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Date

An Analysis of Some Issues in Asset Price

Behaviour

Vladimir Pavlov

A thesis submitted for the degree of

Doctor of Philosophy of

The Australian National University

June 2003

DECLARATION

The work contained in this thesis has not been previously submitted for a

degree or diploma at any other education institution. To the best of my

knowledge and belief, the thesis contains no material previously published

or written by another person except where due reference is made.

Signed --~ ..... :71 ..... k ............... . ~/ 1.1

Date

11

ACKNOWLEDGEMENTS

First and foremost, I would like to acknowledge the help and support of

my supervisor, Adrian Pagan, whose commitment and patience have been

an inspiration to me and shaped my views on what it is to be a real

academic and a professional. I am also grateful to Simon Grant for his

support and comments on various parts of the thesis. Since I joined QUT,

Stan Hurn has been a friend, a mentor and a constant source of

encouragement. I would also like to thank all of my family and friends for

their support (and constant nagging on the topic of getting it over and

done with).

III

ABSTRACT

The equity premium puzzle, described by Mehra and Prescott in 1985, has

baffled financial economists for almost two decades and still lacks a

satisfactory resolution. For an area of economics, whose main focus is on

measuring risks and rewards for risk taking, not being able to explain the

difference in expected returns between two major classes of financial assets

- equity and bonds - is a major challenge. We examine some of the issues

associated with the equity premium and related puzzles. The mam

contribution of the thesis is in establishing that a representative agent

model can be used to approximate the solutions of an over lapping­

generations economy when markets are conditionally complete and then

using this approximation to examine age-related liquidity restrictions as

possible explanations for the puzzle. We conclude that such restrictions by

themselves can not explain the equity premium puzzle.

iv

TABLE OF CONTENTS

PREFACE ........................................................................................................................... 1

LASSET PRICING PUZZLES ........................................................................................... 4

1.1 INTRODUCTION ..................................................................................................... 4

1.2 RISK AVERSION ...................................................................................................... 5

1.3 RISK AVERSION AND EQUITY RETURNS .......................................................... 9

1.3.1 Euler Conditions and Assumptions ...................................................... ......... 9

1.3.2 Mehra-Prescott (MP) Formulation of the Equity-Premium Puzzle. The Risk-

}?ree Rate Puzzle ............................................................................................... 15

1.3.3 The Equity Premium Puzzle ....................................................................... 18

1.3.4 Estimating the Parameters /3, T Using Australian Data .............................. 19

1.4 SOURCES OF THE PUZZLES ............................................................................... 23

1.4.1 Equity Premium Puzzle ............................................................................. 23

1.4.2 Risk-Free Rate Puzzle ............................................................................... 27

1.5 CONCLUSION ........................................................................................................ 29

~A REVIEW OF ASSET PRICING PUZZLES .............................................................. 30

2.1 INTRODUCTION ................................................................................................... 30

2.2 MOVEMENTS IN EXPECTED RETURNS AND THE EQUITY PREMIUM ...... 31

2.3 ALTERNATIVE UTILITY SPECIFICATIONS ..................................................... 41

2. 3.1 Habit Formation Preferences ..................................................................... 42

2.3.2 Relative Consumption ............................................................................... 43

2.3.3 Non-Expected Utility ................................................................................. 45

2.4 CREDIT MARKET EXPLANATION .................................................................... 50

2.5 LIQUIDITy ............................................................................................................. 56

2.6 INCOMPLETE MARKETS .................................................................................... 61

2.5.1 The Effect of Transaction Costs ................................................................. 66

v

2.5.2 Persistence of Individual Income Shocks ..................................................... 68

2.7 CONCLUSION ........................................................................................................ 72

;LAS SET PRICING IN OLG MODELS ........................................................................... 73

3.1 INTRODUCTION ................................................................................................... 73

3.2 CDM ~IODEL .......................................................................................................... 76

3.3 CDM CALIBRATION ............................................................................................. 79

3.4 EXTENTIONS TO CDM ........................................................................................ 86

U,LFinancial Structure ............................................................................... .... 87

~ OLG versus Representative Agent .... .......................................................... 93

3.4.3 Simple Equilibria ...................................................................................... 96

3.5 CONCLUSION ...................................................................................................... 101

;:LCALIBRATION AND EQUILIBRIUM ....................................................................... 103

4.1 IN1'RODUCTION ................................................................................................. 103

4.2 EQUILIBRIUM WITH LOGARITHMIC PREFERENCES .................................. 103

4.3 CALIBRATION AND SCALING .......................................................................... 107

4.3.1 Aggregate Dynamics ..................................................................... ........... 108

4.3.2 Markov Chain Approximation .................................................................. 110

4.3.3 Scaling ................................................................................................... 116

4.3.4 Distribution of Wages ......................................................................... ..... 117

4.4 SOLUTION TO THE REPRESENTATIVE AGENT PROBLEM ....................... 119

4.5 PRICES IN OLG AND REPRESENTATIVE AGENT ECONOMIES ................. 120

4.6 ASSET RETURNS IN RESTRICTED AND UNRESTRICTED ECONOMIES ... 123

4.7 CONCLUSION ...................................................................................................... 125

!LSOLVING ASSET PRICING MODELS ...................................................................... 128

5.1 INTRODUCTION ................................................................................................. 128

5.2 SOLUTIONS .......................................................................................................... 129

VI

5.2.1 Discretisation (DA) ................................................................................ 130

5.2.2 Parameterized Expectations (PE) ............................................................. 132

5.2.3 Direct Approximations (A A) ................................................................... 134

5.2.3 A Simple Solution Algorithm when G is Linear in Z (LA) .......................... 137

5.2.4 Recursive Algorithm (RAJ ....................................................................... 138

5.3 EXAMPLES .......................................................................................................... 142

5.4 CONCLUSION ...................................................................................................... 147

REFEREN CES ................................................................................................................ 150

Vll

PREFACE

In an article published in 1985 Mehra and Prescott argued that the equity

premium in the US appears too high to be explained by a consumption

asset-pricing model (CAPM). The growth of aggregate consumption is too

smooth and not sufficiently correlated with the equity premium to

generate the observed level of the average excess return on equities over

risk-free bonds for a reasonable level of risk aversion. This observation,

termed the equity premium puzzle, has been a subject of much research

since the Mehra and Prescott publication but, despite a number of

important contributions, the equity premium puzzle is still unresolved.

On the other hand, finance in general has been an extraordinarily

successful field of modern economics. Factor pricing models, in particular,

have found numerous practical applications in areas such as capital

budgeting, performance evaluation, investment analysis, risk management,

etc. The failure to find a satisfactory economic explanation of the equity

premium puzzle means that these models, and business tools derived from

them, are built on shaky theoretical grounds, as all factor models and

economic intuition about these models are, in fact, based on Lucas's {1978}

1

consumption pncmg model and its generalizations. As Cochrane (2001)

points out:

IIThis is a point worth remembering: all factor models are

derived as specializations of the consumption-based models.

Many authors of factor model papers disparage the

consumption-based model, forgetting that their factor model

is the consumption-based model plus extra assumptions that

allow one to proxy for marginal utility growth from some

other variables. "

Understanding the fundamental economic factors driving the equity

premium is vastly important. Grant and Quiggin (1998), for example,

argue that, if a large portion of the observed equity premium is the result

of market frictions, then the common practice of using equity market

returns to evaluate the value of public projects can lead to under-financing

of such projects and create inefficiencies.

This thesis exammes some of the issues associated with the equity

premmm and related puzzles. Chapter 1 formulates the puzzles and the

main empirical features of the data that generate it. Chapter 2 provides a

brief overview of the main contributions to the literature. An important

explanation for the puzzles emerged recently and is based around the effect

of liquidity restrictions on asset pricing in models that take into account

2

life-cycle considerations. The mam body of the thesis examines this

explanation under an alternative market structure. Chapter 3 shows that

asset prices in such a model can be approximated using an appropriately

calibrated Lucas model. Chapter 4 uses this approximation in the context

of a calibrated economy to examine the effect of liquidity restrictions on

asset pricing and the equity premium. Chapter 5 concludes with a

discussion of various algorithms used to obtain equilibrium solutions for

asset pricing models.

3

L

CHAPTER 1

ASSET PRICING PUZZLES

1.1 INTRODUCTION

Low volatility of consumption growth and its low correlation with equity

returns mean that, in order to explain the high mean equity premium, the

consumption capital asset pricing model based on time-separable CRRA

preferences has to rely on an extremely high degree of consumer aversion

to consumption fluctuations. This chapter starts with a brief exposition of

the argument about why such high estimates of risk aversion are

implausible (the equity premium puzzle). We then review some of the

related anomalies and discuss the features of the data that a successful

model must address to resolve the puzzles.

4

1.2 RISK AVERSION

Crucial to interpreting the failure of the consumption CAPM to fit the

observed asset returns is to set a prior reasonable degree of risk aversion

and to do that is the concern of this section.

Consider the problem of a consumer facing a potentially adverse

consumption realization. With probability p the consumer enjoys a high

level of consumption C and, with the complementary probability 1 - p ,

consumption drops to 8C,8 < 1. We want to know the maximum amount

that the consumer will be willing to pay to avoid the latter outcome.

Assume that the consumer has expected utility preferences with the form

of the constant relative risk aversion (CRRA) von Neumann-Morgenstern

utility function:

u(c) 1-,

(1.1)

"/ In this specification of preferences the parameter ,= -c u / u I IS the

Arrow-Pratt coefficient of relative risk aversion.

5

The maximum amount that the consumer will be willing to spend to

completely insure against the adverse outcome is determined by the

equality between the utility of the certain (insured) level of consumption

and the expected utility of a consumption lottery:

u(nC) = pu(C) (1 p)u(8C)

where 7fC is the certainty equivalent level of consumption and 1 - 7f is

the price of the insurance contract relative to consumption. Substituting

from (1.1), the maximum amount that the consumer will be willing to

spend on insurance is determined by

1

7f = [p + (1 p)8(1-r)]1-,. (1.2)

Equation (1.2) shows one of the implications of CRRA preferences: the

proportion of wealth that the consumer will be willing to spend to insure

against wealth gambles is independent of the level of wealth. It is also easy

to show that, consistent with intuition, d7f < 0, so that the proportion of d'Y

wealth that consumers are willing to spend to insure against a gIven

negative consumption shock is increasing in risk aversion. Table 1 lists this

proportion for a number of values of the risk aversion parameter and

different loss levels when the loss probability is set at 0.1%. The second

6

Loss

(with p=O.l%) , 8 =0.5 8 =0.2

1 0.1 0.0

5 0.4 0.0

10 4.5 0.1

20 28.1 0.3

30 36.6 1.7

Table 1 Risk aversion and the maximum insurance premium.

and third columns correspond to the cases of 50% and 20% consumption

shortfalls in the bad state.

Introspection suggests that the magnitudes III the second column of the

table point to risk averSIOn parameters above 10 being somewhat

implausible, SIllce a consumer with the relative risk averSIOn of 20 for

example would be willing to give up almost all of the upside potential to

avoid what is essentially a very low probability event.

A large part of the literature and this thesis accept this argument for

believing that large values of , are implausible. Nevertheless, a number of

valid criticisms can apply to the way the argument has been made.

Firstly, introspection is a slippery slope for establishing economic laws or,

as in this instance, deviations from them. The argument does rely

substantially on the rather extreme case of a 50% loss in consumption. In

7

contrast, the values obtained for a loss of 20% of consumption in the bad

state appear quite reasonable. Accordingly, some researchers (Kandel and

Stambaugh [1991]) have argued that the deficiency in the preference model

is simply that it is not general enough to deal with such extreme

outcomes.

Secondly, a perhaps more valid defence of the argument against high

values of the risk aversion parameter comes from the empirical literature.

Estimates of the risk aversion coefficient obtained from macroeconomic

data vary a great deal and are typically very imprecise, but generally point

to a value well under 10 (Hansen and Singleton [1983]; Vissing-Jorgensen

[2002]; Brav, Constantinides and Geczy [2002]; see also the references in

Mehra and Prescott [1985]). While there exists some experimental

literature attempting to quantify risk preference parameters, it is

unfortunately not sufficiently developed to suggest easily identifiable value

for macroeconomic or financial models.

To conclude, this section limits the range of reasonable values for the risk

aversion parameter. In the next section we will discuss the inability of a

dynamic consumption-based equilibrium with low risk aversion to explain

the observed returns on risky assets.

8

1.3 RISK AVERSION AND EQUITY RETURNS

1.9.1 Euler Conditions and Assumptions

We start this section with a discussion of Euler conditions for the dynamic

consumption problem with time-separable preferences which provide the

foundations of the consumption capital asset pricing model (C-CAPM) and

much of the empirical work in the economic literature on asset pricing.

Consider an economy with a single homogeneous consumption good and

competitive asset markets where n perpetual assets are traded. A unit of

asset i provides its holder with a consumption dividend di,t each period.

An asset which provides the same dividend in every possible state of the

economy is riskless; otherwise it is risky. Riskless assets will be referred to

as bonds, while, for the purposes of this chapter, risky assets will be

identified with equity. Where it does not create confusion, the dependence

of the asset payoff on the state vector will be suppressed in the notation.

Consider the investment problem facing an infinitely-lived consumer­

investor with time-separable expected utility preferences:

9

u" (C) = E, {~t3'U(C,)} C = {ct}:to

and the CRRA instantaneous utility function

1 ,

The coefficient !3 captures the time-preference or impatience of the

investor while , determines the risk aversion. In addition to dividends

from current asset holdings, the consumer receives an exogenously

determined consumption endowment et •

The consumer allocates the available funds, equal to the sum of non-

investment et and investment (dividend) income, between current

consumption ct and investments in assets Bt to maximize expected lifetime

utility:

Ct + I:: (Bi,t - Bi,t-l)Pi,t = et + I:: Bi,t_ldi,t' Vt i

where Bi,t refers to the time-t of asset holdings, Pi,t and di,t are the asset

prices and dividends per unit of asset respectively.

10

For the assumed form of utility, if the endowment is not expected to grow

forever faster than the individual rate of time preference, and if asset

markets produce no price bubbles, the individual consumer's problem has

a solution described by the Euler equations (dynamic first-order

condition):

(1.3)

The Euler equation can be interpreted as the balance between the cost of

additional investment in the current period and the expected benefit from

this investment in the future. Investing in an extra-marginal unit requires

a sacrifice of Pt units of consumption in period t, which, for small changes,

results in the utility loss of u I (ct ) Pt. On the other hand this investment

allows extra spending of (PHI + dt+l ) in the following period providing the

utility benefit u' (CHI )(PHl + dHI ). Ignoring corner solutions that would be

ruled out in the equilibrium, the optimal program has to balance the

marginal cost of investment today with the benefit that the consumer

expects to derive from this investment discounted to the current period.

An important aspect of time-separable preferences is that, to achieve

optimality, it is sufficient to consider the trade-off between the current and

the immediate future periods only. In other words, time-separability leads

to rationally myopic behaviour.

Taking into account that u' > 0 we can re-write equation (1.3) as

11

(1.4)

where

(1.5)

IS a random discount factor termed the pricing kernel and

RHI = (PHI + dt+1X, is the vector of total (gross) returns on holding

available assets for one period.

Formulae (1.3) and (1.4) can be easily seen as generalisations of the

textbook constant expected return model under uncertainty. For example,

assuming that asset prices do not grow forever at a rate greater than the

time preference, after rearranging (1.3), making use of (1.5), and iterating

forward one obtains

(1.6)

which expresses the current prIce of an asset as the discounted sum of

future dividends. If the pricing kernel IS constant, (1.6) turns into the

standard present value calculation. The potential benefit of the C-CAPM

is in identifying the factors driving expected returns.

12

To show the versatility of the consumption price formula consider the

problem of pricing discount bonds of different maturities. Denote the price

at time t of a zero-coupon bond maturing in n periods as bt n' Assuming ,

that bonds have unit face value, the price of a bond maturing in one

period is just the expectation of the discount factor

(1.7)

For a bond maturing in two periods

which, after substitution of (1.7) followed by an application of the law of

iterated expectations, becomes

By repeatedly applying the same logic the asset-pricing equation can

generate the discount function for bonds of all maturities

(1.8)

which can be transformed to obtain the complete term structure of interest

rates.

13

Bond pricing equation (1.8) underscores the importance that interest rate

models play in the empirical asset pricing literature; if the pricing theory is

correct, fluctuations in bond rates must directly reveal movements in the

discount factor.

The fundamental pricing equation can be adapted to price any imaginable

asset. In fact, one of the main insights of the consumption asset pricing

theory is that prices of all assets, no matter how exotic, are driven by a

single set of fundamental factors. This also helps to understand the

extraordinary challenge that the theory is facing. So far, despite some

breakthroughs, the empirical asset pricing literature has not been able to

come up with a successful single model for the term structure, let alone a

model that would explain all the multitude of assets including various

stocks, bonds and financial derivatives.

To give the necessary conditions for the consumption/investment problem

empirical substance, it is first necessary to specify a form for the utility

function along with the information structure of the problem. Than a

consumption series must be identified using an observable or imputed

(Restoy and Wei! [1998], Campbell [1993]) series. The necessary conditions

must of course be satisfied at the level of individual consumers, but to test

the restrictions implied by the model at this level of disaggregation would

require access to a panel of individual consumption and asset holdings

14

data, which are very rarely available and, even when available, are very

imperfect (Mankiw and Zeldes [1991]' Marshall and Parekh [2000]).

Much of the empirical literature assumes that aggregation conditions

(Rubinstein [1974]) are satisfied, at least approximately, which allows the

use of some measure of aggregate consumption in place of individual

consumption in the equation (1.4). In addition, since data at the quarterly

frequency are often used, the consumption of durables is likely to exhibit

considerable non-separabilities across quarters, so that the literature,

following the early study of the consumption pricing theory by Hansen and

Singleton [1983], assumes that preferences are separable between non­

durable consumption and the services provided by durables. This

assumption means it is possible to integrate out of the necessary condition

for consumer optimisation that component of utility that depends upon

the services provided by the durables.

