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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
THE EFFECTS OF INCOMPLETE
INSURANCE MARKETS AND
TRADING COSTS IN A
CONSUMPTION-BASED ASSET
PRICING
MODEL
by
John Heaton and Deborah Lucas
January 1992
3379.92.EFA
Working Paper No.
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
THE EFFECTS OF INCOMPLETE
INSURANCE MARKETS AND
TRADING COSTS IN A
CONSUMPTION-BASED ASSET
PRICING
MODEL
by
John Heaton and Deborah Lucas
January 1992
Working Paper No.
3379-92.EFA
January 28, 1992
The Effects of Incomplete Insurance Markets and Trading Costs in
a Consumption-Based Asset Pricing Model
John Heaton*
Deborah Lucas**
Abstract
markets,
Incomplete
financial
coupled
with
undiversif iable
idiosyncratic shocks, have the potential to explain a number of
asset pricing puzzles.
We study a model in which agents have
access to a limited set of securities markets, while facing
aggregate and individual uncertainty.
Trade is limited by the
presence of transactions costs, borrowing constraints, and short
We find a systematic relation between the
sales constraints.
extent and type of market frictions, and their implications for
asset prices and consumption policy. With trading costs or binding
borrowing constraints, the riskfree rate falls and the risk premium
rises relative to the complete markets case, and the term structure
exhibits a positive forward premium. However, with costless access
to either the stock or bond market, agents effectively smooth out
transitory income shocks, and equilibrium asset prices resemble
those in the complete markets case.
*
Massachusetts Institute of Technology and NBER
** Northwestern University and NBER
Please address correspondence to:
Deborah Lucas
Dept. of Finance
Northwestern University
2001 Sheridan Rd.
Evanston, IL 60208
We wish to thank Rajnish Mehra and Latha Ramchand for helpful
Conversations with Narayana Kocherlakota and Jean-Luc
comments.
Vila have also been influential. We would like to thank the IFSRC
at M.I.T. for financial support.
^
1
1.
Introduction
problem
The
predicting
of
consumption
patterns
in
an
environment of random income fluctuations and incomplete insurance
markets has traditionally been the domain of macroeconomists.
Friedman's (1957) permanent income hypothesis provided the insight
that transitory
disturbances
income
capital market transactions.
should be
In the
smoothed through
extensive literature that
develops this idea, it is generally assumed that consumers face an
exogenous
and
constant
interest
Starting
rate.
from
the
perspective of the consumption-based asset pricing model of Lucas
(1976)
,
question
number
a
of
how
of
authors
income
have
recently
fluctuations
and
asked
the
incomplete
related
insurance
markets influence predicted rates of return (Aiyagari and Gertler
(1991), Constantinides and Duffie (1991), Heaton and Lucas (1991),
Huggett (1991), Ketterer and Marcet (1990), Lucas (1991), Telmer
(1991),
Marcet and Singleton (1991)).
Essentially, these recent
papers imbed a permanent income type model in an infinite horizon
general equilibrium setting.
There
number
motivations
studying
the
implications of incomplete markets for asset pricing models.
The
failure
are
of
a
simple
consumption-based
of
versions
asset
of
the
pricing model
for
representative
to
predict
the
consumer
observed
statistical properties of asset returns can be attributed in part
to
the
^
low variability
of
aggregate consumption growth rates.
and Singleton (1983) and Mehra and Prescott
among others, identified the asset pricing
puzzles in empirical investigations of the Lucas (1976)
Hansen
(1985)
,
.
2
Market
breaks
incompleteness
link
the
between
individual
and
aggregate consumption, and in particular allows for the possibility
individual
that
growth
is
volatile
more
than
in
Furthermore, with incomplete markets agents engage in
aggregate.
trade.
consumption
Thus one can address questions about trading volume,
the
impact of transactions costs, and the nature of liquidity trading.
Critics argue
These models have several drawbacks as well.
that the ability to arbitrarily close markets results in a loss of
discipline; small changes in assumptions can produce qualitatively
The models
Complexity is also an issue.
different conclusions.
can only be solved numerically, and very few general results have
been
calibrating
the
Finally,
derived.
the
models
are
requirements
for
escalated by
the
data
often
testing
or
fundamental
heterogeneity
The purpose of this paper
is
to provide
clearer intuition
about the predictions of these models, with an emphasis on showing
which implications are robust and which are not.
To do this we
study a relatively simple, three period, two person model in which
a large number of cases can be easily analyzed and interpreted.
We
are particularly interested in comparing the impact on asset prices
of
different
types
of
trading
constraints,
short
sales
proportional
costs.
