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AN INVESTIG~~TION OF ASSET PRICING PUZZLES WITH CYCLICAL RISK AVERSION AND INTERTEMPORAL SUBSTITUTION

Xian Yang

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosoph- Graduate Department of Economics

University of Toronto

Copyright @ 2001 by Xian Yang

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Abstract

An Investigation of Ase t Pncing Puzzles with Cyclical Risk Aversion and

Intertemporal Substitut ion

,\an Yang

Doct or of P hilosop hy

Graduate Department of Economics

University of Toronto

200 1

Represent at ivc agent models t hat embecl the Lucas-Breeden ( L ticas ( L9fY). Breecien

( 1979) 1 paradigm for explaining asset ret iim differentials are geiierdly regarded as incon-

sistent with the enlpirical data. Difficulties such as the eqiiitv premiurn puzzle (llehra

and Prescott (1985) ). the risk free rate puzzle (Weil (1989)). ctc.. are weII documented

and it hm been shown t hnt t hese puzzles are very robiist (Kocherlekota ( 1996). Ctimpbell

(1996) and Cochrane (1997) provide good surveys). Recently. howevcr. severnl authors

( Campbell and Cochrane (1999). Cordon and St. Amour (2000. '2001) and Bakshi and

Chen (1996) are some examples) have pointed to time-vaqing risk aversion as a potential

source of rnisspecification that may account for these puzzles. However. risk aversion

and intertemporal substitution are intertwined in t hese models. jirst as t hey are in the

additive expected utility model. therefore it is impossible to interpret unambigtiously

which feature of preferences varies over the cycle.

The preferences suggsted by Epstein and Zin (1989) can separate the coefficient of

relative risk aversion (CRRA) fiom the elasticity of intertemporal substitution (EIS)

and d o w average consurnption growth to have a much smaller effect than consumption

volatility on the risk free interest rate. This psper generalizes the model of Epstein and

Zin (1989) by allowing the representative agent to display countercyclical risk aversion

and assesses if such behavior can add to the explmation of various empiricid phenomena

that have been investigated in finance and macroeconomics. such as the Mehra and

Prescott (1985) equity premium puzzle.

I investigate various combinations of state dependent CRRA wit h state dependent

EIS. In the case of constant EIS and time varying CRR.4. my resutts Look very similar

to those generated without state dependence. However. I also investigate the same model

but with time varying E I S and constant C'RRA. I find that a time varying EIS provides

delightfiil resiilts. I also find that time wrying EIS combined with a time varying

CRR.4 leads to even better results. As a hrther check. 1 use my calibrateci preference

parameters to predict the long-term interest rate. The calibrated preference parameters

iead to very sensibIe terni st riict tire predictions.

1 also investigate a similar problem in an open economy Basecl un i l tw-coitntry

general equilibrium model. 1 investigate the asset pricing puzzles from a ciifferenr angle:

i.e. itn analysis of the predictability of excess rates of retirrn on cliscourit horids. eqiiities

and foreign money markets using regession analysis.

Sly work in an open economy setting basically supports Behert. Hodrick and Dnvici

(199'7) conclusion. 1 finci thzit wheri 1 introduce both time mNng EIS and CRR.4 into

my two country modeI. the improved predictability of excess retiirns is insignificant . 51y

resttlts uphold a s t ronger statement: incorporat ing first order risk aversion wit h a simple

pattern for time varying risk aversion and intertemporal substitiition does not help much

either. But rny findings do not rule out the possibility that there could esist a richer

pattern of time varying p and û such that the estirnateci 3s can match the stylized results.

To my family and those who encouraged me to complete this life achievement: especially

to my wife Lin Hua for her constant devotion and support.

1 am indebted to my supervisor Ange10 Melino for his insightful comnients. excellent

instruction and patience. Without his encouragement and help. 1 would not have been

able to complete this interesting thesis topic. 1 thank Xiaodong Zhu. Sliquel Faig and

John Siaheu for their very helpful comments on an earlier draft of mv thesis. and thank

,Xaodong Zhu and hliquel Faig for their detailed cornments on my oral presentation of

the first three chapters in the Macroecononiics Workshop at the University of Toronto.

1 thank Ramazan Gencay for his encouragement and close attention to my every step of

progres. 1 give special t hanks to Larry Epstein for suggesting t hat 1 w r k on this thesis

topic and for his helpful comments. 1 am grnteful t~ Geert Edcae:-i ancl Gmrge Taiichen

for their patience in answeririz my qitestions =md offeriris me t heir source code.

Contents

1 Introduction 1

2 Cyclical Risk Aversion 7

. 2.1 The Representative Agent's Problem . . . . . . . . . . . . . . . . . . . . 1

. . . . . . . . . 2.1.1 Budget Constraint Facing a Representative Xpent Y

3 - 1 2 Preferences of the Represent at ive Agent . . . . . . . . . . . . . . 8

2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Equilibrium wit h Expected Utility for Certainty Eqiiivalent . . . . 14

2-22 Eqidibriiim with Yaari Preferences for Certainty Eqilivalent . . . 15

2.2.3 Long Run Average Eqiiilibriiim . . . . . . . . . . . . . . . . . . . 16

2.3 Eqiiity Premium Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 'Llodel Caiibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Expected Utility for Certainty Equivalent . . . . . . . . . . . . . 19

2.5.2 Yaari Preferences for Certainty Equivalent . . . . . . . . . . . . . 25

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Cyclicai Intertemporal Substitution 57

3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.1 LupectedUtilityforCedainty Equident . . . . . . . . . . . . . 58

3.1.2 Yaari Preferences for Certainty Equivalent . . . . . . . . . . . . . 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Conclusion 61

4 Cyclical Intertemporal Substitution and Risk Aversion 89

. . . . . . . . . . . . . . . . . . . . 4.1 The Representative Agent's Problem 90

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Equilibrium 92

4.2.1 Equilibnum with Ekpected Utility for Certainty Equivalent . . . . 93

4.2.2 Equilibrium with Ynari Preferences for Certainty Eqtiivalent . . . 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Mode1 Calibrntion 9.5

4.3. J Calibration Results of Expected Utility for Certainty Equivalent . 96

4.3.2 Calibration Restilts of Yaari Preferences for Certainty Equivalent 96

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Interpretatiori 98

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Prediction 100

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Conclitsion 101

5 A Two-Country Mode1 106

. . . . . . . . . . . . . . . 5.1 Stylized Facts on Exces Return Predictnbility 108

. . . . . . . . . . . . . . . . . . . . . 5 . L . 1 The foreign exchange market 108

. . . . . . . . . . . . . . . . . . . . . . 5.1.2 The discount bond market 109

. . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 The eqiiity markets 110

. . . . . . . . . . . . . . . . . . . . . . 5.1.4 Implications for rnodeling 111

. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Preference Structure 111

. . . . . . . . . . . 5.3 Budget Constraint Facing The Represent ative Agent 114

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Equilibrium 116

5.5 Data Description Y Calibration and Solution of the Mode1 . . . . . . . . . 118

. . . . . . . . . . . . . . . . 5.6 Implications for excess r e t m predictability 121

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Conclusion 124

6 Four States Closed Economy 132

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Equilibrium 133

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Calibration 134

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Conclusion 136

A Cyclical CRRA and Constant EIS 143

. . . . . . . . . . . . . . . . . . . . . . . . . . . . A . 1 First Order Condit ions 143

. . . . . . . . . .4 . 1.1 Case 1: Evpected Utility for Certainty Equivalent 144

. . . . . . . . A.1.2 Case 2: Yaari Preferences for Certainty Equivalent 144

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Nurnericai Solution 144

. . . . . . . . . A.2.i Case 1: Expected Utility for Certainty Equivalent 1-44

. . . . . . . . A.2.2 Case 2: Yaari Preferences for Certainty Equimlent 149

. . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Approximatesolution 152

B Cyclical E I S and Constant CRXA 154

. . . . . . . . . . . . . . . . . . . . . . . . . . . . B.l First Order Conditions 154

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Numerical Solution 154

. . . . . . . . . B.2.1 Case 1: Expecteù Utility for Certainty Eqiiinlent 155

. . . . . . . . B.2.3 CCwe 2: Yaari Preferences for Certaintu Eqiiiwlent 158

. . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Approximate Solution 161

. . . . . . . . . B.% 1 Case 1: Expected Utility for Certaintu Equivrilent 161

. . . . . . . . B.3.2 Case 2: Yaari Preferences for Certainty Equimlent 162

C Cyclical C RRA and E IS 165

. . . . . . . . . . . . . . . . . . . . . . . . . . . . C.l First Order Condition 165

. . . . . . . . . C.1.1 Case 1: E-tpected Utility for Certainty Equivalent 167

. . . . . . . . C.1.2 Case 2: Yaari Preferences for Certainty Equiden t 169

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Numerical Solution 171

. . . . . . . . . C.2.1 Case 1: Expected Utility for Certainty Equivaient 171

. . . . . . . . C.2.2 Case 2: Yaari Preferences for Certainty Equivalent 173

. . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Approximate Solution 176

viii

. . . . . . . . . C.3.1 Case 1: Expected Utility for Certainty Equivalent 176

C.3.2 Case 2: Yaari Preferences for Certaintg Equivaient . . . . . . . . 177

D Two Country Mode1 179

. . . . . . . . . . . . . . . . . . . . . . . . . . . . D . 1 First Order Condition 179

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Solution Procediire 189

E Prediction of the Bond Yield 196

F Robust Test Results 200

List of Tables

2.1 CalibrationResultsn~thEspectedUtilit?;forCertaintyEqiiivalent . . . 29

3.2 Names of Output Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Group 1 Cdibration Results Summarization . . . . . . . . . . . . . . . . 30

2.4 Calibration Resiilts W - 1 : CRRA. = 45 and EIS = x . . . . . . . . . . 31

2.5 Calibration Results Wl-2: CRR.4, = 45 ancl EIS = 2 . . . . . . . . . . 32

2.6 Calibration Resrilts Wl-4: CRR.4, = 45 and EIS = 0.2 . . . . . . . . . 33

2.7 Calibration Restilts LW-5: CRRA, = 45 and EIS = 0.1 . . . . . . . . . 34

2.8 Calibration Resrilts W1-6: CRR.4, = 45 ancl EIS = 0.05 . . . . . . . . . 35

2.9 Calibration Resiilts W1-7: CRR.4. = 45 and EIS = 1/45 . . . . . . . . 35

2.10 Calibration Results W2-1: Cl?RA. = 20 and EIS = x . . . . . . . . . . 36

2.11 Calibration Results W2-2: CRR.4, = 20 and EIS = 2 . . . . . . . . . . 34

2.12 CalibrationResiiIts tV2-4: CRRAW=20and EIS=0.2 . . . . . . . . . 38

2.13 Calibration Results tt'3-5: CRRA, = 30 and EIS = 0.1 . . . . . . . . . 39

2.14 CaIibration ResuLts W-6: CRR.4, = 20 and EIS = 0.05 . . . . . . . . . 40

2.15 Calibration Results W2-7: CRR.4, = 20 and EIS = 1/45 . . . . . . . . 41

2.16 Calibration Results W3-1: CRR.4, = 10 and EIS = x . . . . . . . . . . 42

2.17 Caiibration Results FV3-2: CRR4, = 10 and EIS = 2 . . . . . . . . . . 13

218 Calibration Resuits \VM: CRR.4, = 10 a n d EIS = 0.2 . . . . . . . . . 44

2.19 Calibration Results W3-5: CRBA, = 10 and EIS = 0.1 . . . . . . . . . 15

2.20 Calibration Results W3-6: CRRA, = 10 and EIS = 0.05 . . . . . . . . . 46

2.21 Calibration Results W3-7: CRRA. = 10 and EIS = 1/15 . . . . . . . . 47

2.22 Calibration Results with Yaari Preferences for Certainty Equivalent . . . 48

3.1 Kames of Output Tables for Cyclical EIS and Constant C R R I . . . . . . 63

3.2 Group 2 Results Simmarization . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 Calibration Resiilts W(E1S)l-1: E(EIS) = sc and CRR-lu? = 45 . . . . 65

3.4 Calibration Results CiT(EIS)1-2: E (EIS) = 2 and CRRAu* = 45 . . . . . 66

3.5 Calibrat ion Results W(E1S) 14: E( EIS) = 0.2 and C R RA tu = 45 . . . . 67

3.6 Calibration Results W(EIS)l-5: E(EIS) = 0.1 and CRR.4u. = 45 . . . . 68

3.7 Calibration Resiilts W(EIS) 1-6: E(EIS) = 0.05 and CRR.4u . = 45 . . . 69

3.8 Calibration Resiilts LV(EIS) 1-7: E(EIS) = 1/45 ancl CRR.-lul= 45 . . . 70

3.9 CalibrationResitltsW(EIS)2-1: E ( E I S ) = x a r i d C R R . 4 ~ * = 2 0 . . . . 71

3.10 Calibration Resiilts W(EIS)2-2: E( EIS) = 2 and CRRrlu* = 20 . . . . . 72

3.1 1 Calibrntion Resdts W(EIS)2-4: E(EIS) = 0.2 and CRR.4w = 20 . . . . 73

3.12 Calibration Resiilts W(EIS)2-5: E ( E I S ) = 0.1 and CRR.4 it. = '20 . . . . 74

3.13 Calibration Results LV(EIS)2-6: E(EIS) = 0.05 and CRRA u. = 30 . . . 75

3.14 CaIibration Resiilts LLr(EIS)2-7: E(EIS) = 1/45 and CRR.4 w = '20 . . . 76

-.-. 3.15 Cdibration Results W(EIS)3-1: E ( E I S ) = x and CRRAw = 10 . . . . r i

3.16 Calibration Results W(EIS)3-2: E(EIS) = 2 and CRR.4w = 10 . . . . . 78

3.17 Calibration Reçiilts W(EIS)3-4: E(EIS) = 0.2 and CRR.4w = 10 . . . . 79

3.18 Calibration Resdts W(EIS)3-5: C E(EIS) = 0.1 and C'RRAu = 10 . . Y0

3.19 Calibration Results W(EIS)3-6: E(EIS) = 0.05 and CRRrlu: = 10 . . . 81

3.20 Caiibration Results W(EIS)4-1: E ( E I S ) = x; and CRR-4w = 5 . . . . . Y2

3.21 Calibration Results W(EIS)42: E(EIS) = 0.5 and CRRAu? = 5 . . . . 83

3.22 Calibration Results W(E1S)M: E(EIS) = 0.2 and CRRAw = 5 . . . . û-4

3.23 Calibration ResuIts W(EIS)45: E(EIS) = 0.1 and CRRAw = 5 . . . . 85

3-34 CaGbration Results W(EIS)4-6: E(EIS) = 0.05 and CRRAw = 5 . . . . 86

3.25 Calibration Results W(EIS)47: E(EIS) = 1/45 and CRRAw = 5 . . . 87

3.26 Calibration Results: Cyclical EIS and Constant CRRA with Yaari Cer-

tainty Equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Calibration Resdts Given Expected Utility for Certainty Equivalent . . .

Prediction Given E~pected Utility Certainty Equivalent . . . . . . . . . .

Prediction Given Yaari Preferences for Certainty Equivalent . . . . . . .

Cyclical CRRA and EIS: Ymri Preferences for Certainty Equivalent . .

The stylized facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bekaert . Hodrick and David (1997) Simulation Results . . . . . . . . . .

Calibrat ion Results for the foreign exchange market . . . . . . . . . . . .

Calibrntion results for the dollar bond market . . . . . . . . . . . . . . .

Cnlibration results for the yen bond market . . . . . . . . . . . . . . . .

Calibration results for the excess dollar return on agaregate wealth . . .

Discrete Endonment Growth . . . . . . . . . . . . . . . . . . . . . . . . .

Calibrat ion Resiilts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Discrete Endonment Growth Back out from Bekaert VAR(1) . . . . . . .

Calibration Result from Bekrrert VAR(1) Parameters . . . . . . . . . . .

Robust Test Rest~lts With E-qected Utility CE . . . . . . . . . . . . . .

Robust Test Results With Yaari CE . . . . . . . . . . . . . . . . . . . . .

List of Figures

Price dividend rut io from Table 2.4 to Table 2-9 ( CRRAw =45) . . . . . 49

EquityreturnsfiomTable3.4toTable2.9(CRRAw=45) . . . . . . . . 50

Risk free rates from Table 2.4 to Table 2.9 (CRMn: =15) . . . . . . . . 51

Average risk free rates and equity premium from Table 2.4 to Table 2.9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (CRRAw =45) 52

Price dividend ratio From Table 2.10 to Table 2.15 (CRRXw =20) . . . . 53

Eq~iityreturnsfromTable2.10 toTable2.15(CRRAw=20) . . . . . . . 54

r r Risk free rates Erom Table 2.10 to Table 2.15 (CRRAw =20) . . . . . . .

Average risk free rates and equity premium from Table 2.10 to Table 2.15

(CRR4w =20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1 U S 10-year Treasury Constant Maturity Rate . . . . . . . . . . . . . . . 105

. . . . . . . . . . . . . . F.l Matching Bond Yield Rrith Expected Utility CE 205

F.2 Matching Bond YieId with Yam Certainty Equivalent . . . . . . . . . . . 205

Chapter 1

Introduction

Representative agent moclels t hat embecl the Liiccas-Breeden ( L i i z ~ 3 (l9ï'S). Breeden

(1979)) paradigm for explaining asset retiirn differentials are generally regarded as incon-

sistent with the empirical data. Difficulties sirch as the eqiiity preniiiim puzzle (Mchra

and Prescott (1985)). t he risk free rate piizzle (Weil (1989)). etc.. are well docrinientect

and it has been shoum tha t t hese puzzles are v e n robiist (Korherlakota (1996). Campbell

( N96) and Cochrane ( l99ï) protide good survevs). Recent ly. however. several aut hors

(Campbell and Cochrane (1999). and Gordon and St-Amour (2000. 20011). among oth-

ers ) have pointed to time-mqing risk aversion as a potential source of mis-specification

that may account for these puzzles.

Campbell and Cochrane (1999) connect time varyïng risk aversion to the price of

equity nsk and conclude that the price of risk not only varies over time but seems to

have a countercyclicd pattern. They find that a tirne varying price of risk can account

for a number of seemingly unrelated empuicai anomalies. An interesting and still open

question is how to build a mode1 that generates substantial countercyclicd variation in

the price of risk.

This point can be understood with the basic Euler Equation:

where J I is marginal utility growth and (R, - r f ) is the excss return on the market

port folio.

Given a power utility Funct ion: i.e. ut(c) = C-?. the basic Euler equation leads to:

h = E(& - rf)/o(&) = rn(gt) . (1.2)

where y is the coefficient of relative risk wersion (CRRA), o ( g t ) is the volatility of

consumption growth and X is the price of risk.

Since t here is not much evidence of heteroskedast icity in the consrirnption data (Ferson

aiid Merrick (1987)). Therefore they focus on the risk aversion piirameter y and point

out the neeci for the representative agent to display countercyclical risk aversion in order

to explain the apparent countercyclical nature of expected excess returns to equity.

In Campbell and Cochrane (1999). the device that generates time v a ~ i n g risk aversion

coefficient in their model is a slow moving e,uternal habit. Dunng trotlghs. consiimption

falls toward the habit level so that small movemeiits in consiimption lead to large vari-

ability in marginal utility. therefore agents reqiiire a high expectecl return to induce thtm

to hold eqttity The opposite occurs diiring expansions.

Their model is admittedly ad hoc. particularly in the w-ay the habit is "reverse en-

gineered" to fit the data: but it leads to rernarkable empirical success. For example. it

generates long horizon predictability of exces stock and bond returns from the dividend-

pnce ratio. yield spread. and short term interest rate: it generates a high average ievel of

stock return volatility; and it generates persistent movements in return volatility. What

is crucial for their story is that they have a slow moving habit which makes preferences

state dependent.

Gordon and St-Amour (2000, 2001) incorporate time-t-arying risk aversion into a

consurnpt ion-based asset pricing model in which attitudes towxds risk are contingent

upon an unobserveci state of the world. They obtain a valuation equation in which

the vector of excess r e t u m on equity includes both consumption risk as well as the risk

associateci with variations in preferences. They discussed how state dependent preferences

cunature leads to rotations of the marginal utility schediile. which increase risks to the

intertemporai marginal rates of substitution ( I M RS). If initial consumption is low and

risk aversion is countercyclical. an unanticipated fa11 in consumption translates into a

larger increase in marginal utility t h m iinder f i d preferences. Conversely procyclical

curmtitre and high consumption lead to a larger dccline in niarginal iitiiity following a

positive innovation in corisuniption. They also introdiice a scaIirig parameter and show

that the ratio of consumption to their scaling parameter plays criicial role in mûtching

the data. Using Bayesian met hods. t hey find that they can provide a good match to pmt

WNII data.

Other aiithors have tnken different approaches to introdticing tirne v a ~ i t i g risk aver-

sion and reach simihr concl~isions.~ Hoaever. risk aversion and intertemporal siibstitii-

tion are intertwined in these models. jiist as they are in the additive expected iitility

model. therefore it is impossible to interpret unnrnbigtiously which Featiire of preferences

varies over the cycle.

The preferences suggested by Epstein and Zin (1989) can separate the coefficient of

relative nîk aversion (C RRA) Frorn the eiasticity of int ertemporai siibstitiit ion ( E I S )

and aHow average consumption growth to have a mtrch smaller effect than consumption

volatility on the risk free interest rate. However? Epstein and Zin (1989) preferences

do not display time W n g risk aversion. But Epstein (1996) points out that reciirsive

utility is ideally suited to modeiing hypotheses regarding variation of risk aversion over

the cycle. Based on the spirit of Epstein (1996). this thesis generaiizes the model of

Epstein and Zin (1989) by allowing the representative agent to display state dependent

'Bakshi and Chen (1996) introduce counter cyciical risk aversion directly through the preferences that depend directly on time varying of wealth.

preferences and assessa if such behavior can add to the expianation of various ernpirical

phenornena that have b e n investigated in finance and macroeconomics. such as the

Mehra and Prescott (1985) equity premium puzzle.

In Chapter 2. 1 analyze the representative agent mode1 with recursive utility sug-

gesteci by Epstein and Zin (1989). Sly mode1 is characterized by a. constant elasticity

of intertemporal substitution (EIS) and time varying CRR-4. The piirpose is to inves-

tigate whet her or not count ercyclical risk aversion. as Campbell and Cochrane ( 1999)

concliided. is the sole explanation of the Mehra and Prescot t (1985) qi i i ty premiiim

puzzle. Sly resuits do not support this concliision. I explore two specific choices for the

certainty eqiiivalent relative risk averse forrn of expected iitility proposed by Epstein and

Zin (1989). and first-order risk averse fiinctional fonn proposed by Ynari ( 1087). For bot h

cases. 1 find that allowing risk at t i tude to vary with the cycle ahile keepirig the E I S

constant Ieads to almost no improvement over the benchmark c'ase in which preferences

are state independent.

In Chapter 3. I present a mode1 with tirne vrirying intertemporal stibstitiition but

constant risk aversion. In this case. 1 find the resiilts are remark~ble. A srnaIl wriatiori

in the EIS results in a dramatic irnprovement over the benchmsrk resiilts i r i both the

expected ut ility certainty equivalent and Yaari cert czinty eqiiivalent cases.

Do the results frorn Chapter 2 and Chapter 3 lead to the concliision that time wrying

EIS alone can explain the apparent counter-cyclical nature of expected excess return on

stock and bond. and the CRRA does not play any roie '! What if time varying EIS is

combined with time vaqing CRRA ? Chapter 4 extends both Chapter 2 and Chapter

3. The specification of preference LI& in this chapter embeds both the time ~~g

EIS and the time mrying CRRA models of preferences as special cases. I find that

time varying E i S combined with the t h e varying CRRA leads to even better resdts.

In the case of an expected utility certainty equivalent, 1 am able to match perfectly the

US historîcal data. In the case of the Yaari certainty equivalent my caiibration results

in parameter values that are inadmissible; therefore, 1 look for an approximate solution.

Even though 1 c m not match the US historical data exact- 1 find rny approximate

solutions are very close t o the observed data and they are a significant improvernent

on the benchmark results of Epstein and Melino (1995). As a fiirther check. 1 use my

calibrated preference parameters to predict the long- t erm interest rate. For bot h cert ainty

equivalents. I find that the calibrated preference parameters lead to very sensible terni

structure predic t ions.

Chapter 5 e-xtends my Chapter 4 to a twcxountry generd eqiiilibrium model and

invest igates whet her time varying countercyclical risk aversion ancl t ime varying E IS

imply sufficient predictability in exces rates of retiim in the foreign exchange niurket

while simultaneously matching the time series properties of interest rates and stock re-

turnq In Chapter 4. 1 tned to resoIve the Mehra and Prescott (1985) eqiiity premiiim

puzzle. In t his Chapter. 1 investigate the asset pricing puzzles from a different angle: i.e.

an analysis of the predictability of excess rates of retiirn on discount bonds. equities and

foreign money markets irsing regression analysis.

Bekaert . Hodrick and David (1997) find t hat in an open economy model. increasing the

anount of first-order risk aversion dramatically increases the variance of risk premiurns in

al1 markets. However. this increased riçk-premium volatility fails to imply a comparable

increase in excess-return predict ability. Their model simulation resiilts can not mat ch the

regression betas on al1 markets. They assert that the predictability ofa set of asset market

excess retums caxmot be fully explainecl simply by modifying preference assurnptions.

My work in an open economy setting basically supports their conclusion. 1 find that

when 1 introduce both time varying EIS and CRRA into my two country model. the

improved predictability of excess ret unis is insignificant . -\1y results uphold a st ronger

statement: incorporating k t order risk aversion with a simple pattern for t h e wrying

risk aversion and intertemporal substitution does not help much either. But my fmdings

do not d e out the possibility that there could e,vist a ncher pattern of time varjing p

CHAPTER 1. INTRODUCTION

and a such that the estimated 0s can match the stylized results given in Table 5.1.

