A New Solution to the Equity Premium Puzzle and the Risk-Free Rate Puzzle: Theory and Evidence

Transcription

1 A New Soluion o he Equiy Premium Puzzle and he Risk-Free Rae Puzzle: Theory and Evidence Hideaki Tamura Yoichi Masubayashi Augus 04 Discussion Paper No.4 GRADUATE SCHOOL OF ECONOMICS KOBE UNIVERSITY ROKKO, KOBE, JAPAN

2 A New Soluion o he Equiy Premium Puzzle and he Risk-Free Rae Puzzle: Theory and Evidence Hideaki Tamura Graduae School of Economics Kobe Universiy Yoichi Masubayashi Graduae School of Economics Kobe Universiy Absrac This paper develops a new mehod for solving boh equiy premium and risk free rae puzzles based on he sandard uiliy funcion. The mehod for solving he equiy premium puzzle in accordance wih Mehra and Presco (985) needs o be simulaneously consisen wih he mehod for solving he risk-free rae puzzle presened by Weil (989). Tha is, he reasonable esimaed values for he degree of relaive risk aversion in he former soluion and for he subjecive discoun rae in he laer soluion need o plausibly fall wihin experienial bounds. This sudy indicaes ha a consisen soluion is possible for he equiy premium and risk-free rae puzzles even when here is a sandard consan relaive risk aversion (CRRA) ype uiliy funcion. This soluion is possible by formularizing he Euler equaion for consumpion, considering he precauionary saving effec. JEL Classificaion Number: E, E44, G Keywords: equiy premium puzzle, risk-free rae puzzle, uncerainy, Euler equaion Correspondence o: Yoichi Masubayashi Graduae School of Economics, Kobe Universiy Rokkodai, Nada-ku, Kobe, , Japan TEL myoichi@econ.kobe-u.ac.jp

3 . Inroducion A cenral issue in he verificaion of he permanen income hypohesis since Hansen and Singleon (98) has been he esimaion of parameers in he Euler equaion for consumpion (he subjecive discoun rae and degree of relaive risk aversion) based on he consumpion-based capial asse pricing model (C-CAPM). However, uncerainy has arisen in he esimaion of he degree of relaive risk aversion using U.S. daa ha depends on he esimaion period, making i difficul o have confidence in hese analyss. Hansen and Singleon (983, 984) repored on he negaive degree of relaive risk aversion when using monhly U.S. daa. In addiion, resuls ofen have been dismissed due o he overidenificaion resricion of J-saisics. Anoher issue is he equiy premium puzzle shown by Mehra and Presco (985), who demonsraed ha he general equilibrium model using sock reurn, rae of reurn on reasury bills, and sample annual consumpion growh rae from 890 o 979 maximizes he resricions imposed on he sock price index and average rae of reurn on reasury bills. Mankiw and Zeldes (99) addressed his problem using he relaional expression derived by applying he Taylor approximaion o he Euler equaion, bu hey also repored ha he degree of relaive risk aversion calculaed using Mehra and Presco s (985) sample is 6.3. In paricular, hey repored an unusually high value of 89 when aking he pos-war sample from 948 o 988. Weil (989) pursued he values of he risk premium and risk-free rae for each combinaion of degree of relaive risk aversion (γ) and elasiciy of iner-emporal subsiuion (/ρ) in he nonindependen and idenically disribued (i.i.d) dividend growh process using Kreps Poreus ype preferences. Specifying a consan relaive risk aversion (CRRA)-ype preference (ρ=γ), he derived a risk premium of 6.37%. Alhough ha specificaion approximaes he acual level wherein β = 0.98, γ = 0, and /ρ = 0.05, he 5.0% risk-free rae a ha ime was high. He also derived a risk premium of 5.7% and a risk-free rae of 0.85%, which are close o acual levels, wherein β = 0.95, γ = 45, and /ρ = 0., bu noed his resul depends on a high value for γ. This inconsisency is referred o as he risk-free rae puzzle. According o he rerospecive review by Mehra and Presco (003), he exising soluion for he equiy premium puzzle based on radiional heory is largely spli among he alernaive

