“Projected Earnings Fixation”
Paul E. Fischer
Mirko S. Heinle
Robert E. Verrecchia
University of Pennsylvania
November 2012
Abstract
We consider the implications that arise from investors believing that investors in the future
will place too much emphasis on earnings information: we refer to this phenomenon as
“projected earnings …xation” (PEF). Our results show that periods of PEF lead to greater
price volatility, and price behavior inconsistent with standard valuation frameworks. We also
show that investors who exhibit PEF achieve a higher ex ante expected utility. The reason
for this is that PEF results in greater price volatility and investors bene…t from the increased
risk premium inherent in stock price. In other words, our analysis extends beyond simply
positing that investors project that other investors place too much emphasis on earnings, to
show why investors might gravitate toward this projection.
We thank workshop participants at Baruch College, Carnegie Mellon University, the Chicago-Minnesota
Theory Conference 2012, the Tel Aviv International Conference in Accounting, and UC Berkeley for helpful
comments.
1
Introduction
Keynes is credited with the observation that short-horizon investors will attempt to predict
future beliefs about a …rm’s equity value, as opposed to making predictions about the …rm’s
fundamental value. As a consequence, beliefs-about-beliefs – and not beliefs about fundamentals –may drive …rm share prices. Various theoretical analyses have demonstrated that
beliefs-about-beliefs can foster pricing bubbles in the marketplace, thereby causing share
prices to rise temporarily above fundamental value and manifest excess share-price volatility. In most of these theoretical models, beliefs-about-beliefs pertains to the …rst moment
(mean) of beliefs: if investors believe that some investors will view share prices more favorably in subsequent periods, prices will rise each period to re‡ect these beliefs. This results
in an equilibrium where shares are overvalued relative to fundamentals.
The purpose of this paper is to consider a variation on Keynes’observation, one that pertains to the second moment (variance) of beliefs. Speci…cally, we consider the implications
that arise from investors believing that investors in subsequent periods will place too much
emphasis on earnings in their beliefs formation. We refer to the phenomenon where investors
in the current period believe that investors in subsequent periods place too much emphasis
on earnings as “projected earnings …xation”(PEF). The chief goal of this paper is to demonstrate that PEF can be self-ful…lling: that is, a belief by current investors that investors
in subsequent periods will place too much emphasis on earnings leads to current investors
also placing too much emphasis on earnings. As a consequence, equilibrium price paths can
exhibit excessive volatility relative to the volatility of underlying economic fundamentals.
The notion that users of accounting information place undue, or disproportionate, em-
1
phasis on accounting measures of performance has a long tradition in accounting (see, for
example, Ijiri, Jaedicke, and Knight 1966 and Ashton 1976). Although undue emphasis
might arise in the context of a variety of measures, research has focused primarily on investors’…xation on earnings. For example, Hand (1990) examines the stock price reaction
to quarterly earnings announcements of …rms that undertook debt-equity swaps, which increased earnings by about 20%. Hand …nds an abnormal stock return at the time of the
announcement, consistent with the earnings …xation hypothesis. In a similar vein, Sloan
(1996) …nds that stock prices act as if investors “…xate” on earnings, to the exclusion of
information contained in the accrual and cash ‡ow components of current earnings. While
much of that literature alludes to cognitive limitations or biases as the driver of …xation, our
analysis suggests that …xation can arise as an equilibrium phenomenon among investors who
are not inhibited cognitively.1
Our intention in coining the expression “projected earnings …xation”is to make reference
to the phenomenon in Psychology of a person denying his own attributes, thoughts, and
emotions, and instead projecting the belief that others originate those feelings. In Psychology
projecting thoughts or emotions onto others allows the person to evaluate them and the
extent to which they are dysfunctional, but without feeling the attendant discomfort of
knowing that these thoughts and emotions are his own. For example, projection allows
a person to criticize another person, while distancing him from his own dysfunction. We
transport this idea into the realm of equity markets by suggesting that a person who exhibits
1
The overemphasis on earnings discussed here is related to the notion that with privately informed
investors, public information (i) facilitates coordination among traders and (ii) shapes higher order beliefs,
both of which lead to an overreaction on public information (e.g., Morris and Shin 2002, Allen, Morris, and
Shin 2006, and Gao 2008).
2
projected earnings …xation projects that investors in subsequent periods will place too much
emphasis on earnings: in other words, investors believe that investors in subsequent periods
will be dysfunctional in the emphasis they place on earnings.