1.3.2 Mehra-Prescott (MP) Formulation of the Equity-Premium

Puzzle. The Risk-Free Rate Puzzle

MP considered a Lucas economy populated with agents having time

separable utility

15

U, (e) = E, {~tJ'u(c,)} C = {cJ:tu

Cl-1' /

and the CRRA instantaneous utility function, u (c) /1 - 'Y .

Under these assumptions on preferences, the pricing kernel becomes

In their economy there is a single firm producing Yt of a non storable

consumption good each period and the growth rate of output YHi'( Xt+l

follows a 2 state Markov chain with the state space {-\ , -\} and time

invariant transition probabilities <Pij. There is a competitive market III

perpetual equity, which entitles its owner to claim Yt units of output.

Since consumption is non-storable everything produced within a period

must be consumed within the same period; market clearing conditions

become

(1.9)

and the stochastic discount factor is

16

(1.10)

MP search for a stationary equilibrium in which prices of equity S (cp i)

and bonds B (ct , i) are functions of the current state and Ct. With prices

quoted ex-dividend, the first-order conditions for the prices of equity S

and bonds B are determined as

S(clli) = (3~¢ij)..? [S(\cpj) ct\]

j (1.11)

B (cn i) = {3~ ¢ij)..?' j

Noting that the price function is linear in c, substitution of S (cll i) = wict

into (1.11) yields a linear system of equations that can be solved for Wi'

Once equilibrium asset prices have been determined, expected equity and

bond returns and the equity premium can computed by taking the

expectation of returns

1

with respect to the stationary distribution of the Markov chain 7r,:

(1.12)

17

1.3.3 The Equity Premium Puzzle

MP calibrated the chain (output growth and 2 transition probabilities) to

match the mean, variance and autocorrelation of the US per capita

consumption. The annual return to short-term riskless borrowing adjusted

for inflation is taken as a proxy for the risk-free rate R: (that yields the

sample average of 0.8%), and the annual return on the S&P500 is taken as

a measure of the return on equity R; (with the sample average 6.98%).1

Requiring the risk-free rate to be reasonably low (less than 4%) and

restricting the parameters to lie in 0<1l<1 and 0<'Y<10, the greatest

equity premium that (1.12) can produce is 0.35% (the observed premium

would require 'Yof about 30, which is judged implausibly high).

1 Strictly speaking bond returns are only nominally riskless. MP argue however that

inflation is almost uncorrelated with consumption and output growth and thus does not

contribute any systematic risk and therefore is priced at zero in equilibrium.

18

r

1.3.4 Estimating the Parameters (3, 'Y Using A ustralian Data

The MP example illustrates the inability of a simple calibrated equilibrium

model to reproduce the historical level of the equity premium with low risk

aversion and a reasonable level of the risk-free rate. This calibration

however does rely on very restrictive assumptions about the structure of

the exogenous driving process and the highly stylised setting of the MP

exchange economy. The aim of this section is to establish what can be said

about preference parameters by exploiting the restrictions implied by the

first-order conditions of the consumer problem (1.4) while making

minimalist assumptions about the factors driving the economy and its

structure. This can be done by interpreting C-CAPM pricing equations as

moment conditions and using the generalised method of moments (GMM).

It will be convenient to rewrite the pricing equations for equity and bonds

in terms of the equity premium and bond returns

Et {mt+l (R: R:)} 0

Et{mt+l(~+l)} 1

Define the deviations from the equilibrium conditions (1.13) as

e~l mt+l b, (3)( R; Rn e:+1 = m t+1 b,(3)(~ + 1)-1.

19

(1.13)

(1.14)

After taking the unconditional expectation, equations (1.13) provide

sufficient moment conditions to identify the time preference and risk

aversion parameters by the generalised method of moments (GMM).

Denoting the vector of instruments as Zt the GMM estimates are obtained

as the global solution for

where V is the variance covariance-matrix of the errors and their cross-

products with the instruments.

The series employed are the quarterly seasonally adjusted chain-volume

ABS per capita consumption, the All Ordinaries accumulation index and

the Total Return Bond index deflated by the consumption deflator; the

last two being used to compute real returns for the period from September

1959 to September 1997.2 Table 2 reports parameter estimates obtained

using two popular sets of instruments; one using two lags of consumption

2 All data sources are listed in the data appendix at the end of the thesis.

20

growth [1 Ct~ /ct_ 1 1 Ct_l~ -1]' and the other using two lags of /ct_ 2

the interest rate z~ = [1 ~-1 R:_2 r . The serial correlation and

heteroscedasticity consistent (HAC) variance-covariance estimator of

Newey-West with 4 lags was used to estimate V.

Using z~

Using z: 1.71

1.72

Table 2 GMM parameter estimates

115.04

115.04

Using either consumption or interest rates as instruments produces

identical parameter estimates. To understand this behaviour of the

estimation procedure note that (1.13) provides sufficient conditions to

estimate the two parameters by the method of moments. Fitting these two

conditions exactly gives i = 115.04 and ~ 1.71. This clearly shows that

the extra available instruments provided by lagged consumption and

returns simply do not provide any information about the parameter values

and are effectively discarded by the GMM.

To understand the relationship of the estimates to the puzzles consider the

parameter values for f and /J that satisfy the two sample moment

conditions

21

(1.15)

(1.16)

Since the time preference parameter fJ is not in the first condition (1.16)

this solely determines the value of the risk aversion parameter. The high

estimate of 1 required to fit the moment condition based on the equity

premium, (1.15), will be shown in the next section to be a consequence of

the low volatility of consumption growth and its low correlation with the

equity premium.

Given the high value of 1 required to fit the sample moment involving the

equity premium the value of fJ that would satisfy the bond return

condition (1.16) becomes greater than one, i.e. to reconcile the high risk

aversion of the representative consumer with the low historical bond

return the C-CAPM must rely on negative time preference. All things

equal, a consumer with such preferences would prefer to delay

consumption, which goes against much of the accepted economic wisdom

for aggregate behaviour.

22

r

1.4 SOURCES OF THE PUZZLES

1.4.1 Equity Premium Puzzle

To illustrate the mam empirical features that contribute towards the

puzzles) consider a simple first-order approximation to the Euler

conditions. Under the homogeneity assumptions the marginal rate of

substitution in consumption between time-periods depends only on the

consumption growth rate Xt+l = ct+1 • We will approximate the MRS using

ct

the first-order Taylor expanSlOn centred on the sample mean of

consumption growth J-Lc 3

where xt = X t - J-Lc is the deviation of the growth rate from its mean value.

3 A similar approximation was employed by Heaton (1995).

23

For the CRRA form of the instantaneous utility this approximation

becomes

(1.17)

Substituting the linear MRS approximation into the prIcmg equation

(1.13) and taking the unconditional expectation gives the equity premium

as

E(Rte - Rn = ~Cov(Xt+pRte - R;)

{tc

= ~Corr(xt+pR; - R;)O"R'_R!'O"X {tc

(1.18)

where Corr (Xt+l' R; - R;) is the correlation between consumption growth

and the equity premium and 0" Ir_Rb, O"x are the standard deviations of the

equity premium and the consumption growth rate respectively.

U sing a first-order expansion under the expectation raises some technical

issues, in particular there is no guarantee that the approximation errors

remain small after integration with respect to the joint density of the

equity premium and consumption growth rate. Table 3 shows however

that, when quarterly data on consumption and returns are used, the linear

approximation (1.17) works sufficiently well for the purposes of our

illustrative example. The sample means of the equity premium condition,

24

Sample Means of Forecast Errors (fJ 0.998 ) Carr (mtl mt )

Exact - mt Linear Approximation - mt

Equity Bond Equity Bond

Condition Condition Condition Condition

1 1.68 -0.16 1.68 -0.16 0.999

10 1.64 -4.30 1.63 -4.82 0.996

20 1.60 -7.84 1.58 -9.73 0.986

30 1.57 -10.30 1.53 -14.39 0.969

Table 3 Performance of the Linear Approximation to the Pricing Kernel

in particular, obtained using the linear approximation are very close to

those obtained using the CRRA form of the discount factor.

The correlation between excess equity returns and growth of the total real

final consumption has been 0.13 in quarterly and 0.24 in annual data. The

level of equity volatility has been between 20% and 30% per annum, while

the consumption series has been relatively smooth with a volatility of 4%

per year. Annual consumption growth averaged 2%. Importantly, the

sample estimate of the average equity premium obtained using real returns

on the All Ordinaries index4 is very close to the 6% obtained for the US.

4 The All Ordinaries accumulation index was deflated by the OPI in quarterly data and

GDP deflator in annual data

25

Substituting these estimates into the approximation gives a range for 'Y of

between 25 and 47 depending on what estimate of the equity-consumption

growth correlation is used.

Equation (1.18) demonstrates very clearly the origins of the inability of the

C-CAPM to explain the level of equity returns and also helps to illustrate

the main thrust of the theories proposed to explain the equity premium

anomaly.

Full sample estimates of the relevant moments basically tell the following

story. Over the period when the relevant economic and financial statistics

are available, the consumption series has been very smooth, at least at the

aggregate level and has had a very low correlation with the equity returns.

If aggregation conditions (Rubinstein [1974]) are satisfied at least

approximately, a well-diversified equity portfolio, while extremely volatile,

is not particularly risky for an average consumer. Therefore, equity

volatility does not translate into consumption volatility, which could be

either due to the ability of households to effectively diversify away equity

risks, which appears unlikely, or that they hold little equity. Disregarding

the latter, the only way to reconcile these facts is to allow for extremely

high risk aversion.

Based on this discussion it is easy to identify the main factors that drive

the equity premium puzzle. There is very little disagreement about the

26

volatility of equity returns a R'-lt ' which is the most easily measured data

component. Examination of the equation (1.18) then leads to the

conclusion that, in order to succeed in explaining high equity returns

without resorting to high values of " a model must produce either a

greater consumption volatility (J x or a higher correlation between ,

consumption growth (or mt ) and equity returns. This leads to two

prominent strands in the research. The behavioural approach seeks to

modify the basic model of preferences while the incomplete markets

argument suggests that smooth aggregate consumption may hide a great

deal of idiosyncratic consumption volatility at the individual level.

In the following chapter we will review some of the popular modifications

of the utility function as well as selected sections of the incomplete

markets literature. The effect of a specific form of market incompleteness

due to borrowing restrictions on young agents is the subject of chapters

three and four.

1.4.2 Risk-Free Rate Puzzle

U sing the same linear approximation for the MRS as in the previous

section and applying it to the pricing equation for bonds produces

27

(1.19)

The correlation between bond returns and consumption growth is 0.2 in

annual data from 1928 to 1997 and is slightly negative in quarterly data

from 1959 to 1997. The annual bond return volatility is roughly 5%.

Substituting the annual correlation estimate into the approximate bond

return condition, and assuming a risk aversion coefficient of 20, the time

preference parameter f3 that would explain the historical real bond return

of between 1% and 2% per annum is about 1.4 7) implying that the

consumer will be willing to substitute current for future consumption.

To understand the source of this "risk-free rate puzzle" consider the

elasticity of intertemporal substitution (EIS) with respect to the gross

interest rate:

The elasticity of intertemporal substitution measures the willingness of the

consumer to substitute consumption between time periods in response to

changes in interest rates. Small values of 't/J indicate that the consumer

would be reluctant to accept even small consumption fluctuations between

time periods. Ignoring uncertainty about future consumption growth, with

CRRA preferences the EIS is equal to the inverse of I' Very large values

28

r

of "y required to reconcile the model with the observed equity premium

would therefore imply a very low EIS. Consequently, the consumer would

be trying to borrow against expected future consumption growth unless

interest rates are very high or utility displays negative time-preference

(/3>1).

1.5 CONCLUSION

This chapter presented a brief overview of some of the anomalies that have

preoccupied financial economists for the past 15 years. Ever since Lucas's

influential publication, the C-CAPM model has shaped economists

thinking about the fundamentals that determine asset prices and returns.

And yet the model has failed empirically in almost any application

involving risk pricing. The next chapter considers some of the popular

models put forward as potential explanations for the puzzles.

29

r

CHAPTER 2

A REVIEW OF ASSET PRICING PUZZLES

2.1 INTRODUCTION

The literature on the pricing puzzles is huge and has become multi-

disciplinary. This review is therefore necessarily selective, concentrating on

the main contributions to the literature.

The chapter starts with a brief exposition of the movements in the

expected excess returns on stocks. The second section reviews the main

behavioural explanations for the puzzles. The chapter concludes with a

discussion of the very influential literature investigating the effect of

market incompleteness on the pricing of risk.

30

2.2 MOVEMENTS IN EXPECTED RETURNS AND THE EQUITY PREMIUM

The commonly quoted estimate of the equity premium is obtained using

data on realised returns. The significance of the equity premium puzzle

hinges to a large extent on accepting this figure as a reasonable estimate of

the expected level of relative stock return. Fama and French (2002) in

particular argue forcefully that the high realised return estimate of the

equity premium over the second half of the last century is the result of

unexpected capital gains, and therefore using this estimate to draw

inferences about the expected level of stock returns can be misleading.

Investigations into the level of the equity premIUm have not only

generated an active academic literature but also some debate among

practitioners:

Academics argue until they are blue in the face about the size of the

equity risk premium - some go as high as 8 per cent; others as low as

zero. But fund managers do not seem to have such difficulty pinning

down a number if a London conference this week organised by Merrill

Lynch and Imperial College, London, is anything to go by. The

overwhelming consensus, on an admittedly unscientific show of hands,

was that the sustainable premium for US equities is 2-4 per cent.

Financial Times, 18 September 1999

31

A number of recent papers (Blanchard [1993], Kortian [1997], Pastor and

Stambaugh [2000], Jagannathan, Grattan and Scherbina [2001], Fama and

French [2002]) explored alternative estimates of the level and the dynamics

of the equity premium.

The Gordon formula provides the common theoretical framework for these

studies. Denote the ex-dividend price of the stock Pt and the end of period

dividend dt,. Assuming that the cash-flows received from holding the

market index are discounted at a constant rate we can write

(2.1)

If dividends are expected to grow in perpetuity at a constant rate

9 Et rl . < R ) from (2.1) the current stock price is "'t+.

Pt (2.2)

Rewriting (2.2) as R = Et dt+fpt + 9 and defining the dividend yield

dYt+l dt+fpt' the expected return on stocks R can be expressed as the

sum of the expected dividend yield and the dividend growth rate g:

(2.3)

32

Expression (2.3) is used almost universally to provide the estimate of the

"fundamental" equity return. Main differences arise in the way different

papers form estimates of the dividend growth g. Fama and French use the

simple sample mean over sub-periods as an estimate of the g, while

Blanchard and Jagannathan et. al. use the weighted average of I-period

ahead dividend growth forecasts, in which case (2.3) holds only

approximately and is obtained from (2.1) by linearization. The discount

formula is typically applied to real (deflated by the Cpr) dividends.

Subtracting the real bond return gives the equity premium.

It is well known (Shiller [1981]' Campbell and Shiller [1987, 1988]) that

(2.1) can not replicate the observed volatility of realised stocks return -

rational forecasts of the dividend growth are simply too smooth to explain

shifts in returns and valuation ratios. The contention among the majority

of financial economists (see e.g. Cochrane [2001]) is that movements III

expected returns must play at least some role III explaining changes III

prices and dividend price ratios.

For the discount formula to produce meaningful inferences about expected

returns two assumptions must hold. First, the expected return is not

modelled explicitly - only forecasts of the dividend growth are used in

reconstructed returns. This effectively restricts the processes for the

dividend growth and the pricing kernel (or equivalently the required rate

of return) to depend on the same information set, so that only the

33

r ,

variables useful for predicting future dividends provide information about

expected return. Second, since (2.3) only holds if the discount is constant,

it is effectively assumed that expected returns move slowly relative to

dividend forecasts. When combined these assumptions, not surprisingly)

produce equity premium estimates that are even smoother than dividend

forecasts.

The first important constituent of the current debate is the conditional

expected return on stocks. Using annual data for the period from 1901 to

1998 (with the exception of the bill yields which are only available from

1928), Table 4 shows estimates of the mean real and nominal dividend

growth rates, dividend yields and returns. The Gordon equity premium

estimate is computed using (2.3) by adding the average dividend yield and

the average dividend growth over a sub-period and subtracting the average

return on bills over the same period. Real returns are computed by

deflating the nominal dividends, as well as stock and bond indices by the

CPI.S In addition to 10-yearly samples, the first and second halves of the

5 This differs from Blanchard who uses V AR forecasts as proxies for expected inflation.

34

period are presented separately, as these correspond to the periods

considered in Fama and French (2002).

The most prominent feature of the table is the apparent decline of the

Gordon estimate of the equity premium in both nominal and real returns

over the period 1970-1998, driven by the dramatic drop in the dividend

growth rate and the unusually low dividend yields of the period.

12~------~-------.--------r-------~------~--------.

10

-0 S <6 >--0 c: 4)

~ .<=:

6 Cl

2~------~------~--------~------~------~------~ Js.nSO Ja.nOO Ja;n20 Js.n40 Js.n60 JanOO

Figure 1 Nominal Monthly Dividend Yields (annualized, %) from 10/1882 to 1/2000.