4 preference srucure (he ime non-separable model ha does no assume separabiliy in relaion o ime and habi formaion), he model ha incorporaes idiosyncraic and uninsurable income risk and models incorporaing a disaser sae and survivorship bias. Soluions no based on radiional heory are affeced by borrowing consrains, liquidiy premiums, axes and regulaion. Of he above soluions, he soluion for boh he equiy premium and he risk-free rae puzzles in accordance wih Mehra and Presco (003) is limied o he alernaive preference srucure (he ime non-separable model ha does no assume separabiliy in relaion o ime and habi formaion). An evaluaion is as follows. Firs, he model ha does no assume separabiliy in relaion o ime miigaes he severe (reciprocal) relaionship beween he degree of relaive risk aversion and iner-emporal elasiciy of subsiuion, which comes ino exisence under he radiional CRRA-ype uiliy funcion, and also inroduces he Kreps Poreus ype uiliy funcion ha has been pariioned respecively, as is exemplified in Epsein and Zin (989, 99). This model expresses he iner-emporal elasiciy of subsiuion as he weighed average of he marginal rae of subsiuion (MRS) for consumpion and marke porfolio. This builds he CAPMha generalizes boh he C-CAPM and Sharp Linner CAPM. However, he inadequacy of any special device o influence an esimaion of he degree of relaive risk aversion implies i will no solve he equiy premium puzzle, alhough i is believed ha he riskfree rae puzzle can be solved by making he iner-emporal elasiciy of subsiuion and degree of relaive risk aversion independen of each oher. Nex, habi formaion deermines he uiliy of curren consumpion relaive o pas consumpion ha is considered o have become habiual or consumpion when benchmarked o neighbours. Consaninides (990) defines uiliy as he difference beween curren and pas consumpion (habi) wih a lag, indicaing ha he equiy premium puzzle can be solved when he weighing of pas consumpion (habi) is increased, even when he coefficien for he degree of relaive risk aversion is low according o he inernal habi model ha calculaes a high degree of effecive risk aversion. In addiion, Campbell and Cochrane (995) presened a model ha was consisen for boh consumpion and asse marke daa incorporaing he possibiliy of an economic downurn using he sae variable ha risk aversion changes in a non-linear way. Abel (990) presened a model ha can solve he equiy premium puzzle while also avoiding he risk-free rae puzzle modelled in accordance o he proporion of consumpion (habi). These habi formaion 3

5 models can solve he equiy premium puzzle, anicipaing a consumer wih an exreme dislike of consumpion risk even when he degree of risk aversion is small. In addiion, i is believed ha he dislike of consumpion risk booss demand for safe asses and reduces he risk-free ineres rae and will be used o solve he risk-free rae puzzle. Alhough hese habi formaion models can solve boh he equiy premium and risk-free rae puzzles given limied assumpions, all are based on mehods ha complicae he sandard CRRA-ype uiliy funcion. Thus, hey can hardly be referred o as generic soluions. The Euler equaion for consumpion ha uses he sandard CRRA-ype uiliy funcion has feaured only wo explanaory variables ill dae: he consumpion growh rae and reurn on asses. On he oher hand, fuure expeced marginal uiliy rises under a heoreical model of precauionary saving due o he increased income uncerainy when U"'>0, and he maximisaion of household firs-order condiion of expeced uiliy. However, his suggess ha an increase in income uncerainy influences he MRS for he iner-emporal indifference curve assumed for esimaing he uiliy funcion parameers. Tha is, he MRS for an indifference curve will always change during opimizaion of muliple periods when here is a relaive change in income uncerainy beween he presen and fuure due o he change in relaive saus of a uiliy funcion ha considers presen and fuure uncerainy. Thereafer, if esimae employs only he consumpion growh rae and reurn on asses wihou correcion of ha influence, here will be an incorporaed bias for he esimaed value of he parameers hemselves, disoring he esimae for he degree of relaive risk aversion. The Euler equaion for consumpion ha is expanded o include income uncerainy as an explanaory variable needs o be formularized o esimae parameers wih a greaer degree of informaion o correc for his precauionary saving effec and hereby esimae parameers correcly. This sudy formularizes he Euler equaion for consumpion (he coefficien of variaion (CV) model) wih hree explanaory variables (consumpion growh rae, reurn on asses, and growh rae of income CV). Using he Mankiw and Zeldes (99) mehod, he esimaed degree of relaive risk aversion can be explained using he wo covariances beween he reurn on asses and consumpion growh rae and he reurn on asses and growh rae of he income CV while clarifying he feaures by comparing resuls of he analysis using he Euler equaion for consumpion (he normal model ) ha uses he wo radiional explanaory variables of he consumpion growh rae and reurn on asses using U.S. daa. 4

6 This paper is organized as follows. Secion idenifies he feaures of he CV model hrough analysis of a model ha uses he same Taylor expansion as Mankiw and Zeldes (99) while also formularizing he Euler equaion for consumpion under income uncerainy. Secion 3 calculaes he degree of relaive risk aversion and subjecive discoun rae for he normal model and he CV model using he U.S. monhly real consumpion growh rae, real reurn on asses, and he income uncerainy index for February 978 o December 00. Secion 4 considers he correcion effec for he degree of relaive risk aversion under he CV model from he perspecive of a wo-period model. Secion 5 presens our conclusions.. Model. Euler equaion for consumpion under income uncerainy This secion specifies he mehod for seing an opimal consumpion model under income uncerainy using he marginal uiliy ha is influenced by he income CV under he sandard CRRAype uiliy funcion o derive he Euler equaion for consumpion ha includes he income CV. The individual s expeced marginal uiliy funcion under income uncerainy can be expressed as follows from equaion (A.6) in Appendix A: [ + 0.5( γ + γ ) CV ] γ U '( C) = C. () However, C represens an individual s real consumpion for period, value of he CV for consumpion a period : CV = ( h/ C) CV represens he square where h represens he sandard deviaion of consumpion a period due o income uncerainy and γ is a parameer represening a consan degree of relaive risk aversion. From equaion (), he individual s expeced marginal uiliy under income uncerainy is he expeced marginal uiliy of he normal model muliplied by 0.5( γ +γ ) CV +. The expeced marginal uiliy rises in proporion o he square value of CV when he CV for consumpion increases. This analysis is a modificaion referred o in Skinner (988). 5