Our
including
frictions,
constraints,
analysis
quadratic
suggests
that
borrowing
costs
with
and
trading
frictions, this class of models has the potential to help resolve
cleanly
Jagganathan
and
(1991)
Hansen
model.
consumption
between
connection
characterize
the
volatility and asset prices.
3
the equity premium and low risk-free rate puzzles and to explain
the forward premium in the term structure.
however,
Calibration results,
are likely to be quite sensitive to the assumed market
structure.
The paper is organized as follows.
the
frictionless model,
markets
critical.
is
agents
costless,
these markets.
we present
2
explain why the assumed timing of
Section
show that:
We
frictions.
and
In Section
3
introduces
various
trading
some
trading
markets
is
substitute almost entirely towards trading
in
(1)
If
in
Hence trading costs influence prices only if they
are present in all asset markets;
(2)
The assumed market structure
has a large and systematic impact on predicted asset prices.
In
the equity premium is larger and the risk-free rate
particular,
lower when only borrowers pay trading costs in the bond market, or
when there is a binding borrowing constraint;
(3)
The implications
of proportional costs are similar to those of quadratic costs.
Section
4
we demonstrate that the model with trading
frictions
predicts a positive forward premium in the term structure.
5
2.
In
Section
concludes.
Asset Pricing with Incomplete Markets and Frictionless Trade
This section presents two simple models that illustrate how
small changes in assumed market structure can significantly affect
predicted asset prices and consumption allocations when
shocks are uninsurable.
(1986)
,
income
The first model is in the spirit of Mankiw
while the second captures the idea behind the infinite
.
4
horizon models of Lucas (1991), Marcet and Singleton (1991), and
Telmer (1991)
These models have the following basic structure.
contains two
(groups of)
income realizations.
income,
no debt.
Y'
.
agents who are distinguished by their
Stochastic income derives from two sources
an aggregate stock dividend,
each period:
Each economy
6,
and individual labor
Initially each agent owns half of the stock and holds
Riskfree bonds are in zero net supply at time
stock price is
p^,
and the bond price is
0.
The
p''.
We assume that labor income is uninsurable in the sense that
agents cannot write contracts contingent on the outcome of future
labor income.
This assumption can be motivated by considerations
such as moral hazard and legal limitations on contracting, or the
observation that complete insurance does not appear to be a feature
of modern economies.^
To the extent that many capital assets have
limited collateral value,
falls
income from these assets conceptually
into the same category as
labor income.
Conversely,
any
future labor income that can be used to secure debt is equivalent
to dividend income.
The intuition behind the model only requires
that some component of income is uninsurable,
regardless of its
For example, Hayashi, Altonji and Kotlikoff (1991) and
Cochrane (1991) offer evidence to this effect for the
U.S. economy.
One could also think of a bad event as
inducing a taste shock that increases the marginal
utility of income.
This appears to have similar
prices.
implications for asset
5
origin.-'
The key ide is that the inability to fully insure against
idiosyncratic
volatility
income shocks
may
much
be
implies that individual
higher
than
aggregate
consumption
consumption
volatility.
2.1
Model
l:
Asset Markets Close Prior to the Resolution of
Uncertainty
We begin with a two period model similar to Mankiw (1986)
time
each agent
i
and bond purchases,
chooses consumption,
b',
Cq'
,
.
stock purchases,
At
s',
to maximize:
U(ci) + ^EUCcj)
(1)
subject to the budget constraints:
ci = yj ^
^
-
b'p" - s*p^
(2)
cj = YJ
+ ll
+ b*
+ s''5i
Market clearing requires:
cj + Cj = Yj +
sW
bW
s2 =
(St
t = 0, 1
(^^
b^ =
Asset prices satisfy:
Of course, identifying the source of uninsurable income
is important for empirical applications of these models.
Due to the limited data on the composition of income,
equating labor income with uninsurable income is a
practical first approximation.
E[U^(c,)]
_ ^
- >^
P
(4)
:
U'(Co)
ps =
^_L^_LLJ..
(5)
U'(Co)
characterize
To
patterns
consumption
implied
and
asset
returns, we must specify the exogenous stochastic process governing
dividends and labor income, and further restrict the form of the
utility
Assume U'''(c)
function.
>
We
0.
implications of the following income process.
equal
have
aggregate
probability
labor
and
output
is
dividend
Y^^
e
each agent receives Y„/2.
high,
agent receives
be
XY^,
in
a
6^/2
probability
[Y^_,
Y„]
q,
as
the
recession.
