals o Economics and Finance 1, 265-
282.
6
Table 1
Tes ing asse p icing Eule equa ions using he Meye and Meye (2005) u ili y unc ion
Rela i e isk a e sion a he
a e age consump ion le el
  -s a is ic
01
:0
e
He


-s a is ic
01
:0
b
He


6 1 14.04 1.85 -1.92
1.5 21.48 1.64 -1.21
2 32.88 1.34 -0.34
8 1 18.72 1.33 -1.42
1.5 28.64 1.07 -0.60
2 43.84 0.68 0.34
10 1 23.40 0.09 -0.92
1.5 35.80 0.55 -0.03
2 54.80 0.09 0.84
Sou ce: Table 3 in Meye and Meye (2005)
7
Table 2
Tes ing asse p icing Eule equa ions using he powe isk a e sion u ili y unc ion
Rela i e isk a e sion a he
a e age consump ion le el
  -s a is ic
01
:0
e
He


-s a is ic
01
:0
b
He


6 3 20.12 1.71 -2.10
4 24.43 1.63 -1.61
5 28.13 1.66 -1.50
8 3 26.52 1.17 -1.16
4 48.86 0.97 -0.01
5 84.40 0.72 0.83
10 3 37.12 0.69 -0.35
4 73.29 0.39 0.84
5 140.67 -0.52 1.06
8