Chapter 2

Cyclical Risk Aversion and the

Equity Premium Puzzle

In this Chapter. 1 describe the representative agent's prohlem wit h recursive preferences

t hat exhibit constant elasticity of intertemporal substitut ion ( EIS) but state clependent

risk aversion and charact erize the soliit ion. To facilit at e com parison wit h ot her papers

in the Iiterature. 1 consicler both an expected titility and Yuri functional lorms for the

certainty equivalent. Section 2.1 sets up the representative agent model. Section 2.2 look

at equilibrium and. in particular. denves the equilibriiun price-diiidend ratio. Section

2.3 introduces the equity premium puzzle. The model calibration is discussed in Section

2.4. Section 2.5 provides the simulation results. Section 2.6 summarizes and concludes

this chapter.

2.1 The Representative Agent's Problem

Consider an idinitely-lived represent at ive agent a-ho receives ut ility froni the consump

tion of a single good in each period. At t h e t. curent consumpt ion is non-stochastic

but future consumption levels are generally stochastic. It is convenient to discuss first

the budget constraint and then the agent's preferences.

2.1.1 Budget Constraint Facing a Representat ive Agent

h u m e there are n assets t hat pay dividends dt = ( d i t , dzt' ..... d,,) and are traded

competitively at prices pt = (p l t , f i t . ....prit). The agent starts euch period with asset

holdings denoteci by q = (lit. zi,. ... . z,,).

Let st represent beginning-of-period wealt h.

Define the portfolio share vector wt = (u.lt. uyzt. .... un,) and the g r o s retilrn vector

rt = ( r i t . r.t. -... rnt ) by

Pit Z t t U',t = - .

Pt "1

and

Then we see that wealt h evolws according to

2.1.2 Preferences of the Represent ative Agent

In this section. 1 wül build a mode1 in the recursive utility hamework by using two

dinerent funct ional fonns for the certainty equivalent. In the first case. the certainty

equivalent will take an expected utility hinctional form. whiie in the second case the

certainty equivalent is taken Eiom Yaari (1987). 1 will introduce state-dependent risk

aversion into both forms and derive their first order conditions.

Recursive Utility

A recursive utility defined on random consumption sequences may be constructed by

means of the following recursive functional relation

where Ut = U(q. Cc+ 1. ... II,) is the intertemporal utility for sequences beginning at t and It

denotes available information. LC' (. . . ) is called the aggregator fiinct ion since it aggregates

current consumpt ion cc wit h <an index of t lie fiit ure ut ility to cletermine the current lit ility.

To generate empirically usefiil results. folloming Epstein (1988). a powerfiil assiimption

is t hat int ertemporal ut ility is homot het ic: specifically. 1 consider a CES a=~gat or

\Vit h the CES aggregator of eq(2.6). the elast icity of intertemporal subst itiit ion is t here-

fore EIS = 1/ (1 - p): 3 is the subjective discoiiiit factor.

Certainty Equivalent ( C E )

The functioii p, = p(Ut+iIIt) in eq(2.5) is the certainty equivalent of the distribution of

htiire utility 4, conditional tipon period t informat ion. If 2 is a random variable wit h

distribution function Fi. it is convenient to write p ( 2 ) rat her t han p ( Fi) . Ir1 general.

the certainty equivalent satisfies

1. p : D - R. D = {cdf's F on [O. XI: F has compact support in (O. x)}.

An interpretation of p is that if one fixes a detenninistic consumption process c =

(qcl, . . . .h) and lets X and X' be two random variables that are time 1 measurable. then

(Q,XC~. X C ~ , ...) t (c~.X'C~. X ' C ~ , ...) e p ( X ) > p (XI) (2.7)

A special case applicable to stochastic programs mises if p has the expected utility form:

pN (i) = h-' ( E h ( i ) ) . (2.8)

for some increasing function h. Then eq(2.5) corresponds to a special case of the struc-

ture studicd in a finite horizon framework by Kreps and Porteus (19%). It is apparent

h-orn eq(2.5) how a degree of separation is achieved bettveen elast icity of intertemporal

substitution and risk aversion. Certainty preferences are determined by IC' alone. Only

risk attitudes are affected by a change in p. Thus it is passible to change the ciegree of

risk aversion wit hout affect ing the preference ranking of deterniinistic.

Furt her parametric Forms for the fiinctional p have beeri presented in Epstein ( 1990)

. In this pnper 1 corisider two different functional forms for p.

Case 1: Expected Ut ility for Certainty Equivalent

In this case. I choose a convenient special ccwe from the class of expected iitility fitnctionnl

forms t O be the cert ainty equivdent :

where Et is the expectation conditional on period-t information. û ( s t ) is a parameter

that depends on an exogenous state variable st. i t can be interpreted as a risk aversion

parameter wit h the degree of risk aversion increasing as û ( s t ) falls: 1 -cr ( s r ) is the measure

of relative risk aversion with respect to consumption gambles of the fom describeci in

eq(2.7) . Thus eq(2.6) becomes

P L = ((1 - ) 4 + O [ E , LI^:;'')]^'^'')^ y O # O < 1, O p p < 1.

bIaximizing Ut subject to the wealth accumulation eonstraint given in eq(2.4) yields

the following first order conditions (see First Order Conditions in Case 1 of Appendiv -4

for the derivat ion).

E~ (IMRS;:, ,A~.,+~) = 1. (2.10)

Et ( + + ) = 1. j = 1.2. .... n (2.11)

where &+ = cc; r, , , = 1:. , ~ ; r , , ~ + , is the gross ret iirn of holding the optimal port folio

w; from time t to t + 1. md the intertemporal marginal rate substitution from time t to

t + 1 (1.11 RSt.t+i) is given by eq(A.5)

The elas3icity of intertemporai substitution ( E I S ) and coefficient of relative risk aver-

sion (CRRA) are 1/(1 - p) and 1 - a ( s t ) respectively. IF the cycle is inclexecl by st.

coitntercyclical risk aversion requires n(s t ) to be high when the state is high and n(s,)

to be low when the state is low.

If a (st) = a (a constant). eq(2.10) and eq(2.11) redtice to the model of Epstein and

Zin (1989): namely

a p-', 2 - 1

I A ~ R s ~ , Z , , = g p g t l l (2.13)

with EIS = 1/ (1 - p) and CRRA = 1 - cr.

If a ( s t ) = a (a constant) = p , eq(2.10) and eq(2.11) reduce further to the Familiar

Euler equations of the expected utility model (see Hansen and Singleton (1983))' i.e.

IMRS;L,, = ,dg:;;

with EIS = 1/(1- p) and CRRA = 1 - p.

Case 2: Yaari Preferences for Certainty Equivalent

In this case, 1 use the following specification for the certainty quivalent function p

corresponding to Y& (1987) dual t heory of choice applied to a binary gamble.

pya (qÇ1) = E; (rt+i) = p7(Sc)l + (1 - p'('')) z h . (2.15)

where O < y (s,) 5 1. 7 (si) = 1 implies that the individual is risk neiitral. 7 ( s t ) = O

implies the individual is extremely risk averse and ranks al1 gsmbles by the worst outcorne

possible. The EIS is still given by 1/(1- p ) . while the coefficient of relative risk aversion

for this class of preferences is given by (see Epstein and Zin (1990)):

Becaiise CRRA is clecreitsing in -[ (si ) . we see t hat a smaller -, (sr) irnplies grenter aversion

to risk.

The individ~id's problem is then

siibject to the wealth accumulation identity given in eq(2.4).

The above problem implies the foIlowing first order conditions (see First Order Con-

ditions in Case 2 of Appendk A For details ).

where the intertemporal marginal rate substitution £rom t h e t to t + 1 (I.\.IRSt.,+l) is

giwn by eq(A.8)

Here? 1 consider a Lucas-style endowment economy (Lucas (1978)). where the represen-

tative agent has a recursive intertemporal utility function. There is a single perishable

consumption good. the total c;iipply of which is descrihed hy the endowment pmces..

c = {c,) E LI+. Equity. interpreted as a daim to the endowment process. is the only

asset in non-zero net supply. Without loss of generality. 1 take equity to be the first asset

in the vector z and 1 riormalize the number of shares of equity in this economy to 1. The

equilibrium is described by the price process p such t hat the goods market anci eqiiity

market clear. That is. the solution to the utility maximizat ion above nit h price p hiu

cc = c i l t and rt = (1.0.0 .... 0) for t 2 1.

The retiirn to the market portfolio . I l t . t+ l . iinder the previoiis i ~ ~ s i ~ m p t ion. eq~ials the

return to wealt h

For convenience. 1 drop the first siibscript and subsequently will refer to the price of

equity and its dividend at time t as p, and dt .

Following hIehra and Prescott (1985). I assume that the endoment proces for con-

sumption is such that the growt h rate gt+ 1 = follows a first-order h r k o v process.

Further 1 assume the exogenous state variable st mentioned in Section 2.1.2 is the con-

sumption growt h rate. Now 1 d l determine the supporting equilibrium pnce process.

1 consider only equilibria where the ey-dividend price of equity is described by the

theinvariant and positive function, p (gt .ct) of the date variables gt and et. It foilows.

because of the hornogeneity of preferences. that the price is linearly homogeneous in

consumption: that ist

Thus 4 2 . 2 0 ) simplifies to

iU . t+ l =

For convenience. 1 use Pt

gt takes only two \dues and

to denote P ( g t ) . 1 also assume that the exogenoiis \xriôble

is distributed as

, gh W.P. iih high state. gt+1 = {

w.p. q low state.

wit h transition probability matrix

where nu is the probability of going frorn state i to stnte j . Sote that this transition

probability matrix yields the marginal probabilities T,, = n h l + r h h and ;q = qi + ;il,, . The conditional espected return on equity (which is the same as the retiirn to the

market portfolio given t here is only one oittside asset) if t ocl-'s ( t inie t ) stat e is i E ( 1 . h }

and t + l state is j E { l . h } will be:

where xij is transition probability [rom eq(2.21) and JI,, is aven by eq( l .22) .

2.2.1 Equilibrium with Expected Utility for Certainty Equiva-

lent

The equiiibrhm condit ion correspondhg to Case 1 of Section 2.1.2 is obtained by substi-

tuting eq(2.22) into Euler equation eq(2.10) and noting st = gt: that provides a relation

that must be satisfied by any eguilibnm price-dividend hinction P (-) . namely

Because there are only two states and ( g r } is Slarkov process. eq(2.26) reàuces to

The risk free rate can be determined froni Euler eqiration (2.11). That is.

I r/.t = Et [I.\IRSE,] '

The risk free rate r, rvill prevail if today's state is i E ( 1 . h ) .

where AI,, is given in eq(2.22). For convenience. below 1 use ci, to clenote n ( g t ) .

2.2.2 Equilibrium wit h Yaari Preferences for Certainty Equiv-

The equilibrium condition corresponding to Case 2 of Section 9.1.2 is obtainecl by stib-

stituting eq(3.72) into Euler equation (1.17) and noting st = gt: that provides a relation

t hat muvt be ~ s t isfied by any equilibriurn price-dividend funct ion P f . ) . narnely

where E' is defined in 4 2 . 1 5 ) . In our econorn. this can be written as

where Yi = 7 (gi), i E (1. h ) .

The risk Eree rate c m be determined from Euler equation (3.18). That is.

The risk free rate r, will prevails if today's state is i E { l . h } .

2.2.3 Long Run Average Equilibrium

Civen r, and .\A denved above. the long run average g r o s risk frce rate is then E (rf) =

0 . 5 ~ + 0 . 5 ~ ~ . The (proportional) equity preniiuni. defined by \ t é i l ( 1989). when today's

state is i is <Pl = M t / r l . I t follows that the long-nin average eqiiity premittm is =

O.5Qt + 0.5Qh.

2.3 Equity Premium Puzzle

5Iehra and Prescott (1985) report that the iinconciitional niean of the risk preniiirrn

observed in the US stock market is much larger thnn that predictecl by a standard version

of the representative agent model. Frorn 1898 to 1978 in the United States. the average

annual real rate of return on short-term bills was 0.80 percent and the average ariniial

real rate of return on stocks was 6.98 percent- Thiis the average equity premium was 618

basis points. They calibrated an asset pricing model wit h t imeadditive isoelast ic u t ility

to see if the mode1 could deliver unconditional rates of return close to the historical

average rates of return on stocks and bills. They used a %point llarkov process For

consumption growth with mean E (gt) = 1.018. standard deviation o ( g t ) = 0.036. and

correlation (gt. g t - i ) = -0.14. They found that the mode1 cannot account For both the

average level of the risk-free rate (0.8%) and the diaerence (6.18%) between the average

rates of retum on equity (6.98%) and on risk kee securities during the period 1898-1978.

The model could not produce more than a 35 basis point equity premium while keeping

the expected nsk free rate at less than or equal to 4 percent per year. This result is

called the equity prernium puzzle.

In the time-additive isoelastic expected utili ty framework. the coefficient of relative

risk aversion and the elast icity of intertemporal substitut ion are i n t e r t ~ h x i . The elas-

ticity of intertemporal substitution is constrained to be the inverse of the coefficient of

reIative risk aversion. A very high coefficient of relative risk aversion (of the order of 10

or 50) makes it possible to replicate the large seciilar risk premium on equity yet. the

implied very Ion. elasticity of intertemporal substitiition also leads to the coiinter-hctiial

prediction of an estremely high risk free rate. Conver~ely~ kt low coefficient of relative

risk aversion leads to a couriter-fnctually low eqiiity premiiim. although it does imply a

relatively low risk free rate.

Wit hin the context of representative agent models. various approaches have been pro-

posed to solve the equity premium puzzle. One approach is to disentangle risk aversion

attitudes frorn intertempord siibstit ut ion using a special class of reciirsive preferences

(e-g. Epstein and Zin (1989). Weil (1989)). The intiiitiori behind this approach is t hat

given the risk free rate is mainly controlled by the magnitude of the elasticity of intertem-

poral substitution. while the risk premium is a reflection of the coefficient of relative risk

aversion. a preference ordering t hat can parametrize the elast icity of int erternporal sub-

stitution and the coefficient of relative risk aversion independent ly should provide the

additional degree of freedom required to replicate both the Ievel of the risk Çee rate

and the risk premium. The results reported in Epstein and Zin (1989) and Weil (1989):

however, were not very encouraging (although Epstein and Melino (1995) do miich bet-

ter). A second approach modifies the preferences by introducing habit formation. Habit

formation leads to a form of state dependent preferences that can generate substantial

variation in the price of risk. A very successN example is Campbell and Cochrane

(1999). However, risk aversion and substitution are intertwined in their model: therefore

it is hard to interpret unarnbiguously which feature of preferences varies over the cycle.

In a third approach, Gordon and St-Amoiu (2000. 2001) use expected utility preferences

but allow the CRRA to be a latent process. Using Bayesian methods. they find that

they can provide a good match to post L V I I data even though the CRRA process show

ve- little variation over the cycle. Again. it is not obvious which feature of preferences

is varying over the cycle in t heir model.

In the following 1 will demonstrate that introducing state-dependent risk aversion in

a simple way does not contribute to soIving t h e eqriity premiuni puzzle.

2.4 Mode1 Calibration

1 assume both points of support for consumption gron-th are equally likely. i-e. ;ri =

?ih = 0.5 in eq(2.23). and choose (gr, g h ) to match the k t two moments of corisumpt ion

growth given for the US historical data: E(g) = 1.015 and s (9) = 0.036. This involves

solving the equation system:

0.5g + 0.5qh = E(g).

0.5 (gr - E ( ~ ) ) ' + 0.5 (gh - E ( ~ ) ) ' = -5' ( g ) .

which implies (under the obvious requirement gl c gh)

These parameters are the same as thase used by Slehra and Prescott (1985), Weil

(1989), Epstein and Zin (1991b) and Epstein and Melino (1995). Mehra and Prescott

(1985) assume t hat the probability of staying in the same state is 0.43 and the transition

probability matrix is symmetnc, that is,

Epstein and Melino (1995) choose the FolIowing transition probability rnatn'c: I d l ülso

use t his transit ion probability matrix to match t heir results.

0.45 0.55 n2=[Il :]=[ r i hh 0.55 0.45 1 . (2.33)

Slehra and Prescot t (1985) est imate the hist oncd average ret uni and standard cle-

viation of equity to be E (JI) = 1.07 and s (61) = 0.165. arid estimiite the historical

average retiirn and standard deviat ion of T-bills to be E (rf ) = 1.008 and s (rr ) = 0.056.

These first two moments with transition m ~ t r i x IIl ' irnply (6. Ph) = (23.467.27-838)

or (?8.951.3?.i3O) and (ri. rh) = (0.952.1.064) or (1.064.0.952) .\mong t hese four

pairs. three imply arbitrage opportunities: therefore 1 chose to calibrate to (9.0,) =

(23.467- 27.838) and (ri. rh) = (1.061.0.952).

2.5 Simulation Results

2.5.1 Expected Ut ility for Cert ainty Equivalent

In this section. 1 present the simulation results when the representative agent has reciirsive

preferences wit h an expected utility cert ainty equivalent . Matching the observed US

historical data requires that we solve the FOC provided by eq(2.27) and eq(2.28) For

parameters (p. p. al. ah) that lie in the admissible set (O 5 0 5 1. p 5 1. ai 5 1. a h 5 1)

and that deiiver the Slehra and Prescott ( 1985) estimates: E ( A I ) = 1.07 and s (JI) =

0.165. E(g) = 1.018 and s(g) = 0.036. as weIi as E (rf) = 1.008 and s (q) = 0.056.

'If transition matrk ii2 is used. (8: Ph) = (23.424,27.886) or (.22.690.29.W1) and (9' Ph) = (22.690,29.00 1) is inadmissible.

Equivalently, given my simple endowment rnodel for consumption groa-th. 1 want to see

if the set 8 can be rationalized by some admissible combination of (3. p. al. ah)? where

if hfehra and Prescott (1985) transition mat* E l is used or

if Epstein and .\.lelino (1995) transition rnatriu II2 is iised.

Exact Solutions Following the niimerical method described in Siitrierical Solution in

Case 1 of Appendk A. 1 obtained the calibration results reported in Table 2.1.

Table 2.1 contains two sets of calibration results nith Expcctccl Utility for the cer-

tainty quivalent. They correspond to the transition matrices 111 anci I l 2 . which are giveri

in eq(2.32) and eq(2.33) respectively. As expected. neither is admissible'. Both sol.oliitions

are inadmissible becatise p > 1. But it is still too e d y to jiirnp to the coricliisiori t hat

time varying risk aversion does not play any roIe in explaining the eqiiity premiiim puzzle.

There might exist some -*nearbya* asset return processes. similar to Epstein and .\.lelino

(1935): that can be rationalized.

Approximate Solutions In the following. 1 use the results in Weil (1989) Table 1

as a benchmark and see whether or not cyclical risk aversion in this model can lead to a

promising improvement in Weil's results.

In rny model, the CRRA parameter is no longer constant but depends on the state

variable gt. For the sake of cornparison, 1 di constrain the average CRRA to the Weil

(1989) coristant CRRA parameter. That is,

*Epstein and hlelino (1995) use a revealed preference argument to show that there is no state- independent p that can be combined with admissible values of LI and p that can rationalize the data.

E(CRRA) = 0.5CRRAi + 0.5CRRAh = Weil's CRRA (C'RRA,). (2.36)

where CRRAi and C RRAh denote the coefficient of relative risk aversion in low and high

economy state respectively. In the case of non-i.i.d. dividend 39wth . N'cil (1989) fin&

that choosing a very high coefficient of relative risk aversion (aroiind 45) and an elasticity

of intert emporal subst itiit ion not as small as the von Yeumann-LIorgenstern woulcl imply

(aroiind 0.10. instead of 1/45 = 0.022) results (for 3 = 0.95) in a (proportional) risk

premium of 5.72% and a risk free rate of 0.85%.

In order for my results to be comparable. 1 chose the valrie 3 = 0.95 that is iisecl

in TabIe 1 of Weil (1989). Given the consumption growth proces. for any pair (ai . ah )

chosen frorn the admissible area and a given intertemporal siibstit i!t ion paramet er p ilseci

in Table 1 of FVeil( 1989). it is relatively straight fornard to solve for the eqtiilibriiim pricc

dividend ratio and risk-hee rate processes frorn q(3.37) and q(2.28). SIy resiilts are

presented in 15 tables: each of them reports the price clividencl ratio in the two differerit

states. the percentage level of eqiiity retums in the two different states. the stanclarcl

deviation of the eqiiity retiirn. the average value of the equity returns betwcen the Iow

and iiigh states. the percentage valiie of the net risk free rate in the two different 33ates.

the standard deviation of the risk free rate. the net percentage valiie of the average

risk free rates between the low and high states. and the percentage vahie of the net

proportional risk premiunl defined by Weil ( 1989).

As one can see in Table 2.2. I divide these 18 tables into 3 groups according to the

values of C R R A , . For instance, the n-th p u p contains Table W n - 1 to Table 6C'n - 7

(n = 1, 2.3): and in the first group (i-e. n = 1): CRRA, = 45. in the second group

(Le. n = 2): CRRA, = 20, and in the third group (2.e. n = 3). CRRIL, = 10. Given a

certain group, i.e. a certain value of CRRA,, the EIS changes Erom table to table. Le.

EIS changes from ao in Table W n - 1 of Group n down to 1/45 in Table CVn - 7 of

the same group, etc.. Although a lot of numbers are presented, the tables reveal several

interesting patterns that are worth noting.

1 start by describing the tables in Group 1 (Le.? Table 2.1. 3.5. 2.6. 2.7. 2.8 and

2.9). For each table? the first line replicates the results of CRRA, = 45 in Weil (1989)

Table 1. The remaining lines of each table report the results when the individual's risk

aversion parmeter a reacts to the states of the econorny. Wlen the economy realizes a

low growth rate. the individual has risk aversion parameter al. and when the economy

realizes high growth rate. the individual has risk aversion parameter ah. In the following

description. 1 link the difference between ûr and ah to the parameter d t hroiigh the

eqiiation d = ah - a , = a , - al. where a , = 1 - CRR.4,. Therefore d reflects the

discrepancy of individuals at tit tides towards the risk between the two st at es. For instance.

given Weil's C R R A , = 45. if d = O. then ai = oh = a, = -44. ancl the corresponding

results in Weil's Table 1 shouid be reprodriced. If d = 4. then 0 1 = -48. and a h = -40.

so the individual is more risk averse in the bad state than in the good state.

A siimmary of Group I results is presented in Table 2.3. In Table 2.3. the 18 rom

are divided into six sections. where each section corresponds to a d i i e of Weil's EIS

parameter from his Table 1. As one c m see. the first row of each section corresponds to

d = O. i.e.. constant CRRA and replicates ?Veil (1989) Table 1 restilts k t s a special case.

When d deviates from zero. cyclical CRR.4 occurs. It tiirns out none of the extensions

deliver any improvement on Wd's benchmark results. In al1 the sections. except for the

cases E I S = 0.05 and EIS = 1/45. risk premia are too smdl and the cyclical CRRA

only makes things worse. When EIS = 0.05, the risk prerniitm can be matched by

increasing the value for d but the risk free rate moves to the undesired direction (too

srnail). In the following, 1 report my simulation results in more detail.

Price Dividend Ratio: The simulation results of the pnce dividend ratio corre-

sponding to different EIS in the tables of Group 1 are plotted against d and are shown

in Figure 2.1. 1 find that when EIS is greater than one. the price dividend ratio in

both low and high economic states dopes downward with respect to d and changes from

counter-cyclical (e.g. when d is less than 11 in Table 2.4) to procyclical (e.g. when d is

greater than 11 in Table 2.1). A counter-cyclical price dividend ratio is not consistent

with the empiricd evidence (e.g. Fama and French (1988)). Therefore 1 only focus on

those cases that generate a procyclical price dividend ratio. When d increases. the price

dividend ratio appears to be inore volatile. however. the risk premium does not increase

very much and risk free rate does not fall enough. so the puzzle is not resolved in this

case. .As long as EIS is greeater than one. 1 find the price dividend ratio shows almost

the same pattern as in Table 2.4.

Lihen EIS is smaller than one. the pnce dividend ratio begins to show a different

response than that when EI S is greater than one. It slopes upward nith d. The price

dividend ratio changes gradiinlly from pr*cyclical (when d is l e s than 10 in Table 2.6)

to coiinter-cyclical (when d is greuter than 11 in Table 2.6). In this case. the best 1 can

get is Weil's results (i.e. the first line in each Table). When d increases. the risk preiniuni

falls and risk free rate increase. there ti no improvement on iveil's results. Giveri an EIS

l e s than one. I fi nd the price dividend ratios show almost the same pattern from table

to table as in Table 2.6.

Equity Return: 'iow I turn to the behavior of the equity return. The simulation

results of equity retums corresponding to different EIS in the tables of Group 1 are

shown in Figure 2.2. i h e n EIS is greater than one. the equity returns. ml in the low

state and mh in the high state. show the same pattern from table t o table. Lt appears to

be counter-cyclical and ml slopes upward: mh slopes downward. X countercyclical equity

return is consistent with econornic intuition. When the economy realizes a low rate of

growth, people wiii feel more risk averse and require a higher expected rate of real return

to hold equity. When the economy realizes a high rate of growth, people Eeel l e s risk

averse than in the business trough and therefore require a lower e-xpected rate of red

re turn,

When EIS is l e s than one, the equity return in the Iow state, ml, slopes downward

and the equity return in high state, mh, slopes upward. as we increased d. The equity

return appears t o be counter-cyclical when d is srnaIl ( eg . when d is iess t han Id in

Table 2.6) and t o be procyclical when d is large (e.g. when d is greater than 15 in Table

2.6). For smaller E H . the pattern observed above does not change from table to table

except the graph of rnh shifts downward and the graph of mi shifts upward: that makes

the equity more volatile and countercyclical. It is hard to imagine that inwstors reqiiires

a lower expected real equity return when the econoniy realizes a low growth rate than

that when the econorny realizes a high growth rate. Looking rit the tables in Group 1.

for those values of d that generate a countercyclicc?l eqtiity return. the second moment

still appears to be too small. State dependent CRRA does not help to explain eqiiity

puzzle in t his respect.