5^
At
.
and
If realized output
increased
is
6^],
probability
Ex ante each agent
contingent on the realization of this income.
low,
{l-k)Y^^; k > 1/2.
of
1,
with
and
a
one
This
becoming
faces the
distribution of future labor income, and cannot write
>.=l/2
low
[0,
e
time
If realized output is
.
while the other receives
interpreted
unemployed
+
Yq
all agents
At time
The aggregate dividend is
(1-q).
aggregate labor income is
can
with
high
income:
consider the
will
same
contract
Notice that the case
corresponds to the standard representative agent model.
Under these assumptions it is straightforward to demonstrate
that
the
equity
premium
may
be
much
higher
than
in
the
7
representative consumer economy.^
wealth and information at time
in the asset markets,
period.
The
0,
of
stocks
substituting income into equations
a
k
it follows that no trades occur
so that consumption equals
price
constant for all
Because agents have the same
and
(4)
bonds
and
(5)
is
.
income in each
then
found
The stock price is
since we have assumed that the stock only pays
dividend in the high income state.
The convexity of the marginal
utility function implies that the bond price increases in
the premium,
U' (0)
=
income.
by
E((S^)/p®
-
1/p'',
increases in
A,.
Thus
X.
If we assume that
00,
the premium becomes arbitrarily large as the share of
A,,
received by the employed agent goes to one.
When the premium is written as a ratio of the stock and bond
returns rather than their difference,
it has been shown for more
general income and dividend processes that the proportional premium
increases in the level of idiosyncratic risk (Mankiw (1986)
(1992)).^
on the
However,
level
interest.
,
Weil
idiosyncratic risk has two offsetting affects
of the premium,
which is the object of empirical
While idiosyncratic risk increases the required return
on stocks relative to bonds, it also decreases the required return
on both stocks and bonds because it creates a precautionary demand
Rietz's (1988) demonstration that the equity premium
puzzle can be resolved with the assumption of a small
probability of extremely low (aggregate) consumption
relies on a similar mechanism. Mehra and Prescott (1988)
argue that disasters of the requisite magnitude are not
supported by the data. It seems more plausible, however,
that individuals face a non-negligible probability of a
very bad shock.
These results also depend on the
derivative of the utility function.
sign
of
the
third
.
8
It should be noted that one can easily find examples
for assets.
which
in
premium
proportional
the
complete markets case)
increases
(relative
to
the
but the level of the premium falls.^
,
This analysis suggests that a model with incomplete insurance
markets
potential
the
has
riskfree rate puzzles.
to
resolve
the
equity
premium
and
However, the next section demonstrates that
even when idiosyncratic risk implies a large premium in this model,
the
result
is
not
robust
to
realistic
changes
in
the
assumed
structure of asset markets when income shocks are transitory.
2.2
Markets Close After the Resolution of Uncertainty
Model 2:
Here we take into account the possibility that agents can use
asset markets to partially offset idiosyncratic income shocks.
To
provide agents with a chance to trade, we extend the above model to
three periods.
At time
agents can trade in one and two period
riskfree bonds, and in the stock.
can be resold, and new
be bought or sold.
1
and
1
At time
1,
old two period bonds
period bonds can be issued, and stock can
The stock pays a stochastic dividend at times
2
As before, agents are assumed to be identical at time
0.
They
potentially receive different shares of aggregate income at time
and time
^
2,
however,
1
due to the employment shock at that time.
For instance, let U(c)=c'V-4, iS=.9, S e [.1, .2], Y e
[.8, 1.2], and normalize consumption to one at time 0.
Each realization (Y, S) has probability .25. The implied
Now at time 1 increase the
equity premium is 12.7%.
idiosyncratic shock, so that Y e [-7, 1.3]. The implied
premium falls to 9.0%, while the percentage premium
increases from 1.09% to 1.10%.
9
Agents choose asset holdings and consumption in each period to
maximize:
+ /3EU(c;)
U(ci)
(6)
+ /S^EUCc*)
subject to the budget constraints:
^'
v' ^
Cq - ^0 "^
cj
= Y;
.
cj = yJ -
where
i,
_i_s
Dq Po " v^'2»,2
- v,'''v^''
°o Po " SqPo
(^^si)5i .b'^ -plbj^
(l-si*s;)52 -bj^
- sjp^
- bj'
the price of the n period bond at time t,
is
purchases of stock at time
t,
and
p^®
assets
are
traded;
b^'^
=
bg'^
is
s^'
net
is the stock price.
As above, since agents are identical at time
no
(7)
is net purchases of the n period bond at time t by agent
b^'"
p^"
'^0
7
=
Sq'
=
0.
0,
in equilibrium
Without
loss
generality, we assume that when aggregate output is low at time
the
first
agent
receives
the
good
idiosyncratic
second agents receives the bad shock:
Y„ > Y^.