Understanding the behaviour of the dividend yields is crucial to

formulating a view of the expected return on stocks. Empirically nominal

35

dividend yields (either real or nominal) are characterised by very high

persistf' nce with the first characteristic root of thE' autoregressive

700 - ---- -- - 70U

600 +---------------------------------~ 600 c:::J 95%

500 ,"i00 c:::J 90%

c::=J 75% 400 400 +- ~ 50% oj

"'-" U}

I

c:::J 25% >-:> ::WO

c:::J 10% 200 200

c:::J5%

100 100 -- J-Stat

0 0

0 50 1UO 150 200 250 300

S

Figure 2 S-period ahead forecast variance divided by sample varia nce (J-Sta t ) .

polynomial close to 0. 99. Not surprisingly, standard ADF statistics with

meaningful lag valup,s do not reject the null of a unit-root in dividend

yields at conventional sign ificance levels. On thE' othr:r hand, the evolution

of dividend yields owr a long historical peri(Jd (Figurp, 1) shows few signs

;;6

of explosive variance behavior associated with unit-root processes. For

example, taking as given the sample standard deviation of dividend yield

changes of 0.22 and the beginning of the sample dividend yield of 6.83%, if

the dividend yield series followed a simple random walk, the probability of

observing a realisation staying between the historical bounds for dividend

yields (2.49% to 10.29%) would be less than 0.01. A similar picture

emerges if we consider the behavior of the n-period ahead forecast

varIance.

Figure 2 plots the behavior of the s-period ahead forecast varIance

normalized by the sample variance of dividend yield changes (Cochrane

[1988], Lo and MacKinlay [1988])

1 T-s

--;;'"C-T---s-l-) {; (dYH8 - dYt )2 (2.4)

against the forecast horizon s together with Monte-Carlo confidence

intervals based on 10000 replications of a Gaussian random walk. If the

process contains a unit root, this variance ratio should grow linearly with

the forecast horizon. In contrast, if the process is stationary this ratio

should converge to a finite number. While short term forecasts (up to 10

years) clearly show the effect of persistence, the behavior of long term

dividend yields forecasts indicates that this persistence appears to die out

too quickly. While empirically the issue of whether dividend yields are

37

stationary or not is still unresolved (Campbell and Yogo [2002]), the field

is leaning towards stationarity of the process.

If dividend yields were to revert to the historical average from the low

level at the end of 2000 it could happen by either an increase in the

dividend growth rate or through a period of flat stock prices. Dividend

forecasts and historical experience (Campbell and Shiller [2001]) appear to

favour the latter scenario, pointing to a period of low stock returns and a

disappearing real equity premium.

It is interesting to note that the realised equity premium estimate paints a

somewhat more sanguine picture for stock market investors. The equity

premium of the last decade (5%), while depressed by historical standards,

is closer to the unconditional estimate of 8.4% than the Gordon estimate.

In fact, the equity premium appears to be exceptionally stable over most

sub-samples with the low premium of 1980-1990 appearing quite

anomalous.

Fama and French (2002), usmg the data on US equity returns and

dividend yields over the period from 1872 to 1999, argue that the full­

sample estimate of the equity premium is overstating the expected equity

return. Their argument is based to a large extent on the observation that,

in the 1872-1949 sub-sample, the Gordon estimate of the real premium of

3.79% per year is close to the estimate from the realised returns - 4.10%.

38

On the other hand over the last 50 years the Gordon estimate of 3.40%,

while very close to the estimate for the first sub-sample, is less than half

the estimate obtained from stock returns - 8.28%. The difference they

argue is due to unexpected capital gains over the period.

In the Australian data the two estimates of the equity premIUm are

remarkably close until the early 70's. Consistent with the US experience,

the decline in the Gordon estimate of the equity premium is also very

pronounced and concentrated in the past 30 yearSj the estimate of the

equity premium for the past 20 years is in fact negative.6 However there is

a very significant divergence between US and Australian data which

deprives the Fama and French story of its punch. Namely, Table 4 shows

that until the 70's both estimates are very close to the historical average of

about 8%. Therefore, if Australian experience is any guide, there is no

particular reason to prefer the latter estimate. And, as has been noted, the

decline of the equity premium is largely a feature of the Gordon estimate.

6 We do not consider the effects of changes in taxation, such as the introduction of the

dividend imputation system and the taxation of capital gains. The dividend series in

particular is not adjusted for dividend imputation. The overall effect of these changes on

the expected returns might have been smalL

39

1901-1998 1901-1950 1951-1998 1930-1940 1941-1950 1951-1960 1961-1970 1971-1980 1981-1990 1990-1998 -----

Dividend Growth 6.0 (12.5) 5.5 (11.3) 6.5 (13.6) 4.6 (16.0) 5.5 (10.5) 7.2 (8.5) 6.3 (16.2) 6.8 (12.8) 7.5 (18.3) 1.5 (12.5)

Dividend Yield 6.0 (1.3) 6.3 (1.1) 5.7 (1.5) 6.0 (1.2) 5.3 (0.8) 6.6 (0.8) 5.9 (0.5) 7.0 (1.4) 5.0 (1.1) 3.8 (0.5)

Stock Return 13.3 (15.9) 12.5 (11.3) 14.1 (19.7) 11.6 (14.2) 14.7 (13.7) 12.4 (17.4) 12.7 (23.2) 16.9 (23.3) 16.0 (23.0) 11.9 (7.4) ~ Bond Return 5.4 (4.6) 2.1 (1.5) 7.0 (4.8) 2.6 (1.4) 1.1 (0.3) 1.4 (1.1) 4.4 (0.7) 8.3 (2.4) 14.3 (2.4) 7.0 (2.5) ;::: ·s

Equity Premium 0 Z

Realised Returns 7.9 (18.4) 10.4 (15.3) 7.1 (19.8) 9.0 (14.5) 13.6 (13.8) 11.0 (17.4) 8.3 (23.3) 8.7 (22.7) 1.7 (22.8) 4.9 (8.4)

Gordon Estimate 6.6 9.7 5.2 8.0 9.7 12.4 7.8 5.6 -1.8 -1.7

CPI Inflation 4.2 (5.4) 2.8 (5.7) 5.7 (4.7) 0.2 (4.9) 7.0 (5.9) 4.7 (6.0) 2.9 (1.7) 10.8 (3.3) 7.7 (2.5) 2.0 (1.4)

Dividend Growth 1.9 (12.4) 2.8 (11.3) 1.0 (13.4) 4.3 (14.2) -1.5 (7.6) 2.8 (10.9) 3.4 (15.9) -3.6 (10.8) -0.1 (17.6) -0.5 (12.2)

Dividend Yield 5.8 (1.3) 6.1 (1.2) 5.4 (1.3) 6.0 (1.3) 5.0 (0.9) 6.3 (0.8) 5.7 (0.5) 6.3 (1.1) 4.6 (1.0) 3.7 (0.4)

Stock Return 9.0 (15.9) 9.7 (11.8) 8.3 (19.4) 11.6 (14.8) 7.3 (12.1) 8.2 (19.9) 9.7 (23.1) 5.8 (22.0) 7.7 (20.9) 9.8 (8.1) ~ Bond Return 0.6 (5.7) -0.7 (7.2) 1.2 (4.8) 2.5 (6.4) -5.6 (5.6) -3.0 (5.7) 1.5 (1.0) -2.3 (3.3) 5.9 (2.2) 4.7 (1.9) Q;)

~ Equity Premium

Realised Return 8.4 (17.3) 10.4 (15.1) 7.1 (18.4) 9.1 (15.2) 12.9 (12.3) 11.2 (16.1) 8.2 (22.7) 8.1 (20.4) 1.8 (20.9) 5.0 (8.3)

Gordon Estimate 7.1 9.7 5.1 7.8 9.1 12.1 7.6 5.0 -1.4 -1.5 ---_.- . - '------- ---- ----_.- -~--.- -----_.-

Table 4 Means of dividend growth rate, dividend yield, inflation and asset returns (annual percentages, standard deviations of the series in parenthesis).

40

2.3 ALTERNATIVE UTILITY SPECIFICATIONS

The explanation for the equity premmm and risk-free rate puzzles must

reconcile the low volatility of measured consumption and its low correlation

with stock returns with the high volatility of the pricing kernel implied by the

observed asset prices. The most straightforward approach is to modify the

functional form of the pricing kernel which leads to a consideration of

alternative and less restrictive preference specifications. While economists are

generally reluctant to play with utility formulations, this research has

produced a number of insights - in particular the effect of habit formation in

preferences has now become a standard tool in macroeconomic models. Time

interdependent preferences, including habit-formation and relative

consumption effects, have a long history in dynamic economics (see e.g. Pollak

[1976, 1970] and the references in Abel [1990]).

We will consider the recursive utility solution of Epstein and Zin, the habit

formation approach of Constantinides (1990) and Campbell and Cochrane

(1999), and the relative consumption formulation of Abel (1990).

All utility specifications seek to augment the utility function with an

additional state variable ht so it becomes U(C) = EoLJ3t u(Ct,ht).

41

2. 3.1 Habit Formation Preferences

In Constantinides (1990) the utility function is a generalisation of the CRRA

class, u{ct,hd 'Y-1{ct - hd' but consumption is measured relative to the term

ht L j ajct- j, where ht represents the habitual level of consumption or the

time varying subsistence level. In this formulation the marginal utility

depends on past (habitual) levels of consumption. An alternative, and

somewhat empirically more convenient specification of a habit, is the function

u(ct,ht )

For a consumer with habit formation preferences, a given level of consumption

is more satisfying when it is achieved quickly. Alternatively, a rapid

consumption decline hurts more than a gradual consumption decline thus

making habit formation preferences asymmetric relative to the standard

CRRA preferences case the consumer enjoys increased levels of consumption

more while becoming extremely averse to negative consumption shocks. Otrok,

Ravikumar and Whiteman (1998) show that, in contrast with time-separable

utility, this property makes agents averse to high frequency fluctuations in

consumption.

Addition of the habit level to the instantaneous utility function increases the

volatility of the MRS by making the argument of Ut more volatile. With habit

42

formation marginal rates of substitution may vary even if the consumption

level is fixed as the consumer develops a habit for a particular consumption

pattern!

The MRS for habit formation preferences with additive habits and a single lag

of consumption in Itt takes the form

fJ(Ct+l - aCt f"'l + fJa(Ct+2 - aCt+l f"'l . ( Ct aCt_ d-"'I + fJa ( Ct+1 aCt) "'I

2.3.2 Relative Consumption

Abel (1990) used relative consumption preferences based on the idea that

social standing may affect the satisfaction that an individual derives from a

given level of consumption. Now the representative agent's utility is made

dependent on the societal level of consumption as well as the level of personal

consumption

00

Ut(c) = EtL fJt+iu(Ct+i,Vt+d, (2.5) i=O

43

This formulation nests habit formation with a = 1, but makes the marginal

utility dependent on the aggregate consumption per capita Ct as well.7 With

the instantaneous utility function u (ct/Vt)l - 7

1 7 ,the marginal utility of

consumption at time t becomes:

In equilibrium, since all consumers are identical and the consumption good is

perishable, individual consumption ct is equal to per capita consumption Ct

and per capita output Yt

(2.7)

Substituting the expression for the marginal utility (2.6) into the pncmg

kernel and taking into account the equilibrium condition (2.7) the kernel

becomes:

7 Due to similarity of the predictions of both theories, vt sometimes, and somewhat

confusingly, is referred to as external (versus internal) habit.

44

yt/Yt I - the aggregate output

growth.

If consumption growth is Li.d, the pricing equation can be used to compute

unconditional expected returns. Abel used). 1, a E {O,l} , various values of

the risk aversion parameter 'Y from 0 to 10, and two distributional

assumptions for X t (two state Markov and lognormal) to examine the

predictions of the model. He found that preferences described by (2.5)

generate greater equity premia under both habit formation (a 1) and

relative consumption (a:;z::; 1 ), although the risk free rate appears to be too

high. Also a fairly high value for 'Y is still needed to force the equity premium

to be more than 600 basis points.

2.3.3 Non-Expected Utility

Epstein and Zin (E&Z,) building on the foundation of Kreps-Porteus temporal

lotteries, consider the utility function of the form:

45

where f.Lt is the certainty equivalent of uncertain future utility given current

information. The most important feature of Kreps-Porteus preferences is that,

unlike expected utility preferences, a consumer with such utility will generally

be sensitive over the timing of the resolution of uncertainty. It is also

interesting to note that E&Z preferences are inconsistent with habit formation

as the former must satisfy payoff independence (Kreps-Porteus [1978],

Corollary 4).

E&Z argue that one of the main advantages of using this form is that it allows

the time preference to be disentangled from the attitude towards risk. In

particular, the form of the aggregator W determines intertemporal

substitution, while JL is responsible for risk aversion. 8 E&Z considered

where Et is the conditional expectation taken with respect to the currently

available information. In this form the parameter 'Y determines intertemporal

8 The form of U determines preference towards the time when uncertainty is resolved (Kreps­

Porteus, 1978).

46

substitution while 0: can be interpreted as the relative risk averSlon

parameter. Expected utility is nested as a special case when 0: = 1) which can

be shown by repeated substitution

With this utility function the Euler equations for an agent who is endowed

with Wo of wealth at birth and receives no labor income during her lifetime

yield asset prices and returns which satisfy:

(2.8) 1

where w 0/, and Rtm is the gross return on the agent's portfolio of assets.

The pricing kernel is now a combination of consumption growth and the

return on the market portfolio. According to E&Z, the model can be thought

of as combining predictions from a consumption based model and the CAPM -

the covariance of the asset return with the return on broad wealth will affect

excess returns.

47

To test the model empirically E&Z (1993b) construct various proxies for the

consumption series; it is worth noting that, unlike most papers in the

literature, they attempt to construct a proxy that takes into account the

services provided by durables. The return on the optimal portfolio is measured

by the value weighted index of NYSE shares, and returns on industry groups

properly deflated are taken to represent Rt. The deflated series of 30 days

Treasury Bills is identified with Rt.

Euler equations were estimated by GMM for different time periods and for

different sets of instruments. Full estimation results are too extensive to

report here but the estimates are very unstable, imprecise and appear to be

very sensitive to the choice of instruments (which is not surprising as their

instruments are likely to be very weak) and consumption series. Another

troubling feature is that the time preference parameter is quite often

estimated to be negative ({3 > 1), although positive parameter estimates are

associated with better precision. The main findings are that the risk aversion

is estimated to be quite low and not significantly different from a logarithmic

specification for J.Lt, while the elasticity of substitution is close to unity.

Kocherlakota (1996) argued that E&Z's results are not robust to the choice of

specification. In particular, he questions the validity of the stock market

portfolio as an approximation to broad wealth. Assuming that consumption

follows a martingale he derives an alternative form of the Euler equations for

the E&Z preferences that are formulated entirely in terms of consumption. In

48

r

this form the E&Z preferences produce the same equation for the equity

return as the CRRA model and can not explain the equity premium puzzle.

Unfortunately, it is difficult to work out how these alternative formulations

were obtained, as the paper to which Kocherlakota refers the reader does not

in fact contain any details of the derivation. In fact, if the representative

agent model is applicable to pricing bonds and equities then the return on

broad wealth is identical to the growth rate of the aggregate consumption

endowment: Ct; ~ 1 = Rt+l' Substituting this into (2.8) reduces the Euler

equations for E&Z preferences to

Et [ ( Ct ~ 1 ) (a - 1) (Rr - Rf ) 1 = 0

[ ( Ct; + 1 )< a 1) b 1 1 Et (J -Ct- Rt

which is formally identical to the conditions for CRRA preferences!

E&Z preferences therefore work by introducing an additional factor into the

pricing kernel. The low a estimate obtained by E&Z might be merely a

consequence of the fact that a use of the stock market return as a proxy for

is likely to overstate the covariance of the

1)( Rr t -1 with the excess return (Rr Rf) in (2.8).

Including other forms of investment, such as bonds and real estate, with

realistic portfolio weights is likely to result in lower covariance between the

equity premium and the pricing kernel.

49

2.4 CREDIT MARKET EXPLANATION

Nominally risk-free government securities have special properties; III

particular, unlike most equities, they can be used to collateralise loans. In

addition, the liquidity and low risk of bonds and bills make them more

suitable than equities for short-term investment. Bansal and Coleman (B&C)

built a model on the idea that the puzzles can be explained by taking into

account the value of non-pecuniary services provided by bonds and bills. They

modify the basic setup of a representative consumer economy by introducing

different forms of transacting. In particular, in their economy a single non­

storable consumption good can be purchased with cash, checks and on credit.

The key to the model is that check purchases must be backed by bonds

deposited with the bank. Purch&Sing goods is costly; the costs to transacting

are determined by the (linear homogeneous) transaction technology

where the total real consumption Ct. = Ct.l + Ct.2 + Ct3 is the sum of the amount of

the consumption good purchased with cash (Ct.l), checks (Ct.2) and on credit

( Ct.3 ). Transaction costs are increasing with the level of consumption so that

[h%c > o. Purchasing goods with cash and checks is cheaper than using credit,

50

r r

so that, for a given level of total consumption ct, purchasing more of the good

with cash and checks decreases transaction costs; a%ctl < 0 and a%ct2 < 0 .

Since the technology is linear homogeneous these restrictions on derivatives

also imply that, for a given consumption level, transaction costs are decreasing

in the shares of goods purchased with cash and checks. Unlike goods markets,

financial markets are assumed frictionless; buying and selling securities is

costless.

Each period the representative agent receives an endowment y(s) , where s is

the single source of uncertainty driving the economy which evolves according

to a discrete-time Markov process. There is a government that targets interest

rates and money growth using open market operations and a printing press.

Apart from money, the government issues zero coupon one-period bonds. In

addition to bonds there is a risky asset (equity) that pays off 8t (s) in real

terms.

The utility function of the representative agent takes the standard discounted

CRRA form but the transactions structure of the economy adds restrictions

on portfolio and consumption decisions made by consumers.

Denote the nominal prIces of consumption, bonds and equity as PllBt,St

respectively and the holdings of bonds and equity - bt,st. The constraints that

agents face become:

51

CI. Cash ~ and dividends are spent on consumption and portfolio

adjustments:

C2. Checks are written against the nominal value of bonds and are cleared

in the same period:

C3. Households can not sell goods on credit:

C4. The budget constraint takes the form

at + 1 = PtYt Pt1fJ (ct, CIt , et;.d + + (~ + PtStDt - Ptclt Btbt - sd St + 1 - ad) + (Bt - PtO],t ) - PtC3t

Holding bonds enables the consumers to optimise their consumption patterns

by balancing the costs of different types of transacting and the opportunity

cost of holding cash.