Risk Free Rate and Risk Premium: In al1 the tables. the percentagci nliies of

net risk free rates in low and high states are denoted by rl% and rh% respectively. Figtire

2.3 plots rl% and rh% for different values of E I S that belong to Group 1. As one can sec.

when E I S is greater than one. the risk free rate displays a procyclical pattern. When

EIS is less than one. the risk free rate moves from couriter-cyclical to prc>-cydical as ci

increases. The relationship between the percentage value of average net risk free rate

(rf %) and the percentage value of net average (proportional) risk premiurn is plot ted in

Figure 2.4. 1 find when EIS is greater than one. and with constant CRR.4 (Le. ti = 0).

the percentage value of net average (proportionai) risk premium is too low and the net

risk free rate is too high. When d increases. rf% rises even higher than the desired value

0.8% and the net percentage value of average (propertional) risk premium (QF%) falls

even lower t han the desired value 6.18%. so t here is no improvement upon Weil's results.

When EIS is lesi than oneo Qw % could start wit h a higher value and the increase of d

can help to puil a"% back to the reasonable region (e.g. consider the case of EIS = 0.05.

when d = 0' Qu% = 8.6%: as d increase to 5. a"% falls to 6.3%): but rf% starts too low

and the increase of d makes it even lower. Therefore state de?endent CRRA does not

help to explain equity premium puzzle in this group of tables.

If we look at tables in Croup 2 and 3. the above obsenrations ail1 go through simi-

larly. In summary. by introducing cyclicd risk aversion into a representative model a i th

recursive utility and e-xpected certainty quivalent . the best 1 cati achieve is Weil ( 1989)

results. Countercyclical risk aversion drives either the risk free rate or the eqiiity pre-

mium to move in the undesireci direction. so that it does not help to explain the .\lehra

and Prescott equity premium puzzle.

2.5.2 Yaari Preferences for Cert ainty Equivalent

In this section. 1 present my simiilation results when the represerit at ive agrnt hi; reciirsive

titility and the certainty equivalent is taken from Ymri (1987). As in Section 2.5.1. I

want to see if the set C3 from eq(2.34) can be rationalized by sorne parameters in the

admissible set (,J 5 1 . p 5 1. O 5 3 5 1. O 5 y,, 5 1). 1 will use the Epstein and .\lelino

(1995) results as the benchmark and see whether or not state-dependent risk aversion

can add to their s t o . Epstein and 5Ielino (1995) assume constant p and y. while in rny

model. the -., parameter no longer has to be constant.

Exact Solutions Following Surnerical Solution in Case 2 of Appendiu A. I find the

solution. corresponding to the Epstein and Melino (1995) transition matrk i12 is (3 =

1.0504, p = 1. y1 = -0.2901. 7 h = 0.8453). This solution is inadmissibte because both *3

and fall in the inadmissible area. therefore 1 look for a -nearbf0 asset return processes.

similar to the one of Epstein and Melino (1995) that can be rationalized. Le.. I look for

approximate solutions.

Approximate Solutions The approximation method used here is a bit different horn

what 1 have used in Section 2.5.1. In that section, I chose EIS and CRRA,: and for

each value of d. solve for pnce dividend ratio Pl and Ph, and see whether or not 1 c m get

close to the first and second moment of US historical risk free rate data, In t his section.

1 set (Pl, Ph) = (23.426,27.889) (which is corresponding to the solution of II?), so that

the first and second moment of equity return are guaranteed to match the US history

equity return data. Then, in order to be comparable nith Epstein and Melino (1995).

1 use the same 3 = 0.99 as they used. As above. we assume that the growth rates at

two different economic states are ( g l . gh) = (0.982.1.054) and their marginal probabilities

are both qua1 to 0.5. These parameter values are the same as those iised bu 1Iehra and

Prescott (1985). Wei1 (1980). Epstein and Zin (1991a) and Epstein and Melino (2995).

Following these aiithors. 1 choose to use transition rnatrix n2 given in eq(2.33) becailse

this transition rnatrk is used in Epstein and 1Ielino (1995) which is n conipromise of

'Llehra and Prescott (1989) estimates with Cecchetti. Lam and hlrirk (1993) estirnates.

For fked 3 and a given value for p. rny approximation solution chooses the rcmaining

parameterç 71 and according to eq(4.63) and eq(A.67) in order to match the first and

second moments of the U S historical equity retiirn data. Risk Free rates ri and r h cari

then be solved direct- from eq(2.31) (thus the first and second moments of the risk free

rate. and the risk premiiim). .\lu results are presentd in Table 2.22.

As one c m see for each p listed in Table 2.22. allowing -/ to van with the cycle

produces dmost no difference from Epstein and Melino (1995)-.s. results. Wit h p =

-41.87. my algorithm chooses = -,h = 0.4057. so my restilts replicate Table 1 of

Epstein and Melino (1995) as a special case. Further when p is set to be greater or

l e s t han the value of -41.87. -,l and yh are chosen to be different reflecting the time

tarying CRR.4. When p is set to be greater t han -41.87. bot h the average and standard

deviation of the risk free rate get worse. CVhen p is set l e s t han -41.87. bot h the average

and standard deviat ion of the risk free rate move in the desired direct ion: unfort unately

they do not move enough before they reach the Limits (-0.3 196%. 0.0868). The risk free

rate is too low and the standard deviation of risk fiee rate is too high. In another words.

state dependent risk aversion alone makes only an extremely modest contribution toward

CHAPTER 2. CYCLICAL RISK AVERSION

the equity premium puzzle in this case.

2.6 Conclusion

The Campbell and Cochrane (1999) model is admittedly ad hoc but it leads to remark-

able ernpirical success. It helps to resolve many of the difficiilt ies t hat beset the standard

power utility model. including: Euler equation rejections: no correlation between mean

consumption growth and interest rates: very high estimates of risk aversion: and. pric-

ing errors t hat are larger t han t hose of the st at ic C-4 P.V. However. risk aversion and

substitution are intertwined in their model. jiist as they are in the additive expected

utility model. and therefore it is impossible to interpret iinanibigtiously which featiire of

preferences varies over the cycle.

Gordon and St-Amour (2000. 2001) use expected iitility preferences but allow the

CRRA to be a latent process. They find that they can provicie a good match to post

W V I I data even though the C RRA process show vent lit tte variation over the cycle.

Again. it is not obvioits which feature of preferences is varying over the cycle in t heir

model.

Epstein (1996) points out that recursive utility is ideally suited to rriocleling hypothe-

ses regarding variation of risk aversion over the cycle. Based on the spirit of Epstein

(1996). t his chapter generalizes the model of Epstein and Zin (1989) by allowing the r e p

resentative agent to have countercyclical risk aversion. M y findings are. to some extent,

in support of Campbell and Cochrane's (1999) as well as Gordon and St-Amour (2000.

2001) claim that countercyclical risk aversion is crucial to generate large time variation in

rïsk premia and c m contribute to an euplmation of the Mehra and Prescott ( 1985) equity

premium puzzle. For the most part, however, 1 find that introducing state-dependent

risk aversion in a very simple way provides at best an extremely modest improvement

for the two certainty equidents examineci in t his Chapter.

I have considered only two different pararneterization of the certainty equivalent.

There are a large number of alternative certainty equivalents that could be explored.

However, the results in this chapter do not suggest that this will be a profitable line of

research .

On the other hand. the state variable used in t his paper is the exogenoiis current

period dividend growth rate. One rnay ask what if CRR-4 depends not oiily on the current

state but on the history of States '? An example would be to asstinie c i ( s t ) = a ( g t . gt- ).

Such a parameterization would seem to be a natural way to capture habit formation in

the rectirsive utility framework.

Of the variotis extensions that could be explored. the remainder of the thesis will

focus on models where the EIS is also time mrying.

Table 2.1: Calibration Results wit h Expected Utility for

Cert aint y Equivalent

solve eq(2.27) and eq('l.28) with b p e c t e d Ctility for the certninty

Transition Mat riu

h

II:,

equivalent and the transition matrices ii and i l2. which are given

in eq(2.32) and eq(2.33) respectively. Seither solutions is admissi-

ble beceuse p > 1.

EIS

Notes: This table reports two sets of calibration results that exact ly

,d

0.7366

0.7893

Table 1.2: 3

Groiip 3

CRRA = 10

P

28.3003

21.1663

Table bL.3 - 1

TabIe LV3 - 2

X/A

Table CV3 - 4

Table W 3 - 5

Table W 3 - 6

Table tt'3 - 7

mes of Output

Group 2

Q I

-23.4768

-32.0408

CRRA = 20

a h

0.8381

0.2537

Table IV2 - 1

Tabie tt'2 - 2

XIA

Table LV2 - 4

Table LV2 - 5

Table W2 - 6

Table LV2 - 7

rables

Group 1

CRRA = 45

Table CVl - 1

Tabte WI - 3

Y/A

Table W l - 4

Table CVI - 5

Table CVl - 6

Table Wl - 7

Table 2.3: Summarized Group 1 Calibration Results for CyclicaI CRRA and Constant EIS

EIS d s W. ~ ( m ) I r j x au'%

1 1 r

Yotes: E IS is elasticity of intertemporal substitution. d reflects the discrepancy of individuals attitudes

O

5

30

towltrds the rkk between low and higk erom>my growth stttteu. In Group 1. the average CRR.4, = -1-4. s,

is standard deviation of equity returns. E ( m ) is average olgoss equity returns. sr/ is s%andcwd deviation of

0.005

0.006

0.0 13

risk free rates, rf % is net average percentage nsk free rate, and <PW% is net average percentage (propertionaf)

equity premium. The 18 raws are divided into six sections, where each section corresponds to a d u e of

Weil's EIS parameter From his TabIe 1. The first row of each section corresponds to d = O. Le.. Wei1 (1989)

1.072 O.Oû1

Table 1 results. When d deviates From zero, qclical CRRA occurs. Mehra and Prescott (1985) estirnates

3.926

3.886 L .O73

E ( m ) = 1.07, s, = 0.165. E ( r f ) = 1.008. +1 = 0.056 and <PW% = 6.18%. It tums out none of the

3.144

3.249 0.00 L

extensions deliver any improvement on Weil's benchmark results.

3.280 1.075 0.00.5 1 4. LOO

Table 2.4: Cdibration Results FiTl-1: ,O = 0.95? CRRA, = 45 and E i S = x

d 4 Ph ml% nah% s, E ( m ) ri% rrh% sr, rj% au''%

32 16.635 16.901 9.115 6.408 0.021 1.079 4.120 4.284 0.001 4.202 3.561

Xotes: f i and Ph are price dividend ratios when the economy realizes a low and a high growt h

rate respectively. mi% and mh% are percentage values of net equity returns when the economy

redizes a low and a high growth rate respectiwly. rr% and rh% are percentage values of net risk

free rates when the economy realizes a low and a high ,(growth rate respectively. The values in bold

face replicate the Weil's (1989) resdts given EIS = x and CRR.4 = 45. Xlso see notes in Table

2.3.

Table 2.5: Calibration Resdts W1-2: 0 = 0.95. C'RRA, = 45 and E I S = 2

32 l7.741 17.882 8.550 6.518 0.014 1.07.5 3.750 -1.586 0.006 4.168 3.236 Notes: Pl and Ph are pnce dividend ratios when the economy realizes a 10%. and a high growth

rate respectively. ml% and mh% are percentage values of net equity returns when the economy

realizes a low and a high growth rate respectively. rl % and rh% are percentage values of net risk

Eree rates when the economy reaiizes a low and a high growth rate respectively. The values in bold

face replicate the Weil's (1985) results given E i S = 2 and CRR.4, = 45. Xiso see notes in Table

2.3.

Table 2.6: Calibration ResuIts W14: B = 0.95. CRRA,,, = 45 and EIS = 0.2

32 43.058 40.388 0.954 7.505 0.046 1.042 0.482 7.310 0.048 3.896 0.326 Sotes: Pt and Ph are price dividend ratios when the economy reaiizes a Iow and a high growth


realizes a low and a high growth rate respectively. ri% and rh% are percentage values of net risk

Eree rates when the economy realizes a low and a high growth rate respectively. The d u e s in boId

face replicate the Weil's ( 1985) results given EIS = 0.2 and CRRA, = 45. Ais0 see notes iri Table

2.3.

Table 2.7: Cdibration Results W1-5: d = 0.95. CRRA, = 45 and EIS = 0.1

d S Ph nit% mh% sm E ( m ) r,% rh% sr/ l x aW%

Notes: 4 and Ph are price dividend ratios when the economy realizes a tow and a high growth

rate respectively mi% and mh% are percentage values of net equity returns when the economy

realizes a low and a high ,gowth rate respectively. ri% and rh% are percentage d u e s of net risk

free rates when the economy reaiizes a low and a high growth rate respectively. The d u e s in bold

face replicate the Weil's (1985) results given E IS = 0.1 and CRRA, = 4-5. .4h see notes in Table

2.3.

Table 2.8: Caiibration Results Wl-6: 8 = 0.95. CRRA,,, = 45 and EIS = 0.05

8 669.088 689.938 4 . 3 2 -0.249 0.032 1.0'20 -1.319 -4.080 0.0% -2.700 4 - 6 9

Notes: fi and Ph are price dividend ratios when the econaniy reolizez a low and a high gn-iwth

rate respectively. ml% and mh% are percentege values of net equity returns wheri the econonly

realizes a low and n high growth rate respectively. rl% and rh%, are per~ent~ige values of net risk

free rates when the economy realizes a Iow and a high growth rate respectively The values in bold

face replicate the Weil's (1985) results &en EIS = 0.05 and CRR:1,, = 4.5. Mso see notes in Table

Table 2.9: Calibration Results W1-7: 3 = 0.95. CRR.4,. = 45 and E I S = 1/45

2 2380.57 2930.03 16.214 -9.300 0.180 1.035 -0.311 -19.400 0.134 -9.806 14..567

Xotes: f i and Ph are price dividend ratios when the economy realizes a low and a high growth


r d i z e s a low and a high growth rate respectively. rl% and rh% are percentage values of net risk

Free rates when the economy realizes a low and a high growth rate respectively. The values in bold

face replicate the iveil's (1985) resuits given EIS = 1/45 and CRRA, = 45. Also see notes in Table

Table 2.10: Calibration Results W2-1: , = 0.95. CRRA, = 30 and EIS = x

d 4 Ph ml% rnh% s, E ( m ) r rh% sr/ r eWrC

10 19.960 20.329 8.539 -3.263 0.023 1.069 3.476 s5.263 0.013 4.370 2.4-Lï

Xotes: f i and Ph are price dividend ratios when the economy realizes a low and a high growth

rate respectivety. ml% and mh% are percentage values of net equity returns when the economy

realizes a low and a high growth rate respectively. rl'% and rh% are percentage values OF net risk

free rates when the economy realizes a Iow and a high growth rate respectively. The values in bold

face replicate the Weil's (1985) results given EIS = x: and CRR.4, = 20. Also see notes in Table

2.3.

Table 2.11: Cdibration Results W2-2: 3 = 0.95. CRRA, = 20 and EIS = 2

d 9 Ph mt% mh% S , E(m) rr% rh% ~ r f rf% (Pw%

19.468 19.637 8.111 5.942 0.013 1.070 3.567 5.960 0.017 4.764 2.18.5 Fj and Ph are price dividend ratios when the economy realizes a low and a high growth

rate respectively. ml% and mh% are percentage values of net equity returris when the econorny

realizes a Iow and a high gowth rate respectively. rl% and rh% are percentage \dues of net rkk

free rates when the econorny realizes a low and a high gr0~3h rate respectivety. The values in bold

face replicate the Weil's (1985) resuits aven EIS = 2 and CRR-4, = 20. Also see notes in Table

Tabie 2.12: Calibration Results CV2-4: /3 = 0.95. CRR.4, = 20 and EIS = 0.2

d 8 Ph ml% mh% s, E ( m ) rl% rh% sr, r f% a"%

20 15.936 14.821 4.518 12.344 0.053 1.084 4.373 12.350 0.0.56 8.362 0.067

Sotes: Pl and Ph are price dividend ratios when the economy realizes a low and a high growth


realizes a low and a high gowt h rate respectively. ri% and rh% are percentage values of net risk

Eree rates when the economy realizes a low and a high growth rate respectively. The values in bold

face replicate the Weil's (1985) results given EIS = 0.2 and CRR-4, = 20. Also see notes in Table

2.3.

Table 2.13: Calibration Results W2-5: d = 0.95. CRRA, = 20 and E I S = 0.1

d 8 ph ml% rnh% s, E ( m ) rt% 'ch% sr, r f % a"%

Notes: f i and Ph are price dividend ratios when the economy redizes a Iow and a high gowth

rate respectively. ml% and mh% are percentage vaiues of net equity returns when the economy

realizes a low and a high growth rate respectively. rr% and rh% are percentage values of net risk

free rates when the economy realizes a Iow and a high ,orowth rate respectively. The d u e s in bold

face replicate the Weil's (1985) resuits given EIS = 0.1 and CRRA, = 20. Also see notes in Table

2.3.

Table 2.14: Cdibration Results W2-6: B = 0.95. CRRA, = 20 and E I S = 0.05

20 9.958 7.126 -1.508 37.755 0.299 1.166 6.571 35.345 0.201 21.108 -4.433 Notes: Pl and Ph are price dividend ratios when the economy realizes a iow and a high growth


reaiizes a low and a high growth rate respectively. r i% and rh% are percentage values of net risk

free rates when the economy realizes a lotv and a high growth rate respectively. The values in bold

face replicate the Weil's ( 1985) results @en EIS = 0.05 and CRR.4, = 20. Aiso see notes in Table

2.3.

Table 2.15: Calibration Results W2-7: d = 0.95. CRR.4, = 20 and EIS = 1/45

Notes: f i and Ph are price dividend ratios when the economy realizes a low and a high growth


realizes a low and a high growt h rate respectively. ri% and rh% are percentage values of net risk

kee rates when the economy reatizes a low and a high growth rate respectively. The values in bold

face repiicate the Weil's (1985) results given E I S = 1/45 and CRRA, = 20. Also see notes in Table

2.3.

Table 2.16: Calibration Results W3-1: d = 0.95. CRR,4,,. = 10 and EIS = .x

8.9 24.660 24.688 6.520 5.337 0.006 1.059 4.117 5.190 0.00.5 4.654 1.223

Xotes: fi and Ph are prke dividend ratios when the economy realizes a low and a high gotvth


realizes a Iow and a high growth rate respectively. rt% and rh% are percentage \dues of net risk

free rates when the economy realizes a low and a high g~owth rate respectively. The values in bold

face replicate the Weil's (1985) resuits given EIS = x and CRRA, = 10. A h see notes in Table

Table 2.17: Calibration Results W3-2: . = 0.95. CRR-4. = 10 and EIS = 2

d 8 Ph ml% m& s,,, E ( m ) ri% r y,/ r f % @"'%

21.471 21.483 7.102 5.980 0.006 1.065 - 1 . 7 -5.835 0.006 3.270 1.213

Pl and Ph are price dividend ratios when the economy redizes a low and a high gowth


realizes a low and a high growth rate respectively. rr% and rh% are percentage values of net risk

free rates when the economy realizes a low and a high =wt h rate respectively The values in bold

face replicate the Weil's ( 1985) results Bven EI S = 2 and C RR.4, = 10. Also see notes in Table

Table 2.18: Caiibration Results W3-4: d = 0.95. CRRA,,, = 10 and E H = 0.2

Sotes: Fj and Ph are price dividend ratios when the economy realizes a low and a high growth

rate respectively. ml% and nih% are percentage values of net equity returns when the economy

realizes a low and a high growth rate respectively ri% and rh% are percentage values of net risk

free rates when the economy realizes a low and a high growth rate respectively The values in bold

face repkate the Weil's (1985) results given EIS = 0.2 and CRR.4, = IO. Aiso see notes in Table

2.3.

Table 2.19: Calibration Results W3-5: 3 = 0.95. CRR.4, = 10 and EIS = 0.1

d 8 Ph ml% mh% Y, E ( m ) rl% rh% .sr, rf% <Pu'%

8.9 5.971 5.912 18.849 18.984 0.001 1.189 16.539 18.858 0.012 17.698 1.04-1

Yotes: fi and Ph are price dividend ratios when the ecoriomy redizes a low and a high growth

rate respectively. ml'% and mh% are percentage d u e s of net equity returns when the economy

redizes a tow and a high ,wwth rate respectively. ri% and rh% are percentage values of net risk

free rates when the economy realizes a low and a high growt h rate respectivel. The values in boId

face repIicate the Weil's (1985) results given EIS = 0.1 and CRR.4, = 10. Xlso see notes in Table

Table 2.20: Calibration Results iV3-6: i3 = 0.95. CRR.4, = 10 and EIS = 0.05

d Ph ml% mh% s, E ( m ) ri% rh% Sr/ rl% 4W%

8.9 3.235 3.170 32.707 34.407 0.008 1.336 30.435 34.294 0.019 32.364 0.913

Notes: Pl and Ph are price dividend ratios when the economy realizes a Low and a high growth


realizes a low and a high growth rate respectively. ri% and rh% are percentage values of net risk

free rates when the economy realizes a low and a high growth rate respectively. The values in botd

face repiicate the Weil's ( 1985) resdts given EIS = 0.05 and CR RA, = 10. .&O see notes in Table

2.3.

Table 2.21: Calibration Results CV3-7: L3 = 0.95. CRR.4, = 10 and E I S = 1/45

d fi Ph ml% mh% s, E ( m ) 9 % rh% s r / rf% 9%

Notes: f i and Ph are price dividend ratios when the economy redizes a low and a high growth


realizes a low and a high growth rate respectively. rl% and rh% are percentage d u e s of net risk

fiee rates when the economy realizes a low and a high growth rate respectively. The values in bold

face replicate the Weil's (1985) resuits given EIS = 1/45 and CRR.4, = 10. A h see notes in Table

2.3.

Table 2.22: Calibration Results for Cyclical CRRA and Constant EIS witli Yaari Pref-

erences fc

P

-5

- 10

- 20

-30

-40

-41.87

-50

- 60

- ï O

- 80

- 90 -X

Notes: T

Certai

EIS

O. 166

0.090

0.047

0.032

0.034

0.023

0.019

0.016

0.014

0.012

0.011

0.000

den t

I h

-0.10

0.122

0.316

0.371

0.402

0.405

0.418

0.429

0.436

0.442

0.446

0.48 1 5 table reports simulation results for cyclical C the

T I -r

l

i change)

and constant E IS = 1/( 1 - p ) wit h Yaari Preferences for certainty equivalent . .\ho see notes in Table

2.3. In order to be comparable with Epstein and SIelinn (1995). L use the same transition matrix

il2 given in eq(2.33) and 3 = 0.99 as they used. 1 assume that the growth rates at two different

economic States are (g l .gh) = (0.982.1. O S ) and their marginal probabilities are both equal to 0.5.

These parameter d u e s are the same as thase used by JIehra and Prescott (198.5). Weil (1980).

Epstein and Zin (1991a) and Epstein and Jlelino (1995)- Yurnbers in bold face replicate the Epstein

and Melino (1995) results. ce1 and ce2 are the certainty equivalent value corresponding to lottery

(50,O.a: 100.0.5) and (70.0.5: 80,O.s) respectivel.

Figure 2.1: Price dividend ratio from Table 2.4 to Table 2.9 (CRRBw =45)

ta4 - 18.2 - rao - 178 . 17.6 - 174 . 172 . 17.0 - -

4u

35 .

30 -

25 .

Figure 2.3: Equity returns from Table 2.4 to Table 2.9 (CRRAn =45)

Figure 2.3: Risk free rates frorn Table 2.4 to Table 2.9 (CREWw =45)

Figure 2.1: Average nsk free rates and equity premium from Table 2.4 to Table 2.9

5.000 Nat AVE

10.0 20

an- 6 0 -

Figure 2.5: Price dividend

23.5 ElSdnt

ratio from Table 2.10 to Table 2.15 ( C R R h * =?O)

Figure 2.6: Eqtiity retiirns froni Table 2.10 to Table 2.15 ( C R R h - =?O)

Figure 2.7: Risk free rates frorn Table 2.10 to Table 2.15 (CRMw =20)

Figure 2.8: Average risk free rates and equity premium from TabIe 2.10 to Table 2.15

( C r n 4 w =20)

Chapter 3

Cyclical Intertemporal Substitution

In Chapter 2. 1 have only modeled the case when the coefficient of rrlative risk aversion

(CRRA) is state-dependent and cotild not find strong evidenc~ to support Campbell

cmci Cochrane (1999). It is natiiral to ask what happens if one moclels the elasticity of

intertempord sirbstitution ( E I S ) to be state-dependent instead of the CRR.4:' In t his

Chapter. for both the expected iitility for certainty equivalent and Yaari preferences for

certainty eqiiiwlent cases (see Appendix B for details). 1 dlow the EIS to be state-

dependent and the CRRA to be constcmt. I find a remarkable improvenient over the

Weil (1989) results. that is. a state-dependent E I S with constant CRR.4 does help to

explain equity premium puzzle.