Y^^
= Y„, and
of
1,
shock and the
Y,^
= Y^, where
The equilibrium portfolio rule and asset prices are found
by solving the first order conditions,
(8)
-(12),
for i=l,2.
pXcyo-^) =^Eo[u'(Y;4-b;V!-sip^)]
(8)
p^u'(Yo4°) =^Eo[u^(Y;.^-b;V!-s;p^)(5,.p?)]
o)
10
PoU'(Yo + 5)
=
(10)
fi%[\j' (Yl^S^il^s]) ^h\')]
py(Y;4-s;p^b;y)
:;+^-s;p^-b;V!)
Market clearing implies that
=
=
^^^^
pE,iu'(Y',^s,ii^s\)^h\')]
^^^^
^Ei[u^(y*+52(1+s;) ^bj^^s]
b^^^
=
-b^^\
and
s^^
=
-s^^
in
each
state.
In general, a closed form solution to the nonlinear system
cannot be found. ^
-(12)
We solve these equations numerically to
illustrate some qualitative features of the model. °
We make the
standard assumption of a constant relative risk aversion
period utility function, U(c) =
advantage
of
being
(8)
relatively
(c^^'^^-1)/ (l-y)
tractable,
,
y > 0*
and
of
(CRRA)
It has the
predicting
stationary prices when the growth rate of income is stationary.
Furthermore,
these
preferences
have
the
desired
property
that
increasing income volatility increases the required return on risky
assets.
''
®
In this case it is
The exception is quadratic utility.
easily shown that prices only depend on the aggregate
endowment, independent of the wealth distribution. It
should be noted, however, that even when prices are
unaffected by the absence of insurance markets, welfare
is reduced.
The model is solved using Newton's Method.
11
Whether
markets
asset
will
be
used to
smooth
consumption
depends critically on the persistence of the idiosyncratic shocks.
Evidence from U.S. panel data suggests that income shocks have both
a permanent and transitory component
(e.g., Carroll
(1991)).
The
ability to self-insure diminishes as shocks become more persistent,
because more persistent shocks have a larger impact on permanent
income and hence on desired consumption.
We consider both cases,
but focus primarily on transitory shocks.
a.
Transitory Shocks
With transitory idiosyncratic shocks, the equilibrium has the
characteristic that agents trade away from the initial allocation
of equal stockholdings and no bonds in order to smooth consumption
at
time
marginal
1.
At
rate
the
Hence market
assets to agent
allocation,
substitution
of
consumption.
initial
between
agent
2
current
has
a
and
higher
future
clearing requires him to sell both
The magnitude of the equity premium will depend
1.
on the residual variability of consumption after trading has taken
place.
To calibrate the model, we assume that /3=.95, and y
5,
7)
.
At time
e
(1,
3,
each agent receives an equal share of dividend
and labor income, with dividends comprising 12.5% of total income.
Dividend
and
labor
income
grow by
2%
on
average
each period.
Aggregate dividends are high or low with equal probability, with
= 1.3 5^
S^^
When aggregate dividends are high, both agents receive
equal labor income.
When aggregate dividends are low at time
1,
.
12
the first agent receives income equal to 1.62 the income of the
second agent.'
Time
labor income is also stochastic in the low
2
aggregate state, and independent of the time
Y^^
if
The lower variance at time
= LlOY^.
the
economy
idiosyncratic
were
to
continue
2
for
1
idiosyncratic shock;
reflects the idea that
another
period,
the
shocks would again be partially offset by
income
trade.
Table
reports the average returns, the value of stocks and
1
traded
bonds
in
variability.
the
low
income
state,
and
time
date, and time
1
demonstrate
than
variability
1
is the terminal
consumption is constrained to equal income.
variability at time
lower
consumption
For comparison with section 2.1, we also report asset
returns and consumption variability when time
results
1
that
1,
in
the
is
lower
no
despite
the
high
individual
These
income
the implied equity premium is substantially
trade
than
case.
income
The
fact
that
consumption
variability accounts
for
the
difference; the coefficient of variation of consumption is about
9%, while the coefficient of variation of income is 15.5%.^°
These
results are qualitatively similar to those of the infinite horizon
models of Lucas (1991), Heaton and Lucas (1991), and Telmer (1991).
The choice for the variability of individual income is
loosely based on data from the Panel Study of Income
Dynamics, as described in Heaton and Lucas (1991)
10
coefficient
For comparision,
the
aggregate consumption is 2.64%.
of
variation
of
13
Table
Y
1.