B&C solve for an equilibrium in which all three forms of payment are used.

By assumption credit is the most expensive form of transacting. Therefore, if

52

credit is used, the Clower constraint (C1) and the checking constraint must

bind, thereby determining the nominal value of consumption purchased with

checks and cash. The rate of change in the nominal transaction cost as real

consumption changes will be denoted 'IjJ~ (.,.,.) = ~i (PtCf;) .

The agent can save in the form of either bonds or money. Buying bonds costs

Bt but also requires cash and such a purchase sacrifices the transaction service

return to cash -e2 (PtCt). Bonds also yield transaction service return -6 (PtCt) ,

making the total cost of a unit bond investment (1 + e2 (PtCt) - e3 (PtCt )). Saving

in cash has unit cost. These effect must be balanced at the optimum which

yields (with rt ):

(2.9)

This condition determines the price level Pt. Solving a rather lengthy set of

Euler equations B&C obtain the following characterization of the equilibrium

(7rHl is the inflation rate). The intertemporal balance condition becomes:

U' (Cf; ) = Et

u' ( ct + 1 ) 1 e2 (PtCf; )

l+el(ptCf;) l+el(pt+lCt+l) 7rt+l (2.10)

The trade-offs summarized by this condition involve, firstly, giving up

consumption at t at a cost of in utility with a transactions cost

economy of el (pted and, secondly, increasing consumption at t+1 to increase

53

r

utility at the rate '/1,1 ~ Ct + 1) ) which costs ~l (Pt + lct + 1) lost in transactions. t+l

The latter cost is partially offset by the cost reduction -~2 (Ptct) due to

increased cash holdings next period. Using (2.9), the equation (2.10) can be re-

written for the real bond return Rt as:

E t3 '/1,1 ( Ct + d 1 + ~l (Ptct ) 1 - ~2 (Pt + 1 Ct + 1 )

t '/1,1 (ct) 1 + €l (Pt + 1 ct + d 1 €2 (ptCd x (2.11)

1 - ~2 (Ptct ) (1 + Rb )j- 1 1 - €2 (Ptct ) + €3 (Ptct ) t + 1 -

A similar condition can be derived for the real equity return

E t3 '/1,1 ( Ct + 1 ) 1 + €d Ptct ) 1 - ~2 (Pt + 1 ct + d Re = 1

t '/1,' (ct) 1 + ~d Pt + 1 Ct + 1 ) 1 - €2 (Ptct ) t + 1 (2.12)

The mam difference from the Euler equations in the Lucas model is the

1 - ~2 (Pt + 1 Ct + 1 ) presence of the term m (2.11), reflecting the role that 1 €2 (ptCd + €3 (Ptct)

bonds play in transactions in the goods market. This term is absent from the

second condition because equity does not play any role in transacting. Higher

transaction service returns to bonds will, for a fixed nominal rate, yield higher

inflation and thus lower real return to bonds.

B&C take the standard CRRA instantaneous utility, specify the transaction

technology of the form

54

(2.13)

and estimate the model usmg GMM on monthly data from Citibase for the

period from January 1959 to June 1991. The real consumption senes IS

identified with the consumption of nondurables and services with the deflator

for this series taken as a proxy for the price level. Currency in circulation and

the series of M3 plus non-bank public holdings of short-term debt instruments

(US savings bonds, Treasury securities, commercial paper and banker's

acceptances) net of currency in circulation scaled by the average ratio of

nondurables and services to the gross national product are identified with the

cash component of consumption and the supply of riskless assets respectively.

B&C's estimates of fJ = 0.998 and l' = 1.49 are within the "acceptable"

parameter region but are estimated quite imprecisely. Crucial to the model is

the proposition that bonds provide transaction services which make bonds

more valuable than the standard model would suggest. For the transaction

service technology specified in (2.13), transaction service return is present and

of hypothesised sign if the parameter k is non-negative. B&C tested and

strongly rejected the restriction of no transaction return to bonds (Ho: k = 0).

They then fitted a V AR to obtain the evolution of the exogenous variables

and simulated the model to obtain average values for the equity premium and

the risk free rate. With the k estimate of 1.23 the model gives an average

equity premium of 2.42% which is below the historic value of 5.02% but,

interestingly, quite close to the "fundamental" value advocated by Fama and

55

French (2002). The real interest rate predicted by the model is too high at 4%

against 1.12% observed in the data. The model was simulated for a range of k

values. At the higher end, with k = 2) the model's predictions for the equity

premium and the risk-free rate are very close to the data at 5.25% and 1.46%,

but no other statistics are reported, so this outcome is hard to evaluate.

2.5 LIQUIDITY

An interesting liquidity based explanation of both puzzles was suggested

recently by Swan (2002). There is an extensive body of empirical literature -

see Swan (2002) - that shows that low liquidity attracts a considerable

premium in financial markets. Assets with low turnover and high transaction

costs tend to have high expected returns. This conclusion survives after

conditioning on commonly used priced factors; in fact, for NYSE stocks, the

liquidity effect appears to be stronger than the widely-documented size effect.

The markets for government bonds, both in terms of their turnover and

average levels of bid-ask spreads, are typically much more liquid than equity

markets. Swan's argument is that the large equity premium is the

compensation that investors demand for the relative illiquidity of most

equities.

56

Swan's contribution is in formulating a representative agent model that allows

for explicit valuation of liquidity. In his model a representative consumer

derives direct utility from trading shares and bonds. The representative

consumer solves a two-period problem

max U(Yt,Yt+l) S,b,Te,Tb

= u( cd + Ed u( CHl + si( 7 e ) P~,Hl

s.t.

Ct = ~ - SP~t b ,

bi ( 7b ) pb,t+l ) }

CtH = ~+1 + sP:'Hl (1- Ce7 e ) sD + b(1 + D - Cb 7 b)

(2.14)

where e is non-investment income, sand b are the holdings of equity and

bonds and P: and pC are the ask-prices of equity and bonds. Equity and

bonds both provide an identical non-stochastic end of period dividend - D.

The function i (.) converts liquidity, measured by turnover per dollar traded r

- into its consumption equivalent.

The novel element of the model is that the equilibrium turnover rates arise

endogenously as the consumer chooses the level of turnover to equate the

liquidity benefit from the marginal unit of investment in equities and bonds to

the incremental transaction costs per dollar of turnover - Ce and Cb:

i '( re) Ce

i'(rd = Cb-

Swan specifies the liquidity benefit function in the form

57

(2.15)

(2.16)

which, when taken together with (2.15), implies that the turnover elasticity

with respect to the relevant transaction cost (ce for equities and CIJ for bonds)

is constant

(2.17)

Substituting (2.16) into the expression for the equity premIUm implied by

(2.14)

(2.18)

Swan obtains the expression for the equity premium in terms of the turnover

rates of bonds and equities

(2.19)

which together with (2.17) forms the testable implications of the model.

Based on the formula (2.19) and estimates of the liquidity coefficient a and

turnover elasticity {3 = - ~ , the paper demonstrates convincingly that there is

a strong relationship between transaction costs and turnover and turnover and

the equity premium. On the empirical side, however, it would be interesting to

58

, j

see how the liquidity based explanation would address the yield differences

between different classes of bonds themselves.

Most importantly, government bonds, that Swan uses for calibrating bond

turnover, are not the only fixed-income instruments available. In 1993 in the

US, for example, there was $2.3 trillion in outstanding Treasury debt, $1.4

trillion of corporate and $802 billion of municipal debt (Fabozzi [1996]). The

liquidity of the latter two markets was much lower than that of the Treasuries

market. While the turnover of Treasury bonds was roughly 11% per day over

1995-1997 (Chakravarty and Sarkar [1999]), the daily turnover of corporate

and municipal bonds over the same period averaged below 3%. Mean volume

weighted bid-ask spreads were also wider in the municipal and corporate bond

markets; Chakravarty and Sarkar calculate them as 22c per $100 for

municipals, 21c for corporates and only 11c for the Treasuries. Over the same

period the yield spread over Treasuries for all corporate bonds was only about

1 % per year (7.35% for corporate bonds and 6.35% for Treasury bonds) and

was much lower for issues with high credit ratings. Moreover, municipal bonds

were actually selling at a discount relative to the Treasuries, yielding only

5.44% per year.

Swan does not provide direct estimates of turnover elasticity for the US, but

empirical literature suggests that a turnover elasticity of one is plausible (the

estimate for Australia provided by Swan is 0.78). Using a unit turnover

59

r !

elasticity the ratio of the equity premmm to the yield spread between

corporate bonds and Treasuries SPt can be expressed from (2.19) as

epti "/SPt (2.20)

Over the period 1980-1991 the average turnover of Treasury securities was

roughly 25 times the turnover of NYSE equities (see Table 1 in Swan [2002]).

The figures reported Chakravarty and Sarkar [1999] suggest that the turnover

of Treasuries is about 3.5 times the turnover corporate bods. Assuming that

these ratios are reasonably stable over time, (2.20) would imply that the

spread between corporate and Treasury yields should be 40% of the equity

premium or about 3% per annum based on the historical estimate of the

premium, which appears implausibly high.

Unlike the equity premium, the long-term movements in which is still an issue

for debate, bond spreads are empirically very persistent and contain a

considerable amount of predictable short-term variation. While not attempted

in this study, the liquidity theory could be tested more formally by examining

whether movements in spreads can be explained by changes in liquidity.

Another difficulty with the liquidity explanation is that it is not quite clear

what the restrictions in (2.19) and (2.17) have to do with the representative

consumer model. The strength of consumption-based models is that they

60

impose restrictions on the joint distribution of consumption and asset returns.

In Swan's model these restrictions do not feature due to a peculiar

specification of the optimization problem where liquidity provides benefits in

the future but not in the current period.

2.6 INCOMPLETE MARKETS

The interpretation of the puzzles relies on accepting aggregate consumption as

a proxy for individual consumption. This aggregation condition holds if

markets are complete, in which case marginal rates of substitution of

consumers are equated state by state so that idiosyncratic shocks are

diversified away and individuals bear only aggregate uncertainty. However, if

markets are incomplete, smooth aggregate consumption series can hide a

greater degree of idiosyncratic variation in disaggregated consumption.

Mankiw (1986) showed the importance of these background risks by pointing

out that, if only a small proportion of consumers bear the consequences of

uninsurable consumption shocks, any level of the premium can be reconciled

with a given volatility of the aggregate consumption series. Mankiw considers

two periods and two equiprobable income states: in the good state aggregate

consumption is f.L, in the bad state consumption falls by 1/Jf.L (1/J < 1) relative

61

to the good state. In addition, the shock is concentrated, so that only a

proportion ). of consumers experience a shock of "": each and these shocks are

independent across the population. There are bonds that pay 1 unit of

consumption independent of the realization of the state and equity that only

pays-off in the good state. With three states for each individual the optimality

condition for the excess return on equities becomes

which can be solved for the equity premium

1f

Differentiating the equity premium

<0 u (It)

62

if preferences exhibit prudence.9 In other words, as the shocks become more

concentrated (>' -+ 0) the premium goes up. In fact, it is easy to show that, if

the utility function satisfies the Inada conditions, so that u' (c) -+ 00 as c -+ 0,

then, as >. -+ 0 1 lim7f(>') -+ 00, and therefore the equity premium can be made

arbitrarily large by making a small enough part of the population bear all the

consequences of a shock.

Unfortunately, it is difficult to relate the simple two period model of Mankiw

to the actual dynamics of consumption and asset returns. Subsequent studies

have shown that, in multi-period dynamic models, the consequences of a given

uninsurable shock can be mitigated considerably if consumers can respond by

selling off their assets. Introducing dynamic trading allows agents to come

surprisingly close to the complete market solution, even in the presence of

transaction costs, by maintaining the inventory of stocks and bonds. 10

9 Preferences that are prudent have U III > O. Under this condition precautionary demand for

savings increases with uncertainty.

10 Unless income shocks are persistent.

63

Mankiw and Zeldes (1991) used the Panel Study of Income Dynamics to

compare consumption of stockholders with that of non-stockholders. They

found that consumption of stockholders is more volatile and more highly

correlated with the return on equity; their correlation estimates reduced "I for

stockholders by 1/3.

Kahn (1990) attempted to endogenize the amount of idiosyncratic risk that

individuals must bear in an economy with asymmetric information. In his

model agents have access to a productive technology such that the output

produced by an individual agent is affected by a binomial state variable e and

the unobservable amount of effort ~; the output Yi of the i - th agent can

take one of two realizations with probabilities P(Yi j) = p(~,e). Since there

are an infinite number of identical agents, and output realizations are

independent across agents, uncertainty over the individual level of output

averages out in the aggregate and, therefore, the aggregate output Yo is

entirely determined by the state of the economy e. Kahn then considers a

central planner's problem of choosing the optimal sharing rule

c~ = asyi + (1 as)Yo' maximizing the utility of a representative agent

U(c,l) = E[u(c) - vel)] (increasing in consumption and decreasing in effort l)

subject to the constraint that the agent who takes aggregate output as given

chooses effort optimally. Imperfect risk sharing induces the agent to deliver

effort in states where it's desirable. Completely insured agents (au = 0) have

no incentive to provide effort.

64

Kahn then takes the CRRA form for u(c) and v(c) and a logit form for p(~,(J)

and computes optimal levels of risk sharing for various parameter values. The

numbers indicate that, to induce optimal efforts, agents must bear 15-20%

more risk (as measured by the standard deviation of consumption) relative to

the range of deviations in the aggregate market output.

In the next step he considers the decentralized outcome conditional on the

optimal level of sharing. Equity and debt are traded in a competitive market

and agents choose their portfolio composition ", so as to maximize utility with

a given sharing level and subject to the constraint

c~ anyi + (1- ae ){1]Ye + (1 1])B}. With the bond price B set to induce a no

trade outcome (1] = 1) this leads to the equity premium

-Cov ( (1 - a) c -"1, Y ) / E ( (1 - a) c -"1 )

The equity premium is determined by the covariance of the risky asset - the

claim on a unit of aggregate output Y - with the marginal utility.

Kahn constructs a synthetic two period model where agents are endowed with

a unit of capital and produce random output (y{);. Numerical solutions

indicate that, for the range of parameter values considered by MP, and for the

amount of idiosyncratic risk consistent with the model, the equity premium

only goes up from 0.8% to 1.2%. From the unreported simulations Kahn

claims that producing higher premiums would require some probability of an

65

"idiosyncratic catastrophe - a decline in consumption greater than 95%." The

main problem with explanations that rely on consumption catastrophes

however (see also Rietz [1991]) is that the possibility of such events will

depress the prices of bonds as well as equity, thereby exacerbating the risk-

free rate puzzle.

2.5.1 The Effect of Transaction Costs

Lucas (1994), Heaton and Lucas (1996) work with the standard CRRA time

separable utility, but the constraints are modified to include transaction costs

and short-sale restrictions. The constraints facing individuals become

c1 + StS1 + Btb: + k( s: + 1,s1iZt) + w( bt + 1,b1iZt) ~ s1 (St + 0t) + bt + el sf ~ Kt,bl ~ Kf,

where the ke .. ) and we .. ) functions represent transaction costs to trading in

stocks and bonds respectively. There are two agents, both with identical

utility parameters but individual income shocks. The state variable Zl

contains the following exogenous processes:

Xt = ~t - aggregate income growth (Yi yi + or is constructed as the

sum of aggregate labor and dividend incomes);

66

dt ~ - the dividend share in aggregate income. (Xt.dt) was calibrated

from a bivariate autoregression using the data from the National

Income and Product Accounts;

1]1 = Yi; is the first person's fraction III the labor income (1]; = 1 - 1]i) . Yi

The process for 1] IS taken to be a first-order autoregression

In 1]£ = Tj + pin 1]L + 17Gb with p and a set to equal their average values

in the panel. 11

The state process is then approximated with an 8 state Markov chain by

moment matching.

Heaton and Lucas then consider model predictions for various cases including

a complete market, a frictionless market and different specifications for cost

functions (with 'Y preset to 1.5). The most surprising finding of this research

is that, for realistic levels of transaction costs, by maintaining an inventory of

assets, the agents can mitigate the effect of consumption shocks and the

economy comes very close to the complete market solution. Since in their

11 HL also consider a specification with heteroscedasticity related to the state of the cycle.

67

model idiosyncratic income shocks are transitory, agents can smooth out

negative individual consumption realisations by maintaining a buffer stock of

assets which are sold off during bad times and accumulated during good

times. Although the model produces sizable risk premia, they are still

considerably below those observed in the data.

2.5.2 Persistence of Individual Income Shocks

Constantinides and Duffie (1996) effectively generalised Mankiw (1986) to a

fully dynamic setting. The common property of the models considered above

is that income follows a stationary process with no (Lucas [1994]) or low

(p = 0.5) (Lucas and Heaton [1996]) persistence. Constantinides and Duffie

construct an example where nonstationary income/consumption processes

can support any given asset prices in equilibrium.

The existence of the pricing kernel is not a unique prediction of consumption

based models. Under much weaker conditions of the absence of arbitrage

opportunities there exists a (possibly non-unique) pricing kernel fflt (see e.g.

Duffie [1994]) so that asset prices are characterized by the condition:

68

Taking an arbitrary aggregate consumption process Ot, let individual

consumption c; be related to Ot VIa c; DitCt. The idiosyncratic shock Cit

follows a very particular process

(2.21)

where Cit is a sequence of normal shocks, independent from Yt as well as

across consumers and time and e-P = f3. In addition, the measure space on the

set of consumers is constructed in such a way that f e;tdJ.L(i) 1.

The key difference between the consumption processes described in (2.21) and

those used in Heaton and Lucas (1996) is in the persistence of the series of

idiosyncratic shocks. Under the assumptions that Constantinides and Duffie

make about the behavior of individual consumption (2.21):

(2.22)

and, using the independence between cit and Yt,

69

Therefore, a shock to individual consumption at time t in the Constantinides

and Duffie model has a permanent effect on all the subsequent consumption

levels and can not be mitigated by temporarily reducing asset holdings as in

Heaton and Lucas (1996).