The mode1 I used here ni11 be the same as the one descnbed in Chapter 2 except for

letting the substitution parameter p depends on state st' that is p (s t ) . With the CES

aggregator of eq(z.6), the elasticity of intertempord substitution is t herefore EIS =

1/ (1 - p(s t ) ) ; procyclical EIS requires p(st) to be low when the economy realize a low

state. The risk aversion parameter no longer depends on the state of the worid. but is

restricted to be a constant .

The Bellman equation for this problem becomes

st.:

where the fiinction p ( ) is the certainty equivalent defined in 2.1.2. Here it is chosen to

take the same functional form as eq(1.9)

in the case of expected iitility for the certainty eqiiiwlent. or to take the same functional

form as eq(2.15)

in the case of Yanri preferences for the certainty quivalent. The first orcler conclitions

for this problem are given in Appendk B. Yote. a and 7 are no longer state dependent.

thus CRRA is constant over time. In the lollowing. 1 report my simulation results.

3.1 Simulation Results

3.1.1 Expected Ut ility for Certainty Equivalent

Exact Solution

In the case of expected utility for the certainty equivalent. 1 show in Xumerical Solutions

in Case 1 of Appendk B that it is impossible to match the data process observed by

Mehra and Prescott (1985) exactly. But, there might e.& some hearbyn asset return

proceses, similar to Epstein and Melino (1995): that can be rationalized. Therefore. 1

investigate the approximate solutions.

Approximate solution

In the following, similar to Section 2.5.1. 1 use the results in Weil (1989) Table 1 as a

benchrnark and see whether or not a cyclical EIS in this mode1 can lead to a proniising

improvement in Ckil's results.

As defined in Appendiv 8.3.1. the approximate solution for this case is given by

choosing Pl and Ph from the optimal consurnption choice first order condition eq( B. 1)

for any given admissible a. pi and ph. to giiarantee a match of the first two moments of the

observed US consumption growth data. It folIows t o calculate the risk free rate from the

optimal portfolio choice eq(B.2). In order to compare to LVeiI's restilts. a is chosen siich

t hat the corresponding C RRA equals the Weil (1989) Table 1 paramet ers: 45.10. 10 and

5. 1 chose i3 = 0.95 that is used in Table 1 of Weil (1989). There are many ~ctmissible pairs

of (p i . ph) to choose. For descriptive purpose. pl and ph are chosen siich that the iwernge

of the two equals to the substitution parameters p, (= 1.0.5. -4. -3. - 10. ancl - 44)

irnplied by Ers,(= x;. 2.0.2.0.1.0.05.and 1 / 4 5 ) in Table 1 of LVeil (1989). Specifically.

p, = 0.5 (pi i ph) . where E I S , = (1 - p , ) - L .

1 have organized the results around parameters d and p,. where pi = p . - ci and

ph = pw + d. For a given parameter p,, d determines the extent of time varying EIS:

i.e., d = O irnplies constant EIS, and d > O irnplies procyclical EIS with EIS increasing

with d. For each p,. 1 look for the d u e of d that best matches the first and second

moments of the risk free rate (and the risk premiiun) as well as the first and second

moments of the equity returns.

1 report the results in 24 tables denoted by Table W( EIS) n - xo where n represents

the group number 1: 2, 3 and 4. which are corresponding to Cireil (1989)'s Table 1

parameters CR& = 45, 20, 10 and 5. Given a certain goup. Le. a certain value of

CR&! x = 1,2, ... ,6 and 7 are corresponding to Weil's Table 1 parameters E IS, = m,

2. 0.2, 0.1, 0.05, and 1/45. For instance, the n-th group contains Table LV ( E I S ) n - 1 to

Table CV (EIS) n - 7. where (n = 1,2.3.4). Table 3.1 Iists al1 the output table niunes.

Each table reports the price dividend ratio in the two difTerent states. the percent age Ievel

of equity returns in the two different states. the standard deviation of the eq~iity return.

the average value of the equity returns between the low and high states. the percentage

value of the net risk free rate in the two different states. the standard deviation of the

risk free rate. the net percentage value of the average risk free rates betrveen the Iow and

high states. and the percentage value of the net proportional risk premiiim defineci by

Weil (1989). From these tables. 1 find that tvith constant CRR-4. by int roducing jiist

a little bit of time varying EIS. 1 can find a drarnatic irnprovenient on the LVeil (1989)

results for certain CRRA and average EIS levels.

Consider the group 2 tables as an example: a surnmnry of the resd ts is preseiited in

Table 3.2. In Table 3.3. rows are divicled into five sections according to different espected

value of EIS. In each section. the first line corresponds to the case of d = O. When

d = 0. the results correspond to constant CRR.4 and constant EIS and replicate the

results of LVeil (1989). Lkhen d increases a bit. al1 the first two moments of the risk free

rate and equity retum move to the desired direction as does the risk prerni~irn. althoiigh

the standard deviation of the risk free rate increases a bit too quickly conipared to the

others. Sirnitar condusions mrr be made regarding the revutts for the other grottps zts

well.

3.1.2 Yaari Preferences for Cert ainty Equivalent

Exact Solution

In the case of Yaari preferences for the certain@ equivalent. the numerical results pre-

sented in Case 2 of Appendix B shows that no admissible solution e-xists for 7. Therefore

CHAPTER 3. CYCLICAL INTERTEMPORAL SUBSTITUTION

1 investigate the properties of an approximate solution.

Approximate Solution

The approximate solution for this case is a little bit different froni the one defined in

Case 1 of the expected utility for certainty quivalent case of this section. Here. given

consumption process. I choose y. pl and ph to match the first and second moments of the

US equity return data exactly. and see if the calculated first and second moments of niy

risk free rate can get close to the observed US risk free rate data.

I present my resiilts in Table 3.26. In Table 3.26. the first line corresponcfs to =

0.4057. pl = ph = -41.87. and it replicates Epstein und Melino (1995). For 1 # 0.4057.

the EIS varies with the state. When -/ = 0.1957. eq(B.54) and eq(B.56) can be solvecf

to get pi = -7.8111 and ph = -7.0475. which implies a procyclical EIS. In this c~ase.

1 match the risk free rate and q u i t y premiiim exactiy. althoiigh the standard deviation

of the risk free rate moves slightty in the wrong direction (up to 0.096 which is worse

given the standard deviation of Epstein and hlelino (1995). 0.094 is alreacly too high).

Therefore I find. in this case. my resiilts improve dramaticdy or1 those reported bu

Epstein and !delino (1995).

3.2 Conclusion

The results given in this and the previous chapters show that it is a time varying pro-

cyclical EIS t hat is required tu mode1 time m g expected exces ret urns to q u i t ies.

EIS reflects the willingness to trade off between the growth and the smoothness of an

individual's determinist ic consumption stream. A low EIS indicates t hat individuals are

less ~?Iling to trade smoothness of t heir consumption stream for consumption g o w t h:

i.et they Iike their deterministic consumption stream to be srnooth. There is not much

empirical research about what cyclicd pattern of EIS should foilom~ but my simulation

resuits reveal a procyclical pattern.

Do the results both in Chapter 2 and in Chapter 3 imply countercyclical CRRA has

no contribution to explain time vaqing risk prernium ? 1 will investigate this in Chapter

4.

Table 2

EIS

X

2

1

0.3

o. 1

1: .\;ames of Out~tit Tables for Cvclical EIS and Constant CRR.4 - L

Group 4 Groiip 3 Groiip 2

CRR-4 = 5 CRR.4 = 10 CRRA = 20

L\*(EIS) 1 - 1

LV(EIS)l - 2

S/A

Li-( E I S ) 1 - 4

IC'(EIS)L - 5

Li-( E I S ) 1 - 6

IC'(EIS)l - T

Table 3.2: Summarized Group 2 Calibration Results for Cÿclical EIS and Constant

CKRA, = 20

E ( E I S ) I n Sm ~ ( m j 7% ,9r 1 rf% ,pw%

I

Notes: E ( E I S ) is average EIS. d determines the extent of time v a ~ i n g EIS. s, is standard deviation of

equity returns. E(m)% is net average percentage \due of the equity returns. w,f is standard deviation of risk

free rates. rf% is net average percentage vatue of risk free rates and 9'"% is net average percentage d u e

of (propertional) equity premium. The rows are divided into five sections. where each section corresponds

t o a value of E(EIS) t hat equals t o Weil's EIS parameter hom his Table 1. The first row of each section

corresponds to d = O. i.e.. Weil (1989) Table 1 results. When d deviates from zero. cyclicd EIS occurs.

3lehra and Prescott (1985) estimates E (m) = 1.07. s, = 0.165. E ( r f ) = 1-008. sr/ = 0.056 and W'%

=6- 18%. It t ums out when d increases a bit. al1 the first two moments of the risk free rate and equity returns

move t o the desired directions as does the risk premium, aithough the standard deviation of the risk free

rate increases a bit too quidcjy compared t o the others.

Table 3.3: Calibration Results W(E1S) 1-1: Cyclical EIS with E ( E I S ) = x and Constant

0.8 18.953 18.390 1.059 1.085 0.026 7.918 1.037 1.067 0.02 -5.203 1.919

Xotes: Fj and Ph are price dividend ratios at low and a high state respectively. ml and mh are ~~ equity returns at tow and a high state respectively. ri and r h are goss risk Free rates at low and

a high atate respectivel- pl = p, - d and ph = p, + d. d determines the extent of time varying

EIS: Le.. d = O imply constant EIS. and d > O implies procyclical EIS with E[S incrc'asing with

d. EtEFS) = ( 1 - p,)-L isthe asengeof EIS. f chose$ =O.!% and E(EfS) = x thnt are oseci in

Tabte 1 of Weil ( 1989). implyîng p, = 1. The nurnbers in boid face replicate Table 1 of iveil { 1989).

Also see notes in Table 3.2.

Table 3.1: Calibration Restilts W(E1S)l-2: Cyclical EIS with and Coristant CRRAw =

Sotes: Pl and Ph are price dividend ratios when the econorny realizes a low and a high growth rate

respectively. ml and m h are gros equity returns when the econorny realizes a Iow and a high growth

rate respectivety. rl and ri, are gros risk free rates when the economy realizes a low i d a high

growth rate respectively. pl = p,, - d and ph = p, t d. d determines the extent of time mrying

EIS: Le.. d = O imply constant EIS. and d > O impliw procyclical EIS with EIS iti~re~asing with

d. E ( E I S ) = ( 1 - p,)-l is the average of EIS. 1 chose 3 = 0.95 und E ( E I S ) = 2 that are usd

in Table 1 of Weil (1989). implying p, = 0.5. The numbers in boId face replicate Table 1 of !Veil

f 1989). A b see notes in Table 3.2.

Table 3.5: Cdibration Results LV(E1S)l-k Cyclical EIS nith E ( E I S ) = 0.3 and Con-

Xotes: Fj and Ph are price dividend ratios at low and a high state rspectively. ml and mh are g o s s

equity returns at low and a high state respectively. ri and r h are goss risk free rates at low and

a high state respectiuely. pi = p, - d and ph = p, + d d determines the extent of tirne kzuying

EIS: Le.. d = O impty constant EIS. and d > O implies procyciical EIS with EIS increaing with

d. E ( E I S ) = ( 1 - p,) -' is the average of EIS. 1 chme 3 = 0.95 and E ( E I S ) = 0.2 that are used

in Table 1 of Weil ( 1989), implying p, = -4. The numbers in bold face replicate Table 1 o f Weil

(1989). Also see notes in Table 3.2.

Table 3.6: Calibration Results W(E1S)l-5: Cyclical EIS with E ( E I S ) = 0.1 and Con-

stant CRRAw = 45

d f i Ph *I mk s, E(rn)% ri r h srf rl% aW% - - --

O 21.40'2 22.531 1.103 1.030 0.0729 6.6202 1.036 0.9809 0.0390 0.8-18 5.70.1

0.1 21.432 22.490 1.100 1.032 0.0706 6.6051 1.056 0.9839 0.0367 0.984 .5..348

0.3 21.491 22.408 1.096 1.035 0.0658 6.3767 1.035 0.9899 0.0321 1.258 .4.238

0.5 1 . 1 22.3'25 1.092 1.039 0.0612 6.5306 1.035 0.9960 0.0274 1.535 4.928

0.7 21.61 1 22.243 1.088 1.043 0.0566 6.52'72 1.034 l.OOu1 0.022'7 1.SlS -1.621

0.9 21.672 22.162 1.084 1.036 0.0520 6.5061 1.034 1.0082 0.OlSO 2.094 -131.5

1 21.702 22.121 1.082 1.0-18 0.0498 6.4964 1.093 1.01 13 0.0156 2.236 .1.162

1.1 21.733 22.080 1.080 L.0.50 0.0476 6.4874 1.033 1.01-1-1 0.0132 2.577 1.010

1.3 21.794 21.999 1.076 1.05-4 0,0434 6.4712 1.033 1.0207 0.008-t 2.663 3.707

1.5 21.855 21.917 1.V72 1.0.58 0.0393 6.454.5 1.032 1.01'70 0.0036 2.9.51 9.40.5

1.6 21.886 21.877 1.070 1.059 0.0374 6.4.51-5 1.032 1.030'2 0.001 1 3.095 3.255

1.7 21.917 21.836 1.068 1.061 0.0356 6.4462 1.031 1.0333 0.0013 3.240 3.lOr',

1.9 21.979 21.7.55 1.063 1.06.5 0.0321 6.437-5 1.031 1.0307 0.0062 3.533 2.807

2 22.010 21.71*5 1.061 1.067 0.0306 6.4:J-11 1.031 1.0.129 0.OOSï 9.680 2.6%

3 22.325 21.314 1 .@12 1.087 0.0266 6.4348 1.028 1.0758 0.0339 1 1.193

4 22-6-48 20.91'7 1.022 1.108 0.0428 6.5001 1.025 1.1 100 0.G601 6.148 -0.234

3 22.980 20.525 1.003 1.129 0.0659 6.6320 1.022 1.1455 0.0873 8.378 -1.623

6 23.320 20.137 0-98.5 1-1-72 0.0912 6.8324 1.019 1.1825 0.1 156 10.076 -2.974

Xotes Pt and Pk am p r i a dividend ratios when the economy realizes a low and a high growth rate

respectively. ml and mh are goss equity returns when the economy realizes a low and a high gowt h

rate respectively rl and rh are gros risk Fr- rates when the economy realizes a low and a high

growth rate respectiwly. pi = p, - d and ph = p, + d. d determines the extent of time mrying

EIS: Le., d = O imply constant E f S , and d > O implies proqclical EIS with EIS increasing with

d. E(EIS) = (1 - p,,,)-' is the average of EIS. 1 chose 3 = 0.95 and E ( E I S ) = 0.1 that are useci

in Table 1 of Weil (1989), implying p, = -9. The numbers in bold face replicate Table 1 of M'eil


Table 3.7: Calibration Results W(E1S)l-6: Cyc1ica.i EIS with E ( E I S ) = 0.05 and Con-

stant CRRAw = 45 d PI Ph mr mh s, E ( m ) % rr r,, sr, rlrC WL'%

Yotes: Pl and Ph are price dividend ratios when the economy reaiizes a low and rr high growth rate

respectively ml and mh are gros equity returns when the economy realizes a low and a high growth

rate respectively. ri and r h are goss risk fiee rates when the economy realizes a low and a high

growth rate respectively. pl = p,. - d and ph = p, + d. d determines the extent of time varying

EIS: Le., d = O imply constant EIS . and d > O impiies procyciical EIS w i t h EIS increasing with

d. E(EIS) = ( 1 - p,)-L is the average of EIS. I chme B = 0.95 and E(EIS) = 0.05 that are used

in Table 1 of Weil (1989), implying p, = -19. The numbers in bold face replicate Table 1 of Weil

(1989). Aiso see notes in Table 3.2.

Table 3.8: Calibration Results W(E1S)l-7: Cyclical EIS with E(EIS) = 1/45 and Con-

stant CRRAu.7 = 45 d f i Ph ml mh .4, E(m)'% ri ph 9 rl% P'%

40 274.92 22.442 0.475 7.355 3.166 291.5 0590 8.670 5.713 363.0 -11.536

Notes: Pl and Ph are price dividend ratios at low and a high state respectively mi and mh are goss

equity returns at low and a high state respectively. ri and rh are goss risk free rates at low and

a high state respectively. pl = p, - d and ph = p, + d. d determines the extent of time mrying

EIS: i-e.. d = O irnply constant EIS. and d > O implies procyclical EIS with EIS increasing with

d. E ( E I S ) = (1 - p,)-L is the average of EIS. 1 chose 9 = 0.95 and E(EIS) = 1/45 t hat are used

in Table 1 of Weil (1989), irnplying p, = -44. The numbers in bold face replicate Table 1 of \Veil


Table 3.9: Calibration Resultç W(EIS)%l: CycIical EIS with E ( E I S ) = x and Constant

Xotes: Pl and Ph <are price dividend ratios at low and a high state respectively. rni irricl rnh are gros

equity returns at Iow and a high stnte respectively. ri and rh are g r o s risk free rates at Iow and

a high state respectively. pl = p, - d and ph = p, + d. d determines the extent of time varying

EIS: Le.. d = O impty constant EIS. and d > O implies procyclical E I S with EIS increasing with

d. E(EIS) = (1 - p , ) -L is the average of EIS. I chose J = 0.95 and E ( E I S ) = x that are wed in

Table 1 of Weil (1989). implying p, = 1. The nurnbers in bold Face replicate Table 1 of Weil (1989).

,4so see notes in Table 3.2.

Table 3.10: Calibration Resiilts W(EIS)2-2: Cyclical EIS with E ( E I S ) = 2 and Constant

0.4 19.337 22.100 1.162 0.989 0.136 7.530 1.088 0.93.5 0.109 1.196 6.286 Notes: Pi and Ph are price dividend ratios when the economy realizes a Iow and il high g o w t h rate

respectively. ml and nrh are goss equity returns w hen the economy renlizes a low and n high growt h

rate respectively. ri and rh are gros risk hee rates when the economy realizes a low and a high

growth rate respectively. pl = p . - d and ph = pw + d. d determines the extent of time varying

EIS: i.e.. d = O imply constant EIS. and d > O implies pracyciical E I S with EIS incre'asing with

d. E ( E I S ) = (1 - p,)-L is the average of EIS. 1 chose i3 = 0.95 and E ( E I S ) = 2 that are useci

in Table 1 of Weil (1989). implying p, = 0.5. The numbers in bold face repliciite Table 1 of \Veil


Table 3.11: Calibration Results W(EIS)2-4: Cyclical EIS with E ( E I S ) = 0.2 and Con-

3.*5 7.709 1'7.965 1.95-5 0.750 0.826 35.242 1.315 0.530 0.556 -7.7.53 4.5.110 Notes: f i and Ph are price dividend ratios itt Iow and a high state respectively. ml and mh are grcm

equity returns at low and a high state respectively. t r and t h are g r o s risk hee rates at low and

a high state respectively pl = p , - d and ph = p . + d. d determines the extent of tirne varying

EIS: Le.. d = O impty constant EIS. and d > O impties proc~clical EIS with EIS increclsing with

d E ( E I S ) = ( 1 - is the average of EIS. 1 chose i3 = 0.95 and E ( E I S ) = 0.2 that are used

in Table 1 of iveil (1989), implying p, = -4. The numbers in bold face replicate Table 1 of Weil

(1989). A b see notes in Table 3.2.

Table 3.12: Caiibration Results LV(EIS)2-5: Cyclical EIS with E ( E I S ) = 0.1 and Cori-

5 4.90*5 13.339 2.26.5 0.73.5 1.043 49.986 1.420 0.487 0.659 -4.6.54 55.186

Xotes: Pl and Ph are price dividend ratios when the econorny recllizcs a low and a high growt h rate

respectively. ml and rnh are goss eqtiity returns when the economy reatizes a low iind a high grotvth

rate rspectively. rl and rh are gros risk Free rates when the economy realizes ri low and high

growth rate respectively- ppr = p, - d and ph = pu. + d. d determines the extent of time mrying

EIS: i.e.. d = O impIy constant EIS. and d > O implies procyclicd EIS with EIS increczsin,o with

d. E(EIS) = ( 1 - p , ) - l is the average of EIS. I chose j = 0.95 and E(EIS) = 0.1 that are iised

in Table 1 of Weil (1%9). implying p, = -9- The numhers in bald face replicate Table 1 of Weil


Table 3.13: Calibration Results W(EIS)2-6: Cyclical EIS with E( EI S) = 0.05 ancl

Constant CRRAu, = '20 d fi ph mi mh s, E(mj% ri r h .s,f r f% (PU'%

Sotes: fi and Ph .are price dividend ratios at low and a high state respectively. ml and mh are goss

equity returns at low and a high state respectively ri and rh are goss risk free rates at low and

a high state respectively. pl = p, - d and ph = p . t d. d determines the extent of time wrying

EIS; i.e.. d = O imply constant EIS. and d > O implies procyclical EIS with EIS increasing with

d- E(EIS) = ( 1 - is the average of EIS. 1 chose d = 0.95 and E(EIS) = 0.05 that are used

in Table 1 of Weil (1989). implying p, = -19. The numbecs in bold face repticate Table 1 of Weil

(1989). A h see notes in Table 3.2.

Table 3.14: Calibration Results W(EIS)%T: Cyciical EIS with E ( E I S ) = 1/45 and

Notes: Pl and Ph 'are price dividend ratios at low and a high state respectively. rnl and rnh are gw

equity rcturns at low and a high state respectively. rl and r h are gros risk free rates at low and

a high state respectively. pl = p, - d and ph = p . + d. d determines the exterît of time varyirig

€ES: Le.. d = O imply corntant EES. and d > O irnpfies prucycfim! EIS +th EES incrwtsing nrith

d. E(EIS) = ( 1 - p,)-l is the average of EIS. 1 chose 3 = 0.95 and E ( E I S ) = 1/45 that are useci

in Table 1 of Weil (1989). implying p, = -44. The numbers in bold face replicate Table 1 of Weil

(1989). AIso see notes in Table 3.2.

TabIe 3.15: Calibration Results W(EIS)3-1: Cyclical EIS with E ( E I S ) = x and Con-

stant CRRAu? = 10 d Pi ph ml rnh Y, E ( m ) % ri rh $ 9 r f% @w%

Xotes: Pl and Ph a re price dividend ratios when the economy realizes a low and a high growth rate

respectively. ml and mh are gros equity returns when the economy realizes a low and a high growth

rate respectively. ri and rh are gros risk free rates when the economy realizes a low and a high

growth rate respectively. pl = p, - d and ph = p . -+ d. d determines the extent of time varying

EIS: Le.. d = O impky constant EIS? and d > O imphes procychcd EIS with E I S incre.lsing 6 t h

d. E(EIS) = ( 1 - p,)-l is the average of EIS. 1 chose d = 0.95 and E(EIS) = x t hnt are used in

Table 1 of Weil (1989)1 implying p, = 1. The numbers in bold face replicate Table 1 of Weil (1989).

Also see notes in TabIe 3.2.

Table 3.16: Calibration Results W(EIS)3-2: Cyclical EIS -5th E( E I S ) = 2 and Constant

Notes: Pl and Ph are price dividend ratios when the economy reiilizes a low and a high growth rate

respectively. mi and mh are poss equity returns when the economy realizes a low and a high growth

rate respectivety. ri and rh are gross risk Free rates when the economy realizes a low and a high

g w t h rate respective!. pr = p, - d and pn = p,. t d. d determin- the extent of t in= v ~ r y t n g

EIS: i.e.. d = O impty constant EIS. and d > O implies procpclical EIS with EIS increasing with

d. E(EIS) = ( 1 -p , ) - l is the average of EIS. 1 chose a = 0.95 and E(EIS) = 2 that are useci

in Table 1 of Weil (1989), implying p, = 0.5. The numbers in bold face repIicate Table 1 of Weil

(1989). .iUso see notes in Table 3.2.

Table 3.17: Calibration Results W(EIS)M: Cyclical EIS with E ( E I S ) = 0.2 and Con-

stant CRRAw = 10 d 4 Ph ml r n h s , E ( m ) % ri r h sr, rr% @"%

1.637

1.950

.> .y- ,., I I

2.6 11

2.972

:3.339

3.72 1

-1. 1 1.5

4-52.!

4.946

05.38 1

5.830

6.292

6.768

7.2.57

7 .ï6O

8.276

8.Y0.5

9.348

IO. -1 74

2-1.666

44.169

67.6 17

Notes: Pi and Ph are price dividend ratios a t low and a high state respectively. ml and mh are goss

equity retunis at low and a high state respectively. rl and rh are goûs risk free rates at low and

a high state respectively. ppt = p, - d and ph = p, -t d. d determines the extent of time varying

EIS: i.e., d = O imply constant EIS, and d > O implies procyclical EIS with EIS increasing with

d. E(EIS) = ( 1 - is the average of EIS. 1 chose ,O = 0.95 and E(EIS) = 0.2 tha t are used

in Table 1 of Weil (1989), implying p, = -4. The numbers in bold face replicate Table 1 of Weil


Table 3.18: Calibration Results W(EIS)3-5: Cyclical EIS with E(EIS) = 0.1 and

Sotes: 4 and f i are price dividend ratios when ttre ecanomy mfizes a Icmr and s high ,(grwth rnte

respectively. mi and mh are g r o s qui ty returns when the economy realizes a low and a higb growth

rate respectively. rl and rh are ~~ risk hee rates when the economy realizes a Iow and a high

growth rate respectively. pi = p, - d and ph = p, + d. d determines the extent of time varying

EIS: Le.. d = O imply constant EIS. and d > O implies procyclical EIS with EIS increasing with

d. E(EIS) = (1 - p,)-L is the average of EIS. 1 chose 3 = 0.95 and E ( E I S ) = O. 1 that are used

in Table 1 of Weil ( 1989), impiying p, = -9, The numbers in bold face replicate Table 1 of Wei1

(1989). .&O see notes in Table 3.2.