7 The muli-period opimal consumpion model ha uses he individual s expeced uiliy funcion (equaion (A.5) in Appendix A) under income uncerainy is se as follows: i= 0 i max E[ β U ( C )], () + i s.. N j j+ j= j= N q A + C = ( qj + dj) Aj + Y. (3) However, β is he subjecive discoun rae ( 0<β < ), q j is he price of asse j a period (j=,,,n), d j is he dividend from asse j a period (j=,,,n), A j is he quaniy of asse j held a period, Y is he non-asse income for period and E [ ] is he condiional expecaion operaor based on informaion available a ime. There are N asses in he economy, and individuals selec cash flow for each asse and consumpion ha maximizes he presen discouned value of heir expeced uiliy derived hrough consumpion from he presen ( = 0) o he fuure. maximizaion: Solving he above opimizaion problem resuls in he following firs-order condiion for U '( C ) q + + j+ d j+ E[ β ( )] = 0. (4) U '( C) q j Here, he rae of reurn rj+ on asse j is defined as r j+ = ( qj+ + dj+ )/ qj ( q d ) / q j+ + j+ j in equaion (4) can be subsiued for ( + j+ ), so r. Therefore, by replacing his hen subsiuing equaion () ino equaion (4), he individual s Euler equaion for consumpion under income uncerainy when here is a CRRA-ype uiliy funcion can be expressed as follows: E γ C ( γ + γ ) CV+ [ β (+ r j+ C + 0.5( γ + γ ) CV )] = 0 (j=,,,n). (5) Transforming + γ +γ CV in he expeced marginal uiliy of equaion () using a 0.5( ) 6

8 linear approximaion of he exponenial funcion for he Taylor expansion formula resuls in + 0.5( γ + γ ) CV exp[0.5( γ + γ ) ]. Therefore, he middle erm of equaion (5) can be ransformed as follows: CV + 0.5( γ + γ ) CV + 0.5( γ + γ ) CV + exp( CV exp( CV + ) ) 0.5( γ+ γ ) Applying he ransformed middle erm o equaion (5) leads o he following for he Euler equaion for consumpion. The ransformaion also adds he growh rae of he exponenial of he squared value of he CV for consumpion as an explanaory variable while he coefficien for he degree of relaive risk aversion 0.5( γ ) consumpion. γ + is applied as an index of ha growh rae of he CV for γ 0.5( γ+ γ ) C exp( ) + CV+ β (+ ) = exp( ) rj+. (6) C CV This is he Euler equaion for consumpion under income uncerainy formularized wih he hree explanaory variables (consumpion growh rae, reurn on asses and he growh rae of he CV for consumpion).. Analyzing he model for deermining he degree of relaive risk aversion and subjecive discoun rae Mankiw and Zeldes (99) applied he Taylor expansion of he wo variable funcions o he Euler equaion wih wo variables from equaion (7) as explanaory variables (consumpion growh rae and reurn on asses). This led o he relaional expression among he equiy premium, degree of relaive risk aversion and covariance beween reurn on asses and consumpion growh rae when consanly abbreviaed as expressed as equaion (8). i C γ E [(+ r )(+ g ) ] = + ρ, (7) i C E[ r] r γcov( r, g ). (8) i 7

9 C However, g ( C / C) and he subscrip for ime have been abbreviaed. In addiion, = + i r represens he rae of reurn on risk asse i, and r represens he rae of reurn on he risk-free asse. E[ r i ] E[ r] represens he equiy premium, and ρ represens he ime preference rae (equivalen o ( / β ) ). In accordance wih equaion (8), he degree of relaive risk aversion for he normal model is defined as he equiy premium divided by he covariance beween he reurn on asses and he consumpion growh rae. Tha is, γ ( E[ r] r) / σ, (9) NM i C σ =Cov( r, g ). ic i ic In addiion, he relaional expression for he risk-free asse ha can be used in he resuls of equaion (8) can be expressed as follows: r C C ρ + γe[ g ] 0.5( γ + γ ) Var( g ). (0) Subsiuing ρ = ( / β) ino equaion (0), and solving for he subjecive discoun rae β resuls in he following equaion: C β /(+ E[ r] γe[ g ] + 0.5( γ + γ ) σ ), () σ =Var( g C ). c c The subjecive discoun rae for he normal model is obained by subsiuing γ NM as deermined from equaion (9) and subsiuing he average, variance and covariance for each variable ino equaion (). On he oher hand, he CV model formularized in he preceding paragraph uses hree explanaory variables (consumpion growh rae, reurn on asses and he growh rae of income CV). This can be expressed in he same way as equaions (7) and (8) for he normal model as follows from equaion (B.6) in Appendix B: 8