To study the role of PEF in pricing, we posit a parsimonious, overlapping generations
(OLG) model common to the literature. Identical investors with constant absolute risk
aversion (CARA) utility functions live for two periods. These investors are savers (buyers)
in the …rst period of life, and consumers (sellers) in the second period of life. The investment
opportunity set includes a risky asset and a risk-free asset, where the risky asset generates
earnings each period that are paid out immediately as a dividend and the risk-free asset yields
a …xed return each period. The earnings of the risky asset follow a simple, one-period, autoregressive time series process. Within the context of this model, the only news that arrives
each period is the most recent earnings information. To establish a benchmark, we …rst
characterize a steady-state linear equilibrium where the equilibrium price is a linear function
of earnings, and the intercept and coe¢ cient on earnings both remain unchanged over time;
this is consistent with a stable earnings process. We depart from the existing literature
by also characterizing alternative linear equilibria where the coe¢ cient on earnings and the
intercept term vary over time.
Our analysis shows that when current investors project that investors in the future will
place too much emphasis on earnings information –in e¤ect, current investors exhibit PEF
– current investors themselves place too much emphasis on current earnings. As a consequence, equilibria can arise where investors’response to earnings information becomes more
exaggerated than the variability of the fundamentals would suggest. In addition, our analysis
demonstrates that the reaction to information about stochastically stable fundamentals can
3
vary over time, with, for example, some periods of PEF and other periods with a steady-state
equilibrium that re‡ects investors responding in a fashion that is consistent with the stable
variation in fundamentals. Periods of PEF lead to greater price volatility and price behavior
inconsistent with standard valuation frameworks. Steady-state periods, on the other hand,
result in pricing patterns consistent with standard valuation frameworks; accordingly, we
refer to steady-state price as the asset’s fundamental value.
By way of motivating why investors might gravitate toward projecting that investors in
the future will place too much emphasis on earnings information, we show that investors’
ex ante expected utility is higher when they exhibit PEF. The reason for this is that PEF
results in more price volatility, and investors’ex ante welfare increases as volatility increases.
In other words, our analysis extends beyond simply positing that investors project that other
investors place too much emphasis on earnings to show why investors might gravitate toward
this projection.
Our study contributes to the literature on excess stock price volatility initiated by LeRoy
and Porter (1981) and Shiller (1981), two papers that argue that stock prices exhibit excessive
volatility relative to fundamental cash ‡ows and discount rates. One strand of the literature
on excess volatility that is related to ours studies sunspot equilibria: this literature includes,
but is not limited to, Azariadis (1981), Cass and Shell (1983), Peck (1988), Jackson and
Peck (1991), and Jackson (1994). In sunspot equilibria, prices are determined in part by
stochastic events unrelated to economic fundamentals (so called “sunspots”). In overlapping
generations models, security prices are a function of sunspots because investors believe they
will be a function of sunspots. As a consequence, sunspots create volatility that is unrelated
to underlying fundamental volatility. Similar to sunspot models, in our model investors’
4
projected earnings …xation is sustained because investors believe it will continue into the
subsequent period. Unlike sunspot models, however, projected earnings …xation is based on
an economic fundamental –namely, earnings –and thus is not entirely spurious.
Another strand of the literature on excess volatility related to ours relies on the introduction of noise trade, or some other form of irrational trade, to drive excess volatility.2 Within
this literature, the study most related to ours is Spiegel (1998); this paper also employs an
overlapping generations framework and focuses on the role of alternative excess volatility
equilibria. Spiegel (1998) shows that, in addition to an equilibrium where security prices are
driven solely by fundamentals, a small amount of noise in security supply (or noise trade)
can lead to high volatility equilibria where the noise is the primary driver of security prices.
Unlike the analysis in Spiegel (1998) and other studies that rely on noise trade, the volatility
in our study is attributable to how investors respond to earnings information, as opposed
to how they respond to noise: in other words, our model does not introduce an additional
source of noise (additional to fundamental uncertainty).3
The remainder of the paper proceeds as follows. In the next section we describe our model
and characterize the benchmark steady-state equilibrium. In Section 3 we identify another
class of linear equilibria characterized by projected earning …xation, and characterize how
equilibrium prices behave when investors exhibit PEF. In Section 4, we show the existence of
additional equilibrium paths and discuss how the equilibrium can revert back to the steady
2
Studies that rely on noise trade, or some other form of irrational trade, to drive prices away from
fundamental values include: Spiegel (1998), Delong, Shleifer, Summers and Waldmann (1990A) and (1990B);
Allen and Gale (1994); Bhushan, Brown, and Mello (1997); Abreu and Bunnermeier (2002) and (2003); and
Watanabe (2008).