Table 3.19: Calibration Results W(EIS)3-6: Cyclical E I S with E ( E I S ) = 0.05 and

Constant CRR.4w = 10 d f i Pi, ml m ~ , s r E ( m ) % rt rh s r / rj% P"3

18 1.834 17.990 6.873 0367 4.569 271.983 1.9.56 0.181 1.255 6.836 232.490

'iotes 6 snd f i are price dividend ratios when the emnomy rertkes a low and a high ,gowth rate

respectiveIy. ml and mh are goss equity returns when the economy realizes a low and a high growth

rate respectiveiy. ri and rh are goss risk free rates when the economy reslizes a low and a high

growth rate respectively. ppi = p, - d and ph = p, + d. d determines the extent of lime varying

EIS; i.e., d = O imply constant EIS , and d > O irnpties procyclical EiS with EIS incresing with

d. E(EI.9) = (1 - p,)-l is the average of EIS. 1 chose 4 = 0.95 and E ( E i S ) = 0.05 that are used

in Table 1 of Weil (1989), implying p, = - 19. The numbers in bold b c e replicate Table I of Iveil


Table 3.20: Caiibration Results W(EIS)&l: Cyclicat EIS with E ( E I S ) = x and Con-

Notes: Pl and Ph 'are price dividend ratios when the economy realizes a Iow and a high growth rate

respectively. mi and mh are gros equity returns when the economy realizes a low and a high g o w t h

rate respectiveiy. ri and r h are goss risk free rates when the economy realizes n 10% and a high

growth rate respectivelj-. ppr = p , - d and ph = p,, + d. d determines the extent of tinie vmying

E f S : i.e.. h = O imptp constant EIS. rvld d > O impties pmcychnt EfS n+th EIS increCasing with

d. E(EIS) = (1 - p,)-L is the average of EIS. I chose 3 = 0.96 and E(EIS) = x that are wd in

Table 1 of Weil (1989). implying p, = 1. The numbers in bold face replicate Table 1 of Weil (1989).

Also see notes in Table 3.2.

Table 3.21: Calibration Results W(EIS)42: Cyclicnl EIS with E ( E I S ) = 0.5 and

Constant CRRAw = 5 d 8 Ph ml mh S, E ( m ) % q r h .sr! r aWrC

Xotes: Pj and Ph are price dividend ratios when the economy realizes a iow and a high growth rate

respectively. mi end mh are gros equity returns wheti the economy realizes a low aiid a high gowth

rate respectively rt and rh are goss risk Free rates when the economy rerrlizes a low and a high

growth rate respectively. pl = p,, - d and ph = p . + d. d determines the extent of tinie vzirying

EIS: i.e.. d = O imply constant EIS. and d > O iinplies procyclical EIS with EIS increiwing with

d. E(EIS) = ( 1 - p,)-L is the average of EIS. 1 chose J = 0.95 and E ( E I S ) = 0.5 thnt are used

in Table 1 of Wei1 ( 1989). implying p, = 2. The numbers in bold face replicate Table 1 of Weil

(1989). AIso see notes in Table 3.2.

Table 3.22: Calibration Results W(EIS)44: Cyclical EIS with E ( E I S ) = 0.2 and

Constant CRRAuv = 5 d f i ph ml mh sm E ( m ) % rl r h Srf r / % QW"rC

Xotes: Pl and Ph are price dividend ratios when the economy realizes a low i d ii high growth rate

respectively. ml and mh are gros equity returns when the economy realizes a low and a high growth

rate respectively. ri and rh are gros risk free rates when the economy realizes ü low and a high

growth rate respectively. pi = p, - d and ph = p, + d. d determines the extent of time v a l i n g

EIS: i-e.. d = O imply constant EIS. and d > O implies procyclicai EIS with EIS in~re~ls ing with

d. E ( E I S ) = (1 - is the average of EIS. 1 chose 3 = 0.95 and E(EIS) = 0.2 thüt are wed

in Table 1 of Weil (1989)? implj-ing p, = -4. The numbers in boId face replicate Table I of Weil


Table 3.23: Calibration Results W(EIS)4-5: Cyclical EIS with E ( E I S ) = 0.1 and

O

O . 1

o. 2 0.3

0.4

0.5

0.6

O.?

0.8

0.9

1

1.1

12

1.3

1.4

1.5

Notes: Pi and Ph are price dividend ratios when the econorny redizes a low and a high growth rate

respbively. ml and rnh are gros guity returns when rhe economy realiaes a low and a high uowth

rate rmpectively. rl and rh are goss risk free rates when the econorny realizes a low and a high

growth rate respectively. pl = p, - d and ph = p, + d. d determines the e-xtent of time varying

EIS: i.e.. d = O imply constant EIS. and d > O implies procycIica1 EIS with EIS incresing with

d. E(EIS) = (1 - is the average of EIS. 1 chose 17 = 0.95 and E ( E I S ) = O . 1 that are used

in Table 1 of Weil (1989). implying p, = -9, The numbers in bold face replicate Table 1 of Weil


Table 3.24: Calibration Results W(E1S)CG: Cyclical EIS with E(EIS) = 0.05 and

Sotes: and Ph are price dividend ratios when the economy reolizes a low and a high growth rate

respectively. r n r and rnh are g r o s equiby FebWW when 6he wonomy reitlizw a k w atd cl hhigh g o w € h

rate respectively- ri and rh are gros risk free rates when t h e economy realizes a low and a high

g o w t h rate respectively. pi = p, - d and ph = p, + d. d determines the extent of time varying

EIS: Le.. d = O imply constant EIS, and d > O implies procyclical EIS with EIS increasing with

cf. E(EIS) = (1 - p,)-' is the average of EIS. I chose 3 = 0.95 and E ( E I S ) = 0.05 that are used

in Table 1 of Weil (1989). implying p, = -19. The numbers in bold face replicate Table 1 of Weil


Table 3.25: Calibration Results W(EIS)-L7: Cyclical E I S with E( E I S ) = 1/45 and

Constant C'RRAw = 5 d 8 Ph mi mtl s, E ( m ) % rl r h -%I rf% Q>"%

Notes: Pl and Ph are price dividend ratios when the economy realizes n low and A high growth rate

respectively. ni1 end mh are g r o s equity returns when the economy renlizes a Iow and a high growth

rate respectively. rl and rh are gros risk free rates when the economy reaIizes a low aird a high

growt h rate rwpectively. pl = p, - d and ph = p , + d. d determines the extent of time varying

EES: t-e-. cf = O irnpty constant E I S . and d > O irnpties pro-ctinl EFS with EIS m d g with

d- E ( E I S ) = ( 1 - p,)-L is the average of EIS. 1 chose 3 = 0.95 and E ( E I S ) = 1/45 that are useci

in Table 1 of Weil (1989). implying p, = -44. The numbers in bold face replicate Table 1 of \Veil


Table 3.26: Calibration Resuits: Cyclical EIS and Constant CRRA with Yaari's

Preference for Certainty Equivalent, ,B = 0.99 f Pi Ph 9, E(m) ri rh s , ~ E r ) +"X wl r.i.2

Xotes: s, and E(m) are standard deviation and mean of the market return respectively. ri and rh are

gros risk bee rates at low and high stat respectively. sr, and E(rf )% are standard deviation and the

mean of percentage value of net risk 6ee rate. aW% is net average percentage (propertional) equity

premiums between the two economy states. 7 is risk aversion parameter. pl and ph are eiasticity

ofintertemporai substitution parameters at low and highstate respectively. I chose d = 0.99 which is

used inEpstein and Melino (1995). The first line replicate their results. SIehra and Prescott (1985)

estimates E (m) = 1.07, s, = 0.165, E ( r f ) = 1.008, sri = 0.056 and aW% = 6.18%. ce1 and

ce2 are the certainty equivaient value corresponding to lottery (50,0.5: 100.0.5) and (70.0.5: 80,0.5)

respect iveiy-

Chapter 4

Cyclical Int ert emporal Substitut ion

and Risk Aversion

In Chapter 2.1 bui1d a mode1 where the representative agent has recursive preferences and

investigate whether time varying risk aversion can contnbute to the explanation of the

equity premium piizzle. By separating the risk aversion parameter from the intertemporal

substitution parameter. 1 find that introdticing tirne varying risk aversion done does not

contribute to the explanation. 1 also modelled t ime mrying intertemporal siibstit ut ion

with constant risk aversion. In this case. 1 find the results are remarkable. Introducing

just a bit of time W n g E i S resdts in dramatic improvement over the benchmark

resdts in bot h the e.xpected utility certainty equivalent and Yaari preferences certainty

equivalent cases. Do the results Erom Chapter 2 and Chapter 3 lead to the conclusion

that only t h e varjing EIS matters to explain the apparent counter-cyclicai nature of

eupected excess return on stock and bond. and that a state dependent CRRA does

not play any role in this conte-xt ? What if time v a m g intertemporal substitution

combineci with time varying risk aversion c m contribute to e,uplaining various empirical

phenornena that have been investigated in finance and macroeconomics ? In order to

answer these questions, this chapter extends Chapter 2 and Chapter 3 by dowing bot h

the risk aversion parameter and the intertempord substitution parameter to vary wit h the

state. The specifîcation of preference used in this chapter embeds both the time varying

intertemporal substitution and the time vaqing risk aversion modek of preferences as

special cases.

4.1 The Representat ive Agent's

1 use the same model built in Chapter 2 except for the

aggregator takes the form

Problem

following differences. The CES

Notice t hat the elasticity of int ertemporal substitut ion (EIS) depends on the exoge-

nous state variable st ; EIS (st ) = l / ( 1 - p ( s r ) ) .

Similar to Chapter 2 . 1 consider two different fiinctional forms for certainty equivalerit:

expected utility and Yaari preferences.

Case 1: Expected Utility Certainty Equivalent

In this case, the certaiinty equivalent p,, is the same as eq(2.9) and 1 -a (s t ) is the measiire

of relative risk aversion with respect to consumption gambles. Thtis (4.1) becomes

Mauirnizing Ut subject to the wealt h accumulation constraint given in (3.4) yields the

following first order conditions (see First Order Conditions in Case 1 of Appendix C for

the derivat ion).

CHAPTER 4. CYCLICAL INTE~CTEMPORAL SUBSTITUTION AND RISK AVERSION 91

where l&+i = w;ro = x:, wrtriVt+ 1 is the gros ret uni to holding the optimal port folio

w; h m time t to t + 1, and the intertemporal marginal rate substitution from time t to

t + 1 (1111 RSt,t+l) is aven by eq(C.15)

where at is the consumption wealth ratio.

The elast icity of intert emporal substitution ( E I S ) and coefficient of relative risk aver-

sion (CRRA) are 1/(1 - p (st)) and 1 - a (st ) respectively. If the cycle is indexed by s t .

a procyclical EIS requires p (st ) to be high when the state is high and p (so to be low

when the state is low: countercyclical CRRA requires ~ 4 . 9 ~ ) to be high wheri the state is

high and a(s t ) to be low when the state is low.

If p(st) = p (a constant) and note that the budget constraint eq(C.2) implies xt,l =

(xt - c t ) ilIt,t+l = (1 - n t ) ~ ~ l r , t c l x t : therefore

then the eqiiations eq(4.2) and eq(4.3) rediice to eq(2.10) and eq(2.11) of rny Chapter 2

results. namely

Case 2: Yaari Preferences for Certainty Equivalent

In this case. certainty equivalent ha is given in eq(2.15) and the EIS and CRR-4 are

given by 1/( 1 - p (st ) ) and ($'" ' - p) / r-ectively-

The individual's problem is then

CHAPTER 4. CYCLICAL INTERTEMPORAL SUBSTITUTION AND RISK AVERSION 92

subject to the wealth accumulation identity given in (2.4).

The above problem implies the following first order conditions (see First Order Con-

ditions in Case 2 of Appencik C for further details)

where the intertemporal marginal rate substitution from time t to t + 1 (I.\I RStSt+ 1 ) is

given by eq(C.32)

i f p (st) = p ( s ~ , ~ ) = p (a constant). by using eq(4.5). t hen eqtiations eq(4.7) and

eq(4.8) reduce to eq(S.17) and eq(2.18) of Chapter 2, nümely

4.2 Equilibrium

The equilibrium condition is the sarne as the one describeci in Chapt er 2. In equilibrium.

the representative agent will hold one share of the stock and consumes al1 of his dividend.

t hus eq(2.4) becomes:

q = ( q + p t ) * 1

or equivalently, the consumpt ion weait h ratio c m be writ ten

CHAPTER 4. CYCLICAL INTERTEMPORAL SUBSTITUTION A N D RISK AVERSION 93

where from 4 2 . 2 1) , Pt = pt/ct, the price dividend ratio.

Using eq(2.22) to replace and eq(4.12) to replace n t . it is seen that in the

equilibrium for t his economy. the repreçentative agent's intertemporal marginal rate sub-

stitution from time t to t + 1. given by eq(4.4) . simplifies to

and the representat ive agent 's intertemporal marginal rate çtibst it tit ion given by eq(4.9)

4.2.1 Equilibrium wit h Expected Utility for Cert ainty Equiva-

lent

The equilibrium condition corresponding to Case 1 of Section 1.1 is obtained by substi-

tuting eq(2.22) and eq(4.12)into Euler equations eq(4.2) and eq(4.3) . and noting st = g,:

that provides a relation that must be satisfied by any equilibrium price-dividend function

P (.) . namely

Becaw there are only two states and {gt ) is Markov process, eq(4.15) reduces to


(see Numerical Solutions in Case 1 of Appendiv C.2.1 for hrther details).

The risk free rate can be determined from Euler equation (4.3). That is.

Using eq(2.22) and eq(4.12). the risk free rate r, will prevail if todq's state is i E

{ I . h} .

(agaiii. see hmerical Solutions in Case 1 of Appendiu C.2.l for further details).

4.2.2 Equilibrium with Yaari Preferences for Certainty Equiv-

alent

The equilibrium condition corresponding to Case 3 of Section 4.1 is obtained by suhsri-

tuting eq(2.22) and 44 .12) into Euler eq~iation eq(4.i) and eq(4.Y). and noting st = gt:

that provides a relation that must be satisfied by any equilibrium price-dividend function

P (a). namely

In our economy: this can be written as (see 'iurnericai Solutions in Case 2 of Appendiv

C.2.2 for furt her det ails)

CHAPTER 4. CYCLICAL INTERTE~~PORAL SUBSTITUTION AND RISK AVERSION 95

where pi = p (!Ji), and ri = 7 (gi) .

The risk free rate can be determined from the Euler equat ion (4.8)

The risk free rate r, that will prevail if today's state is i E (1 . h } .

(again see Numerical Solutions in Case 2 of Appendi'r C.2.S for fiirther details.)

4.3 Model Calibration

In this section. I calibrate rny mode1 to soIve the Mehra and Prescott (1985) eqtiity

prernium puzzle. Similar to Section 2.4. 1 calibrate to the consiimption proces with

transition matrk eit her IIl or I l 2 given in eq(2.33) and eq(2.33) and at tenlpt to match

either 8 = ( (A Ph) = (23.467.27.835): ( r i . r h ) = (1.064.0.952) } or 8 = ( (4. Ph) =

(23.424.27.886); (ri. r h ) = (1.0611.0.952) }: that is. plug 8 into tiie equation -stem

determined by eq(4.16) and eq(4.17) to solve for (pl. ph, a l . ah) in the cûse of an Expected

Utility certainty equivalent: or piug 0 into the equation system determined by eq(4.19)

and eq(4.30) to solve for (pl. ph. -fr . uh) in the case of a Yaari certainty equivaient . Cleariy.

in each case there are four equations and four unknowns: one suspects that there should

exist solutions that match the empirical observations exactly. But we are only interested

in the admissible solutions; that is. solutions such that ( p l 5 1. ph 5 1. 01 5 1. o h 5 1 )

for the first case; or (n 5 1. ph 5 1, O 5 R 5 1. O 2 7 h 5 1) for the second case. Epstein

and Melino (1995) show that counting parameters and equations can be misleading. They

consider fixed ,O and p but are nonparametric in the certainty equivalent p and show t hat

there is no admissible solution.


4.3.1 Calibration Results of Expected Utility for Certainty Equiv-

In this case 1 choose ,O = 0.98 and the transition rnatrix to be III given in eq(2.32).

These are the same d u e s used by Weil (1989) Table 2. The equation systcm cq(4.16)

and eq(4.17) can be solved to deliver the two sets of solutions giveri in Table 4.1 (see

Numerical Solution in Case 1 of Appendk C.2.1 for details.)

In Table 4.1, both sets of soliitions imply a procyclical elasticity of intertemporal

substitut ion. The first implies countercyclical risk aversion: the second is inadmissible

but it implies procyclical risk aversion.

1 focus on the first solution. Yotice that it implies risk aversion changes dramatically

wit h the state of the economy while attitudes toward intertempord substitution are fairly

constant. Xonetheless. contrasting with the results from Chapter 2 shows t hat even stich

smdl variation in p helps a great deal in fitting the historicat data. More intuitive

elaboration will be given below following the disctission of the restilts obtaineci with a

Yaari certainty equivalent.

4.3.2 Calibration Results of Yaari Preferences for Certainty

Equivalent

In this case. for the same reason given in Chapter 2. 1 choose to Lise .3 = 0.99 and

the transition matrix II2 given in eq(2.33). These values are used by Epstein and MeIino

(1995). Given Yaari preferences for the certainty equivalent : the equat ion systern eq(4.19)

and eq(4.20) deliver the solution (pr , ph 31, Y ~ ) = (12.1981,16.4239.0.2242.0.9441). (see

Numerical Solution in Case 2 of Appendiv C.2.2 for deta&.)

This solution implies countercyclical risk aversion and procyclical elasticity intertem-

poral substitution, but it is inadmissible because both pi and ph are greater than one.

Therefore 1 look for an approximate solution (dehed in Approximate Solution in Case

CHAPTER 4. CYCLICAL INTERTE~IPORAL SUBSTITUTION AND RISK AVERSION 97

2 of Appendk C.3.2) that is generated by a t h e varying EIS and CRRA to see if it

c m dominate the solution produced by Epstein and Melino (1995). To this end. similar

t o Chapter 2.4: 1 set (P (gc) . P ( g h ) ) = (23.426.27.889) with transition matriv equals to

II2: so that the first and second moments of the equity return are guaranteed to match

the US historical equity return data. Specifically, for given values of pi and ph. I choose

the values of 71 and ?,, to satisfy eq(C.57) given in Approximate Solution in Case 2 of

Appendk (7.3.2. 1 then solve for the risk free rates rl and r h from eq(C.58). The ecl-

uity premium and the long-run average equity prerniiirn defined by Weil (1989) has been

describeci in Chapter 3. There are many admissible pairs of (pl . ph). For descriptive

purpose. 1 have organized the results aroitnd parameters d and p. where pl = p - cf and

ph = p + d. For given parmeter p. d deterniines the extent of time vaqing EIS: i.e..

d = O imply constant EIS. and d > O implies procyclical EIS with EIS increasing with

d. For each p. 1 look for the value of d that best matches the first and second moments

of the risk free rate (and the risk prerniiirn). 5fy resiilts are presented in Table 4.4.

From Table 4.4. we see that introdticing ri procyclical time varying EIS dong with

state dependent risk aversion leads to a substantial irnprovernent in fit. LVhen p = 41.87.

the d = O resiilts correspond t o those in Epstein and Slelino (1995). .\llowing the

parameter d to increase: equivalently. by introducing procyclical EIS. 1 get a better fit.

although 1 can not get a perfect match. For example. p = -10 and d = 0.7 matches the

average risk free rate. but the standard deviation of the risk free rate is stilI too high:

p = -10 and d = 1.61 matches the standard deviation of risk free rate but the average

risk free rate is too high.

To summarize, I use a simple mode1 based on recursive utility preferences. with t m

different certainty equivaient hinctional forms: Ewpected Utility and Yaari. In the first

case, with procyclicaI EIS and countercyclical CRRA, 1 get a perfect match to the

observed US historical data. In the second case, even though 1 caonot get a perfect

match, my approximation solutions are a significant improvement compared to the se-

CHAPTER 4. CYCLICAL ~ N T E ~ E ~ I P O R A L SUBSTITUTION AND RISK AVERSION 98

lected benchmark results (of Epstein and Melino (1995)). Further, I find that these

results are robust to various changes in the caiibrat ion ( s e Appendk F .)

4.4 Interpretation

Hansen and Richard (1987) show that under very general assumptions there exists a

stochast ic discoiint factor Qt + sat iskng:

This implies substantid t ime variability in excess ret iirns can be achieved only if t here

is siibstantial time variation in the conditional second moments of the joir,t (Qt+ ,. R t + l )

process. which in turn reqiiires substantial volatility in the marginai rate of substitution.

From our Euler equation. we see that to generate a high return for equity in excess of the

risk free rate. al1 we need is a high variance of intertemporal marginal rate of substitution.

Evisting models based on traditional expected ut ility preferences and the represent iit ive

agent assumption have trieci to link the risk premium to economic fiindamentels biit have

been unable to explain the exces return predictability observed in equity markets. bond

markets and foreign excliarige markets. The failiire of these efforts is in large part due to

their inability to generate a sufFiciently high variance for the resulting stochastic discoiint

factor that matches the data.

Campbell and Cochrane (1999) claim that this high ri an ce is a result of counter-

cyclicd risk aversion. The device that generates time varying risk aversion coefficient in

their mode1 is a slow rnoving e-xternal habit which is usehl for addressing the risk free

rate puzzle, but do not generate sufficient additional risk to reconcile the high premia

with reasonable prices of risk. Gordon and St . Amour (20001 200 1) &O c l a h that state

contingent risk aversion helps to explain equity premiwn puzzle because Iow (high) level

of consumption and countercyclical (procyclical) risk aversion implies that consumption


shocks generate Iarger fluctuations in marginal utility, against which the agent will hedge

in choosing his optimal portfolio. Thus state contingent risk aversion can justify high risk

preniia without requiring excessive risk aversion. Ako it hm the potential for explain-

ing cyclical movements in ret unis. Furt hermore. countercyclical risk aversion. as well

as the additionai volat ility in the intertemporal marginal rate of substitut ion induce a

precautionary dernand for risk free mets . which would jiistify high bond prices without

requiring negative cliscounting of future utility. However. risk aversion and substitution

are intertwined in their models. just as they are in the additive expected utility model.

so it is impossible to interpret unarnbiguously which feature of preferences mries over

the cycle. By separating risk aversion from elasticity of intertemporal substitution in

rny model. I finù by introdiicing coiinter-cyciical risk aversion itself can not provide a

t hroiigh explanat ion of equi ty premium puzzle. the t iine nrying procyclical elast icity

of intertemporal substitution. EIS. also pl-ç an important role E I S reflects the

dlingness to trade off between the growth and the smoot hness of an individual's deter-

ministic consumption stream. A low EIS indicates t hat individiials are less wiIling to

trade smoot hness of t heir consurnpt ion st ream for consumption growt h: i.e. t hey like t heir

deterministic consumption stream to be smooth. There is not miich ernpirical research

about what cyclical pattern of EIS should foIlow. but my simulation results reveal a

procyclical pattern. Now let me start with an Etiler equation with rectirsive utility. e-g.

eq(213): and rewrite here

where T = a / p : and Or.t+l = (i\&t+l - rrt).

It is shown by Epstein and Zin (1991) that this Euler equation can be approximated

by :

CHAPTER 4. CYCLICAL INTEFWEMPORAL SUBSTITUTION AND RISK AVERSION 100

Exploit ing (4.23) in the decomposition of the condit ional covariance: we have

therefore. the risk premium can be expressed as

It is evident from eq(4.26) that in the non-expected iitility mode1 considercd here.

the risk premiuni of an =asset is relnted to both the covariance of its retiirn wit h the

market portfolio and the covariance of its retiirn wit h the growth rate of consunipt ion.

State dependent EIS and CRRA generate two effect: p effect and r effect. Procyclical

EIS increase corv (394' . Q , . , , ~ ) but not a lot in the expected valiie Et (3gf::) . thiis in-

crease Et ), I cd1 this as p effect. Both procyclicczl EIS and countercyclical C'RRA

drive r change drarnntically. since cm? (.\Il:, . @t.t+l) > EL ( b f ~ : ~ ) . t hi~s Et ( @ t . l + l ) a h

increase. 1 cal1 this as T effect. These two effect together result in a higher equity pre-

mium. 'CVhile in the eq(1.2) . Campbell and Cochran (1999) only uses the ve- simple

form implied by eq(2.8) . t herefore can not captures these two efFects.

4.5 Prediction

In this section, 1 use my Chapter 4 calibration results to predict the yieId-tematuritu of

a ICI-year discount bond and par-coupon bond. The n-year bond yield-t-maturity are

derived in Appendix E. Using Y, to denote the yieId of 10-year discount bond and ct to

denote the yield of IO-year par-bond, given state of economy at issue t h e is i E ( l . h ) :

CHAPTER 4. CYCLICAL ~UTEEYTEPUIPORAL SUBSTITUTION AND RISK AVERSION 101

the prediction results of the yield-ternaturity of 10-year discount bond and par-coupon

bond are presented in Table 4.2 and Table 4.3 separately accorcling to different cases in

cert ainty equident hnct ional forms.

From the historical data. the observeci US. average nominal 10-year bond yield is

about with annual compounding (see Appendk E). Taking account of an average

inflation rate of 4%: the real yield is about 3%. The predictions in both crws are close

to the historical data.

4.6 Conclusion

Campbell and Cochrane (1999) among others ' point out the need for the representstive

agent to display countercyclical risk aversion in order to explain the apparent corinter-

cyclical nature of expected excess returns on stocks and bonds.