10 i C γ ecvsq 0.5( γ+ γ ) E [(+ r )(+ g ) (+ g ) ] = + ρ, () i i C i E[ r] r γ Cov( r, g ) 0.5( γ + γ ) Cov( r, g ecvsq ). (3) ecvsq However, g (exp( CV )/ exp( CV )), and he subscrip for ime has been abbreviaed. = + In he normal model shown in equaion (8), he degree of relaive risk aversion was explained only hrough he covariance beween he reurn on asses and consumpion growh rae. The degree of relaive risk aversion when exended o he CV model is explained by boh he covariance beween he reurn on asses and he consumpion growh rae and he covariance beween reurn on asses and he growh rae of income CV. Equaion (3) can be rewrien as he funcion for degree of relaive risk aversion γ, resuling in he following: i C i i ecvsq Cov ( r, g ) γ E[ r ] r+ 0.5Cov( r, g )( γ + γ ). (4) i ecvsq In addiion, applying Cov ( r, g ) = 0 o equaion (4) resuls in he following equaion, which is he same as he equaion for deermining he degree of relaive risk aversion for he normal model (equaion(8)): i C i Cov( r, g ) γ E[ r ] r. (5) Under normal marke condiions when he equiy premium exceeds zero, he covariance beween reurn on asses and he consumpion growh rae is posiive and he covariance beween reurn on asses and he growh rae of he income CV is posiive or negaive, he relaionship for deermining he degree of relaive risk aversion under he normal model and he CV model based on equaion (4) and equaion (5) is shown in Figure. Figure 9

11 NM Figure deermines he degree of relaive risk aversion for he normal model by he level of i C γ ha resuls in he sraigh line wih he gradien of Cov ( r, g ) on he lef-hand side of equaion (5) being consisen wih he equiy premium of E[ r i ] r. On he oher hand, in he CV model, when he covariance beween he reurn on asses and growh rae of income CV is negaive, he righ-hand side of equaion (4) will resemble (a) wih a curve ha shifs o he righ of segmen E[ r i ] r due o he monoonic increase funcion of γ + γ when > 0 relaive risk aversion for he CV model will be deermined by he level of γ. Thus, he degree of γ ha inersecs CV_a i C wih he sraigh line wih he gradien of Cov ( r, g ) from he lef-hand side of equaion (4). In addiion, when he covariance beween reurn on asses and he growh rae of he income CV is posiive, he righ-hand side of equaion (4) will resemble (b) wih a curve ha shifs o he righ from segmene[ r i ] r. Therefore, i will be deermined by he level of γ ha inersecs wih CV_b i C he sraigh line wih he gradien of Cov ( r, g ) from he lef side of equaion (4). The equiy premium puzzle noes ha he covariance beween reurn on asses and consumpion growh rae obained from acual daa is exremely low relaive o he equiy premium. Therefore, he degree of relaive risk aversion γ NM derived from equaion (5) is an exremely large value exceeding wha is considered normal (wihin 0), or he heoreical model only parially explains he equiy premium wihin wha is considered normal for he degree of relaive risk aversion (wihin 0). On he oher hand, under he CV model using equaion (4), he esimaed resul can be grealy improved over he normal model, even when covariance beween reurn on asses and he consumpion growh rae is exremely small because he degree of relaive risk aversion can be deermined wih he γ being much lower han CV_a γnmwhen covariance beween reurn on asses and growh rae of he income CV is a large negaive value. Solving equaion (4) for γ means he deerminan level for he degree of relaive risk aversion for he CV model can be defined as follows: γ σ σ σ σ σ r) (6) ( ic 0.5 iv) ± 4( ic 0.5 iv) 8 iv( E[ ri] CV σiv i C i ecvsq σ =Cov( r, g ), σ =Cov( r, g ). ic iv 0