3
A …nal study worth noting is Dow and Gorton (1993), which shows that prices can deviate from
fundamentals as a consequence of fund managers being given incentives to take on risk. Such incentives are
not present in our model.
5
state. Welfare considerations are addressed in Section 5. In Section 6 we extend the model
to consider spillover e¤ects. Section 7 concludes
2
Model
Consider an overlapping generations model where a continuum of atomless investors invest
in a one-period, risk-free asset with a rate of return, r, and shares of an in…nitely-lived risky
asset. At the beginning of each period, the principal and interest for the risk-free asset are
paid, and the risky asset’s earnings are disclosed and paid out as a dividend. After the
interest and dividend payments, the investment market opens and investors form portfolios
of new, one-period risk-free asset claims and shares of the in…nitely-lived risky asset.
Investors live for two periods and have wealth w to invest in the …rst period of life.4 In the
…nal period of life investors liquidate their investments and consume their wealth. Investors
are risk averse, and have preferences characterized by a negative exponential utility function
with coe¢ cient of risk aversion , where
= 0 corresponds to the case of risk neutrality.
Hence, at time t an investor in the …rst period of life chooses the quantity of shares, q, to
maximize the expectation of
1
exp
h
q(~
et+1 + P~t+1 ) + (1 + r) (w
qPt )
i
1 ,
(1)
where Pt and P~t+1 are the time t and t + 1 share prices, “~” combined with a time t + 1
subscript denotes a random variable from the perspective of an investor at time t, and the
absence of “~” denotes either a realization of that variable or a …xed parameter. Because
the single-period investment horizon might seem arti…cial, note that we can obtain identical
4
Note that we assume that each period investors arrive at the market with a constant amount of wealth.
Thus, the high volatility equilibria in our model are not due to changes in the amount of capital available
for investment (i.e., there is no in‡uence of volatility on production or vice versa).
6
results by allowing each investor to live (and trade) for a …nite number of periods and
maximize his expected utility in his …nal period of life, which is when investors liquidate
their portfolios.
There is 1 risky asset share per-capita in each generation, which yields an earnings/dividends
per share that follows a time series process of the form
e~t+1 = e + ~"t+1 ,
(2)
where e is a positive constant that represents the permanent component of earnings and
~"t+1 is the variable component of earnings in period t + 1. Henceforth we refer to ~"t+1 as
“abnormal earnings.”Abnormal earnings has the following time series property:
~"t+1 = "t + ~t+1 ,
where
(3)
2 (0; 1) is a persistence parameter and ~t+1 is the “abnormal earnings innovation”in
period t + 1. The innovations are independent and identically, normally distributed random
variables with mean 0 and variance v.
2.1
Linear Equilibria
We de…ne an equilibrium to our game as the following three conditions: (i) each investor
chooses his demand by maximizing his expected utility conditional on his expectations; (ii)
expectations are rational and met in equilibrium; (iii) markets clear. We initially focus on
equilibria that have the feature that end-of-period price can be written as a linear function
of abnormal earnings:
P~t+1 =
t+1
+
7
"t+1 .
t+1 ~
(4)
While we do not require that intercepts and slope coe¢ cients,
and
t+1
t+1
respectively,
be constant over time, in the next section we study an equilibrium where this is the case
(i.e.,
t+1
=
and
t+1
=
). We refer to the latter as the “steady state equilibrium.”
In any equilibrium where price is a linear function of abnormal earnings, a new investor’s
maximization of his expected utility function in period t yields a demand of qt , where qt is
de…ned by
qt =
e + "t +
t+1
+
t+1
v 1+
"t
(1 + r) Pt
2
.
(5)
t+1
As re‡ected in eqn. (5), an investor’s demand increases in the expectation of next period’s
earnings/dividend, e + "t , and share price,
t+1
+
t+1
"t , and decreases in the risk-free
return that an investor forgoes to invest in shares of the risky asset, (1 + r) Pt . Note that
demand is more sensitive to current abnormal earnings when abnormal earnings persistence,
, is greater because the expectation of earnings/dividend and price are more sensitive to
current abnormal earnings when
is greater, and is greater in magnitude as investors’risk
aversion decreases.