This paper generdizes the mode1 of Epstein and Ziri (1989) by alIowing the repre-

sentative agent to display countercyclical risk aversion ni t h t ime ~ ~ n g elast icity of

intertemporal siibstitut.ion. In Chapter 2. 1 show t hae coiinter-cyclical risk aversion alone

can not provide an explanation of the SIehra and Prescott (1985) equity premiiini piizzle.

which does not support Campbell and Cochrane (1999). By contrast. in Chapter 3. I

show that introducing a state dependent elasticity of intertemporal subst itiit ion. keeping

the C R U constant. produces a dramatic improvement over the benchmark resdts. Fur-

themore: in Chapter 4. 1 introduce both time varying CRRA and EIS into my mode1

and get remarkable results. 1 can match the o b s e d US historical data perfect ly in the

e-xpected utility certainty equivaient case. In the case of Yaari preferences For the cer-

tainty q u i d e n t , even though 1 c m not get a perfect match, the improvement over the

benchmark results is substantid, 1 can also match long term interest rates. In surnm;uv.

a t h e varying procyclicai elasticity of intertemporal substitution plays an important

'Gordon and St-Amour (2000, 2001). Bakshi and Chen (19961, etc.

role in leading to empirical success. EIS reflects the willingness to trade off b e t m n the

growth and the smoothness of an individual's deterministic consumption stream. A low

EIS indicates that individuals are l e s wiIIing to trade smoothness of t heir consumption

stream for consumption growt h; i.e. t hey like their determinist ic consirmpt ion s t rearn to

be srnooth. There is not much empincal research about what cyclical pattern of EIS

should follow. but my simulation results reveal a procyclical pattern. 1Iy resiilts. to

some extent. also support the conclusion of Campbell and Cochrane (1999) that a time

varq-ing countercyclical CRR.4 plays an important role in expiaining the eqiiity premiiim

puzzle, but contrary to the Campbell and Cochrane (1999): it is not the major effect.

-4 moderately time varjing procyclicaI EI S conibined with a very strong coiintercyclicai

risk aversion Ieads to bet ter empirical success.

CHAPTER 4. CYCLICAL INTEFYTEMPORAL SUBSTITUTION AND RISK AVERSION 103

1 Solution Set 1 1 (-2.2236 1 -1.6767 1 -23.1778 1 0.8274) 1

Table 4.1: Cdibration R e d t s Given Expected Utility for Certainty Equivaient

Table 4.2: Prediction Given Elvpected Ut ility Cert einty Eqtxivalent

1 4.271 2.741 4.334 2.60.5

,i3 = 0.98

Solution Set 2

Sotes: Y, denotes the yieid of 10-year discount bond: c' denotes the yield

of lsyear par-bond, given state of economy nt issue time is i E ( 1 . h ) .

Ph PZ

Table 4.3: Prediction Given Yaari Preferences for Certaintv Equivdent

Notes: pl and ph are elasticity of intertemporal substitution (EIS) parani-

eters when the economy realizes a low and a tiigh growth rate respectively.

al and ah are coefficient of relative risk aversion (CRR.4) parameters when

the economy renlizes a low and a high growth rate respectively. Both sets

of solutions are mlved froni equation srstem eq(4.16) and eq(4.17) and

both irnply a procyciiccrl EIS. The first implies countercyclical CRR.4:

the second is inadmissible but it implies procyclicnl C R RA.

(-1.1171 1 -0.8643

1 1 Discount Bond Yield Par-Bond Yield 1

ar

Notes: Y, denotes the yield of 10-year discount bond: c' denotes the yield

of 10-year par-bond, given state of economy at issue time is i E (1 . h ) .

a h

4.3395 0.2071)

Tddc 4.4: Cyclicd C R R A t i i i t l EIS: Ymri Prefcreiicw for C~rtaiut~y Eqiiiviilc~it

CHAPTER 4. CYCLICAL I N T E ~ E M P O R A L SUBSTITUTION AND RISK AVERSION 105

LOI L/L

t 01 C/9

101 C/S

COI CIP

COI L I € COI CE

LOI C l I

001 LE C

OO/C/C C

001 1/0 C

OO/ LI6

OO/ LI8

001 CIL

001 t/9

O01 L E

001 CIP

OUI Cl€

001 LE

Chapter 5

A Two-Country Mode1

Bekaert. Hodrick and David (199'7) introduce preferences e-xhibit ing first-order risk aver-

sion into a general eqtdibritim two-country rnonetary rnodel. They find that first-order

risk aversion substmtially increases excess retum predictability in eqiiity. bond and for-

eign exchange markets. However. this incresed predictability is instifficient to match the

data. These authors conclude that the observed patterns of exces ret iirn predictability

are unlikely to be explaineci purely by time-varying risli premiums gentrateci by highly

risk averse agents in a complete markets econoniy. When 1 introduce both a time varying

coefficient of relative risk aversion (CRRA) and elasticity of intertemporal substitiition

(EIS) into my mode1 cw in Chapter 4. 1 c m match the observed US historical data per-

fectiy in the expected ut ility certainty equivalent case. In the case of Yaari preferences for

the certainty equivalent. even though 1 cannot get a perfect match. the improvement over

the benchmark resuits is substantial, Further, although long term ret u m m r e not used

to calibrate the model. 1 can also match the k t two moments of long term interest rates.

Therefore one possible e-xplanation for the M u r e of Bekaert. Hodrick and David (1997)

is that their representative agent does not display a state dependent EIS or C'RRA,

which, according to my Chapter 4 is crucial in euplaining the apparent countercyclical

nature of expected excess returns to equity.

This Chapter extends my Chapter 4 to a two-country generd equilibriu

investigates whet her t ime varying countercyclical risk aversio~ and t irne

imply sufficient predictability in excess rates of return in the foreign exch

m model

varying EIS

lange market

while simultaneously matching the time series properties of interest rates and stock re-

t u m . In Chapter 4, I t k d to resolve the hlehra and Prescott (1985) equity prernium

puzzle. In this Chapter. 1 investigate the asset pricing puzzles From a different angle: i.e.

an analysis of the predictability of excess rates of return on discount bonds. equities and

foreign money markets using regression analysis. Focusing on predictability introciiices

a variety of new puzzles and emphasizes the cowriability in asset retiirns. Consider the

exchange rate as an example. In a risk neutral world. the preriictect change in the spot

rate shoiild move one for one with the fonvard preniium. Biit regrecsing the redized

change in the spot rate on fomard premiuni typically fields a regression beta t hat is

significantly negative. This is one of the longest standing puzzles in the interniitional

finance area. Similar phenomena that point to large predictable movernents in excess

returns occur in the equity and bond markets. Bekaert. Hodrick and David (1997) find

that in an open economy model, increasing the amoiint of first-order risk aversion dra-

matically increases the variance of risk premiums in al1 markets. However. t his increased

risk-premiiim volatility fails to imply a comparable increase in exces-retiirn predictabil-

ity. Their model simulation results can not match the regession bet<as on al1 markets.

Here 1 ail1 analyze whether time W n g countercyclicai risk aversion hehavior and time

varying EIS imply sufficient predictability in exceçs rates of return.

My model foliows clwely the setup in Bekaert et al (1997). Aset prices and exchange

rates are deterrnined by the optimal choice of a representative agent. Each country is

inhabited by an inhitely-lived representative agent who receives utility from the con-

siunption of home and foreign consumption goods. The model also assumes that pur-

chasing power parity (PPP) holds. This assumption is unrealistic because. the ernpirical

data display rnarked deviations from PPP, at least in the short run. This deviation

makes agents from different countries inherently different from each other because they

face difFerent relative pnces for consumption bundles. Although 1 believe that this is

an important drawback. 1 maintain this assumption so that cornparisons with Bekaert

et al (1997) c m be made that highlight the effects of time v a ~ n g risk aversion and

intertemporal substitution.

In section 5.1. 1 list stylized facts on excess return predictability. In section 5.2. 1

descnbe the represent ative agent's preference structure ait h state depenclent risk aversion

in a two country rnodel. In section 5.3. 1 describe the budget constraint . In section 5.4.

1 look at the equiiibriiirn conditions in the setting of n simple Lucas asset pricing niodel

to see nrhether or not cyclical risk aversion and time varying EIS can accotint for the

predictability of observed excess returns. Section 5.5 gives t lie mode1 calibrat ion and

soiut ion. 1 concliide t hereafter.

5.1 Stylized Facts on Excess Return Predictability

In this section 1 will highlight the preùictability of excess rates of retiirn on discount

bonds. equities and foreign money markets reported by Bekaert et al (1997). Thcir

results are consistent nlth the evidence documented by the empirical studies of Harvey

(1991). Bekaert and Hodrick (1992. 1993). and Solnik (1993). among otliers.

5.1.1 The foreign exchange market

Assume the home country is the US and the foreign c o u n t ~ is Japan. Let st denote the

log of the spot exchange rate at tirne t of dollars per yen. and let ft denote the log of

the fornard exchange rate of dollars per yen quoted at time t for date t + 1 transactions.

Using interest rate pari& the continuously compounded excess dollar rate of return boom

an uncovered inwstment in the Japanese money market is st+l - ft. A cornmon way

of testing the predictability of this excess rate of return is to regress it on the fornard

CHAPTER 5. A TWO-COUNTRY MODEL

premium:

si+, - ft = ors + a, (fi - + (5.1)

The nul1 hypot hesis of an unpredictable excess rate of return implies J,, = 0'. Bekaert

et al (1997) estimate this equation using monthly observations on the dollar-yen exchange

rate from J a n u a l 1976 to December 1989. The first row of Table 5.ldisplays their

regression results for eq(5.1) : the dope coefficient of -4.016 is significantly negative. rhe

R2 of the regression is 0.22. and the standard deviation of the fit ted value of the exces

return. is 12.355%. The above statistics indicate that t hese excess retiirns are qiiite

predictable and that foreign exchange risk premiurns are qiiite variable.

5.1.2 The discount bond market

Let i t be the continuously-compounded, nomindly risk-free interest rate at time t and let

i tS2 be the continiiously-compounded nominal yield to mat iirity on a two-periad risk-free

zero coupon bond. Let the one-period continuously compounded holding period return

on a tweperiod discount bond realized at tinie t + 1 be ctenotecl as ht + lm-. Denote f bt as

the forward premium in the bond market. Using the definition of the yielct to mat urity.

we obtain

fbt = -Zi ts + 24. (5.2)

The bond market's excess holding period r e t u regression on the fornard premium

is

ht+1.2 - i t = Orb + h b f b t + Et+ l -

If excess holding period retunis are unpredictable' ab should be zero.

'This is quivalant to the nuil hypothesis that the predicted change in the exchange rate. St+ - St , increases one-for-one with the forward premium,

Bekaert et al (1997) estimate this equation using monthly observations on three

month and six-month Eurodollar and Euroyen interest rates from October 1975 to June

1990. The first row and second row of Table (5.1) display their regression results for

eq(5.3) for the US dollar and Japanese yen discount bond markets. For both the dollar

and the yen markets, the estimate of ,Otb is -0.45. and both are significantly negative.

The R2 for the US market is 0.03. and the R' for the yen market is 0.09. The standard

deviations of the fitted mlties of the excess ret unis in the two markets are 0.3 18% for the

US and 0.3% for .Tapan.

5.1.3 The equity markets

Define the equity excess rate of return as an eqiinllyweighted average of the dollar excess

rate of returns on US and Japan equities:

where r:+, ( r z , ) denotes the one period dollar (yen) cont inuoiisly-cornpoiinded ret uni

in the equity market of the US (.Japan). Bekaert et al (1997) construct t his weighteci

average excess return using mont hlÿ observation on t hree month dollar excess returns

to US equities and three month doI1ar excess rettirns to .Japanese equities from J a n u a ~

1976 ta December L989, and r e g e s this weighted average m e s s return on the three

month fornard premium in the dollar-yen foreign exchange market as in the folIon+ng

regression equat ion:

If equity excess rate returns are unpredictabie. dm. shouid be zero. The fourth row

of Table 5.1 display their regression resdts of 45.4); the estimateci &, is -3.453 with

a standard error of 0.816, and the R2 is 0.139.

5.1.4 Implications for modeling

Following the decomposition of forward premiums introduced by Fama (19M) . a sig-

nificant negative estimate on the coefficient of fomard premium implies that the risk

premium is very volatile. As is well known. substantial variahility in risk premiiims

requires substantial volatility in the IdlRS. One way of generating a highly volatile

IMRS is to assume that agents have a high degree of risk aversion. Behert. Hodrick

and David (1997) investigate along this direction and concludes t hat introducing pref-

erences exhibiting first-order risk aversion in an open economy frarnework siibstantiatly

increases excess return predictability in eqi~ity. bond and foreign exchange markets. How-

ever. this increased predictability is insufficient to match the data and therefore these

aiit hors conclude t hat the observed patterns of e sces return predictability are iinlikely

to be explained piirely by time-varying risk premiums generated by highly risk averse

agents in a corriplete markets economy Their best simrtlation resiilts are given in Table

5.2'. In t hi?; paper. I use a mode1 similar to Bekaert . Hodrick ancl David ( 1997) except

risk aversion and intertemporal substitution are allowed to be state dependent. I try

to simulate the regression .3s for the foreign exchange market. bond market and eqiiity

market to see if I can match Table 5.1.

5.2 The Preference Structure

Epstein and Zin (1991b) examine a variety of preferences that exhibit first-order nsk

aversion, including Gd's ( 199 1) disappoint ment aversion preferences. Disappoint ment

aversion was developed to accommodate the .4llais paradox within a parsimonious ex-

tension of e-xpected utilit . Under first-order risk aversion. the risk premium for a small

gamble is proportional to the standard deviation of the gamble. Given the smoothness

*I find e m r in the source provideci by professor Bekaert. After correction, the number should be smailer in absolute value than those reporteci in Table 5.2.

of consumption data, the standard deviation of the consumption growth rate is consid-

erably larger than its variance. Therefore? the role played by first-order risk aversion in

generating a sizeable equity premium is intuitive. Bekaert? Hodrick and David (1997)

follow Epstein and Zin (1991b) and mode1 the representative agent's preferences over

current and uncertain Fut use consumption t O incorporate disappoint ment aversion. They

concluded t hat the o b s e d patterns of excess return predictability are unlikely to be

explained purely by t ime-varying risk premiums generated by highly risk averse agents

in a complete markets economy. But given my Chapter 4 resuIts. it is interesting to

drill d o m further on this topic by introducing time varying EIS as well as time varying

CRRA into a similar two-coiintry mode1 environment as describec1 in Bekaert. Hodrick

and David (1997).

h u m e <r; (6) is the cunsumption good produceci in country x (country g) in period

t which sells at p ice Pz (PO. Let At denote the agent's wenlth at the beginning of

penod t. and It denote the vector of exogenous state variables which span the agent's

information set in period t. F ina l l~ let the iitility of .4 in state It be U (.-Il] It ) and clefine

it recursively by the following frrnctional relation

where Ut = U(c$ C;, , . .. .: 6. i$'+ i . ...l Ic ) is the intertemporal ut ility for honie and foreign

consumption sequences beginning at t. U; (.. .) is called the aggegator function since it

aggregates current consumption c$ and c( ai th an index of the hiture utiiity to determine

the current utility. The Function pt = p(Lrt+, 11;) in (5.5) is the certainty equivalent of

the distribution of future utility L&+, conditional upon period t information. (see Section

2.1.2 for details.)

Let q be a randorn variable with probability distribution P,. As in Epstein and Zin

( 199 1b) , wit h preferences t hat display disappoint ment aversion. the cert ainty equivalent

of P,. denoted p (P,), is detineà by the implicit relation:

where 4 : R -t 92 is continuoust increasing, and concave, and O (1) = O. The properties

assurnecl for d ensure that p is a well-defined certainty equivalent sat iskng that for any

probability distribution P, and any real number B.

Thcre are infinitely rnany possible specifications for d t hat p o s e s the required p rop

ert ies and t hat t herefore deliver a hornogeneous cert aint y eqiiiwdent . Beknert et al ( 1997)

assume q5 is defined by

and

If A = 1. the preferences described by eq(5.6) correspond to expected utility nit h

coefficient of relative risk aversion equal to 1 - a. If .A differs from unity. eq(5.6) c m be

interpreted as follows. Those outcornes below the certainty quivalent are disappointing.

while those above the certainty equ iden t are elating. If A < 1. the elation region

is down-weighted relative to the disappointment region. In this paper. 1 extend this

specification by assurning that preferences are state dependent. Based on my results

fiom chapter 4, 1 assume that the parameter A remains constant: but now I mil1 allow

nt = a (Gt ) where Gt3 indexes the state of the world.

3Gt is a vector of exogenious variables defineci in Appendk D.2. 1 change notation because St is now used to denote exchange rate.

To generate ernpirically useful results? a powerful assurnption is that intertemporal

utility is homothetic. 1 follow mast of the literature and consider a CES aggregator4

where q > 0. ,d is the subjective discount factor of the agent. Again as an extension

fkom Bekaert et al (1997)1 1 tvill allow the intertempord parameter p to va- with the

state Gt. Le.. pt = p(Gt).

1 choose q ( . . .) as the fiinctional form

Wit h the CES aggregator of (5.10). the elmticity of intiïtetnporal substitution be-

tween curent ut ility (F)' (8') '-%and the certainty equivaletit of fut lire iitility. p (Ut, l l lt )

is given by EIS = 1/ ( 1 - p (et)). Therefore p determines the optimal tracie-off between

present and filt tire ut ility.

The maximization of eq(5.10) is sribject to the budget constraint which is given below.

5.3 Budget Constraint Facing The Representat ive

Agent

Let il.I;,, (i\f&,) denote the amount of currency x (currency y) acquired by the agent in

period t. 1 refer to currency x as the dollar, and currency y as the yen. In addition to

currency. agents can hold n capital assets that pay dividends dt = (dit. dZt - - - - - dnr ) and

are traded competitively at pnces pt = (pl t .pu: .... pnt ). Let qmt be the d u e (in units of

c") of the representative agent's investment in asset i. chosen at t - 1. and held to the

JThe aggregator function used here is a bit diaerent h m the previous chapters. It is the same as Bekaert et al (1997) in order to facilitate cornparisons with their results.

beginning of time t.

Let At denote the agent's wealth at the beginning of period t . and let Itdenote the

vector of exogenous state variables which span the agent's information set in period t .

Let St denote the exchange rate (dollar per yen). Slonies are incorporated into the

mode1 using the transaction cost technologies of 'rlarshall (1992) and Bekaert (1996).

Money is demanded by agents because consumption transactions are costly. and increas-

ing real balance holdings decreases these transaction costs. Consumption of ct involves

transaction costs measured by

denominatd in iinits of 8. Siniilarly. consiirnption of @ involves transaction costs

measured by

denominatd in units of cy.

Define rt = ( r i t . r ? ~ . .... rn t ) to be vector of g ros retiirns to thecapital assets i = 1. .... n

wit h

The representative agent's beginning-of-period wealth At (in units of cf) is then.

The budget constraint for the representative agent in units of consurnption good F

Denote by Et the representative agent's total consumption in time penod [t' t + 11 l e s

the transaction cost saved by holding real money balances

where 4, for j = r. y. represents the derivative of cùJ (& mi) wit h respect to the kth

argument. It is shown in eq(D.55) and eq(D.57) of Appendk D that the optimal path

of 4 wil1 sat isb

Also denote r t + ~ to be the market portfolio return defined by eq( D.48): ther; wealt h

can be shown to evolve according to eq(D.85)

5.4 The Equilibrium

The eqaiiibfiwn is dncribed by the price function p : [!.<lin - R. and the prornï

pair (P(. P:):, such t hat the goods market. money market and equity market clear.

In order to denve equilibrium asset prices. consumption goods prices and the exchange

rate' 1 need to solve the representative agent's decision problem in eq(5.5) subject to

the budget constraint eq(5.16) and the definition of wealth eq(5.15). The first order

conditions of this rn-ation problem are given as belod (please refer to Appendix

D for details)

=When both a and p are constants, these first order conditions reduce and David (1997). \men d = 1, these resdts reduce to m y Chapter 4 difference induced by the different agpegator function.

to those of Bekaert. Hodrick resdts except for the slight

E ( I f R S + 1 ) = 1

E (IL\IRS~+~ r,.t+llIt) = 1 . Vi = J. y. 1, .... n

where the intertemporal marginal rate of substitution is @en by

and

"th g;+, i /ci. i = x. y and ai is thc corisumption wealth ratio given by Ct /Ai.

The market portfolio return r,, 1 is . by properly nianipiilating eq( D.48). equivalent

to eq(D.89)

The real returns on holding dollar and yen cur-rencies are aven by eq(D.40) and

eq(D.41). which are reproduced here:

The equilibritm exchange rate is given by eq(D.32)

The nominai risk fke continuously compounded dollar and yen interest rate are

1 $ = in- l + SSt ?

5.5 Data Description, Calibration and Solution of

the Mode1

The endowments and money supplies of the two countries are exogenoiis. In order for

my resiilts to be comparable to Bekaert. Hodrick and David (1997). 1 ilse the snme data

for these exogenous processes as t hey iised.

The US and Japanese money supplies are quarterly .\Il series from [nternatiarial

Financial Statistics (IFS Series 34). Growth rates are deseasonalized by reqessing on

four dummies. .4s in Bekaert. Hodrick and David (1997). I use the following calibration

procediires for the endownients. 1 calibrate the endowments of the ttvo coiintries to

consumption data to the Nondurables and Senices from the O ECD Qiiarterly Nationid

Accounts. The Japanese data include the Semi-durable category. lis this category is

included in the US Yondurables series. Per capita data were derived by using linear

interpolations from annual population series ( IFS Series 994 6 .

The transaction cost technology parameters are considered part of the exogenoits

environment and are calibrated from the model's implications for money demand. See

Marshall (1991) for more details.

The four exogenous processes in the vector Gt are approximated by a first order

Markov chah in which each variable c m take four possible values. implyïng a state space

wit h 256 possible values. The Markov chah is calibrated to the estirnated VAR using the

Gaussian quadrature method of Tauchen and Hussey (1991). The parameters of the first

61 thank Professor Bekaert for providing me with their VAR(1) results: for my purposes. it is equivaient to having their raw data

order VAR irnplied by t his Markov chain approximation are virt ually indist inguishable

from those of the estimated VAR. This is evidence that the discrete approximation is

unlikely to distort the economic implications of the model.

Given this exogenous proces. appendk D shows that the three Euler equiltions

eq(5.19) and eq(5.20) (for i = x and y) can be expressecl in terms of {Gr + l . ~ f +

u:. uL1. at . Q ~ , ~ } . More specifically.

where II\.IR&+~ is given in eq(5.21) wïth

where Jf+, and II:+, are the (exogenous) endomnent growth rates of US and .Japan froni

time t to time t + 1 and t,f and uf are the consumption-velocities of US ancl Japan: Le.

$ = blk,/(4P/). j = x. y.

The market port folio real retum is given by eq(D.90)

The real ret unis. inclusive of marginal transaction cost savings. from holding dollars

and yen. are given by eq(D.80) and eq(D.81)

where K: and K! are the (exogenous) money growth rate of US and Japan €rom tirne t

to tirne t + 1.

The nominal risk kee continuously compounded dollar and yen interest rate are given

by eq(D.82) and eq(D.83)

Thus. the t hree unknown endogenous processes { vf . cf. nt } can be Folinci by solving

the t hree Eiiler equations eq(5.29) eq(5.30) and eq(5.31) simult aneous!y. Siiice the state

space is discrete. the Euler equations can be solved exactly for the 256 valires of each

endogenous variable. A more detailed description of the sollition procedure is given in

Appendiu (D.2). The only approximation introdiiced so far is in the initial discretizntion

of the driving processes. Following Bekaert. Hodrick and Dmld (1997). to simpli- the

calculations. 1 treat the growth rates of 6 and c$' as exogenoiis in the soliit ion algorithm'.

rather than the gron-th rates of' output in the two coiintries. This enables us to solve

the system of equations eq(5.29). eq(5.30) and eq(5.31) reciirsively: the 256 elements of

a, are solved from eq(5.29) first. then given t hese values for at. the 256 elements of r f

are found by solving the 256 equations represented by eq(5.30). Finall. given values

for {at , $1: the 256 elements of v,Y are found by solving the 256 equations represent ed

by q(5 .31)~ . Having solved for {L?. zf' . a i ) , the remaining endogenous variables can be

computd using (5.34-5.38).

'1 have done some caiculations using endogenous growth rates of cf and <. I h d the results are not si,glificantly diaerent from the Bekaert et al (1997) approximation method.

'One way to calculate the solution with endogenous growth rates of cf and c$ is to caicuiate 9;+1 and $+,ushg eq(5.32) and eq(P.33) at the end of this loop and then iterating. In practice. it took many iterations to converge to the "Mi solution'.

The forward premium can be obtained from the covered interest rate parity condition.

That is

where Ft is the dolIar/yen 90-day fonvard rate and St is the dollar/yen spot exchange

rate. Then. excess fornard returns. defined as (Stii - Fr) /Sr. can be determined by

eq(5.39) along ~ l t h eq(D.84)

The models are solved for a variety of parameters chosen by Beliaert. Hodrick and

David (1997). The quarterly subjective discount parcmieter 3 is FLved at 0.!16°.'s and set

The choice of d (the weight on t? in the current-period iitility) is irrelevant. since

1 examine rates of depreciation. rather thnn tevels of exchange rates. The remnining

parameters are ~ r i e d over the foilotking grid: il E {1.0. 0.85. 0.7. 0.53. 0.40. 0.25).

Et(pt+,) E (0.50. -0.33. -4.0. -9.0). Et (a t , i ) = -1.

5.6 Implications for excess return predictability

1 first discuss t h abiiity of the model to replicate the predictability of excess returns

documented in Section 2 of Bekaert, Hodrick and David (1997). 1 also focus on three

measiues of predict ability: the siope coefficient in the excess ret urn regressions analogous

t o eqs. (1) (3) and (5) of Bekaert , Hodrick and David (1997); R2. measured as the ratio

of the variance of the expected excess return t o the variance of the realized excess return:

and the standard deviation of the expected excess return. Ail three statistics can be

computed exactly given the discrete Markov chain driving process.