Asset Returns, Trading Volume, and Consumption Volatility
14
risk lowers the risk-free rate relative to the complete markets
The
case.
results here
suggest that
although a precautionary
demand may substantially lower the riskfree rate,
required return on equity as well.
it
lowers the
One must therefore be careful
not create a "high equity return puzzle" while explaining the low
risk-free rate puzzle.
these
From
calculations,
insurance markets alone are
it
appears
that
limitations
on
insufficient to explain the equity
premium or the low risk-free rate puzzles for low values of the
risk aversion coefficient when income shocks are transitory.
b.
Permanent Shocks
In
this
model,
a
permanent
idiosyncratic
shock
represented most easily by constraining labor income at time
equal the realization of labor income at time
a low aggregate time
a
1
realization occured.
1,
be
can
2
to
in the event that
As expected, we find
marked decrease in trading volume, and an increase in the equity
premium.
These results are consistent with those of Constantinides
and Duffie
(1991)
,
who derive a class of permanent income shock
processes for which there is no trade and a potentially high equity
premium in an infinite horizon model with many agents.
To summarize, if agents have no access to asset markets, or if
idiosyncratic
income
shocks
are
permanent,
standard
utility
specifications can produce a high equity premium and a low riskfree
.
15
rateJ^
However, with access to frictionless asset markets agents
effectively smooth out transitory income shocks, and implied asset
prices are similar to the representative consumer case.
We now
turn to the question of whether more moderate frictions in the
asset markets can help to explain asset prices when income shocks
are transitory.
3.
Asset Pricing with Incomplete Markets and Trading Frictions
In this
income
shocks
constraints,
"short
section we modify the above model with transitory
to
and
sales
allow
for
trading
costs. ^'
constraints"
"trading cost"
is
borrowing constraints,
are
"Borrowing
fixed
trading
short
sales
constraints"
limits,
essentially a tax on trades.
As
and
while
a
one might
expect, predicted asset prices are sensitive to the assumed form of
the market friction.
Here we hope to provide intution about the
systematic differences
in
the
implications of the various cost
structures
For the base case,
trading costs are assumed to take the
quadratic form:
^^
This suggests that the two period models of Mankiw (1986)
and Weil (1992) can be interpreted as representing
permanent shocks in longer horizon models.
^'
Aiyagari and Gertler (1991) examine the equity premium
and riskfree rate puzzles in a related model with a
continuum of heterogeneous agents, and proportional or
Their analysis differs in that
fixed trading costs.
risk,
and the government controls
aggregate
there is no
so as to peg the interest
liquid
assets
the supply of
Lucas
rate.
Heaton and
(1991) calibrate an infinite
horizon version of the model of this section.
16
where the function
Ks(S,p^)
= 7r^(p^S)2
KbCb,?")
= 7r''(p'^)2,
I^j^q)
iridicates that the cost
borrower but not the lender.
for tractability.
^^^^
or
is
paid by the
We use the quadratic form primarily
However, it also reflects the idea that as more
assets must be sold, agents first exhaust their more liquid assets.
section
In
3.d
below
proportional costs.
we
study
the
alternative
assumption
of
Two cost specifications are considered in the
In the first, which we call the "symmetric case", we
bond market.
assume that the borrower and lender each pay an equal trading cost.
Motivated by the substantial spread generally observed between the
borrowing and lending rate for consumers, the second specification
assumes that only the borrower bears the transactions cost.
In the
tables below, we report the cost parameters, and the percentage of
the proceeds of the asset sale absorbed by the costs:
or
Tr^p^s
Now agents choose consumption and asset holdings to maximize
subject to the constraints:
(6)
,ri
'
Cq = Yq +
cj
= y; +
c[=
Y^ +
^0
-
- i_i1
bg po 1
{l^si)s,
i_i2
^ hi'
(2+si+s;)52
2
2
bgPo
-
s
/i_i1
_1\
SgPo ' K^i(bo ,Po)
i
- pibj'
+ bi^
- sjp^ - K
+ bi^
-
/i_i2
_2.
K^2(bo 'Po)
i(b;\p;)
-
-
k3(s;,p^)
,_i
_s.
K3(So,Po)
^^"^^
17
s[ > S,
(15)
b;">B^,
n
=
l,2.
The determination of asset prices depends on whether the short
sales or borrowing constraints in equation (15) bind.
In states in
which both agents are unconstrained, prices are determined by the
marginal rates of substitution of both agents.
If a constraint
binds, the price is determined by the marginal rate of substitution
of the unconstrained agent, and the quantity equals the constrained
value.
The first order conditions for the unconstrained agents are
similar to
-
(8)
(12)
but now include the trading costs.
