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Permission is given 1111.' ,i'fl..-to the University Librarian or his representative to allow persons other than students or members of staff of the University to consult my thesis only for the purposes of private study and research.
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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