Constantinides and Duffie show that, with individual consumption processes

defined in (2.21), marginal rates of substitution are independent of the

consumer's identity i and reproduce the deflator process Tnt:

MRS;/t+1 MRS[/t+l mt, 'Vi, j. So the pre-specified price process Pt is indeed an

equilibrium price process for this economy.

The result can be interpreted in relation to the asset pricing puzzles by noting

that the Euler equations can be expressed as

This expressIOn can be simplified further by usmg the law of iterated

expectations and noting that Ct +1 and Rjt+1 are known at time t + 1

70

1 Et [.6( etc: 1 r'l' Et + 1 {exp['I'( tit + 1Yt + 1 ~ Y1i1 )l}Rjt + 1]

= Et [.6( etc: 1 r'l'exp( '1'2 .; 'I' Y; + 1 )Rjt + 1]'

(2.23)

It is straightforward to show that under (2.21) 109(~!::~~) - N(_Y~l ,yl+l)'

Therefore yl+l in (2.23) is the variance of the cross sectional distribution of

consumption growth In general, ignoring the term et+l = exp ( "l ;'l' Y;+l 1 would

bias the estimate of '1'. If, as in the example considered by Constantinides and

Duffie, inequality increases in economic downturns, when aggregate

consumption growth is low (i.e. ~t+l co-varys positively with consumption

growth), 'I' would be overestimated.

The Constantinides and Duffie model is particularly interesting because of its

parsimony. It "explains" price behavior in terms of consumption very simply

as there are no capital market imperfections, money, non-separabilities in

utility, etc. The question remains of whether this specification is empirically

plausible. In particular the model implies a divergent cross-sectional income

distribution. Some agents must suffer the consequences of bad consumption

shocks, from which they never recover due to persistence of the consumption

processes (c: follow random walks according to (2.21»). Deaton and Paxon

(1993) find some evidence in favour of diverging cross-sectional income

distributions using pseudo-panels constructed from the PSID.

71

An additional complication resides in the no-trade nature of the equilibrium.

This is innocuous in representative agent models but is much less so in this

model. Constantinides and Duffie encourage us to think of idiosyncratic

consumption processes as post-trade consumption allocations, but this

interpretation may not be viable. It would require either the security holdings

to be fixed, which is an unreasonable restriction, or consumption to be

supported by optimal trading strategies, which may well be incompatible with

the given structure of the consumer problem.

2.7 CONCLUSION

This chapter reviewed some of the most influential attempts to explain asset

pricing anomalies. Although the literature has contributed greatly to

understanding the economic fundamentals of asset returns a full explanation

of the puzzles is still lacking.

Incomplete market models cast some doubt on the usefulness of aggregate

consumption series for asset pricing. The important unresolved question is the

nature and the amount of idiosyncratic variation in individual consumption

and chapters 3 and 4 consider the effect of one important source of this

variation.

72

CHAPTER 3

ASSET PRICING IN OLG MODELS

3.1 INTRODUCTION

A number of recent papers (Constantinides, Donaldson and Mehra [1998],

Farmer [2001]) explored the effects of intergenerational transfers and

incomplete participation in insurance markets on asset pricing. The former

paper, in particular, presented a simple and economically convincing

explanation of both a low risk free rate and a high equity premium.

Constantinides, Donaldson and Mehra (CDM) examined an OLG economy

with three generations: the young, the middle aged, and the old. Agents'

claims on consumption are derived from three sources: the exogenous return

73

on their human capital (wage); ownership of a safe technology paying b units

of consumption per period; and shares in a risky technology paying an

uncertain amount of consumption dt . CDM show that restricting the young

from borrowing can simultaneously produce a high return to equity and a low

return on the safe investment. This outcome can be attributed to the relative

size of the wage endowments and the properties of the equilibrium price

process.

Furthermore, they claim that the effect of the borrowing constraint is likely to

be of considerable importance and will remain a significant contribution to the

explanation of the puzzles even in a model with more than 3 generations.

The CDM argument about the directions of the effects on the relative levels of

returns on the available investment opportunities is very convincing. However,

in order to make a numerical solution to this quite complex dynamic general

equilibrium problem feasible, the paper relies on a large number of simplifying

assumptions and a fairly loose calibration procedure.

This chapter examines the effect that changing some of the assumptions of the

CDM model has on average returns.

The asset market in the CDM economy IS incomplete along a number of

dimensions. Firstly, the young generation IS not directly represented in the

insurance market before they are born and, as a result, the young face risks

74

against which they have no insurance. Incompleteness also arises because, at

least if one were to interpret the model literally, the structure of the economy

with the implied time period between trades of 25 years imposes rather severe

limitations on portfolio rebalancing. And, lastly, the economy is driven by an

exogenous Markov process with 4 states, while only two independent assets

are available.

We intend to show that the market incompleteness built into the model by

itself imposes significant restrictions on the composition of available trades,

leading to an over-concentration of risks when the borrowing restriction is

imposed. Much of the presentation in this and the following chapter relies on

a simple argument that suggests that the numerical results obtained by CDM,

in particular the magnitude of the effects, come not from the imposition of the

liquidity constraint but from the market incompleteness related to the latter

two effects, namely, the calibration and the asset structure.

A problem with the interpretation of the CDM results is that it is not clear

whether it is the life-cycle or the market incompleteness that is responsible for

the rather dramatic conclusions. To examine how sensitive the results are to

changes in these fundamental assumptions we modify the asset structure by

making insurance markets conditionally complete.

This chapter presents some results on the convergence of the OLG economy

equilibrium to that of a representative agent economy.

75

3.2 CDM MODEL

CDM considered an exchange economy populated in each period with three

generations of consumers. The economy is driven by a single source of

uncertainty that follows a discrete-state homogeneous Markov chain with M

states S {SJ:1 and the transition matrix II = {7r (Sj I sJ Lj=l,M with the

elements corresponding to the probabilities of transition from the state Sj to

the state S j' The realisation of the factor determines the distribution of wages

and dividends: w; = Wi (St),dt = D(St)- here i = 0,1,2 refers to one of the

three generations.

The economy is equipped with a very simple asset structure consisting of two

traded assets. The assets represent shares in a risky technology (equity)

paying a stochastic dividend dt and a safe technology (a bond) returning a

fixed amount of consumption b every period and in every state. Both

securities are perpetuaL The aggregate endowment of resources available for

consumption in each period is equal to the sum of the aggregate wage, the

dividend and the bond coupon:

76

The economy is an overlapping generations, pure-exchange economy with a

single composite consumption good which is a numeraire for all prices. The

decision problem of the young is to select a trading strategy e specifying how

much of each asset to hold in the portfolio in each future period for every

possible history to maximize the lifetime utility:

m:x Et I: f3iu (c;:~ (0») i=0,2

(3.1)

where c; stands for the consumption of the i-th generation at time t and the

conditional expectation is taken with respect to the information available to

the agent.

The agents receive exogenously determined wage endowments driven by the

aggregate state St and, in the most general setting, can invest in n assets

paying state contingent dividendsdt = d(St)' There is no bequest motive so

the old simply consume their endowments and the young start their life with

no inheritance. The agents allocate current period wealth, consisting of their

labor income and the value of their asset holdings, between consumption and

portfolio investment:

d /1i-1 tUt_l' (3.2)

Here 0; (O;,j )j=1,,.. is the vector of asset holdings of the i-th generation and

Pt is the vector of asset prices. Agents live for N periods, inherit no financial

77

wealth - ()~ 0 - and receive no wage income when old. Without loss of

generality, the total supply of financial assets can be normalized to unity in

each period.

When making decisions consumers are assumed to have observations on the

complete history of prices and asset demands, although in the stationary

equilibrium defined below it is sufficient to have observed only the realisation

of the current state vectorxt

, which includes current asset prices and the asset

holdings carried over from the previous period (or alternatively asset prices

and the distribution of wealth across generations).

Equilibrium: A (stationary) equilibrium is a collection of price processes P (x) ,

security holdings e (x) defined on their respective domains

Q Q Q - Q x Q such that P and e satisfy agent p' ()' x - P (J

optimality, clearing financial markets - 2::i (Ji (x) = 1, \/x - and

the aggregate resource constraint 2:::1 ci (x) = C (x), \/x in all

states.

CDM show that stationary equilibria exist under usual conditions on the

instantaneous utility function. Equilibrium consumption allocations and asset

holding satisfy first-order conditions that represent the balance between the

utility cost of current consumption forgone by investing in an extra unit of an

asset and the utility benefit of the expanded future consumption possibilities

that are brought about by this investment:

78

(3.3)

In what follows the analysis will be restricted to stationary equilibria and

whenever it does not cause confusion the dependence on the current state will

be suppressed into the time subscript.

3.3 COM CALIBRATION

CDM solve the model with three generations of consumers. The annual

subjective discount factor (3 is set to 0.96 and a homogeneous Markov chain

with 4 states is calibrated so that the joint process of aggregate income and

wages {y(s)=b+W(s)+D(s),W(s)} fits:

1. The average share of income going to labor,E[:l E {O.6,O.69}, where

W = WO + WI is the total labor income which is distributed between the

young WO and the middle-aged w l;

WO 2. The average income share of the young,-- E {0.16,0.2};

E(y)

3. The average share of income going to pay interest on debt, ~( = 0.3 ; E y)

79

4. The coefficient of variation of the 20-year wage income of the middle

a(w1)

aged, E (w l ) 0.25;

5. The coefficient of variation

. a(y) {0202~}' mcome, E (y) E .,. v ,

of the 20-year aggregate

6. The 20-year auto-correlations and cross-correlation of the labor income

of the middle-aged with aggregate income,

The calibrated model is solved numerically by fixed-point iterations. The

mean equity premium and risk-free rate implied by equilibrium relationships

are compared with these of a constrained economy in which young agents are

restricted from borrowing from the middle-aged.

CDM demonstrate that the borrowing-constrained economy produces a

significantly higher equity premium and a lower risk-free rate, thereby making

a step towards a joint explanation of the equity premium and the risk-free

rate puzzles.

The CDM paper explains the results by the differences in attitudes to risk

between the young and the middle-aged and the resulting differences in their

desired asset holdings over the life cycle. The middle-aged have to rely on

80

r ! i

their financial wealth when old and are therefore unwilling to sacrifice the

certain consumption payoff offered by the safe instrument for a high but

uncertain equity return. The young on the other hand derive most of their

middle-age consumption from wages and find high yielding equity an

attractive investment alternative to bonds. In the unconstrained equilibrium

the young's demand for equity drives down the equity return. At the same

time the young borrow to finance current consumption, pushing up the

equilibrium return on bonds. In contrast, the constrained economy equity and

bond returns must accommodate a reduced demand for loans and equity,

thereby generating lower bond returns and a higher expected return to

holding equity.

Despite the obviously stylised structure of the economy, CDM argue that the

effect of the borrowing constraint is of considerable importance and is likely to

survive modifications of the basic assumptions of the modeL In particular,

CDM hypothesise that the results in a model with more generations are likely

to be quantitatively similar:

" ... we may increase the number of generations from three to

sixty, representing consumers of ages twenty to eighty in

annual increments. In such a model we expect that the

youngest consumers are borrowing-constrained for a number of

years and invest neither in equity nor in bonds; thereafter they

invest in a portfolio of equity and bonds, with the proportion of

81

equity in their portfolio decreasing as they grow older and the

attractiveness of equity diminishes." (CDM [1998])

They also discuss a number of possible modifications to make the model more

realistic and conclude:

"We suspect that in all these cases the primary message of our

paper will survive: the borrowing-constraint has the effect of

lowering the interest rate and raising the equity premium."

(CDM [1998])

The economic argument in the paper relies heavily on the portfolio

composition implied by the model. It is difficult to assess how plausible this

implication is as empirical evidence on changes in life-cycle asset holdings and

attitudes to risks is rather slim. Existing evidence, in fact, suggests a

relatively flat age profile of the share of equity holdings for those households

who own equity.12

12 See the introduction by Guiso, Haliassos and Jappelli in Guiso et. al. (eds.), 2002.

82

Theory also provides relatively little guidance. Investment recommendations

are traditionally tilted towards a greater proportion of high-yielding risky

assets - equity in particular - for the young, and more conservative investment

options for the old, and appear to be based on a purely probabilistic idea that

early investment losses could be recouped if the investment horizon is long

enough. The fallacy of this argument was demonstrated by Samuelson who

pointed out that, with time-invariant investment opportunities, the standard

optimal consumption problem with HARA-utility produces portfolio

compositions that do not change as the individual gets older. With time­

separable preferences and complete markets, the solution to the dynamic

portfolio allocation problem involves choices based on a trade-off between

current and the immediate future period, conditional on the information

available to consumers at time t.

These results however are not directly applicable to exchange economies with

interacting heterogenous agents, because in the latter the set of investment

opportunities arises partly endogenously. This contrasts with the case of the

standard optimising representative agent models where investment

opportunities are driven by exogenous factors with pre-specified dynamics.

More recently, Gollier and Zeckhauser (2002) investigated the relationship

between risk-taking and the horizon length more explicitly. They examined

the conditions that would insure that young investors in an economy with

exogenous state prices are less risk averse than middle-aged investors. If the

83

markets are complete and the risk-free rate is zero, these conditions require

that the risk tolerance (the inverse of the coefficient of absolute risk aversion)

is convex in the end-of-period wealth. In our general equilibrium scenario the

distribution of wealth arises endogenously and the above results are difficult

to verify directly. But, importantly, the authors note that the attitude to risk

is ambiguous even in such a stylised setting.

Furthermore, Gollier and Zeckhauser (2002) suggested that:

"Guiso, Japelli, and Terlizzese (1996) tested the relationship

between age and risk-taking in a cross-section of Italian

households. Their empirical results show that young people,

presumably facing greater income risk than old, actually hold

the smallest proportion of the risky assets in their portfolios.

The share of risky assets increases by 20% to reach its

maximum at age 61."

Investment horizon is only one of the factors affecting investment decisions of

the young. It can be argued that more important considerations driving risk­

taking behaviour are related to labor income and its co-variation with the

aggregate dividend and stock returns. The peculiarity of the young's position

in relation to their labor income and human capital endowment could affect

investment behaviour in a number of ways, and the overall effect on their

ability to absorb risks can be ambiguous. Without attempting a complete

84

characterisation, two offsetting effects can be seen to be at play. On the one

hand, as pointed out by Gollier and Zeckhauser, the young are facing greater

income uncertainty against which there is very little tradable insurance. On

the other hand, the young have potentially greater scope than the old and the

middle-aged for adjusting their labour supply in response to adverse

realisations of equity returns. The first effect is likely to depress the demand

for equity of the young, while the second may attenuate the utility loss in the

states with low equity return and will tend to work in the opposite direction.

We also find the CD M claim regarding the effect of the borrowing restriction

on the equity premium in an economy with a larger number of generations

very intriguing. Although no formal justification is offered one may

hypothesise that it probably originates in the idea that, typically, most gains

from trade come from trading between agents with the most dissimilar

endowments. In the context of the model this would mean that the young

would mostly trade with the middle-aged even if trading was allowed to take

place more frequently. This reasoning would disregard the fact that at least

some, if not most, of the rationale for trading in the exchange economy comes

from hedging demands. If trades occur more often the young can gradually

transfer their wealth across time as well as across states of nature.

Lastly, the effect of the borrowing restriction, when taken together with the

calibration procedure, results in an over-concentration of equity risks that

exacerbates the dividend shock suffered by the middle-aged in the low

85

r i

dividend state. Trading with the young is the only diversification vehicle

available to the middle-aged; in the borrowing-constrained economy when no

trading possibilities exist their portfolio composition is directly determined by

the relative supplies of equity and bonds. The equity risks in particular are

determined by the ratio of the aggregate dividend over 25 years (a flow) to

the current stock of wealth. If markets were open more often, even with the

existing asset-structure, by trading with each other the unconstrained

generations could create synthetic portfolio insurance that would make equity

more attractive. This effect creates another puzzling feature of the model; it is

somewhat difficult to rationalize the borrowing restriction, when trading with

the young is in fact in the best interest of the middle-aged.

In short, although we still expect the CDM conjecture regarding the direction

of the effects to hold under more general conditions, we suspect that the

introduction of additional generations is going to have a considerable effect on

the size of the equity premium.

3.4 EXTENTIONS TO COM

The main topic that will be investigated in this chapter is the effect of the

number of trading generations or, alternatively, the period between trades, on

86

the size of the equity premium. The main conclusion of the analysis is that

the results obtained by CDM follow mostly from the presence of market

incompleteness rather than the effects of the life-cycle. Once the markets are

completed the equity premium and the risk-free rate in the OLG economy are

indistinguishable from those obtained in a representative agent economy.

The CDM model is modified here in a number of ways. Market completion is

the most important modification and is introduced to isolate the effect of the

market structure from the effect of the borrowing restriction and the effect of

births and deaths. To analyse the effect of the borrowing restriction we also

simplify the calibration procedure to obtain a series of calibrated economies

consistent with the short-term dynamics of the aggregate wages and income.

The dynamics at a lower frequency are then obtained through time

aggregation.

We will start with a discussion of the market structure and the differences

between the OLG economy and the representative agent benchmark. We then

describe the calibration procedure and conclude with a series of simulation

exercises using the calibrated economy.

9.4.1 Financial Structure

With only two assets (stocks and bonds) available in the CDM version of the

model the budget constraint (3.2) becomes:

87

(3.4)

where B?, B;,e and P:, P: are time t holdings and prices of equity and bonds

respectively. Under the standard monotonicity assumption on the utility

function these constraints will be satisfied with equality.