,
For
Note that
reference, the Appendix lists these modified equations.
the returns of interest are now those realized between times
1
and
2.
Trading
at
time
1
in
either
market
can
be
eliminated by setting the appropriate constraint in
binding (i.e.,
cost
tt"*
or
TT*
B^
= 0,
or
S^
= 0)
sufficiently large.
,
effectively
(15)
to be
or by making the transactions
One might suppose that either
mechanism for restricting trade would have similar implications for
asset prices in this extreme case.
1
As illustrated by Propositions
and 2, however, predicted asset prices are sensitive to the form
of trading frictions.
In particular,
either a binding borrowing
constraint or transactions costs borne entirely by the borrower
tends to lower the implied lending rate relative to the case in
which both the borrower and lender directly pay a cost.
18
Proposition
time
1;
s'.
Assume that there is no trading in the stock at
1.
= 0.
(a)
p^^
will be lower in the case in which
and the cost is symmetric than in the case in which
p.^
is the same
or if
B^
Proposition
2.
cost,
1;
b^'^
= 0.
if
tt''
= « and only borrowers pay a
B^
tt^
= 0.
=
«>
(b)
transactions
= 0.
Assume that there is no trading in the bond at time
Then
p^®
will be lower in the case in which
tt*
=
<»
the transactions cost is symmetric than in the case in which
and
=
S^
0.
(The proofs of propositions
1
and
2
are in the Appendix.)
These results suggest that the predictions of this class of
models may be very sensitive to the assumed cost structure.
examine
the
equilibria
implications
for
a
of
variety
less
of
extreme
we
costs,
parameterizations
To
calculate
and
market
structures.
a.
Trading Costs in the Stock Market
Participation in the stock market is often thought to be more
costly than participation in the bond market for a typical investor
or borrower.
One way to represent this idea in the model is to
impose a trading cost in the stock market only
(tt''
= 0,
w^
>
0).
19
Table
Y
2.
Asset Returns, Trading Volume, and Consumption Volatility
with Costly Trading in Stocks and Frictionless Trading in
Bonds
20
Similar results obtain (but are not reported here) when there
in bonds and free trading in stocks.
is costly trading
agents
case
cease
to
trade
consumption by selling stock.
in
bond
the
market,
In this
smooth
and
Asset prices again remain largely
unaffected in the absence of binding short sales constraints.
b.
Trading Costs in Both Markets
From
the
above
results,
costs
consumption volatility,
simultaneously.
Table
3
it
must
appears
be
that
imposed
reports results
for a
in
to
increase
both markets
range of
small
costs, under the assumption that only the borrower bears a cost in
the bond market.
Table
Y
3.
Asset Returns, Trading Volume, and Consumption Volatility
with Costly Trading in Stocks and Bonds, n^ = .2,
Asymmetric Transactions Costs in the Bond Market
21
Table
4
Asset Returns, Trading Volume, and Consumption
Volatility with Costly Trading in Stocks and Bonds,
TT'
Symmetric Transactions Costs in the Bond
2
Market
,
Y
.
22
reduce the volume of trade.
that higher costs
This
^*
indirect
effect increases consumption volatility, and hence the predicted
premium.
For
dominates.
results
must
It
appear
parameters
the
to
be
be
considered,
stressed
fairly
that
general,
the
direct
while
these
we
not
do
effect
qualitative
consider
the
reported level of returns informative.^^
One might expect that higher trading costs would cause the
premium
to
increase
Table
aversion.
4
more
steeply
for
higher
levels
of
suggests that this need not be the case.
risk
An
offsetting influence is that with higher risk aversion, agents are
willing to incur higher transactions costs to smooth consumption.
The net affect of risk aversion on the interaction between the
premium and transactions costs is therefore ambiguous.
c.
Borrowing or Short Sales Constraints
In this section we look at the impact of borrowing and short
sales constraints on asset prices.
be
thought
of
as
representing
The borrowing constraint might
type
the
suggested by Stiglitz and Weiss (1981)
,
of
credit
rationing
who show that consideration
of moral hazard may cause lenders to deny credit even if borrowers
promise a high interest rate.
The short sales constraint reflects
^*
The analysis of Aiyagari and Gertler (1991) examines the
direct effect only, since there is no aggregate risk to
differentiate stocks and bonds in their model.
^^
For quantitative implications of a related model,
Heaton and Lucas (1991)
see
23
the
that
idea
individuals
may
find
short-selling
stock
prohibitively costly.