Because there is 1 share of the risky asset per investor, market clearing for period t
requires that qt = 1 for all investors, which implies an equilibrium price of
Pt =
e + "t +
t+1
+
"t
1+r
t+1
v 1+
2
t+1
.
(6)
The pricing condition implies that the price at time t is equal to the discounted expected value
of next period’s earnings/dividend plus the price at which shares will be sold next period,
e + "t +
t+1
+
t+1
"t , minus a risk premium, v 1 +
2
t+1
(assuming that investors are
risk averse). The risk premium arises because the dividend investors receive and the price
at which they will sell their shares are both determined by the realization of next period’s
8
earnings, e~t+1 . The equilibrium pricing condition, eqn. (6), implies that a linear equilibrium
is de…ned by any at and
t
that satisfy the following two conditions for all t:
t
t
1 + t+1
1+r
t+1 + e
=
=
and
v 1+
1+r
(7)
2
t+1
.
(8)
Note that the possibility of a time-dependent earnings response coe¢ cient implies that the
intercept term can also vary over time. This happens because the risk that time t investors
bear depends on the price reaction of time t + 1 earnings, and therefore the risk premium
that time t investors demand.
2.2
Steady-State Equilibria
First we establish a steady-state benchmark equilibrium by characterizing an equilibrium
where the coe¢ cient and intercept are stable over time; this approach is consistent with
some antecedent research that employs a overlapping generations framework in conjunction
with the requirement that the process driving fundamentals be stochastically stable over
time.5 Speci…cally, we restrict attention to an equilibrium of the form
Pt =
+ "t .
(9)
For such an equilibrium to exist, the equilibrium pricing condition, eqns. (7) and (8), implies:
=
=
(1 + )
and
1+r
+e
v(1 + )2
.
1+r
Solving the two equations above yields the following observation.
5
See, for example, Allen and Gale (1997) and Spiegel (1998).
9
(10)
(11)
Observation 1. There exists a unique, steady-state linear equilibrium of the form Pt =
+ "t , where
, and
1+r
e
v (1 + r)2
.
=
r r (1 + r
)2
=
(12)
(13)
The steady-state coe¢ cient and intercept have some intuitive properties. The coe¢ cient
on abnormal earnings increases in the persistence of abnormal earnings and decreases in
the risk-free rate, r, which e¤ectively serves as the discount rate. The intercept has two
components, one tied to permanent earnings, e, and one tied to the variation in payo¤s
associated with the evolution of abnormal earnings. The component tied to the permanent
earnings, re , is merely a capitalization of permanent earnings, which is decreasing in r. The
second component tied to the variability of payo¤s,
v(1+r)2
,
r(1+r )
captures the “haircut” in
price for risk. The haircut is larger in magnitude (i.e., more negative) when earnings are
more persistent or the discount rate is lower because the uncertain earnings innovation has
a greater impact on future price when earnings are more persistent or the discount rate is
lower. Obviously, it is also increasing in magnitude if the variance of the innovation, v, is
larger, or the degree of risk aversion, , is larger.
3
PEF Equilibria
The steady-state linear equilibrium is simple and intuitive; it relies on an assumption that
each generation believes that the subsequent generation’s response to next period’s abnormal
earnings is characterized by the steady-state value . When we consider other beliefs, however, we open up the possibility of equilibria with time-varying volatility, which we demon10
strate below. To do so, we initially consider a class of equilibria where current investors
exhibit projected earnings …xation (PEF).
3.1
Characterization of PEF Equilibria
PEF describes the phenomenon where investors in the current period project or believe that
investors in subsequent periods will place too much emphasis on earnings. We operationalize
PEF as follows. Given that abnormal earnings exhibit some persistence, if an investor in
period t projects that investors in period t + 1 will place too much emphasis on earnings, a
period-t investor’s expectation of price at t + 1 will be larger if period-t abnormal earnings,
"t , are positive and smaller if "t is negative. Because a period-t investor’s expectation of
future price places more emphasis on "t when period-t + 1 investors place more emphasis on
t + 1 abnormal earnings, it follows that a period-t investor’s demands will also place more
emphasis on "t ; this, in turn, implies period-t price will also place more emphasis on "t .
We build o¤ of this partial-equilibrium intuition to characterize equilibria where the price
response, or emphasis, on abnormal earnings grows each generation. In other words, each
generation’s “over-emphasis” on earnings becomes a rational response to the even-moreexaggerated emphasis on earnings of subsequent generations.