Consider the model's implications for the dope coefficients in the excess ret urn regres-

çions reproduced in Table 5.1 from Bekaert et al (1997). Since my model is an extension

of their model. I can replicate their resuits using the same set of parameters as t hey

iised wit h rny intertemporal and risk aversion parameters. p and a. set to be constant.

My resuits are displayed in the Tables 5.3, 5.4 and 5.5 for the foreign exchange niarket

and the doI1ar and yen discount bond markets. respectively. Table 5.6 displays the dope

coefficient when the excess return to the aggregate walth portfolio is regrcssed on the

foreign exchange forward premium.

Bekaert. Hodrick and David (1997) cannot match the dope coefficients and R' esti-

mated from observed data. 5Iy model with t ime varying risk aversion and intertemporal

substitution has more degrees of freedom. I t may be possible by siiitable choice of 2-56

pairs of (a (Gt ) . p (Gi))to match both the dope coefficients and R2 est i m u t ~ l from ob-

served data exact Iy. However. tractability and interpretczt ion are improved if Ive rest rict

the link between the state and the preference parameters. In t his paper. I t rieci the case

in which ût and pt V a r y according to

1 - eh+'? where d = L): + J < E (1.9788.2.05011) . (5.43)

where J;+, and d;+, are the endowment growth rates of the US and Japan from time t

to time t + 1. The funetional form is be chosen to ensure that max (a t ) 5 1: of course

the same condition must hold for pt. The above firnctional fom can generate different

c w e s for different choice of parameters ko and kl. Figure 5.1 gives examples of both

kl > O and k1 < O for some ko > 0.

Given both p and a follow eq(5.43), I tried different combinations of pattern of p

and a, and 1 report my calibration resdts for foreign exchange market. bond market

in the US., bond market in Japan and aggregated market in Tables 5.3. 5.4. 5.5 and

5.6. These results do not match Table 5.1 of the stylized facts reported by Bekaex-t,

Hodrick and David (1997). Even though the cyclical variation of p accompanied by the

cyclical variation of o do help to drive the ,Or, of the foreign exchange market towards

the historical value of -4.09 (the coefficient is 3 times larger than t hat found by Bekaert .

Hodrick and David (1997)). st il1 the resulting level d u e s are too small: the best 1 can

get is -0.11. Another difficulty is that it is hard to improve the regresion 3s in the

four markets simultaneousiy. So rny calibration results are basically consistent Nit h

the Bekaert . Hodrick and David ( 1997) concl~ision t hat simply changing preferences t O

incorporate first order risk aversion does not help to e-xplain the predictability of excess

retiirns. bIy results uphold a stronger statement: incorporatlng first order risk aversion

with a simple pattern for time varying risk aversion and intertemporal stibstitiition does

not help miich eit her. But rny findings do not rule out the possibility that t here coiild

exist a different pattern of time varying p and a such t hat the estimated Js con match the

stylized results given in Table 5.1. Because the computations are very time consiiming.

1 was unable to pursire richer specifications for a (G t ) and p (G,).

Interestingly. I find when 1 m q a o n . I can hardly improve the est imated dope

coefficients. W e n both p and a w q t especially when p is state dependent. t here is a

significant improvement . although as I mentiond. the level value for the 3s are st il1 too

small. For example. when A = 0.25 and when both a and p v a s according to eq(5.43)

with different parameters (ko. ki): the ,ds in the FX markets and -4ggregated Equity

Market, are increased by a factor of 3 tirnes and 10 times respectively. of which alrntist

80% of the improvement is due to the tirne k a t i o n of p. This is consistent witith my

hdings reported in the previous chapters, i.e. time varying intertemporal substitut ion

pla-ys a relative more important role than time ~~g risk aversion in explaining the

equity premium puzzle.

5.7 Conclusion

Bekaert, Hodrick and David (1997) concludes that high levels of risk aversion can explain

on1y a smaii fraction of the predictability of excess returns found in the data. They

ôssert that the predictability of a set of asset market excm returns cannot he fiilly

explainecl simply by m o d i l n g preference assumptions. M y work in an open economy

setting supports their conclusion. 1 find that when 1 introdiice bot h time varying EIS

and CRRA into my two country model. the improved predictability of excess retiirns

is insignificant. But my findings do not mle out the posibility t hat t here coiilrt exist

a different pattern of tirne varying p and û such that the estimated 3s can match the

stylized results given in Table 5.1.

Table 5.1: The stylized facts

variable 1 constant on fp t on f b( on / b r

Dependent

-0.450 (O. 129)

CoeL on Coef. Coef. Coef. R'

sion results from eq(5.1), eq(5.3) and eq(.5.4). The data are monthly obsemtions

rr- , - ij

on quarterly rates. Al1 rates are rneasured as percentage points per annuni. The

21.540 -3.543 O. 139 (~.m) (0.816)

logarithms of the doliar/yen spot and forward exchange rates are denoted .st and

S o t s : This table is c i t d from Bekaert et 4 (L997) Table 4. It reports reges-

fL. hf+ ,2 ( h z l.2) denotes the one-penod continuously compouiided holding pe-

riod return on a twceperiod dollar (yen) discount bond realized a t tirne t + 1.

if ( i l ) denotes the continuously-compounded dollar (yen ) spot interest rate nt

tirne t. fb: ( j b y ) denotes the forward premium in the dollar (yen) bond market.

r E L denotes the one period dollar continuously-compounded return to the awe-

gate wealt h portfolio. The numbers in parentheses are standard errors. which are

heteroskedasticitj--consistent and are corrected for the serial correlation induced by

the overlap in the data using the method of Xewey and West (1987).

Table 5.2: Bekaert. Hodrick and David (2997) Simulation Resiilts

Dependent

variable

3. 4 and 5. The logarithms of the dollar/yen spot and foward exchange rates are

Cod. Coef. Coef. ~2 Paramet ers

on f p t on f b: on f b l

i 3 C+L - t

denoted SI and jt. h:+l + 2 ( h E ,.,) denotes the one-period continuowly conipounded

.A = 0.25 -0.085 0.00042

p = 0.5

holding period return on a tweperiod dollar (yen) discount bond redized ikt tirne

Xotes: This table is citecl irnd summarized frorn Bekiert et al (1997) Table 2.

t + 1 à: ( iy ) denotes the continuously-compounded dollar (yen ) spot interest

rate at tirne t. fb f ( j b l ) denotes the fonvard premiurn in the dollar (yen) bond

market. r,W, denotes the one period dollar continuously-compounded ret urn to the

aggregate weal t h port folio.

cycllcal -3

cycllcal -9

cyclical

Table 5.3: Citlibrot ioii Rcsiilts of t lic /i,, For t lic forcigii cxdiiiiip iiitirkct. mgrcssioi,

4 cycllcal -1 c yclical -1 cyclical -1 cyclical 4 cyclical -1 cyclical -0.0086 -0.01 82 -0.066533 -0.1 19 0.01 49 -0.0307

-0,0088 -0.01 82 -0.0673 -0.120 0.01 5 -0.1052 -0.0066 -0.01 51 -0.055978 -0.094 0.021 5 -0.01 39

-0.0067 -0.01 81 -0.0626 -0.1 16 0.01 0 -0.01 40

Tihle 5.4: Ctililmt.io11 rcsiilts of tlic /il, for tlic tlolliir discoiiiit l~oiitl iiiiirkct rcgrwsioii

0.5 cycllcal -

-0.33 cycllcal

-3 CYCI~CSI

-9 cycllcal

A= 1 -1 cy clical

-0.00003 -0.00055

-0.00003 0.00001

-0.00001 0.00002

0.00000 0.0001 8

A= 0,55 -1 cyclical

-0.001 19 -0.00099

0.00028 -0.00068

0.00090 0.00061

-0,001 14 -0.001 47

A=0,85 -1 cyclical

-0.00003 0.00003

0.00001 -0.00026

O. O0007 -0.00023

0.0001 3 0.00031

A=Oa4 -1 cyclkal

-0.00073 -0.00086

0.00035 0.00301

0.001 17 -0.00308

0.00038 0.001 76

A=0.7 -1 cyclical

0.00038 0.00099

0.00056 0.00028

0.00081 0.0001 5

-0.00052 -0.00055

As0.25 -1 cycllcal

-0.00080 -0.00031

0.00059 0.00063

0.001 28 0.001 28

0.00024 0.001 49

Figure 5.1: Functional Form of 1 - ek0+hLd

This figure shows dinerent curves for different choice of

parameters and kl -

Chapter 6

Four States Closed Economy

Wly do state-dependent preferences work su well in the closed economy model of Chapter

4 but so poorly in the open econorny model of 5 :' There are many factors t h ~ t codd be

blamed. The open economy mode1 has more =sets. and more goods. It also has a richer

state space. The exogenous processes are approximated by ti first-order '1Iarkov chain in

which each variable can take four possible valries insteaci of t w as ir i Chapter 4. Here

1 investigate a closed economy without nominal nioney. virtually identical in setup to

that of Chapter 4. but ailon: the exogenous processes to foIlow a first-order Markov chain

in which consumption growth can take four possible d u e s . The representative agent

has recursive preferences with an expected utility certainty eqiiivalent. This economy

extends my Chapter -4 results by using four states of the world instead of two. On the

other hand. it is a special case of the open economy mode1 of Chapter 5 with .4 = 1.

6 = land >u; = O. I want to investigate whether 1 cari match the JIehra and Prescott

( 1985) est imates in t his closed economy environment.

The FOC of this representative agent -s problem is as follows

CHAPTER 6. FOUR STATES CLOSED ECONOMY

where the intertemporal marginal rate of substitution k given by

and

where gt+l = q+,/ct is the consumption-wealth ratio. The market portfolio return.

rt+ 1. in t his economy sat isfies

which can be so1ved from eq(6.1) .

The risk Free rate c m be cdculated from eq(6.2)

Because 4 = t$ = q. eq(5.18) implies wealth evolves as follows

6.1 Equilibrium

Using the equilibrium condition of eq(4.12) . i.e. al = 1/ (1 + Pt) . the FOC conditions

are the sarne z s t hose in Chapter 4.

6.2 Calibration

Mode1 calibration requires that 1 solve the FOCS provided by eq(6.8) and eq(6.9) for

parameters (P,. rji. ai. pi. i E (1.2.3. -1) ) that lie in the admissible set { P , > O. a, 5 1.

pi 5 1. '1, > O, Vi = 1.2.3.4}. and t hat deliver the Slehra and Prescott (1985) estinlates:

E ( M ) = 1.07 aiid s (Bf) = 0.165, as well as E (rj) = 1.008 and s (rj) = 0.056'. For the

exogenotis process of consiiinptioii growth gt. instead of using the '1Iehra and Prcscott

(1985) estimates, 1 chose to calibrate to the US. constimptiori data given in Chapter

5.5. The exogenous process gt is approxirnated by a first order 1Iarkov chain in which

gt c m take four possible valiies. The 5Iarkov chain is calibrated to the estirriateci first

order VAR iising the Gaiissian quadrature method of Tauchen and Hussey (1991). The

state values. state probabilities and transition mntrku are given in Table 6.l.Xote that

the valiies in Table 6.1 imply the mean and standard clevintion of cliiarterly consiimpt ion

growt h are E (9) = 1 .O055 and a (g) = 0.0054.

Given consiimption growth takes on four values. 1 have to solve for adniissible ( P, . rj, a,. p,. where i E { 1.2.3. -4) } from the following equat ion system '.

'These are annual vaiues: 1 convert to quarterly vaiues in my catculation by dividing net returns and volatility by 4.

2Eq(6. 10) and 4 6 . 1 1) have the same form as Chapter 4 eq(C.30) and eq(C.3 1 ) except the number of states is four.

where B = 0.96°.25: q,. ir,, and g, are given in Table 6.1. Because there are 16 unknowns

but only 12 equations. there are many solutions. Theoretically. given any choice of ad-

missible price-dividend ratios P,'. one can solve for the rernaining 12 iinknowris h m

these 13 equations. In practice. the constraints on O , and p, excludes most of the solu-

tions. (Only for certain choice of admissible price dividend ratios Pt. can 1 get aclmissible

solutions on the paramet ers {r,, . a,. p, . where i E { 1.2.3.4) }. ) 1 present a soliition in

the Table 6.2.

IVith the Table 6.3 solution. the 5lehra and Prescott (1985) estimates of the average

equity premium and risk free rate can be matched exactly. The solut ion also implies a

coitriter cyclical CRR-4 and procyc1icd I.11 RS. This calibrat ion resiilt agrees wit h my

Chapter 1 concliision: i.e. to some extent. it supports the conclusion of Campbell and

Cochrane (1999) that a time varying coii~itercyclicai CRR.4 pltys an role in expliiining

the equity premium puzzle. but contran; to the Campbell and Cochrane ( 1999). it cloes

not play the major role. .A rnoderate time varying procyclical E I S combineci nith e v e n

strong co~intercyclical risk aversion leads to better enipirical siiccess. Further. it shows

that the conclusions of Chapter 4 are robiwt to a slightly richer specification for the stete

space.

Sote. the consumption data iised in Table 6.2 are collected according to the d e x r i p

tion in Bekaert. Hodrick and David (1997). Professor Bekaert also prot-ided me Rith

their V.ilR(1) results. fiom which. 1 can back out the marginal consumption g o w t h and

the transition matrix as in Table 6.3.

Given Table 6.3 and the choice of admissible price ciividend ratios Pa, t he correspond-

ing solution for the rest of the parameters {rl,. a,. & where i E { 1.2.3.4) } is presented

in Table 6.4. The results are similar to those reported in Table 6.2.

%ot aii choices of the price-dividend ratios can be rationalizeù because some will generate arbitrage apport uni ties.

6.3 Conclusion

In this Chapter, 1 investigated a closed econorny environment similar to Chapter 4.

that differed in the number of states of consumption growth- The mode1 can solve the

Mehra and Prescott (1985) equitÿ premium puzzle exactlÿ and a solution is obtained

nith a counter cÿclical CRRA and a slightly procyclical IMRS. This cdibration resiilt

agrees with my Chapter 4 conclusion: Le. to some extent. it supports the concliision of

Campbell and Cochrane (1999) that a time varying countercyclical CRR-4 plnys an role

in mplainitig the equity premium piizzle. but contra- to Campbell and Cochrane (1999).

it cioes not play the major role. X moderate tinie varying procyclical EIS combiried with

a very strong countercyclical risk aversion leads to better empiricai sticcess. Fiirther. it

shows that the conclusions of 4 are robiist tu a slightly richer specificntion for the stnte

space. Therdore further investigation of the difficiilties reported in Chapter 5 c m hais

on nominal money and trade.

Table 6.2: Calibration Resiilts

Solution

Table 6.1: Discrete Endowment Growt h

Table 6.3: Discrete Enclowrnent Growt h Back out €rom Bekaert VV4R( 1)

State Value

State Value State Probability

State Probability

Table 6.1: Calibration Result from Bekaert VA4R( 1) Parameters

Transition Sfatrix

Solut ion

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Appendix A

Cyclical CRRA and Constant EIS

A.1 First Order Conditions

Here 1 solve for the first order conditions of the following problcni given ari cspectctl

certainty eqiiivalent fiinctional form (Case 1) ancf Yaari certairity eqiiivdent fiinc-

tional form (Case 2). The agent's objective is to choose fe<asible consiirnption and

portfolio shares to rn~~uimize titility. The Bellman eqiiation for this problem is

J ( x J J = mau ((1 - 3) cf + 3 p ( J ( r t + l . ~ t + l ) ) p ) ' . (A. 1 ) {ct . w }

Because in Appendk C I hâve solved a more general case wït h bot h EIS and C RRA

are state dependent. the FOC can be simply taken from t here by set t ing p (st ) = p

(a constant) and applying budget comtra.int eq(2.4) .

APPENDIX A. CYCLICAL CRRA AND CONSTANT EIS 144

A.1.1 Case 1: Expected UtiIity for Certainty Equivalent

ahere the intertemporal marginal rate substit~ition from time t to t + 1 (1.U RSt.t_l)

is taken from eq(4.6)

A. 1.2 Case 2: Yaari Preferences for Certainty Equivalent

where the intertemporal marginal rate substitut ion from t ime t to t + I (131 RStSt, )

is taken Erom eq(4.10)

A.2 Numerical Solution

A.2.1 Case 1: Expected Utility for Certainty Equivalent

Rewrite eq(2.27) and eq(2.18)

APPENDIX A. CYCLICAL CRRA AND CONSTANT EIS

where i E { I . h ) .

hIultiple ri/-\Ah to eq(.4.9) and siibtract eq(.4.10) gives

where

(A. 12)

(.A.l3)

(A. 14)

(A. 16)

(A. 17)

APPENDIX A. CYCLICAL C R U AND CONSTANT E r s

where

w here


where

where

and


Equate eq(.-1. 18) to eq(A.27)

where

Equation eq(A.32) and eq(A.37) implies

Solve for p, we have


where

A.2.2 Case 2: Yaari Preferences for Certainty Equivalent

Renrite eq(2.30) and eq(2.31)

From eq(.4.-ll). we have

1-' 1 - 1 , 1-' 1 1 - 1 1

b3!gh ' JG . ~ ~ ~ h f rZdô (g[ p-x - gh

Equivalent ly

1 1-1 1

-3 , 1 - J.gh

" 11 - 1-I 1 1 - I 1

3: (, Mt; - g,, hll;)


From eq(A.42), we have

Equate eq(.4.U) to eq(A.47) @ves

rearrange to get


The above eqiiation mitst hold for both i = I and i = h. that implies

APPENDXX A. CYCLICAL C R m AND CONSTANT Ers

or

w here

I solve p from eq(.-l.58). and solve 3 from eq(A.53) . it follows 7, c m be solved froin

eq(A.45).

A.3 Approximate Solution

Here 1 describe the approximate solution when Y& preferences is taken for cer-

t ainty quivalent. From ey(A.4 1). the 10%- and high economic stat e gives

From eq(A.60), we have

(A. 6 1 )

solve for n/l gives

Side note:

From eq(A.59) . ive have

Side note:

Soh-e for '71, @es

Appendix B

Cyclical EIS and Constant CRRA

B.l First Order Conditions

Here 1 derive the FOCS of the mnvimization problem given in eq(3.1) sitbjcct to

eq(3.2) . Because in Chapter 4. 1 have solved a niore general case with both EIS and

CRRA are s tate depen~ient. the FOC can be simply taken froni there by setting

cl ( s t ) = a (a constant) in the c ~ s e of Expected Utility for certainty eqiiivdent:

or by set ting :, (s,) = y (a constant) in the case of Yaari preferences for certainty

equivalent .

B. 2 Numerical Solut ion

Here I derive the numerical solution for the case of t ime --hg E IS Fcit h constant

CRRA and prove the o b s e d US historical data can not be matched exactly by

any admissible combination (PI a, pi: ph) for the case of E-upected Utility for certainty

equivalent : or (3.7: pi. f i ) for the case of Yaari preferences for certainty equivalent .

APPENDIX B. CYCLICAL EIS AND CONSTANT CRRA

B.2.1 Case 1: Expected Utility for Certainty Equivalent

In equilibiium. the FOCS for optimal consumption choice and portfoIio choice are

taken from eq(C.30) and eq(C.31) by setting a (st ) = a:

% [(1 - 3 ) ( 1 +P,)]g x 5 3 $' = 1 j = { l . h ) [(l - 3) pl]:

C ru$ ï~ [(l- 3) (1 + <)le r ' 1 j = { l . h ) [(1-J)P,]? 'JII.]

where i E { l . h } .

Multiple r,/.\Ill to eq(B. 1) and subtract eq(B.2) gives

% [(1 - 3)(1 + ph)]: I L - 1 7ilh.3 & = .\Cd

[(l - 4 RI (* - k) If i = 1. we have

9 [(l - J ) (1 + p h ) ] &

O g r = - 4 l h - [(1 - 3 ) 41"

where .Alh is given in eq(.4.24) . Take nature log on bot h side @es

If i = h. we have

APPENDIX B. CYCLICAL EIS AND CONSTANT C R m

where Ahh is giwn in eq(A.28). Take nature log on both side, we have

hlultiple r,/rlhh to eq(B.l) and subtract eq(B.2) gives

* [(1 - 3) (1 + R ) ] E r ri 7TIlJ - = - - 1.

[(1 - 3) P,]Z -v h

n-here .4il is gven in eq(,-l.l5) . Take nature log on both sirie. WC have

where -Ahl is given in eq(A.19). Take nature log on both side. we have

(B. 10)

(B. 11)

(B. 12)

(B. 13)

Use eq(B.6) subtract eq(B.8): we have

APPENDIX B. CYCLICAL EIS A N D CONSTANT CRRA

Use eq(B.14) subtract eq(B.12): we have

Eq(B. 15) and eq( B. 16) together implies

(B. 15)

(B. 16)

(B. 17)

(B. 18)

1 - I h 1 d rL 1 - i h (B. 19)

Sirice the risk fÎee rate ((ri' rh) E 0) irnplied hom observed US historical data does

not satisfjr eq(B.22), it implies that there does not exkt an admissible solution that

can solve for the FOC equations eq(B.l) and eq(B.2).

APPENDIX B. CYCLICAL EIS AND CONSTANT CRRA 158

B.2.2 Case 2: Yaari Preferences for Certainty Equivalent

In equilibrium, the FOCS for optimal consuniption choice and portfolio choice are

taken from eq(C.11) and eq(C.42) by setting a ( sJ = a:

Multipiy on bot h side of eq(B.24) . and siibtract eq(B.2-l) from eq( B.23) gives

rearrange to get :

- 1 - 3 l + f i rt - .&,

h 4 r, ( M c 1 - J I t h t E {i. h } . (B.36)

(1-3) (1+4) Pl

take nature log on both side

solve for y :



solw for pi we get

Mtiltiply on both side of eq(B.14) . and subtrnct g( 8-24) from eq(B.2:3) gives

APPEND~X B. CYCLICAL EIS AND CONSTANT CRRA


rl - .vil ln [(1 - 3 ) (1 + Ph)]+ln

Pt Pf Ph rl ( -\ 4 h - -\ 111 )

solve for ph we get

take nat lire log on both side

sohe for -y :

Eq(B.35) subtract eq(B.27)gim

APPENDIX B. CYCLICAL E r s AND CONSTANT C R U

Eq(B.30) subtract eq(2.24) gives

1 rh '.\lhh 4 1+Ph 1 ( J l h l -+\th h J ln - = - In [(1 - Li) (1 + Ph)]-ln ---

1 + P i pi ( 1 3 (1 + ] + rh-. \ Ihi .

1 - Ph ( Jthh -.\!hl J

Eq(B.41) plus eq(B.42) gives

Eq(B.43) irnplies

where

Since there is no admissible -/ that satisfies eq(L3.U) to rat ionalize the set C3. t herc

fore it implies there is no exact match For the obsemd US historical data.

B.3 Approximate Solution

B.3.1 Case 1: Expected Utility for Certainty Equivalent

The approximate solution for this case is given by choosing f i and Ph fiom the

optimal consumption choice first order condition eq(B. 1) for any given admissible

ût pl and ph: to grantee the match of the first two moments of the observeci US

consumption growth data. It foliows to calcuiate rïsk free rate Fiom optimal portfolio

choice eq(B.2). Rewrite eq(B. 1)


solve Ph as a funct ion of f i .

Substitiite (B.48) into eq(B.49). 1 can get a non-linear eqtiation for 9. SoIve for Pl

from t his non linear equation t hen pliig it in eq(B.48) . 1 cnn get Ph. It hlls t hotigh

to solve risk free rate fiom eq(B.2)

B.3.2 Case 2: Yaari Preferences for Certaine Equivalent

The approximate solution for this case is a little bit different h m the one defined

in Case I of this section. Here. 1 choose pl and ph to mat ch the 6rst and second

moments of the US equity return data and the first and second moments of the US

consumption growt h data exactly, and see if the calculateci first and second moments

of my nsk free rate can get ciose to the observeci US risk free rate data. Rewrite

optimal consurnpt ion choice condit ion eq( B.23)

Therefore pliig eq(B.54) in eq(B.56) . 1 c m get a non linear fiinction for pi. Solve pl

frorn this non linear fiinction and plug in eq(B.54) . 1 can solve for ph. It follows to

solve risk free rate from optimal portfolio choice condition g(B .34 )

Appendix C

Cyclical CRRA and EIS

C.1 First Order Condition

Here 1 solve for the first order conditions of the follotving problerri given an expected

cert ainty eqiiivalent fiinct ional form (Case 1) and Ynari cennint y quivalent fiinc-

tional form (Case 2): for fiirt her details. plecase refer to Epstein ( 1988). The agent's

objective is t O choose felisibIe consumpt ion and port folio shares t O nii~~irriize rit ility

The Bellrnan eqiiation for t his problem is

The homogeneity of J (rtJt) and the linearity of xt+l on (xt .c t ) . by eq(C.2) irnplies

that the solution bas the form:

APPENDIX C. CYCLICAL CRRA AND EIS

where A (II) denotes an It-messurable random variable.

Subst ituting eq(C.3)hto eq(C. 1) gives:

This méuiniization problem can be decomposed as the folloning two problems:

Consirmption is chosen by:

a Portfolio choice can be described bu:

Ciise 1 : (Expected Utility for Cert ainty Eqiiivalmt )

(C. 5)

CCue 2: (Yaari Preferences for Cert ainty Equinlent )

where L = (1 : 1: .. . .1)' and El is the conditional expect ation operator defined in

eq(2.15) .

ive first solve the consumption problem for a r b i t r a i pi . Again. the homogeneity of

eq(C.4) implies t hat the optimal consumption policy can be n-ritten as

c; = a(It)xt (C. 7)

where a ( A ) denotes an Il-rneasurable random variable. For convenience we use a, =

o (4) below. Note that at can be thought of as the consumption wealth ratio.