12 The condiion for solving he degree of relaive risk aversion under a posiive risk premium by he discriminan D= 4( σ 0.5σ ) 8σ ( E[ r] r) from equaion (6) is iv ic iv i ecvsq σ = Cov( r, g ) 0. Reurn on asses and he income uncerainy index mus condiionally have zero or negaive covariance. Tha is, a requiremen for solving he CV model is choosing he appropriae choice of income uncerainy index i.e. income uncerainy falls when reurn on asses rises and rises when reurn on asses falls. Subsiuing ρ = ( / β) ino equaion (B.5) in Appendix B and solving for he subjecive discoun rae β resuls in he following equaion: iv i C β /(+ E[ r] γe[ g ] + 0.5( γ + γ ) σ + 0.5( γ + γ ) E[ g ecvsq c { 0.5( γ + γ ) } σ 0.5γ( γ γ ) ) + 0.5( γ + γ ) + (7) v σ cv σ =Var( g C ), ( ecvsq C ecvsq σ =Var g ), σ = Cov( g, g ). Subsiuing he c v cv γ CV deermined from equaion (6) and he average, variance and covariance for each variable ino equaion (7), we obain he subjecive discoun rae for he CV model. ] 3. Empirical analysis using he U.S. income uncerainy index and concurren soluion of he equiy premium and risk-free rae puzzles 3. Daa and processing mehods For calculaing he degree of relaive risk aversion and subjecive discoun rae we used monhly daa for he real consumpion growh rae, real reurn on asses (for equiies, he S&P 500 Index ; for U.S. governmen bonds, he secondary marke rae for 90-day Treasury bills) and he income uncerainy index from February 978 o December 00. Real consumpion uses he aggregae amoun of nondurable goods and services per person. The following four indices are used for he income uncerainy index daa. Labour Share (spline conversion o monhly) (CV). Reciprocal of Uni Profi (spline conversion o monhly) (CV). Unemploymen rae UNRATE (CV3). The reciprocal of he Universiy of Michigan consumer confidence index UMCSENT (CV4).

13 Labour Share and Uni Profi are no released monhly, so quarerly daa are convered o monhly daa hrough non-linear inerpolaion using he cubic spline funcion. I is bes o use he figures for each income uncerainy index adjused for he average value and sandard deviaion benchmarked o he CV for consumpion. However, no coninuous long-erm U.S. daa exis for he CV for consumpion. We have used original daa ha has no been adjused for he average value and sandard deviaion. For he observaion period, Figure demonsraes he movemen of CV CV4 indexed o 00 a he beginning of he period. Figure Figure shows ha CV4 moves almos hree monhs o more han six monhs ahead of CV3 and a abou he same ime or much earlier han CV and CV. In addiion, he exen of movemen is much smaller. 3. Calculaions of he degree of relaive risk aversion and subjecive discoun rae 3.. Trends in Risk premium The average value for he risk premium is needed o calculae he degree of relaive risk aversion, and a period displaying a posiive average value mus be chosen o achieve a sable resul. Figure 3 illusraes he movemen in average value of he risk premium calculaed for each monh of he observaion period. We se February 978 as he firs monh and se he end monh as each monh afer he nex monh of he firs monh up unil December 00. Figure 3 illusraes ha he average value of he risk premium was seadily posiive for each period when he end monh was on or afer January 985. Figure Calculaions The calculaion of he degree of relaive risk aversion considers he observaion period and calculaes he covariance for each variable and he average of risk premium for each period. To obain

14 he degree of relaive risk aversion, he resuls are subsiued ino equaion (9) for he normal model and ino equaion (6) for he CV model. Table presens he descripive saisics for each variable used in his calculaion. Table In addiion, he calculaion of he subjecive discoun rae considers he observaion period and calculaes he average, he variance and he covariance for each variable for each period. [Remark ] Togeher wih he resuls from calculaing he degree of relaive risk aversion for he normal model, hese calculaions are subsiued ino equaion () o obain he subjecive discoun rae of he normal model. In a similar fashion, ogeher wih he resuls of calculaing he degree of relaive risk aversion for he CV model, hese are subsiued ino equaion (7) o obain he subjecive discoun rae of he CV model. Table presens he descripive saisics for each variable used in his calculaion. Table Figures 4 7 illusrae he resuls of calculaing he degree of relaive risk aversion and subjecive discoun rae for he observaion period for each income uncerainy index (CV CV4). The movemen in he resul of he degree of relaive risk aversion and subjecive discoun rae calculaed for he normal model and CV model (considering he case ha he average values for he income uncerainy index from CV o CV4 are 0., 0.3[Edior] and 0.5) are illusraed in he graph. Figure 4 Figure 5 Figure 6 Figure 7 These graphs illusrae he resuls of he soluion for each period wherein he equiy premium has a sable posiive value for he end monhs of January 988 and beyond. Average values of he soluion for each period are shown below he graph. 3