Lemma 1. For any
0 and
0
2 <, there exists an equilibrium pricing function
Pt =
t
+
with the following coe¢ cient and intercept for all t
1+r
t
=
+
t
=
+ (1 + r)t (
(14)
t "t :
0:
t
, and
0
11
) + v (2Jt + Kt ) ,
(15)
(16)
where
=
1+r
equilibrium, Jt =
and
=
t
(1+r)t
1
1
t
e
r
1+r
v(1+r)2
r(1+r )2
(1+r)t
1+r
, and Kt =
Note that the equilibrium de…ned by
interpret
are the coe¢ cient and intercept in the steady state
= 0 and
2t
(1+r)t+1
2
0
2t
.
is the steady-state equilibrium. We
=
> 0 as a phenomenon where investors place increasingly greater emphasis on
abnormal earnings because the price response to abnormal earnings,
t,
increases over time.
As noted above, such equilibria arise when generation-t investors project or believe that
generation-t + 1 investors will “over-emphasize” earnings; this, in turn, causes generation-t
investors to “over-emphasize”earnings in period t. The over-emphasis on earnings, which is
captured by the coe¢ cient on
in
t
(i.e.,
1+r t
), increases by a factor of
1+r
over any two
periods. This increase is necessary to sustain the equilibrium. For example, suppose that
the over-emphasis in period t + 1 is given by
(i.e.,
t+1
=
+ ). A period t investor who
impounds the t + 1 price movement in his period-t demand expects that a fraction
of this
period’s earnings surprise persists. Therefore, in period t the discounted value of the asset
that pertains to the projected over-emphasis is given by
1+r
.
The fact that the emphasis on abnormal earnings grows implies that the variance of price
also grows over time. In equilibrium, risk averse investors are compensated for additional
variation in payo¤s with a higher expected return or risk premium. In the equilibrium characterization of many modeling frameworks, the risk premium is achieved through a reduction
in the current price. In contrast, in the equilibria involving investors exhibiting PEF, the
risk premium associated with increasing variance is achieved through an incremental upward
drift in price. Formally, the intercept in the pricing function impounds the upward drift captured by the term v (2Jt + Kt ), which is increasing over time. Hence, when investors
12
exhibit PEF, price drifts upwards relative to investors not exhibiting PEF. This result is a
consequence of an in…nite time span: when the time span is in…nite, whether today’s price
is lower or tomorrow’s price is higher is irrelevant, provided that price behaves in a fashion
consistent with an equilibrium. Finally, note that the initial value for ,
0,
seems somewhat
ad hoc in that the initial price level is not relevant to investors –what matters to investors
is the dividend payout and the change in price level. As a consequence, any deviation from
the steady state
must be compensated for with the risk-free return each period, which is
why the deviation is multiplied by a factor of 1 + r each period. This part of the upward
drift is similar to the one that explains the existence of …rst-moment bubbles in prior work:
investors are willing to pay a higher price if they expect next period’s investors to pay an
even higher price.6
3.2
Price Variation with PEF
Price variation in period t equals
2
t v,
which implies it is driven by the variation in abnormal
earnings, v, as well as the coe¢ cient on abnormal earnings in the pricing function,
discussed previously, the steady state
t,
1+r t
As
, is increasing in the persistence of abnormal
earnings, , and the discount rate, r. In an equilibrium where investors exhibit PEF,
+
t.
t
=
, the coe¢ cient is greater than the steady-state coe¢ cient, and it grows at an
increasing rate over time. Not surprisingly, the wedge between the steady-state coe¢ cient
and the PEF-induced coe¢ cient is tied to : we interpret as the extent of PEF. The growth,
6
We should note that we have restricted attention to cases where
> 0, which is consistent with
investors exhibiting PEF. We could also consider the possibility of investors “under-emphasizing”abnormal
earnings by allowing < 0. While initially leading to equilibria in which volatility is lower than the steady
state volatility, investor “under-emphasis” would ultimately lead to a high volatility equilibrium in which
investors respond negatively to abnormal earnings because t becomes increasingly negative when < 0 and
t is su¢ ciently large.
13
however, is determined by the risk-free rate and the degree of persistence. In particular, the
rate of growth in the abnormal earnings response is higher when the risk-free rate is higher
and the persistence of earnings is lower. To understand why, note that an investor at time
t over-emphasizes abnormal earnings by
if and only if his over-emphasis is supported by a
su¢ cient expected over-emphasis by time-t+1 investors. Because a time-t investor discounts
expected payo¤s by r, the su¢ cient expected over-emphasis must be larger if the discount
rate is higher. Further, because the over-emphasis pertains to an expectation of the reaction
to realized t + 1 abnormal earnings, the t + 1 over-emphasis can be lower if the abnormal
earnings at time t are more persistent. Formally, we have Corollary 1.