Substitute eq(C.7) into eq(C.4):

The FOC of this equation is:

These latter two equations combine to yield:

Thus:

(C. 10)

Equation eq(C.9) shows that the optimal consumption tvenlt h ratio is an invertible

function of A ( I t ) . W e exploit this feature in our solution of the portfolio choice

problem below.

C. 1.1 Case 1: Expected Utility for Certainty Equivalent

Now subst it ute eq(C. 10) into eq(C.5); the port folio choice problem becomes

(C. 11)

Substituting eq(C.11) into eq(C.8) , we get

APPENDIX C. CYCLICAL CRRA AND EïS

where ~ \ f ~ , ~ + ~ = uv;rt+i = x:=, ~ f : ~ r ~ . ~ + ~ is the gros retiim of the optimal portfolio

u*; . Eq(C. 12) can b e rearranged t O get :

Pt * t ' - 1 p ( * t + , ) - ' P I , * t '

0 f . r t , - 4 * t - 1

f t = 1. (C. 13) 1 Yote that eq(C. 13) can be wi t t en iri the t o m

w here

(C. 14)

(C. 1.5)

Note. the budget constraint eq(C.2) implies xt+l = ( x - ) M . + =

( 1 - at ) Mt.t+lrt: therefore at+l = = gt+lnt/ [(l - a,) 3&.t+& Using this

identity. we can rewrite eq(C.15) into a varietj- of equivalent expressions.

The FOC of eq(C.11) with respect to w, gives

(C. 16)

Multiple by w, for dl i, and sum over i = 1, .... n. we get:

Selecting ~,J+I to be the risk Eree rate. r j t . which is tirne t memurable (note n-e do

not use the notation r ~ , , + l ). eq(C.17) can also be wx-itten as

by iising cq(C. 14) . we get :

Et ( I . U R S ~ ~ + ~ ~ ~ ~ ) = 1

nhere 1.11 RS:;, is given by eq(C. 15) .

C. 1.2 Case 2: Yaari Preferences for Certainty Equivalent

Siibsti tiite g ( C . 10) into eq(C.6): the port folio choice problem hecornes

Substitute eq(C.19) in eq(C.8) . we can get

(C. 19)

where Mt.t+i = = ~ ; ~ r , , ~ + ~ is the goss return of the optimal portfolio

Eq(C.20) can be remangeci to give

APPENDIX C. CYCLICAL CRRA AND E r s

where:

Alternatively . iising the iclentity nt+l = q+l/rt+l = gtAiar/ [(l - n t ) .Ut.t,l]. we can

rewrite eq(C.22) into a tcariety of eqirident expressions.

The FOC of eq(C.19) with respect to a, gives

Slultiple by u?, for al1 i. and siim owr i = 1. .... n. we get:

( 1 4 . * t - 1 ) - '

O ( + < - 1 ) 0 ( - * * - 1 )

E; ( 1 - 3 ) Qtcl ( - K . c + I -r, .t+d = O i Selecting r,,+l to be the risk hee rate. r,,. which is tinie t measitrable (riote we do

not use the notation r ~ - ~ + ). eq(C.24) can also be written as

by uçing eq(C.21) . we get:


C. 2 Numerical Solut ion

C.2.1 Case 1: Expected Utility for Certainty Equivalent

In equilibrium. given two st ates 1 and h. eq(C. l-l) and eq(C. 18) can be w i t ten as:

where i . J E ( 1 . h } and the intertempord marginal rate s~ibstitii tiori froni

to time t + 1 = j (I.IIRSt,J) is giveii by eq(4.13)

Thils eq(C.27) and eq(C.28) can be wit ten as

C ïiij/9 9,"' = 1. j= ( l .h) [(I - d ) pi]%

(C.21)

(C.%)

time t = 1

(C.29)

where i E ( I . h } .

Divide eq(C.30) by ~1.1,~ and subtract eq(C.31) to obtain

APPENDIX C. CYCLICAL CRRA AND EIS 172

If i = 1. then eq(C.32) implies

If i = h. then eq(C.32) implies

Divide eq(C.30) by -\far and subtract eq(C.31) to obtain

If i = I . then eq(C.35) irnplies

If i = h.then eq(C.35) implies

Given pi. irnply al from eq(C.33). thus (pl .nl) irnplies p h from eq(C.36) : ph irnplies

ah from eq(C.37). it follows t hat (ah. ph) implies pi from either eq(C.3-l) . .A solutions

to the equabion syst+em eq(C.30) bo eq(C.35) are pt.9 such rhat pt = pi.

C. 2.2 Case 2: Yaari Preferences for Cert ainty Equivalent

In equilibrium, given two states 2 and h. eq(C.14) and eq(C.18) can be written as:

APPENDIX C. CYCLICAL CRRA AND EIS 1 74

where i. j E {i. h ) and the intertemporal marginal rate substitution from time t = i

to time t + 1 = j ( I L ~ I R S ~ , ~ ) is given by ~ ( 4 . 1 4 )

Equivalentlv eq(C.38) and eq(C.39) can be written as

SIiiltipIy on both side of eq(C.41) . and sitbtract eq(C.42) from eq(C.41) givcs

rearrange eq(C.43) to get:

solve for 71:

APPEND~X C. CYCLICAL CRRA AND EIS

solve for pl we get

.Ilitltipty -\Ill on both side of eq(C.12). ancl siibtrztct eq(C.42) froni eq(C.41) gives

(C.49)

resrrange eq(C.49) to get:

- 1-3 Pt 1 + Ph rl - .\Ill

-7t , i E { l . h ) (1 - .JI* (1 + ph)& p,rt (-\[th --\Id)*

solve for ph we get


solve For 7 h :

Given pi. imply -/l from eq(C.46). t hus (m. 71) implies ph From eq(C.52): ph iniplies

11, from eq(C.55). it follows t hat ( T ~ . p h ) implies pi from eq(C.4X) . A sollit ions to

the equation -item eq(C.41) and eq(C.42) are pis siich that pi = pi.

C.3 Approximate Solution

We Iook for the approximate solution only when oiir calibration resiilts blls in t h e

inadmissible area in eitlier expected utility certainty eqiiivalerit case or Ymri cer-

tainty ecpivalent case given in Append~u C.2. Approxirnate soliit ions st ated here

only calibrate the first and second moments of eqiiity ret urn and chose input param-

eters pl and ph ta minimize the deviation of the first and second moments of risk

free rates (and therefore risk premium) From obserwd US historical values.

C.3.1 Case 1: Approximate Solution with Expected Utility

for Certainty Equivalent

l i e have noticed that when the transition matrix II shift, out solution may faIls in

the inadmissible area. In this case, 1 will look for the second best solution and see

APPENDIX C. CYCLICAL C R m AND Ers 177

if there always exists such a solution that dominates the constant I M R S solutions:

such as solution Unplied by Weil (1989). From eq(C.30). we have

w here:

Given p,. n, c m be solved from non-linear eqiiation system eq(C.56) . it follows t hat

we can solve for the risk free rate from eq(C.31) (and t lierelore eqriity premiiini)

C.3.2 Case 2: Approximate Solution wit h Yaari Preferences

for Certainty Equivalent

From eq(B.23) we have

which impiies

solve for yi we have

Use y, solved above. we can solve for risk free rate r, from eq(B.34) . tha t is

Appendix D

Two Country Model

D.1 First Order Condition

Here i solve for the first order condition of t he following problextr givcn expecteci cer-

t ainty eqiiivalent fitnct ional form. for flirt her details. plewe refer to Epstein ( 1'388).

The representative agent solves the mcrvimization problem given in eq(5.5) anci

eq(5.6) subject to eq(5.16)for optirnnlly chosen (<.cf. Mt,,. .\I:&,. z,,~) . The h+

mogeniarity of the preferences implies that 6. cf . .\IL,. .\fi,, . :,+ 1 and L' a,e ho-

mogeneous in -4. It follows that they can be mit ten as

where 5: @, m f , mp, z, and zl are functions defined on the space of information

vectors It .

Let A denote the amount of wealth the representative agent can invest in the two

currencies and the n capital assets. after pwchasing consuniptions $ and and

paying al1 transactions costs:

for convenience let

so and 2- are the real ~moiints invest eci in the ttvo currencies. iincl roet, i and

r- l .t+ are the real retums excluding transaction cost swings from t hese invest rnents.

The budget constraint. eq(5.16). implies

define portfolio weights

(D. 12)

(D. 13)

Using the above equations, the value h c t i o n can be written:

Let ut; denote opt imally chosen port folio weights. t hen eq( D. 13) irnplies

(D. 1.5)

( s, l=y S J q - s, P; *ir: u!: + l .At - ( c f + uf ) - - (<+$)) = - - - -- ( D . 17)

Pt" pi= F;= P?

Accorcling to eq( D. 16) and eq(D. 17). given opt imally chosen port folio weights

{ "~+ l ) l - , . the choice of cf and determines .\ I:+, / P( and .\If, ,/ P:. So 1 can

write:

(0.18)

(D. 19)

(For notational convenience: the dependence of rn' and r n Y on {u;,+,)L -_ is not

explicitly stated.) The first order conditions of eq(D. 14) Mth respect to < is

where

and

a At arnu st PL - = - (1 + cf

Similarly. the first order conditions with respect to cf is

where

By differentiat ing eq( D. 16) and eq(D. 17) . the partial derivat ives of rn' anci mY \vit h

respect to t? and P can be computed. LVhen these are stibstittited into eq(D.20)

and eq(D.23). one obtains:

and

For j = r. y. let &it denote the derivative of (LJ (4. mi) with respect to the kth

argument. Euleras theorem for linearly homogeneous functions implies that

APPENDIX D. TWO COUNTRY ~ ~ O D E L

and

Using eq(D.27) and eq(D.28) and eq(D.25) and eq(D.26) in first order conditions

eq(D.20) and eq( D.23) . one obtains the follottlng:

and

where -4; denotes At emliiiit ed at the opt irnall-chosen valiies for consiirnpt ion.

money. and asset holdings. Eq( D.29) and eq(D.30) imply t hat the marginal trnns-

action cost adjiisted marginal utilities of the two goocls are eqiiatd:

Eq(D.3 1) implies

which characterizes the equilibrium exchange rate. as a function of equilibrium

prices. consumptions, and money holdings.

Substituting eq(D.29) into eq(D.14) one obtains

Using eq(D. 1) and eq(D.2) and th^ fact t hat uf and are hotnogeneous functions

of degree zero. one obtains

By substituting eq(D.31) into eq( D.34) one obtains

Eq(D.35) gives a closed forni expression for the ~ d t i e frinction in terms of eqiiiiibriiim

consrtmption. nioney holdings. and mal t h.

1 non. continue m i t h the derivation of the first order conditions wit h respect to the

portfolio weights rn,,+l. i = -1. O.. ..... n. Given optimal consumptions cf' and 6'.

the uector of portfolio aeights that maximize the value fiinction m u t mitximize

(D.36)

Using eq(5.15) and eq(D.6). mauimization probiem in eq(D.36) is quivalent to

ri)

+ y -. L Q G C

111 111 III III

APPENDIX D. TWO COUNTRY MODEL

Xotice t hat

and

Define the retum on optirnally invested aggregate wedth as

Finally. let

so from eq(D.36)

Using this new notation. eq(D.42) cm bbe written

Now 1 need to solve for v ( I t + J and p: in terms of observables.. I turn first to p;.

Eq(D.51)- eq(D.30) and eq(D.31) Mpiy

Eq(D.53) can be simplifieci by using Euler's theorern. Euler's theorem implies

Substitute eq(D.5-l) and eq(D.7) into eq(D.43).

Together. eq(D.55) and eq( D.56) irnply

1 i p = -4, - - ( I + u,.;,,c: ri

Substitiite eq(D.59) into eq(D.53) to get the following e'tpression for p( :

Next- a ( I t + I ) is given by eq(D.45)

APPENDIX D. TWO COUNTRY MODEL

Eq( DA$), eq(D.58) and eq(D.59) together imply

where g;+, = c;+,/c;. i = r. y.

Using eq(D.59) and eq(D.60) in eq(D.52) I obtain the following restriction on the

difference between any two retiirns:

5liiit iple bot h side by

The resiilting expression can be Fvritten as

Another equilibrium condition is given as follows by using eq(D.50) , eq(D.36) and

eq(5.7) as well as eq(D.43) and eq(D.59)

Substittite eq(D.58) into eq(D.63) and use eq(5.7). 1 get:

Using ecl(D.49) and eq(5.T). the above can b e simplifieci to

Using definition eq(D.62). the above c m bc n-ritten as

Eq(D.65) and eq(5.6) imply

Et [o (Zt+l)] = 0-

Eq( D.6 1) and eq(D.66) are the equilibrium condit ions.

D.2 Solution Procedure

1 numerically soIve the Euler equations eq(D.61) and eq(D.66) for the endogenous

variables t$ and $ and rt+l. I use a finite state Markov chah to approximate the

exogenous driving process as in Tauchen and Hiwey (1991). and I solve the mode1

exactly for this approximate driving process. Here 1 describe the solution procedtire

in some detail.

Let e; denote the total output of good x at date t. let ef denote the out put of good y

a t time t . and let :\If+, and :\If+, denote the supplies of dollars and yen respect ively.

clvailable for use in mediat ing transactions at t ime t. (These money stocks are dateci

t + 1 becaiise it is w u m e d that the Ioss in value from inflation accrties to the agent

in t + 1.) Let Gt denote the vector of growth rates of outpiits and nioney supplies

in the two coiintries:

ex e l A I L l .\ILl G . - - Y . 7 1 e;', e t f J I ,

It is assumed t hat {e;. ey . .if;,, . JI!+ } is an exogenoiis proces nhose latv of mot ion

is knom.

In equilibrium. the output of each good must either be consumed or useci as trans-

action costs. iising eq(5.12) and eq(5.13)

.Usa in eqiiilibrium. eq(5.16) holds with eqitality. Define 4. j = x. y by

1 can write consumption growths. marginal transaction costs. and inflation rates as

functions of {Gt. u;. vy. rt+i }. Since

where

From eq(5.12) and eq(5.13). 1 have

and from the definit ion of ? j = x. y

PL1 - - - -- 1 4 Pt' .f cy,, &-

where

Thus using eq(D.53). eq(D.75) and eq(D.76). the eq(D.40) and eq(D.41) c m be

expressed as

or equeverently

and the norniinal risk free continuously compotinded clolIar and yen interest rate

eq(5.27) and eq(5.28) are

i: = In 1

1 + A - ( 1 - V ) - [~f]" if = In

1

l + C . ( ~ - ~ ) . [ ~ i Y I C '

Using eq(D.32). eq(D.75) and eq(D.76), 1 have

In a single good nonmonetaq mode1 descnbed in previous chapters, the market

return can be expresseci as a h c t i o n of the consumption wealth ratio and the

growth rate of consuniption. It is convenient to express rt+l in a siniilar way. To do

so. define Et m At - Ar, and let consurnption wealth ratio ët /At be denot ed as al .

Eq( D .49) t hen irn plies

-qt+[ = (.qt - C t ) - rt+l.

Simple manipidation of the above giws

Eq(D.57) implies

t htis

which iniplies the market return

Also eq(D.87) implies both

and

Substitute eq(D.91) and eq(D.92) into eq(D.62) and simplify to get

where g; = cl,+,/c~. i = X . y.

Given eq(5.8). eq(D.66) and eq(D.61) can be written as

P t > = 1 (D.94)

IIt) = O. i . j = s ..y. l..... n (D.95)

where intertemporal marginal rate substitut ion is given by

atS-1 z-1

- - 1.4 ( Z ) . , 1 1.u,11'-~'" 1 + IL$ D f - 1 2-1 (D -96)

W . 4 (Zt+l) V t ) a ut':;' (1 - a r ) 1 + uf''+~

By multiplying uf to eq(D.95) and sum over jand use eq(D.94). 1 can get

By using eq(D.69) -eq(D.ï6) and eq(D.88) in eq(D.93). eq(D.40). eq(D.41) and

eq( D.86). 1 can write the endogenous processes r,,,,, . r . Z and r,,, as a

hnction of {Gt+,. u:. v;,,. u!. cf+,. a, . and a t , , } . It follows that the tiiree ecpation

system consisting of eq(D.94) and eq(D.97) with i = x and i = y. c m be expressecl

Y Y in t ems of {G,+, . L:. u:+~. . c ~ + ~ . ut. nt, }. More specifically.

Let these three eqiiation system be denoted as

where f is a know Function. my next step is to find a stochastic process { c f . r). n t }

which satisfies eq(D.98) for the given G1 process. .As in Ti~uchen and Hu- (1991).

1 approximate Cc by a finite state Slarkov chah usuig Gaussian quadrature.

Appendix E

Prediction of the Yield to

Maturity of a Long-Term

Par-Bond

A one-yenr tliscoiint boncl price at time t : B: can be priced by iising

where I.\IRS is one-year discount factor derived from Appendk -4.

If we plug in the calibration results in this equation. we get one-year bond price at

time t. and t herefore the risk free rate observeci at time t which is exactly the risk

free rate we used to calibrate our mode1 in Section 4.3. Lve c m use our calibration

results to predict the price B!") and yield to maturity Y;(,("' of a n-year discount

bond. or e,uplicitly,

and the ykld to maturity en' can be soived from equation

APPENDIX E. PWDICTION OF THE BOND YIELD 197

where m is compounding frequency and TF is the tirne factor from valriation time

t to the bond matiirity time T.

The above two equations can be written in matriv form:

w here

APPENDIX E. PREDICTION OF THE BOND YIELD 198

I = (1,1)= and IiI1RSij = IhlRSY if Espected utility certainty equivalent is useci;

I11.I RSij = IM RS? if Yaari certainty equivalent is used.

A par-coupon bond p-s a certain coupon amount c' at each coupon p a v e n t date

given state of economy iso at the time of issue. i E 11. h }. The coupon amoiiiit is just

big enough to make the bond prke value at par. i.e. pin' = bond notionxl value.

An n-period par coupon boncl sat isfies the following condit ion:

E~ ( I M R S ~ . ~ , ~ * (f l f ; ' ) + c ) ) . if 1 < n: p/"' = *\i = {

In oiir two stcltes economy. the above eqiiation c m be written as

or in mirtrix form:

where

c = c' or c! depend on current states 1 or h.

Equation (E.6) can be expandeci to deliver

APPENDIX E. PREDICTION OF THE BOND Y~ELD

For different cases of current states, c takes the form

and

Appendix F

Robust Test Results

Hcre. 1 provicie robitst test res~ilts in Table F.l and F.2. They are the rcsiilts corre-

sponding to the transition probability mat n'c

Table F. 1 is the robiist test resiilts with expected iitility certainty eqiii\xlerit. Table

F.2 is the robust test results with Yaari certainty equivalent.

In each table, considering the columns,

-3' column corresponds to the subjective discount factor 3 of the represent agent:

-E (rf)' and 's ( r , ) columns contain the average risk free rate and the standard

deviation of the risk free rate respectively;

'rL' and ' rH' columns represent the risk Eree rate rl and Th respectivel_v. which are

Mplied by the input pair (E ( rJ) . s ( r f) ):

' E (9)' and 's (g)' columns contain the average consumption growth rate and the

standard deviat ion of consumpt ion growth respec tively;

'gL' and 'gH' represent the consumption growth rate gr and gh respectively. which

are implied by the input pair ( E ( g ) . s (g)) . ' E ( M ) ' and .s ( A l ) ' columns contain the average equity return and standard devi-

at ion of eqiiity return respectively:

' f i * and 'Ph' columns contain price dividend ratio when economy realize a low and

high corisumption growth rate respectively. They are irnplied by the input pair

(E (-11) . s (-11)):

'pl' and ' p h ' coliimns contain the calculated parameters pl and ph respectively:

.nl' and *ah' coliimns contain the calciileted parameters al aiid crh respectively:

*y1 ' and ' coliimns cont ain the calciilated y1 and -!,, respect ively:

.CL' cmd .CH* coliimns contain the predicted IO-j-ear par-bond yield giveri toclay's

state 1 or h respectively:

'YL' and 'YU' columns contain the predicted field-tematiirity of a 1 0 - ~ a r ciiscotint

bond:

The robiist test is carried out by shifting 3. s ( r f ). E ( r ! ) E (9). s (9) ta compare the

calibratecl results with the ones calctilated fiotn the benchmark valiies: 3 = 0.98.

s ( r f ) = 0.056. E ( r f ) = 1.008. E (9) = 1.018. s (9) = 0.036: E (JI) = 1.07. and

s ( M ) = 0.165.

The rows of each table are divided into five sections corresponding to each shifted

variable. 3 section represents the shift of 3 from its benchmark value 0.98: s ( r J )

section represents the shift of s ( r f ) from its benchmark value 0.056: E ( r f ) section

represent the shift of E (r,) Erom its benchmark vaiue 1.008: E ( g ) section represent

the shift of E (g ) kom its benchmark value 1.018: s ( g ) represent the shift of s ( g )

from its benchmark d u e 0.036 (The values of these shifted variable are in bold

face).

Frorn Table F. 1 and F.2, we can see shifting O. s ( q ) E (rf ) E (9). or s (9 ) has

no significant effect on our calibrated parameters (pi . ph. ai. ah) (in expected ut ility

certainty equivalent case) or (pi y ph, 71 -th) (in Yaari certainty eqiiivalent case). (Note.

the bold lace values are corresponding to the beiich~nark calibration results). These

can also be easily seen in both Figure F.1 and Figure F.2. Ali the ciirves of calibrated

parameters with respect to the shifting parameters are ven flat in a reiuonable range

and t herefore are considered to be vecy stable.

0 . m (o.ow,l.ooe) 1.084 0.052 (O 0 . w (O.OM,l.ooe) 1.064 O 952 (O 0.002 (o.ose,i.ooa) i .084 O osz (O

p 0.oiio (0068,1.008) 1.064 0.962 (O 0.978 (0.056,1.008) 1.064 0.952 (O 0,870 (U.OSB,1.008) 1.064 0.952 (O. 0.874 (O O58,1.008) 1.084 0.062 (O o . s a (O Oae,l.ooe) 1.084 O Ba2 (O 0.98 (O.OSS,l~) 1.W 0053 (O O m (O.OW,1.006) 1.064 0.952 (O P 0.W) ~om.1.~) 1.065 0 0 1 (O 0.m (O.OW,l.ooci) 1.w 0.850 (O 0.98 (o,ooe,i.m) 1.067 o.erg (O OB0 (O.oa0,l.OQI) 1.068 0.948 io 0.M (0.0ôl.l.tW~ t.Oô9 0.917 {O 0.08 (O.os6.1.006) !.Mt O 94s (O O oe (o.oss,imos) t . ~ oaio (O

p 0.88 (0.OSû.l.W) t.W 0.951 (O J 0.90 (o.osa,i.aos) t . asm (O

0.M (O.oss,r.oool 1.065 O953 (0 0 000 (O.OM,l.008) 1.064 0.B52 (O o.wo (o+ose,i.ooe~ r.w 0 . 9 5 ~ (O

3051 -2 587 -23 118 0826 0657 0764 4274 2741 4 33) 2805 2818 -2368 -23155 0627 0657 0764 4274 2741 4334 2801 2640 -2136 723163 0827 0657 0764 4274 2741 4334 2605 2432 -1 006 -23 171 0027 0057 0764 4274 2 741 4 334 2605 2224 -1.677 -23.178 0.U27 0.667 0.764 4274 2.741 4.331 2A05 2014 -1448 -23181 0626 0057 0764 4274 2741 4334 2605 IBO4 -1210 -23190 OB28 0657 0764 4274 2741 43U 2605 1593 4090 -23196 0828 0657 0704 4274 2741 4 3 2845 1381 -0 762 -23201 0828 0 657 0764 4274 2 741 4 334 2605 2242 -1688 -23432 1058 0659 0765 4247 2725 4305 2590 2 . m . i b n . z a i n 0 . ~ 7 o.an o . 7 ~ 4174 2 . 7 4 . w 2 m s 2207 -1888 -22927 0593 0858 0763 4301 2757 4382 2610 2190 .1658 .2267B 0355 O654 0762 4327 2773 4390 2632 2175 -1047 -22433 O113 O652 0761 4352 2788 4418 2648 2160 . 1 6 3 22lW) .O132 0651 0759 4377 2803 4445 2859 .2 146 4 1 629 2 1 940 -O 302 0640 O758 4401 2817 4471 2 872 ,2304 -1704 a 3 1 2 0111 oeea o n 5 4133 2801 4 i m 2480 .2 328 -1 753 -23 026 0 351 0 663 0 771 4 160 2648 4 238 2514 2 274 -1 714 -23 549 0 690 O 660 0 768 4 227 2 685 4 285 2 658 a . 2 ~ 4 - i b n -sin o a 7 onn o.= 4274 2.741 4.334 PMU .2176 -1841 -22814 1065 O654 0761 4321 2788 4383 2650 .2 148 +le13 ,25069 1006 0 760 0861 4340 2782 4399 2641 .2 186 -1 644 -24 095 0 284 O 702 0807 4 282 2724 4 341 2 505

Bond Prlcs Bond V k M PL PH Y- rij BL OH YLK YW

-1 1.300 -4.500 0.0010 0.9400 0.6401 0.6561 4.5537 4.3382 '

Figure F.1: - - Matching - - Bond Yield with Expected Utility Certainty Equivalent - . - -- - . A-

l

Year

- -YL -- YH -CL ..... CH

This figure match bond yield with Expected Utility Certainty Eqtiiva-

lent

Figure F .2: 5Iatching Bond Yield wit h Yarri Certaintu Eqiiivalent

This figure match bond yield with Yarri Equivalent

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