15 Comparing Figures 4 7, he graph for he CV model is no shown in Figure 6, which uses i ecvsq CV3 becauseσ = Cov( r, g ) > 0, and he real roo condiion for he soluion in equaion (6) iv is no me. Therefore, CV3is noncomplian for he income uncerainy index. As is eviden in Figure, since here is a change hree o six monhs afer he oher income uncerainy indices (CV, and 4), he lag in his iming can be aribued o he collapse of he normally anicipaed negaive correlaion. The degree of relaive risk aversion for he CV model is lower han he normal model due o he upper graph in Figures 4, 5 and 7 and he average value of he soluion. In addiion, an even lower degree of relaive risk aversion is obained if he average values of he income uncerainy index are increased o 0., 0.3 and 0.5. This indicaes he CV model ha considers he negaive covariance relaionship for he reurn on asses and he income uncerainy index can solve he equiy premium puzzle ha poins o an abnormal value for he degree of relaive risk aversion. In addiion, he subjecive discoun rae for he normal model repeaedly rises and falls subsanially and holds he level beyond afer Augus 00 (boom graphs in Figures 4, 5 and 7) and he average value of he soluion, whereas he subjecive discoun rae for he CV model has levelled a slighly below. In addiion, i has fallen due o a rise of 0., 0.3 and 0.5 in he average values of he income uncerainy index. This levelling is pronounced for CV4 in Figure 7. This indicaes he CV model ha considers he negaive covariance for he reurn on asses and income uncerainy index can improve he risk-free rae puzzle ha indicaesan abnormal value for he subjecive discoun rae. Comparing Figures 4, 5 and 7 and examining he relaive meris of income uncerainy indices indicae ha he degree of relaive risk aversion and he subjecive discoun raes are more suiable values for CV4 han CV and CV, and he bes as an income uncerainy index. This is aribued o he negaive covariance of he reurn on asses and income uncerainy index in Table being he larges for CV4. 4 Explanaion of he CV model using he wo period model based on Mehra and Presco (985) daa Aggregaing he uiliy funcion in equaion () as i=0, for jus he wo periods, and 4

16 replacing he oal uiliy wih Z resuls in he following equaion: Z U C ) + β U ( C ). (8) = ( + The Euler equaion, equaion (6), represens he firs-order condiion of expeced uiliy maximizaion derived from he muli-period opimal consumpion model. Tha condiion is he MRS of he iner-emporal indifference curve dc +/ dc which is he same as he gradien of he budge consrain line r ). To obain his MRS, replace he complee differenial of equaion ( + + (8) wih zero o arrive adz U ( C) dc + β U '( C ) dc = 0. Doing so generaes he following equaion: = ' + + dc dc + dz= 0 U '( C) C = = βu '( C ) βc + γ γ + [+ 0.5( γ + γ ) CV [+ 0.5( γ + γ ) CV ]. (9) ] + Transforming he CV iem for equaion (9) using he Taylor expansion formula for he same index funcion as in Secion means he MRS for he indifference curve under he wo-period model for he CV model can be represened as follows: dc dc + dz= 0 C = β C + γ exp( CV exp( CV + ) ) 0.5( γ+ γ ) (0) On he oher hand, he MRS for he indifference curve under he wo-period model for he normal model can be represened by he following equaion ha eliminaed he CV iem for equaion (0) by subsiuing exp( 0) = for he numeraor and he denominaor in equaion (0) for he CV for consumpioncv o remain zero. dc dc + dz= 0 C = β C+ γ. () When he MRS for equaion (0) is he same as he gradien for he budge consrain line, i can be ransformed o derive he consumpion Euler equaion under income uncerainy(equaion (6)). 5

17 Now, in he case ha income uncerainy rises from period o period + so ha CV + > CV, he MRS for he indifference curve under he normal model will no change due o equaion (). However, in he CV model under equaion (0), he hird iem on he righ side, which is he growh rae for he consumpion CV exp( CV )/ exp( ), will be below. Therefore, he CV + decline in he MRS produces a change in he focus on fuure consumpion for he enire indifference curve. On he oher hand, if income uncerainy falls from period o period + so ha CV > CV+, he MRS for he indifference curve under he normal model will no change due o equaion (). However, in he CV model under equaion (0), he hird iem on he righ side, which is he growh rae for he consumpion CV exp( CV )/ exp( ), will exceed. Therefore, he rise CV + in he MRS promps a change in he focus on presen consumpion for he enire indifference curve. Figure 8 shows he correcion effec using he wo-period model in he esimaed value of he degree of relaive risk aversion due o he CV model based on he Mehra and Presco (985) daa used o indicae he U.S. equiy premium puzzle. Figure 8 In Figure 8, he solid line represens he budge consrain ha corresponds o he maximum value (.40649) and he minimum value (0.7354) for he annualized rae of he reurn on socks from 949 o 979. The doed line rising righward shows he exen of he maximum value (.04080) and minimum value ( ) for he annualized consumpion growh rae (cons) fixed a he corresponding subjecive equilibrium poin. Thus, he magnificaion of he sandard deviaion for he sock reurn relaive o he sandard deviaion for he consumpion growh rae is /0.07=.05x. The exen of change in he consumpion growh rae is very small in comparison o he large change in he budge consrain line. The indifference curve for he normal model under his consrain is illusraed by he exremely large convexiy in he doed line in Figure 8. The degree of relaive risk aversion for he corresponding CRRA-ype uiliy funcion is also exremely large. On he oher hand, under he CV model ha considers he negaive covariance relaionship for reurn on asses and income uncerainy index, a rise in reurn on asses coincides wih a decline 6