Now assume that, in addition to equity and bonds, individuals can also buy

and sell insurance claims on consumption in any given aggregate state (Arrow

securities or digital options in finance terminology). The payoff on simple

securities is 1 in the state on which the security is written and zero in all

other states:

d =8 S = A {I, if St = S S S ( t) 0, if St += S (3.5)

Denote the price at time t of the claim on one unit of consumption in the

state s at time t+ 1 by >-ts and its time t holding by the i-th generation as

i Zts' The total consumption payoff received at time t from the portfolio of

Arrow securities carried over from the previous period is zi - 1 and the t -l,St

budget constraint becomes:

88

The equilibrium conditions can be rewritten:

1. Euler conditions for equity and bonds:

U1(C:)

U1(C:)

Et {fJul(C;! i)(p; + 1 + dt + 1)} Et {fJu I ( c; ! i) (pZ 1 + b)}

2. Euler conditions for simple securities:

(3.6)

(3.7)

(3.8)

here 'if (St Is) is the probability of transition from the current state into s.

3. Financial market clearing:

Eot,b = l,EOi,e = 1 (3.9) i i

4. Insurance market clearing:

(3.10)

89

A number of simple observations follow directly from (3.5)-(3.10) Firstly, the

conditions (3.7) and (3.8) taken jointly imply that equity and bonds are

redundant securities in an equilibrium, since bond and equity can be priced,

once the prices of simple securities are determined, with the formulae

P~ L\8(pf+l +dt +1) 8

pf = L \8 (pf + 1 + b) s

(3.11)

Secondly, equilibrium asset holdings are not defined. In particular, take

{ 0t' Zt} to be a part of an equilibrium allocation and consider any arbitrary

redistribution of securities {60; ,6Z;}. If the following conditions are satisfied

{O; +60:,z; +6Z:} is still an equilibrium allocation supported by the same

price process Ats (x):

(3.12)

If (3.12) holds the redistribution preserves the equilibrium in the financial

market while still satisfying the aggregate resource constraint.

The agents are indifferent between holding a portfolio of bonds and stocks or

a portfolio of simple securities replicating its payoff state by state. This

observation implies that stocks and bonds can be redistributed arbitrarily and

90

allows us to concentrate on the analysis of the allocation of state-dependant

consumption claims.13 Under (3.11) consumption can be rewritten

i _ i + i-I + d oi-t.e + bOi-l,b _ pe (Oi.e _ Oi-l,e) _ pb (Oi.b _ Oi-l,b) _ '" \ Zi ct - W t Zt-l.91 t t-l 1-1 t t t-l t t t-l L...,; ''t8 ts

s

9

and

where V:. s is the claim on consumption at time t + 1 in the state s held by

the members of the i-th generation and is defined on the same state space as

the equilibrium price functions pi + 1 (Xt,s) and p; + 1 (xt,s).

Therefore, with complete markets the problem simplifies to that of finding the

consumption demands V;.s and state-prices that jointly satisfy the Euler

conditions:

13 Alternatively the net supply of insurance claims could be frxed at zero.

91

I { i + i-1 " ). ;} \ U Wt Yt-l,1It, - L;- ''t,sY/,s ''t,B

a I { ;+1 +; " \ i+l} ( I ) =,.."U Ws Yt ,9 - L;- ''t+l,sYt+1,s 1T St S

and which clear the markets I: c; Y (8) :

o Yt,s Y:'a = 0 ,

The state vector includes state prices and the distribution of wealth.

(3.13)

(3.14)

Given the equilibrium state-price relationship any asset in the economy can be

priced from arbitrage considerations and the unconditional means of returns

on any asset can be computed using the implied distribution of the state-

vector. For example, the asset representing the next period dividend claim

returns on average

(3.15)

where 1T* is the stationary distribution of the exogenous Markov chain driving

the aggregate values and F (.) is the unconditional distribution of asset-

holdings.

92

r

8.4.2 OLG versus Representative Agent

In this section pricing III an OLG economy with complete markets is

compared to that in a representative agent economy where a single consumer

is the sole recipient of the aggregate consumption endowment.

If the instantaneous utility function u (c) is the same for all agents and is

homogenous in consumption, (3.13) implies that consumption growth is the

HI Ct+l .

same for all agents -i = of" Let C;' denote the aggregate consumption of c

t

all generations except i. Since

the common consumption growth

Although we can not prove this result formally, in any economically plausible

equilibrium the shares of consumption going to the youngest and to the oldest

should converge to zero uniformly for all states as the number of generations

N grows. A large share of consumption going to the young must be paid for

93

with a large proportional transfer to compensate the older generations later in

life. This transfer would mean either a large consumption shortfall at some

later date or a faster rate of consumption decline than the discount factor,

which would be inconsistent with the desire to smooth consumption over the

life cycle.

If consumption shares going to the first and last generations shrink as the

trading restriction is relaxed and the consumers are allowed to trade more

frequently, a (Xt' St+l) -+ 1, and the differences between the common growth

rate of consumption for generations 1 to N-l and the aggregate endowment

growth would be close to zero for large N. The main implication of this is that

the prices of contingent claims in the OLG economy can be expected to be

close to these in the representative agent economy

\. = /38 (8,+1' x,r "(8, Is) = iJrr(s, I S+'(8,+I'x/;r::ilf' -= a (St+l1 Xtf' >"t~s

where 1 - 'Y is the order of homogeneity of the utility function and ,\~s is the

state-price in the representative agent economy. In fact, the above expression

indicates that, if under plausible conditions a (Xt' St+l) -+ 1, the convergence

rate towards the representative agent equilibrium is at a faster rate for more

risk averse economies.

94

The second interesting implication of the convergence arguments is that, in

order to keep state-prices close to those in the infinite horizon economy, the

variance of asset-holdings conditional on the realization of the state must

decline with n.

Although the presentation here relies on the discrete-state Markov process for

the exogenous driving factors the logic is likely to follow through in a

continuous setting. Thus, in a continuous state model, under reasonable

conditions we would expect individual consumption to converge fairly quickly

to the stationary solution of the representative agent model and stay close to

it for most of the individual's life. Hence, for a dominating proportion of

consumers the growth rate of consumption will closely follow the fluctuations

in the aggregate endowment.

The most important conclusion that comes out of the discussion in this

section is that the only variable that can contribute to the explanation of the

equity premium puzzle is the wealth composition of the unconstrained

consumers and, more specifically, the increased proportion of the dividend

relative to wage in the aggregate portfolio when the borrowing constraint is

imposed.

The remainder of this chapter demonstrates that, although OLG state-prices

are expected to converge to representative agent prices, equilibria in these

economies are in general different for any finite N.

95

3.4.3 Simple Equilibria

We first show that, in general, the equilibria in OLG economies are different

from the equilibrium in the representative agent economy. Namely, we

demonstrate that, without additional restrictions, there are no simple

equilibria where asset holdings and state-prices are functions of the aggregate

state only.

Under the maintained assumption of homogeneity of the instantaneous utility

function the Euler conditions imply

ci + 1 (s ')

c'L (s)

j 1 ( ') c . S ,Vi,j;V{s,s'!7r(s!s'»O}.

cJ (S) (3.16)

Ci+

1 (s ')/ Denote 8 (s, s ') = lei (s). Assuming further that all states communicate

(all elements of the probability transition matrix are non-zero), for j i + 1

ci + 1 (s') ci 2 (s') ---:--~ -. -7 8 (s ',8 ') 8 (s, s) 8

ci (s) c'L + 1 (s)

8(8,8') = 8 yes') yes)

(3.17)

96

r

Therefore if simple equilibria exist they can be indexed by a single parameter

6. For a given 6 the consumption of the first generation is determined from

the market clearing conditions as

N 1 N - 1 ill - 6N E ei

(8) = e (8) E 6 = e (s) = y (8),

i=l i=O 1 6

while consumption of generations 2 to N can be calculated using (3.17) as

In order for the computed consumption allocation to be an equilibrium it must

be supported by holdings of traded claims on contingent consumption, in

other words transfers within a given state must be consistent with the price

structure implied by 6. Formally, there must exist yi (s) such that

ei (8) = wi (s) + yi -1 (s) - E...\(S,SI)yi (Sl), Vs,sl,i Sl

Alternatively, using

...\(S,SI) = /31r(sl Sl). = ,87f(sls')6-1' -y-[ei + 1 (s ')j-' ( (s 1)]-' cl(s) yes)

(3.18)

and denoting the share of Xi in output as g~ = {Xi <j{(s) L, (3.18) can be

rewritten in the recursive form for g~ :

97

(3.19)

Provided that the transition matrix II is non-singular (3.19) can be solved

for g;, but in order for a given {j -equilibrium to be supportable by state-

transfers yi the system must also satisfy two boundary conditions reflecting

the requirements that the young start with no assets and the old leave no

inheritance

o N-1 N gy = O,gy = c ,

which in general requires that additional restrictions must be imposed on g~.

It is instructive to examine the allocations that satisfy (3.16) and where the

ex-ante (from behind the veil of ignorance) utility of the young generation is

maximized. We shall call these allocations - ex ante optimal equilibria.

Rewrite the utility of an agent of the first generation in terms of {j noting

that ci (8)

U ( Ct ) Et_1 L f3~ u (c;!n = i=O,N-l

(3.20)

where -h is the degree of homogeneity of the utility function and the

dependence of the discount factor on the number of trading periods (or

98

equivalently the time period between trades) is explicitly recognized in the

notation.

Further, assuming for simplicity that output is driven by a series of stationary

and independent shocks (3.20), can be rewritten:

u(a,) = E("(Y){~ 8'l" bf,;-l (~: J = E(u(Y))1«8) (3.21)

If u (c) is negative, as in the case with the CRRA utility function, then utility

is maximized by 8" = arg min { I{ (8)} .

First, evaluate the derivative of I{ (8) at 8 = 1 :

SinceO < f3 ~ 1, the first two terms in the curly brackets are negative and for

large n will dominate the last term. Therefore, for large n , which corresponds

to more frequent market opening, the derivative is positive at 8 = 1.

Consider the derivative at 8 E [O,i3~ :

99

k'(8) =

(fJ;{h r fJn8h (fJ;{h -l)n + fJn8

h [1- (fJ;{h rn] (1- 8n+1) = -h + (3.23)

8(fJn - 8hr (1- 8)

1- (fJ;{h r+1

(1- 8n+1t-1 8nn(8 -1) + 1- 8n

+h 1- (P;{.) (1- 0)'-' (1- 0)'

For large n the sign of the first term m the sum IS determined by

[fJ;{h - 1] > 0 and the second term is dominated by (8 - 1) < O. Therefore

the derivative is negative. The utility function is continuous in 0 on [0,13';;; 1;

hence there is a minimum 8* on [fJ;;/ h, 1) and 8* ~ 1 as n ~ 00 and fJ ~ 1 .

(3.17) then implies that in an ex-ante optimal equilibrium the consumption

shares of all generations converge uniformly to zero. Moreover, since

8* E [Qn1 / h, 1), . f h t' b 11 . th 1 fJ 1 wage s ares per genera IOn ecome sma WI arge n, so

does the solution to (3.19).

Although in general there are no simple equilibria, ex-ante optimal equilibria

can be viewed as approximate equilibria in the sense of the definition proposed

by Bernardo and Judd (1998). In particular, in our framework their conditions

for an c -rational expectations equilibrium can be stated as:

1. Decisions are nearly optimal:

100

(3.24)

2. Markets nearly clear:

"'{ i-1 '" >.. i} Y Y t- 1,s - L: t,sYt,S

<C (3.25) b + D(s)

In the sequence of ex-ante optimal equilibria rationality conditions (3.24) are

satisfied exactly and the discrepancy between the individual consumption and

available resources in (3.25) is due to the difference between the terminal

condition and the forward solution to (3.19) which converges to zero as the

period between trades shrinks. Therefore we can construct a sequence of cn -

rational equilibria with cn ~ 0 as n ~ 00 .

3.5 CONCLUSION

This section examined some of the arguments about the effect of finite life and

borrowing restrictions related to life-cycle on the pricing of risky assets. The

main conclusion is that if sufficient assets are available so that the trades of

all generations other than the very young are unrestricted then any differences

101

between asset prices in OLG and representative agent models are likely to be

small and shrink towards zero if consumers are allowed to trade more often.

In the following chapter the magnitude of the pricing differences between the

OLG and the corresponding RA models is examined in a simple calibrated

economy.

102

CHAPTER 4

CALIBRATION AND EQUILIBRIUM

4.1 INTRODUCTION

This chapter builds a simple calibration scheme that allows us to examine a

number of the effects discussed in chapter 3. In particular we will consider the

performance of the representative agent approximation to pricing in the OLG

economy, the effect on the number of trading periods on this approximation

and the equity premium and average bond returns implied by portfolio

composition changes caused by borrowing restrictions.

4.2 EQUILIBRIUM WITH LOGARITHMIC PREFERENCES

Since analytical solutions to OLG models with uncertainty are in general

unavailable the equilibrium map must be approximated and solved for

103

numerically. These approximations can become extremely involved even for

moderate dimensions of the state vector. There is however a specific case

where the mean equilibrium returns can be computed straightforwardly by

Monte-Carlo simulation.

Consider the economy with N-generations and assume that the felicity

function takes the logarithmic form so that u (c) = In c and that consumers

only receive a single exogenous payment in the first period of their lives. This

payment can be thought of as the discounted sum of future wages. The first

order-conditions (3.13) for holdings of simple securities become:

\,$ f3

7r (St Is) ,i = 1

Wi 2:\ .• Y;,s i 2: '+1

t Yt,s \+I,sY;+I.s s (4.1) \. 7r (St Is)

=f3 ,i = 1,N-l i-I 2:\sY;,s

i 2:\ ;+1 Yt-l,St Yt,s +1,sYt+1,8 s

These equations can be solved recursively starting from N - 1. First, taking

into account the terminal condition Y~ 0 solve the first-order condition for

i = N - 1 to obtain the demands for simple securities. The FOC takes the

form

A yN-1 = f37r(s I S)[yN-2 t,s t,s t t-l,s(

~ AN-I] ~ t,BYt.S (4.2)

104

Summing both sides of (4.2) over s and, taking into account that

'L:.7f (St Is) 1, the current value of transfers from the next to last

generation to all other generations is

Combining (4.2) and (4.3) gives the demands

N-l Yt,s

fJ N-2

( I ) Yt-l,s, 7f S S ----t 1 + fJ -\,s

Substituting (4.4) into the first-order condition for i = N - 2 produces

,\ t,. N-3 'L:' N-2 Y - A Y t-l,St t,lJ t,s

9

(4.3)

(4.4)

(4.5)

We note that the FOC written as above has the same form as (4.2) but with a

larger discount factor O',N_2 = fJ (1 + fJ). We can therefore continue with

recursive substitutions until we get to the FOC of the first generation, which

IS:

(4.6)

105

with LtN _1 = f3 as the initial and y~. = wt as the terminal conditions for the

recurSIOns.

Due to the form of preferences) unknown future prices dropped out of the

demand for securities, which turns out to depend only on the variables known

at time t and is proportional to the current wealth. This makes the

equilibrium very easily computable since in order to obtain the solution for

the current prices and quantities we do not need to specify an explicit

approximation scheme for the equilibrium map. Unfortunately though, this

property is specific to log-preferences.

With (4.6) the market clearing condition simplifies to

f3 N-2 yi _",,-1 -W + L t,s + yN-l = b + d (s) 1 + f3

1 a ;=1 1 + f3; t,a

(4.7)

Solving the optimality conditions

(4.8)

Expressing y; in terms of Yti=ll and Yt1 state-prices can be eliminated from ~8 ,St,8

the equilibrium equations and the system can be rewritten in the form of a

non-linear first-order auto regression Yt = F(Yt.l'S):

106

(4.9)

where a are weights depending only on fJ and are obtained by substituting ,

Y;,8' i = 1,2, ... ) N 1 into (4.7). The easiest way to compute unconditional

moments of returns is to simulate the dynamic system (4.9) forward until it

settles into a stationary equilibrium and use the realisations of demand to

compute simulations of state-prices and equity returns.

4.3 CALIBRATION AND SCALING

The main objective of this section is to develop a flexible approximation to

the dynamics of exogenous variables that can be easily scaled to accommodate

different assumptions about the frequency of trading.

Instead of relying on a fairly involved numerical procedure based on fitting a

Markov chain to match arbitrarily selected moments of the data-generating

process we start by building up an approximation based on the short-term

dynamics of the aggregate output and wages. This is then scaled to an

107

appropriate time-horizon. The next step involves using Tauchen and Hussey's

(1991) quadrature method to fit a Markov chain to approximate the dynamics

estimated in the first step. The additional benefit of this method is that it

allows elimination of the arbitrariness in the choice of the number of states of

the driving process as the number of states in the approximating Markov

chain can be easily adjusted until it fits the system with the desired degree of

accuracy. The downside of the scheme is that it relies on the adopted scaling

law and the ability of the assumed linear system to approximate the data

well.

4.3.1 Aggregate Dynamics

Unlike CDM who solve the model under the assumption that aggregate wage

and income are stationary in levels we assume that income (aggregate

consumption) follows a random walk. Growth of the aggregate income is

assumed to follow a white noise process. We also assume, somewhat contrary

to the empirical evidence, that wages and income are cointegrated, so that the

wage share in income follows a stationary AR(l) process. In addition, we

assume that contemporaneous innovations to income growth and wage share

are uncorrelated. The system used to approximate the aggregate dynamics

becomes:

108

(4.10)

where cy , Cw are independent Gaussian innovations and

[::]- N (0, I)

The model has the problematic implication that the wage share can fall

outside the [O,lJ segment with non-zero probability. The sample mean and

standard deviation of the wage share in income however are such that this

probability is negligibly small and, using a more careful formulation (e.g.

modelling log wage share as a logit transformation), does not affect either the

numerical results or the conclusions of the chapter.

To estimate the model the aggregate wage wt

was identified with the

compensation of employees series from the national accounts and the total

mixed income was used to fit the equation for the growth of output. Since we

do not explicitly model the population growth (the model can be modified

trivially to allow for a non-stochastic trend in the population) both series are

deflated by the total number of employees.

