The equity premium puzzle and decreasing relative risk aversion

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