18 in fuure income uncerainy under household senimen. Therefore, he increase in he MRS shifs he focus on presen consumpion for he enire indifference curve. The indifference curve indicaing a rise in MRS is no consrained by he budge consrain corresponding o a minimum value (0.7354) for reurn on socks. Because i is possible o genly approach somehing like he solid line in Figure 8, here is no increase in convexiy of he indifference curve even when he consumpion growh rae flucuaes only slighly. The degree of relaive risk aversion for he CRRA-ype uiliy funcion can be esimaed wihin he normally anicipaed realm. Reurn on asses and income uncerainy generally rend owards a negaive correlaion. When an excessive reurn degree of relaive risk aversion is esimaed because he flucuaion in consumpion is small relaive o he flucuaion in reurn on asses, applying he CV model improves esimaion resuls. 5 Conclusions A ypical aemp o solve he equiy premium puzzle uses a non-crra alernaive preference srucure (he ime non-separable model ha does no assume separabiliy in relaion o ime and habi formaion). However, he generalizaion and specializaion of hese uiliy funcions have no yielded generic soluions o he puzzle. Thus, from he viewpoin of parsimony, i would be preferable if he equiy premium puzzle could be explained using a simple CRRA uiliy funcion. To his poin, daa indicaes ha our model is useful in solving he equiy premium puzzle and risk-free rae puzzle wih he CRRA uiliy funcion. Moreover, i could ac as one of he generic soluion because here is no need o assume paricular household behaviours such as habi formaion. This sudy performed wo asks o solve he puzzle. Firs, in addiion o he reurn on asses when deciding he opimal consumpion pah over muliple periods, we assumed ha households accouned for income uncerainy. Similar o he exensive accepance of precauionary savings arising from income uncerainy, our assumpions have already been acceped by several researchers. Second, we formulaed he effec of precauionary savings and wihdrawal of hese savings ha arise from he change in income uncerainy by incorporaing hem ino he Euler equaion. Doing so enabled us o esimae he CRRA uiliy funcion parameers for households, simulaneously accouning for boh he profiabiliy of financial asses and income uncerainy. 7

19 Unil now, dynamic opimizaion of consumpion across periods and precauionary savings have been discussed as separae opics. We believe ha unifying hem wihin one Euler equaion is essenial o he developmen of consumpion heory and o sable esimaions of generalized mehod of momens (GMM). This unificaion forges a close and compaible relaionship beween heory and realiy and emerges as a principal way o solve he sagnaion seen in applying consumpion heories such as C-CAPM, which has been affeced by his puzzle over he pas 30 years. 8

20 Appendix A: Derivaion of he individual s expeced marginal uiliy funcion under income uncerainy Firs, we ake C o represen real consumpion for an individual s period, and ake U ( C ) o represen he insananeous uiliy funcion ha has addiive separabiliy for ha poin in ime. Nex, we represen he level of wavering in consumpion due o income uncerainy as sandard deviaionh, and assume an uncerain siuaion in which case here is a 50% probabiliy ha individual consumpion will increase by only h and a 50% probabiliy i will decrease by only h. The uiliy ha akes accoun of he uncerainy in he individual consumpion level C a his ime is as follows. U ( C) = 0.5U( C h) + 0.5U( C + h) (A.) Here, aking he exen of reducion in he uiliy level for he consumpion level C due o he income uncerainy as C, h) following equaion. ( ρ, ( C, h) = U( C) U ( C) ρ and equaion (A.) resul in he U C) ρ ( C, h) = 0.5U( C h) + 0.5U( C + h) (A.) ( Taking he Taylor expansion up o he second order erm of he U C h) and U C + h) ( ( on he righ hand side of equaion (A.) and subsiuing hem respecively ino equaion (A.) once again means ha ρ C, h) can be expressed as follows. ( ρ (A.3) '' ( C, h) = 0.5U ( C) h (CRRA) ype. Now, we specify he uiliy funcion as he following consan degree of relaive risk aversion γ U ( C ) = C /( γ), γ, = ln( C ), γ =. 9

21 However, γ is a parameer represening a consan degree of relaive risk aversion, and / γ represens he iner-emporal elasiciy of subsiuion. The second derivaive of he CRRA ype uiliy funcion is C, so subsiuing his ino equaion (A.3) means ha ρ C, h) can be γ+ γ/ ( expressed as follows. ρ ( h C ) γ ( C, h) 0.5γC / = (A.4) However, ( h / C) represens he square value of he coefficien of variaion for consumpion a period, so hereinafer we express his as CV. Subsiuing he CRRA ype uiliy funcion and equaion (A.4) ino U ( C) = U( C) ρ( C, h) means he individual expeced uiliy funcion under income uncerainy can be expressed as follows. U γ [ 0.5( γ γ ) CV ] γ ( C) = C /( ) (A.5) Taking he derivaive of equaion (A.5), he individual s expeced marginal uiliy funcion under income uncerainy can be expressed as follows. 3 U '( C) γ = C [ + 0.5( γ + γ ) CV ] (A.6) Appendix B: The Taylor expansion for he hree variable s Euler equaion of he coefficien of variaion model From equaion (), arge equaion for he Taylor expansion can be expressed as follows. i C ecvsq i C γ ecvsq) 0.5( γ+ γ ) f ( r, g, g ) = (+ r )(+ g ) (+ g ) (B.) On equaion (B.), by aking he Taylor expansion of he hree variable funcions up o he second i C ecvsq order wih r = g = g = 0 or hereabous and subsiuing he derived derivaives from a 3 CV / C = h/ C is used in he developmen from equaion (A.5) o equaion (A.6). 0