109

Fitting equations (4.10) to quarterly aggregates (157 observations from

September 1959 to September 1998) using simple OLS regression the following

specification for the exogenous driving processes was obtained:

br = 0.038 + 0.027.:::

b; = -0.038 + 0.935b::-1 + 0.019.::; (4.11)

Given the dynamics of the aggregate income and wages, the dividend can be

inferred from the model as dt = Yt - wt • The growth in the aggregate dividend

can be re-written in terms of the income growth and the wage share as

(4.12)

It may be worth noting that the actual share of dividends paid on stocks is a

rather trivial proportion of total income with the maximum of 2.2% over the

period from September 1966 to September 1998 (ABS).

4.3.2 Markov Chain Approximation

The method that is used here to construct a Markov chain approximation is

described in detail in Tauchen and Hussey (1991). For the white noise income

110

process it reduces to replacing the continuous random variable £: with an n-

state rule with the states corresponding to the nodes of the Gaussian-Hermite

quadrature rule X~H rescaled by the sample standard deviation ;;11 and shifted

to accommodate the mean of the process. Then the probabilities are obtained

by an appropriate normalization of the quadrature weights W~H

11 2~ GH 8; = (JlIXj + J.L

1, ... ,n. (4.13)

By construction the quadrature approximation will exactly match the

unconditional sample moments of the income growth process up to the n-th

order.

(4.14)

Calibration of the autoregressive process for the wage share is somewhat more

involved and requires rescaling the transitional density of the process

(Tauchen and Hussey [1991)). Using w(x»> O,x E lR as the weighting kernel

the expectation of g(xt ) with respect to the conditional density f{xt I xH) of

the process can be rewritten

(4.15)

111

and approximated with the quadrature

This approximation replaces the autoregressive process with an n-state

Markov chain with the states determined by the nodes of the Gaussian

quadrature for the weighting function and the transition probabilities

determined by the corresponding weights

1 w:H(W)f(sj Is;) K (Sj) 'CD (s.)

(4.16)

where K (8) is the normalizing constant

'" CH() f{s 1 8 ) K(S) = ~w, tv I

s, I 'CD (8)

Possible choices for the weighting function are the stationary distribution <;>f

the autoregression and the distribution of the innovation process c~. Although

Tauchen and Hussey advocate using the latter we found that the former

works somewhat better in our application. The possible explanation for this

discrepancy is that the stationary distribution assigns more weight to the tails

of the distribution which is likely to be important in approximating the very

112

persistent process for the log wage share. Consequently, for the same number

of quadrature nodes results in a better approximation.

An alternative is to use the normal density as a weighting function but treat

the standard deviation of the density as a free parameter. On the basis of the

overall performance of the chain (J r:;; = 2.5 appeared to provide the best

approximation.

For this choice of the weighting density the Markov chain approximation for

the log wage share takes the form

w 2 aw GH(r;:;) a s. = --~-=-2 Xi + -1--~ • 1 p - p

(4.17)

here n (x; It, (J) is the normal density with the mean of It and the standard

deviation of (J evaluated at x.

The final Markov chain representation for the wage-output process is obtained

by combining the two sub-chains. In other words the process is described by

the collection of n2 states and the n2 x n 2 probability matrix

113

oy (s .. ) = OY (sY) k=lXJ J

OW ( Sk=iXj ) = OW (Sn (4.18)

Table 5 illustrates the ability of the chain to reproduce selected moments of

the data. It appears that the chain performs quite well, although a large

number of states is required to accommodate the persistence of the wage

share. By coincidence, the number of state (n = 8) adopted here is the same

as that used in Heaton and Lucas (1996).

Generally the approximation appears to provide a reasonable fit to the data

with the exception of the significant autocorrelation of the wage process at the

semi-annual frequency. It also tends to overestimate somewhat the standard

deviation of the wage share and consequently the volatility of the wage

growth and the correlation between the wage share and the income growth.

These discrepancies are however fairly small.

114

Data AR

Mean/StDev log-wage share:

-0.570

0.052

-0 .571

0.052

Mean/StDev log-income growth:

0.005

0.015

0.005

0.015

Mean/StDev log-wage growth:

O.OOS 0.005

0.017 0.020

Autocorrelations:

Lag Wage Growth

1 -0.lR1

2 0.233

3 -0.069

Correlation log-wage/income growth:

0.658 0.751

Parameters of the autoregression:

Estimated

0.967

-0.019

Markov

-0 .571

0.046

0.005

0.015

0.005

0.020

Income

Growth

-0.205

0.045

-0.044

0.772

Markov

0.964

0.021

Table 5. Selected moments of the data and Markov approximation

4.3.3 Scaling

Since the decisions facing consumers in economies with different frequencies of

market openings are different the results obtained in these economies are not

directly comparable and there does not exist a uniquely consistent way of

recalibrating the exogenous process to a given frequency. Conceptually, a

consistent procedure would treat the trading frequency as an explicit

constraint and obtain solutions under different assumptions about the

duration of each trading period. This would however complicate numerical

solution considerably without offering any clear benefits, as the trading

frequency would still be exogenously imposed.

Instead, two simpler alternatives were considered based on the idea of treating

the series of calibrated economies as approximations to the underlying

dynamics. Namely, the calibration procedure described above could be applied

to either n-yearly aggregates or the original process re sampled every 4*n

periods (the source data are quarterly). The latter approach was preferred as

it does not alter the information structure of the problem; in particular the

AR representation in (4.10) is invariant to such re-sampling.

The annual discount factor is assumed to be equal to 0.98, which is consistent

with the estimates presented in Chapter 1. The discount factor at the

frequency of n years is the n-th power of the yearly discount f3n = f3n .

116

4.9.4 Distribution of Wages

The last step requires postulating a rule that determines the allocation of the

aggregate wage endowment between generations. Since there is no exogenous

market incompleteness little is lost by assuming that wage growth rates are

perfectly correlated across generations. This effectively assumes that all wage

related risks are shared by all generations. Since there is no idiosyncratic

component, the wage distribution is described completely by a collection of

generation-specific weights. The weights are specified to adhere to the

following wage-experience profile14

W (Exp) = 4.85 + O.05Exp - O.086Exp2 (4.19)

Once individual weights are determined the wage attributed to the i-th

generation is w; wiwt where wt

is the aggregate wage at time t. The weights

are obtained as a solution to the system

14 I am grateful to Mathew Gray for suggesting this specification. It was estimated from

Census data.

117

r

;=1 (4.20)

j W (i) 1 w =--w

WeI)

Here N W is the number of periods until retirement and the first condition in

(4.20) ensures that the weights sum up to 1.

The main difficulty before (4.9) can be used to simulate prices is to find a

reasonable calibration for the discounted sum of future wages

t+NW-l

n1 = W O + '" '" \ WT

-t+2 t t W L..; ''7,$ T (4.21)

T=t

The discounted value of the wage endowment depends on the future prices

that need to be solved for. We expect however that state-prices in an OLG

model with complete markets are close to the state-prices that would be

obtained in the corresponding Lucas economy at least if the trading period is

short enough. Under this assumption it appears reasonable to approximate n~

using the representative agent prices to discount future wages. Hence, the

approximation to the discounted wage process becomes

(4.22) s

118

here ,X * is the price of a unit of consumption in the state s at time t in the t,s

representative agent economy and

n~s = Q,n > NW.

4.4 SOLUTION TO THE REPRESENTATIVE AGENT PROBLEM

The equilibrium price system must leave the representative agent content

with consuming the non-storable aggregate income. With CRRA preferences

state prices must satisfy

(4.23)

Income innovations are assumed to be independent across time and hence

there is no time subscript. Aggregate dividend and the risk-free investment

can be priced using (3.11)

e; = ""' ,X* (s.) dH1 (s s.) d L.J 'd t',

t 1 t

(4.24)

119

( 4.25)

The bond price and therefore the bond return are constant; the latter is equal

to rb* = X*. To obtain the average stock return (4.24) has to be rewritten in

terms of stationary variables

(4.26)

The unconditional expectation of the stock return is then obtained as

(4.27)

where 7fw is the stationary distribution of the wage share sub-chain.

4.5 PRICES IN OLG AND REPRESENTATIVE AGENT ECONOMIES

To simulate the OLG economy forward the system (4.9) is expressed in terms

of stationary variables by rewriting it in terms of the share of the i-th

generation demands for securities in the aggregate consumption

120

(4.28)

The economy was simulated with 3 generations (the implied period is 20 years

and consumers retire in the third period) and with 6 generations (assuming

the period length of 10 years and retirement in the 5th period). Results are

presented in Table 6. A typical price simulation is shown in Figure 3.

It is clear that the prIces in the economy converge very quickly to

representative consumer prices; with 6 generations average prices are virtually

indistinguishable from the RA prices. In fact, even with as few as 3

generations the economy follows the RA prices very closely confirming the

main proposition of the previous chapter.

It may appear that this result depends on the particular assumptions made in

approximating the wealth dynamics. To examine the robustness of this

conclusion state prices were re-examined by directly imposing the exogenous

wealth dynamics in the form of an ARC!) and AR(2) and simulating under

different assumptions about the parameters of this process.

121

3 generations 6 generations

1.500,... .... -----.----·------,

1.900·

1.400

1.700

1.300

1.500 1.200

1.300 1.100

1.100 1.000

0.900 .L-_______ ... ~ .. ~_~ ... _. __ ___' 0.900 ... --... --------~----'

Figure 3 Growth rate of consumption of the first generation (blue line) and the

aggregate consumption (red line).

Total returns Annualized returns

Bond Equity Bond Equity Premium

3 generations

OLG 54.475% 57.212% 2.941% 3.062% 0.121%

Representative agent 54.437% 57.094% 2.940% 3.057% 0.117%

6 generations

OLG 24.274% 25.259% 2.940% 3.048% 0.108%

Representative agent 24.273% 25.246% 2.940% 3.047% 0.107%

Computed from a simulation of 5000 observations.

PA ( )1 / 6.T Annualised returns are computed as~ = T; + 1 1, .6. T -period between trades

in years.

Table 6 Mean returns in OLG and RA economies

122

The results of these experiments corroborate the story above; in all the cases

we considered OLG prices quickly get very close to RA prices as the number

of trading generations grows.

4.6 ASSET RETURNS IN RESTRICTED AND UNRESTRICTED

ECONOMIES

The previous section established that asset-prices in an OLG economy with

complete markets are close to prices in a representative agent economy with

identical preference parameters. Therefore, a high equity premium in the

borrowing restricted economy can only be explained by the differences in the

portfolio composition. In the CDM economy the borrowing restriction

effectively implies an exclusion restriction; the "young" do not participate in

trading and do not hold financial assets. Imposing this exclusion restriction

tilts the portfolio composition of the unconstrained consumers towards holding

a higher proportion of risky assets and makes consumption of the

unconstrained generations more volatile.

123

Wage share Equity Aggregate Dividend

Stock Bond Stock Bond

l' =1 ~ 0.0 2.929% 2.948% 2.943% 2.947%

0.2 2.962% 2. 98!=)% 2.925% 2.931%

0.4 3.073% 3.101% 2.895% 2.901%

l' =2

0.0 4.838% 4.861% 4.822% 4.831%

0.2 4.985% 5.013% 4.752% 4.763%

0.4 5.451% 5.483% 4.630% 4.643%

l' =4

0.0 8.621% 8.602% 8.401% 8.419%

0.2 9.247% 9.221% 8.116% 8.140%

0.4 11.228% 11.156% 7.617% 7.644%

Returns annualized

Table 7 Mean returns in restricted economics for different shares in the aggregate wage

for the unconstrained generations

Table 7 presents the results from representative agent economies with different

risk aversion parameters and different shares for the constrained generations

in the aggregate wage. Increasing the risk aversion parameter drives up both

the equilibrium equity and risk-free returns due to the inherent link in the

CRRA preferences between risk aversion and the elasticity of inter-temporal

substitution. More importantly , none of the specifications is able to produce

an equity premium that is above a few basis points, effectively reproducing

the original Mehra-Prescott story.

124

The average levels of equity returns reported in Table 7 pertain to the case of

pricing the aggregate dividend obtained in our model as the difference

between the aggregate income and wage. This series has different properties

from the All Ordinaries dividend series. To examine whether the conclusions

are robust to the choice of the series used to represent dividends, the model

was applied to price a dividend stream represented by a Markov chain fitted

to the dividend paid on the All Ordinaries index. The results of these

simulations are quantitatively very similar to the ones reported in Table 7.

The largest equity premium obtained was 1.11% (under a risk aversion

parameter equal to 4 and with the wage share of restricted generations equal

to 0.4) and this was associated with an annual risk-free rate of 11.5%!

4.7 CONCLUSION

A rather large number of results on asset pricing in OLG models have recently

appeared in the literature. Most of them rely on a rather skeletal asset

structure generally comprising of equity shares and a riskless instrument (a

bond). In this chapter we examined the polar case where asset structure is

complete, in the sense that it does not place any exogenous restriction on the

state composition of possible trades and portfolios.

125

f

The main conclusion of this chapter is that the asset pricing results obtained

with simplified asset markets should be treated with a great deal of caution.

Asset structure appears crucial in the determination of the rates of return on

risky assets. In particular, under the assumption that asset markets are

conditionally complete and that portfolios are readjusted with a reasonable

frequency, we found that asset prices converge very quickly to the prices that

would be observed in the representative agent economy.

A second result of the chapter is that liquidity restrictions are not by

themselves sufficient to explain asset pricing anomalies.

These conclusions of course do not imply that OLG models offer no insights

into the determination of the risk premia. The exchange economy examined

here involves a number of extremely restrictive conditions on the specification

of the exogenous process and the asset structure. In particular, the line of

research that appears most promising is to examine how asset market

incompleteness can arise endogenously in these models due to the structure of

the market interactions between agents. For example, the young may have

relatively little information about the quality of their human capital

endowment compared with the middle-aged. This information asymmetry may

have interesting implications for the asset structure.

Another interesting research topic is to explicitly examine the interaction of

the constraints of a legislated minimum saving ratio and the restricted access

126

to retirement savings (a feature of retirement systems of a number of

developed economies including Australia) and the liquidity constraint on the

young generation.

127

CHAPTER 5

SOLVING ASSET PRICING MODELS

5.1 INTRODUCTION

Restrictions on the dynamics of prices Zt and asset returns by representative

agent pricing models take the form: 15

(5.1)

-_._------

15 No explicit form of the state process X t is specified here, although it is clearly important

for convergence properties of the algorithms below. It is only assumed that the state process is

stationary and ergodic. As usual, random variables are in upper case while lower case denotes

realisations.

128

r

The solution involves a search in the class of deterministic stationary solutions

expressing the endogenous variables Zt, and consequently the conditional

expectation, as time invariant functions of the realization of the exogenous

state variable Zt = Z (Xt). Then we have

(5.2)

The solution is therefore the fixed point of the pricing equation (5.2).

Analytically the model can only be solved in a limited number of cases. The

need for simple and efficient solution methods arises particularly in

applications involving parameter estimation, where a solution (or some

function of it) must be computed repeatedly for a range of parameter values

(Bansal, Gallant, Hussey and Tauchen [1995]).

5.2 SOLUTIONS

In rare circumstances it may be possible to find a semi-analytical solution by

directly exploiting the fact that Z (x) solves for a fixed point. If Z E q>, q> is

complete and the operator T (Z) is a contraction on q>, then the solution

could be approximated by taking a convenient member of this set and

iterating analytically until some pre-specified accuracy is achieved. In most

129

practical circumstances however analytical iterations are not feasible and the

problem must be solved numerically. A number of numerical techniques have

therefore been used in the context of asset pricing equations, some of which

are discussed below.

5.2.1 Discretisation (DA)

The procedure involves replacing (5.2) with a discrete scheme which can be

fine-tuned to produce an approximation to the exact solution. The simplest

way to do this is to use a flexible approximation for Z (x) selected from an

appropriate finitely parameterized family of functions. For example, by taking

a partition of the state space {Xi }:l' Z can be approximated using indicator

functions defined as

I (x) = . II, if 0 :S X < 1

0, otherw'tse

Taking the approximation Z(x) ~ Z = 2:Z(xi )I(::") , equation (5.2) can be

replaced with

130

where Pr(xj I Xi) = llF(xj I Xi) is the probability of transition from Xj to Xi'

For an alternative interpretation of the method it can be noted that it

involves substituting a discrete-state Markov approximation for the exogenous

driving process (typically a VAR).

This approximation, while suitable for problems with small dimensions of the

state vector, can behave poorly in larger problems. In addition, a large

number of nodes may be required to achieve a desired degree of

approximation. Tauchen and Hussey (1991) suggest replacing it with a

computationally more efficient procedure involving Gauss-Hermite

quadratures by rewriting the pricing equation as

f(x I xt ) fD(x)G(Z(X),x) w(x)dx z(x)

w(x)

and replacing the integral with a quadrature rule with respect to the

weighting function w.

Under suitable conditions, as the number of nodes of the quadrature increases,

the approximate solution will converge to the exact solution. Discretisation is

very efficient for small linear problems. The downside of this method however

is that, unless G(.,.) is linear in the first-argument, it still requires finding a

numerical solution of a system of non-linear equations, which, in practice,

131

limits the practical number of nodes, thereby potentially making the

approximate solution quite imprecise.