Table
bonds
5
reports the results of limiting the maximum number of
issued by varying the level of the borrowing constraint,
With a sufficiently
while permitting costly trading in stocks.
restrictive
borrowing
constraint,
premium and a low riskfree rate.
model
the
predicts
a
large
Recall that when the borrowing
constraint binds, the bond price is determined by the marginal rate
of substitution of the lender, who in equilibrium must be content
to lend no more than the maximum allowed.
The interest rate must
therefore fall relative to the frictionless case to reduce desired
lending.
Although
these
results
reservations about this case.
are
encouraging,
we
have
several
First, the restriction on borrowing
required to generate a high premium may be unrealistically severe.
For cases in which the reported equity premium is high, borrowers
would willingly pay a substantial spread over the market rate.
instance,
For
for the parameters in the third column of Table 5, the
interest rate implied by the marginal rate of substitution for the
constrained borrowers is 11.3%, while the market rate is only 4.4%.
Secondly,
the
specification.
results
are
quite
sensitive
to
the
constraint
Small changes in the borrowing limit in Table
cause large changes in implied prices.
5
In experiments not reported
here, we find that the opposite assumption of a binding short sales
constraint in the stock market and costly trading in bonds tends to
produce a negative premium,
since limitations on the amount of
24
stock sold increase its price relative to the unconstrained case.
It
seems then,
that an accurate empirical measure of borrowing
constraints must be found before it can be persuasively argued that
borrowing constraints resolve the equity premium and low risk free
rate puzzles.
Table
Y
5.
Asset Returns, Trading Volume, and Consumption Volatility
with Costly Trading in Stocks, and Binding Borrowing
Constraints in the Bond Market
25
Solving the model with proportional costs is complicated by
discontinuity
the
in
the
marginal
cost
functions.
The
Euler
equations (A1)-(A5) can be solved as before to determine prices and
quantities under the assumption that agents trade in both markets.
It must then be checked whether given these prices, agents prefer
to trade in both markets, one market, or not at all.
Table
Y
6.
Asset
Returns,
Volume,
Trading
and
Consumption
Volatility, with Proportional Costs in the Stock Market,
and a Short Sales Constraint in the Bond Market
26
premium is similar in both cases, but the level of the returns is
higher for proportional costs.
The differences can be explained by
the fact that with quadratic costs the marginal cost is double the
average cost, which discourages higher trading volume for similar
In these examples,
levels of average costs.
the equity premium
appears to be largely determined by the trading costs in the stock
The
market.
higher
level
overall
of
returns
in
case
the
of
proportional costs reflects a lower precautionary demand due to
more effective smoothing.
In simulations not reported here, we considered proportional
costs
both
in
constraint.
marginal
without
markets
For many
trading
imposing
parameterizations,
costs
at
the
motivates
zero
exclusively in the lower cost market.
of equilibrium prices that
a
binding
borrowing
discrete
agents
jump
in
trade
to
This produces a multiplicity
support no trade in the higher cost
market.
4.
Incomplete Insurance Markets and the Term Structure
Related asset pricing puzzles arise in the literature on the
term structure of interest rates.
calibrated with aggregate U.S.
The representative agent model
time series data has difficulty
matching the statistical properties of the term structure
^*
^"^
.
One
Backus, Gregory and Zin (1989) demonstrate this failure
by calibration, and a number of recent papers further
examine this issue. Donaldson, Johnsen and Mehra (1990)
consider the term structure implications of the one-good
stochastic growth model. Although the model matches some
characteristics of the data, it does not explain the
forward premium. In a model with idiosyncratic shocks of
.
27
aspect of this is the failure of the standard model to predict the
positive average forward premium.
In this section we show that the
model
shocks
with
uninsurable
income
and
trading
frictions
generates a positive forward premium J^
In our model, the implied forward rate is the one period rate
on time
1
investments that can be locked in at time
by buying a
two period bond and selling a one period bond of equal present
value:
period
f^
rate
time
on
available at time
1.
The expected future spot rate is the one
= PqVPo^ ~ !•
1,
1
investments
that
based on their time
agents
expect to have
information:
The forward premium can then be written as PqVPq^
The forward rate and expected future spot rate at time
by solving equations (Al)
Table
-
positive
trading costs.
constraints.
.
are found
reports the implied forward and spot rates for the
7
a
~ ^^(l/p^^)
-
(A5)
parameterizations used in Tables
generates
Eg(l/p^^)
forward
3
and 4.
The model consistently
premium that
increases with bond
Similar results can also be obtained with borrowing
Notice that when trading costs are zero, we get the
very high frequency relative to aggregate shocks,
Mehrling (1991) shows that the forward premium will be
positive
aggregate
shocks
dominate
the
if
the
idiosyncratic shocks.