22 separae calculaion ino i resuls in he following approximaion equaion. i f( r, g C, g ecvsq ) = + r i C i C γ g γrg + 0.5( γ + γ )( g ecvsq + 0.5( γ + γ ) g + 0.5( γ + γ )[0.5( γ + γ C ) ) ]( g ecvsq C ecvsq i ecvsq 0.5γ ( γ + γ ) g g + 0.5( γ + γ ) rg (B.) ) Subsiuing he resul of equaion (B.) ino equaion () leads o he following. i C i C i C C E [ r] γ E[ g ] γ{ E[ r] E[ g ] + Cov( r, g )} + 0.5( γ + γ ){( E[ g ]) + Var( g + 0.5( γ +γ ) E[ g ecvsq ] C ecvsq + 0.5( γ + γ ){0.5( γ + γ ) }{( E [ g ]) + Var( g ecvsq C ecvsq 0.5γ ( γ + γ ){ E [ g ] E[ g ] + Cov( g, g C ecvsq )}) i ecvsq i ecvsq + 0.5( γ + γ ){ [ r] E[ g ] + Cov( r, g )} ρ E (B.3) )} )} i C Ignoring each iem E [ r] E[ g ], ( C E [ g ]), ( ecvsq C ecvsq E [ g ]), [ g ] E[ g ] E, i ecvsq E [ r] E[ g ] wih a comparaively small value, replacing wih 0, and solving for E [ r i ] resuls in he following equaion. i C i C E[ r] ρ + γe[ g ] + γcov( r, g ) 0.5( γ + γ ) Var( g 0.5( γ + γ ) E[ g C ] 0.5( γ + γ ){0.5( γ + γ ) } Var( g ecvsq ecvsq C ecvsq i ecvsq + 0.5γ ( γ + γ ) Cov( g, g ) 0.5( γ + γ ) Cov( r, g ) (B.4) ) ) The rae of reurn on he risk free asse is deermined o have no relaionship o he consumpion growh rae and he growh rae of income coefficien of variaion, so subsiuing i C i ecvsq Cov( r, g ) = 0andCov ( r, g ) = 0 ino equaion (B.4) resuls in he following equaion. r C C ρ + γe[ g ] 0.5( γ + γ ) Var( g ) 0.5( γ + γ ) E[ g ecvsq 0.5( γ + γ ){0.5( γ + γ ) } Var( g ecvsq ecvsq + 0.5γ ( γ + γ ) Cov( g C, g ) (B.5) ) ]

23 Subracing equaion (B.5) from equaion (B.4) leads o he following equaion relaing o he equiy premium, he degree of relaive risk aversion, he covariance beween he reurn on asses and he consumpion growh rae, as well as he growh rae of income coefficien of variaion. i E[ r] r i C i γ Cov( r, g ) 0.5( γ + γ ) Cov( r, g ecvsq ) (B.6)

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28 Table Descripive saisics for he calculaion of he degree of relaive risk aversion 7

29 Table Descripive saisics for he calculaion of he subjecive discoun rae 8

30 Figure Deerminaion of he degree of relaive risk aversion i i ecvsq i C E [ r] r+ 0.5Cov( r, g )( γ +γ ), Cov ( r, g ) γ γ + γ (b) E[ r i ] r i C Cov ( r, g ) (a) γ γ CV_a γ NM γ CV_b CV model Normal model CV model i ecvsq ( Cov ( r, g ) < 0 ) i ecvsq ( Cov ( r, g ) > 0 ) 9

31 Figure Movemen of he US income uncerainy index(cv-4, Feb-978=00) Figure 3 Movemen in he average value of he risk premium 30

32 Figure 4 The resuls of he calculaion of he degree of relaive risk aversion and subjecive discoun rae (Income uncerainy index: Labor Share CV) 3

33 Figure 5 The resuls of he calculaion of he degree of relaive risk aversion and subjecive discoun rae (Income uncerainy index: The reciprocal of Uni Profi CV) 3

34 Figure 6 The resuls of he calculaion of he degree of relaive risk aversion and subjecive discoun rae (Income uncerainy index: The Unemploymen Rae UNRATE CV3) 33

35 Figure 7 The resuls of he calculaion of he degree of relaive risk aversion and subjecive discoun rae (Income uncerainy index: The reciprocal of he Universiy of Michigan consumer confidence index UMCSENT CV4) 34

36 Figure 8 The correcion effec in he esimaed value of he degree of relaive risk aversion C + Y + Normal model CV model i ecvsq ( Cov ( r, g ) < 0 ) Y max ~min max ~min C (s.dev. 0.07) (s.dev ) cons socks 35