5.2.2 Parameterized Expectations (PE)

Another popular solution method was suggested by Marcet (1988) and

involves replacing the infinite dimensional problem in (5.2) with a finite

dimensional one by choosing a subset of functions <.Pn (0) in <.P characterized

by a finite dimensional parameter vector 0 E R n •

Most often <.Pn (0) consists of members of some polynomial family. Denote Ok -

the value of the parameter vector on the k-th iteration of the algorithm and

4>(X,Ok) E <.Pn (0) - a member of <.Pn (0). The algorithm involves a series of

simple steps:

1. Draw a realization from the exogenous driving process XT = {Xt }~=1;

3. Update 0:

132

0* = argmin} Et 1 {G(Z: + l'xt + 1) - 4> (Xt,O)t

argmin} Et = 1 {G(4)(Xt + I,Ok),Xt + 1) 4>(Xt ,O)t

using Ok+l = Ok + a(O* - Ok)' a E [0,1] to stabilize iterations;

4. Iterate steps 3,4 until convergence is achieved.

Two convergence criteria have been suggested:

(5.3)

1. Based on the coefficients of the projection (Marcet and den Haan

[1994]):

" Ok + 1 - Ok 11< €, where € is some given small number;

2. Based on the simulation of the endogenous state (Bansal, Gallant,

Hussey, Tauchen [1995]):

where € and T] are given small numbers.

133

5.2.3 Direct Approximations (AA)

The Marcet method is really nothing more than a simple way to iterate

towards the fixed point of the pricing equation USlllg a series of

approximations. Clearly the algorithm stops when within pre-specified

tolerance:

In other words ¢(xpBml must be close to a fixed point of

Intuitively, the method selects Bm or a member of <I>n (B) - the set of

approximating functions - that comes as close as possible to an optimal

16 Of course, for this condition to be satisfied exactly <I> n (B) must contain Z.

134

predictor for G(4)(xt+llem),xt+1)' which IS a defining property of conditional

expectation.

Equivalently (5.4) implies that em :

Thus the whole algorithm can be seen as a convenient computational way to

find a member of <.P n (e) that is as close as possible to the solution in mean-

square.

Instead of relying on the Marcet algorithm, the same result can be obtained

by solving either (5.5) directly or

e* (5.6)

denotes the gradient of 4> (Xp e) with respect to e. The solution to the

program in (5.6) is the parameter vector that is close to satisfying the first-

order conditions to (5.5). Generally, (5.6) and (5.5) have different convergence

properties.

135

This direct approach was used by Heaton (1995) and Bernardo and Judd

(1998), where it is motivated somewhat differently by the property of the

conditional expectation that, given random variables X,Y,Z, which are

defined to satisfy appropriate measurability conditions, Z = E (Y I X) if and

only if for all bounded continuous 9

E(g(X)(Y Z)) = O.

The only advantage that the PE method has over the direct method is that it

avoids the need to compute numerical gradients; otherwise the convergence of

PE can be quite slow. In fact, for an arbitrary starting point PE may not

converge at all. Therefore, if analytical gradients of the kernel are available or

are easy to compute or, alternatively, if a well performing non-gradient based

optimisation algorithm (simplex, NeIder-Meade, etc.) can be found, the direct

method is clearly preferable to PE, as it is at least locally convergent.

Example 1. Direct approximation is very convenient when G is linear in Z

and ¢ IS linear III B , e.g. often III asset pricing models

G(Zt+l,xt+l) = m(xt+l)(zt+l + dt+l)' Taking ¢(xjB): RS ~ R to be polynomial

expansions of a fixed degree n : ¢ (x, B) = L j B (j) w (x, j), with

j = {Ul ... j.) I i < k ~ j, < jk;Lji ::::; n} and w(x,j) standing for basis

monomials w (x, j) = xiI ... xIt, the problem becomes that of finding B such

that:

136

(5.7)

linear system that can be solved in one step. 8* depends on the moments of

X t of order up to n and its cross-moments with m (Xt+l)'

5.2.3 A Simple Solution Algorithm when G is Linear in Z (LA)

Under more general conditions the optimization problem in (5.5) can become

quite large and the parameterized expectations method becomes a viable

alternative. For models linear III Z an even simpler simulation based

algorithm can be suggested.

3. Starting from ZT+l) produce a sample z; G(z;+llXt+l);

4. Estimate the mean of z; conditional on xt using any parametric or

nonparametric estimation technique.

137

To see how this algorithm approximates z(xt } assume that ZT+l is an unbiased

estimate of Z (XT +l ). Then

Z (XT) = G (z (XT+l)'XT+1) + cT+l;

E (CT+1 I XT+1) = 0

in other words G[Z(XT+1),XT+1) provides an unbiased estimate of Z(XT)' Then

that every point on the path z; is an unbiased estimate of the corresponding

point on the solution. The last step employs some form of averaging to extract

Z(Xt I XT)' which under stated conditions on x converges to z(xt }·

The obvious advantage of this procedure is that it makes it much easier to

evaluate the accuracy of the solution; using the same sequence of simulated

estimates one can start from a simple approximation and add terms until the

desired precision (e.g. on the basis of Marcet and den-Haan criterion) is

achieved.

5.2.4 Recursive Algorithm (RAJ

Here we suggest a simple method that can exploit the stochastic nature of the

problem. The method is applicable when paths of exogenous variables can be

easily simulated backwards or forwards.

138

An inefficiency of the Marcet solution shared by many other methods that

rely on Monte-Carlo simulation to approximate an expectation is that it may

require a significant number of simulations even at the points on the path of

iterations far from the solution to the problem. Stochastic approximation is a

family of methods developed to deal with such situations. It replaces the

strong requirement of deterministic algorithms to generate steps strictly in the

direction of the solution with the relatively mild requirement that a move

should on average be in the right direction. The time saving is achieved by

replacing a precise expectation estimate that requires Monte-Carlo or some

other computationally expensive procedure for evaluation with a noisy

observation.

More formally, if the conditional expectation of ¢(XpOk) is computable, the

following deterministic scheme can be expected to converge under fairly

general conditions:

To put it in the form of stochastic approximation rewrite this expression as:

Ok + 1 Ok ck [¢(XtlOk) - G(¢(Xt + l'°k),xt + 1)] +

+ck [G(¢(Xt + 1,Ok),xt 1) - Et {G(¢(xt + 1,Ok),xt 1)}} =

Ok + ck [¢(xt,Ok) - G(¢{Xt + 1,Ok)'Xt + 1)] + M t

139

where E(Mt I Om,m ~ k) = 0, M;s thus form a martingale relative to the

history of parameter iterations. If M t averages out for an appropriately

selected c sequence (c", ~ 0 and I: c" ~ 00) the limited behavior of the

sequence is described by the ODE:

which has 0 as a stationary point. This allows one to replace the

deterministic scheme with the stochastic scheme:

For the type of linear problems considered in Tauchen and Hussey (1991),

approximating the decision rule with polynomials ¢(xpO) = OWt , where W t is a

vector of basic monomials of degrees up to n, the following simple algorithm

based on recursive least squares converges to the same 0 as the procedures

above:

2. Guess 0TH;

140

3. Starting with some initial variance covariance matrix VT+1) iterate over

the realization by generating Zt-l = G(¢(xt+1,O),Xt+l) and then update

V-I - V-I t-I - t

(5.8)

4. Stop when II 0t - 0t_l II /01 0t II +1]) < c, where c,1] are some given small

numbers;

Intuition for the step 3 is quite straightforward: 0t is updated based on the

discrepancy between least squares OWt _ 1 and the structural form Zt-l

forecasts. In this scheme least squares weights can be

replaced with some other sequence, satisfying cl.; -+ 0 and I:: ck -+ 00, and

better tailored to a particular problem at hand. The trade-off in selecting ck is

that between speed of convergence and accuracy of solution. In general

parameter update is based on 0t_l = 0t + ct (Zt-l ¢ (xtA)), e.g. the linear

approximation to NLS can be used.

This approach has a number of advantages. Computationally, inversion of a

TxT matrix or numerical minimization required for the least squares step of

the Marcet procedure is replaced with a series of simple iterations.

141

Convergence of parameterized expectations iterates is usually quite sensitive

to the choice of initial condition. Marcet and Lorenzoni (1998) suggest

selecting the initial condition by constructing a homotopy that can exploit

dependence of the solution on parameter values. Iterations are started from a

simple model with known solution which is gradually transformed into the

model of interest. This approach may however involve heavy computational

cost and is not guaranteed to work. A recursive algorithm is much easier in

this respect, since it is likely to converge if the single starting value ZT+l is

close to the conditional or unconditional mean.

The real advantage of the above procedure however is its flexibility. Due to

simplicity of individual steps more general approximations can be easily

incorporated into this framework.

5.3 EXAMPLES

The methods discussed in this chapter are applied to solving for asset price

processes m a Lucas (1978) economy under alternative specifications of

preferences. Agent optimality conditions (1.3) imply

142

Under Lucas' assumptions the Euler condition of the agent problem can be

treated as a restriction on the asset price dynamics.

The two utility specification of preference are contained in the following:

1. Time-separable, constant relative risk aversion:

2. Habit-formation:

1-, 1 Ut = EtL.rf of3i 8!t_~ 8t = ct aCt _ 1

The consumption process is approximated with a first-order autoregression

fitted to the growth rate of real quarterly per capita consumption of food and

other nondurables from the National Income Forecasting model database on

143

DX (series VNEQ.AK90 _ CFO and VNEQ.AK90 CND, December 1959-

December 1997, 153 observations):

L).Ct = 1.13- 0.13L).ct _1 + l/t O.OR 0.08 ' ,

and 1/t - N(0,0.012). Parameter values are set at fJ = 0.97, 'Y = 2,0: = 0.3.

For the purposes of this exercise it is assumed that the growth rate of the

aggregate consumption is perfectly correlated with the growth rate of the

aggregate dividend. Rewrite the first-order condition in terms of the price-

dividend ratios:

Pt _ E jmrs dt + 1 [Pt + 1 + III d - t t+l d d t t t + 1

For a real riskless bond the pricing equation becomes

Solution:

1) Generate a simulation of consumption growth {L).Ct}~=l' T=30000.

Construct implied marginal rates of substitution {mrst }~=1"

144

2) Adopt an approximation for the bond prIce function: a quadratic

function is used to approximate b (xt ) • To solve for the bond pricing

3a) Solving for the dividend yields using Marcet or least-squares

minimization:

a. Take second order approximation to %t

b. Generate C* = mrs tit+! (( PHi)* + 1) t+1 Ii. (~+! and regress them on

{I, xI' x;} until convergence of simulations is achieved (within

0.01%). Alternatively, compute et = (;;:)*

and minimize L e;.

3b) Solve for the dividend yields using recursive least-squares:

a. Starting with ZT+l = 0.2 and VT +1 = 1 iterate (5.8).

3c) Solve for dividend yields usmg the linear algorithm LA. LA is

illustrated using third order polynomials to approximate Z (x) and

145

runmng least-squares and the Nadaraya-Watson nonparametric

estimator with Gaussian kernels to extract information about the mean

(P) (x) = t K(Xt - x) Pt It K(xt - x). d t=1 h dt 1=1 h

Figure 4 illustrates approximations to the state contingent rule for

(Pt +;< + J* based on a single simulation. We can see that the rule selected

by the recursive algorithm is close to the least-squares minimizing one. The

non parametric estimator becomes wiggly far from the mean where it does not

have enough data points for adequate averaging, but is close to the third-

order polynomial within quite a wide range around the unconditional mean

(about ±5 standard deviations). Variations in this range can easily be

smoothed with appropriate bandwidth selection.17

17 We select h = D.gAn-Iff>; A = min (u, (R/1.34)) , R-interquartile range (see Pagan

and Ullah [1998]).

146

Clearly, all algorithms produce solutions that are very close to each other.

The oscillations of the non-parametric version of LA on the boundaries of the

state are caused by sparse observations in the tails of the unconditional

distribution of the driving process and can be smoothed out by additional

simulations or by a using a variable bandwidth kernel estimate.

It is interesting that, with the parameter values adopted, the habit-formation

model implies much less volatile dividend yields than the time-separable

model.

5.4 CONCLUSION

This chapter discussed some popular algorithms that arise in asset pncmg

models and suggested simple modifications. Convergence properties were

illustrated using a popular asset pricing modeL

We may note here that the solutions discussed are taken out of the context of

estimation where they normally appear and are thus likely to be suboptimal

(discussion of some issues related to this observation can be found in Bansal,

Gallant, Hussey, Tauchen [1994]). The objective of estimation is to select the

law of motion of the exogenous variables Xt from a parameterized family

147

f (xt I Xt _1, a) using some estimating equation. The full problem can be

formulated as selecting the parameter vector a conditional on the sample

values of x and Z:

Q ({Xt }~=l' {Zt }~=l;a) = 0

s.t.E{G(Zt+l'XH1 ) I Xt} = Zt·

Developing algorithms for effectively solving these problems is a challenge for

future research. As evaluations of the above objective itself would often

involve some simulation averagmg, exploiting ideas of stochastic

approximation seems to be a promising path to follow.

148

pt/dt

, 1\ !

i \ !

3.06 " :.. . I \. : : ~ I

,.~ ...................................... , ••• _ ••••••••••••••••........................................................ - ...... •• ••• ••• •••• •• •••••••• 1 .... • ....... • •••• • •• _ ....... .

3.04

3.02

3

2.98

2.96

2.94

2.92

2.9

2.88

r ..... ·· .... · .. ····· .. · ..... r .... ···· ....... ········ .... r .. · .. · .......... · ........ ··r ...... · ... · ....... ··· .. · .. 1

-0.15 -0.1 -0.05 o 0.05 0.1 0.15

Xt

Legend

3.9 !

3.8

3.7

3.6

3.5

3.4

3.3

3.2

! ~ I 1 • \

········ .. ········ ..... ·l················· .. ···· .. ·····f .. ···· .. ······················1························· ..... 1.· ........................... 1 ........... 1. .............. . : i ! I ~ \

3.1 ........ ·· ............. ·,·· .. ·· .......... · .. · .... · .. ·+ .. _·· .. ·· .... ·· .. · .. · .. · .. 1 .. · ... · .. ··· .. · ... ··· ...... · .. l· ...... · ... · ..... · ..... · .. ···.l ..... ·· .. · .. j .... · ........ .. ; , ! ' i' : : ! ~ ! \

3 -0.15 -0.1 -0.05 o

Xt

0.05 0.1 0.15

......... E2. Nadarya-Watson .......... Recursive algorithm ................... E2. Least squares -- Marcet

Figure 4 Approximations for price-dividend ratios for the time-separable (left pane) and habit formation (right pane) models.

149

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158

DATA APPENDIX

Consumption Measurement

Measuring consumption growth even on the aggregate level is perhaps the

most delicate issue. One problem (as Mankiw's results suggest) is that

representative consumption theorems that rely on rather stringent conditions

of either homogeneity of preferences or the distribution of endowments or both

(Rubinstein, 1974) may not describe the data even approximately. Time

aggregation can also complicate inferences, especially if preferences have a

temporary dependent component (habits). Finally, separation assumptions,

implicit in using consumption of nondurables as a proxy for the model

consumption variable, may be inadequate. We do not attempt to address

aggregation problems in this study; the contention of much of the literature

on the topic appears to be that time-aggregation is unlikely to be the source of

the puzzles. Some of the modifications involving durables were discussed

previously.

Next, the timing of consumption can be selected somewhat arbitrarily. We

attribute consumption over a period to the beginning of the period, which,

since national account measures are aggregated quarterly, involves lagging

measured consumption once. This convention appears to fit naturally within

159

the decision structure of CCAPM-type models and, in addition, produces

higher correlations with stock return and interest rate measures.

Data Sources

The data are compiled from various Australian Bureau of Statistics (ABS)

and National Income Forecasting (NIF) model database tables. The total

return series comes from the Global Financial databasel8 and the DataStream.

Whenever there was a choice, seasonally adjusted series were used.

Specific sources for the most commonly used series are listed in the following

table.

1. Real consumption (food, other nondurables and services Ct ) and the

associated deflators (~). Source: NIF database on DX (September

1959-December 1997).

18 www.globalfindata.com

160

Series:

Private final consumption expenditure: Food

(VNEQ.AK90_ CFO);

Private final consumption expenditure: Other non-

durables(VNEQ.AK90 _ CND);

Price indexes: Private final cons exp: Food

(VNEQ.AI90 _ PCFO);

Price indexes: Private final cons exp: Other non-durables

(VNEQ.AI90_PCND).

2. Adult (15 and over) population Nt' Source: NIF database on DX, Series

VNEQ.UN NAP - Labour market: Civilian population: Aged 15 years

& over (March 1964-March 1998). The population series was extended

to obtain per capita consumption for the period from September 1959

to December 1997. Although a number of population estimates are

available for the period covered by the table, the population series was

backcast from the trend value. Linear and exponential specifications for

the trend were tried using a linear trend specification. The reason for

using synthetic rather than actual observations is that there appears to

be a great deal of variability in population estimates across different

data sources. Without performing a more involved analysis of the data

the only reliable component that can be extracted from this series is

the trend and, possibly, average growth rates. Fortunately though, the

161

variability of the real consumption series, measured by the coefficient of

variation, is over 3 times that of the population growth (1.01 against

.33), and the measurement error in the population series IS

quantitatively unimportant. In particular, the correlation between the

real consumption growth rates obtained from actual and reconstructed

data - which are the series of interest for all the applications in the

paper - are of the order of 0.99.

3. Stock indices, dividends and the risk-free rate: Monthly series for

various industry indices are obtained from the DataStream, monthly

stock market accumulation and all ordinaries indices and dividend

yields are from the Global Financial Database (GFD in turn refers to J.

Lamberton, "Security Prices and Yields, 1875-1955" and official ASX

publications as original sources). Dividend yields are quarterly from

December 1882 to December 1955 and monthly thereafter.

Total return on the bills index from the Global Financial Database

(TAUSBIM series) was used to construct a proxy for the risk free rate

(monthly, June 1928 - May 1998). The risk-free rate proxy measure is

likely to be contaminated with error due to term-structure effects and

other measurement errors, but, due to the relative variability of stock

and bond returns, for the applications in this paper that involve cross­

correlations between stock returns, equity premium and consumption

growth rates, these errors are likely to be negligible. Real bond and

162

stock returns are constructed from respective indices deflated by the

consumption price deflators.

Yearly series used in the paper on a number of occasions are constructed from

the series in the Australian Yearbook.

163

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