17
Because Treasury securities pay a fixed nominal return,
most models of the term structure incorporate money. For
instance, Backus, Gregory and Zin (1989) add a cash-inadvance constraint to the Mehra and Prescott (1986)
model.
Since our results are suggestive rather than
quantitative, for simplicity this analysis is in real
terms.
28
standard result that with low risk aversion, the forward rate is
approximately equal to the future spot rate.
Table
Y
7.
Foward Rates and Expected Future Spot Rates, n^=.2
.
29
Since in equilibrium two period bonds are in zero net supply, there
is no direct affect of transactions costs on their price,
there is for the one period bond at time
1.
while
The question remains
of whether the prediction of a relatively large positive forward
premium will obtain in a long horizon model that treats bonds of
different maturities more symmetrically.^®
5.
Conclusions
Incomplete financial markets have the potential to explain a
number
asset
of
pricing
puzzles
because
the
presence
of
undiversif iable idiosyncratic risk affects the precautionary demand
for
assets
uncertainty.
which
individuals'
and
attitutde
towards
aggregate
In this paper we examine a three period model
individuals
have
access
to
a
limited
set
of
in
securities
markets, while facing aggregate and individual uncertainty.
The
agents also face restrictions on their trading due to quadratic or
proportional transactions costs, borrowing constraints, and short
sales constraints.
This
analysis,
coupled
studies discussed earlier,
with
the
results
suggests that there
of
is
the
a
previous
systematic
relation between the extent and type of market frictions, and their
implications for asset prices and consumption policy.
The first
important regularity is that when agents have unrestricted access
^®
Another issue that is better examined in an infinite
horizon model is the differential volatility of returns
We plan to examine these questions
across maturities.
using the model in Heaton and Lucas (1991)
30
to any financial market,
idiosyncratic
income
shocks
frictionless securities.
to
representative
the
they effectively smooth out transitory
trading
by
primarily
in
the
The implied asset prices are then similar
consumer
case.
effective
trading
asset prices vary predictably
frictions in all markets, however,
with the assumed market structure.
With
As the cost parameter increases
with quadratic or proportional costs, the riskfree rate tends to
fall and the risk premium tends to rise.
costs,
well
as
predictions.
as
their
level,
heavily
The incidence of the
influence
the
pricing
In particular, the premium is larger and the riskfree
rate is lower when only the borrower directly bears a transactions
cost in the bond market.
Finally,
imposing a binding borrowing
constraint in the bond market has the most dramatic affect in the
direction of explaining the asset pricing puzzles.
The
results
reported
in
this paper are
suggestive of the
potential importance of market incompleteness for explaining asset
market behavior.
Clearly, though, many empirical and theoretical
issues remain unresolved; we mention only a few.
What constitutes
"reasonable" magnitudes for the frictions in this type of model is
an open question.
Since the predictions of the model are most
binding borrowing constraint,
promising under the assumption of
a
endogenizing
perhaps
fundamental
this
constraint,
credit market
imperfection,
by
modeling
a
more
would make this a more
persuasive explanation for rate differentials.
Finally, we have
focussed primarily on the predictions about first moments.
Whether
31
trading frictions will help to match higher moments of asset prices
remains to be investigated.
:
32
Appendix
First order conditions for the maximization problem
=
pXcyo^^)
/3Eo[u'(Y;-ii-b;v;-s;p^-Kb(b;\p;) -k3(s;,p^))]
PoU'(Yo
(?;
-
^^^^
^Eo[u'(Y;H-ii-b;V!-s;p^-Kb(b;\p;) -k3(s;,p^)) (5i+p^)S^^^
=
poU'{Yo*t)
(7)
+
^)
=
(A3)
0%[\j'{Y[^S2{'-*s\)^h\')]
!!^^^)u^(Y;.il-s;p^b;v;-K,(b;^p!) -k^o^p?))
=
(A4)
=
(a5)
^Ei[u'(Yj+62(-J+s;)+b;^)]
(p^ +
!!!_!i:!L)u'(Y;-^-s;p^-b;V!-Kb(b;\p;) -K,(s;,p^))
^Ei[U'(Y*+52(i+s;) +b;^)52]
Proof of Propositions
As
> Y^.
from
00
-<
tt''
00,
b -
1
and 2.
and
(1)
7r''(p^^b)^
-*
Let y/ =
Y„,
(since
p^^
when income is strictly greater than
limit as
TT
-
00
of
(3K/3b)/p^'.
0)
.
and
Y^^
= Y^; Y„
is bounded away
Let x equal the
Then with symmetric trading costs
and without a borrowing constraint, the bond price must satisfy:
33
p!
.
34
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