Corollary 1. Assume the equilibrium price path is characterized by PEF, i.e.,
> 0: The
growth in the variation in prices along that path is increasing in the risk-free interest rate,
r, and decreasing in the degree of persistence in abnormal earning, .
3.3
Expected Price Levels with PEF
While an important characteristic of PEF is greater price volatility, PEF results in price
levels that di¤er from the steady-state price level. More interestingly, the expected price
level only di¤ers if investors are strictly risk averse.
Corollary 2. Assume the equilibrium price path is characterized by some PEF, i.e.,
> 0:
The time-0 expectation of price at time m > 0 is greater for the PEF path than for a steadystate path beginning at the same initial price if investors are strictly risk averse, and the
same if investors are risk neutral.
Corollary 2 holds because, when prices are more volatile, risk investors are compensated with
greater expected payo¤s, which is achieved in equilibrium by an upward drift in prices. To
14
better understand why Corollary 2 holds, note that in any equilibrium the expected return
earned by an investor buying at time t equals
"
#
v 1+
e~t+1 + P~t+1 Pt
E
j"t = r +
Pt
Pt
2
t+1
.
(17)
The …rst term implies that investors’expected return is at least equal to the expected return
investors can earn by investing in the risk-free asset. When investors are risk averse, the
second term captures the additional risk premium they earn because returns are variable.
Because the variability of asset returns is strictly higher when investors exhibit PEF (relative
to the steady state), it follows that the expected returns must be higher.
From an empirical perspective, Corollary 2 also implies that periods of high volatility are
associated with higher price-to-earnings ratios (on average) despite the fact that earnings
are no more informative about the …rm’s future cash ‡ows. This observation must be viewed
with caution, however, because a price path could involve periods of high volatility and high
price level, followed by periods of steady-state volatility and steady-state price level. Here,
one might associate the higher price level with the steady-state volatility once the price path
returns to steady state.
3.4
Numerical Example
To illustrate how PEF a¤ects the path of prices, price levels, and price volatility, we consider
a numerical example and plot di¤erent price paths for randomly generated data. Speci…cally,
we illustrate the steady-state price path and a path with PEF (i.e.,
=
1
> 0) to illustrate
how both paths play out for a case with risk neutral investors versus risk averse investors.
Figure 1 displays these four price paths for the generated data, using the following parameter
values:
1
= 0:2,
= 4,
= 0:9, and "0 = 0. The paths with risk averse investors are based
15
upon a coe¢ cient of risk aversion of
= 0:005. The value for constant earnings is normalized
such that initial prices for the case with risk averse investors equal zero.
— Insert Figure 1 around here —
When investors are risk neutral, the …gure illustrates how volatility increases in t at a
fairly high rate. This increase in volatility can also be seen algebraically in eqn. (15). As
discussed above, investors only increase their weight on the earnings surprise when they
expect the next period’s investors to do the same. However, in order for investors in period t
to “over-emphasize”earnings by , it is necessary that investors in period t+1 over-emphasize
earnings by more than ,
1+r
> , because t-period investors discount the expected over-
emphasis. This implies that the over-emphasis has to increase over time, which leads to
increasing price volatility. Furthermore, with risk neutral investors, high volatility price
‡uctuates around the steady-state price without drift to become either higher or lower. In
other words, the increase of
t
is a mean-preserving spread of prices.
When investors are risk averse, they demand a risk premium as speci…ed in eqn. (11).
The risk premium implies that the prices with risk averse investors start at a lower value in
comparison to the prices with risk neutral investors. Due to the fact that
t
and the associ-
ated volatility in prices increases over time, when investors exhibit PEF, they are rewarded
with a risk premium that increases over time. As is evident in the …gure, the increasing
risk premium causes substantial drift in prices above the steady-state price. Algebraically,
the risk premium is convex in
t+1 ,
which implies the risk premium grows faster than the
coe¢ cient on abnormal earnings that determines it. Hence, in contrast to the risk neutral
case, the increase of
t
is far from a simple, mean-preserving spread of prices.
16
4
Time-Varying PEF
The class of linear equilibria with PEF (i.e.,
> 0) introduces the plausibility of greater
price volatility relative to fundamentals volatility, but these equilibria also have the unappealing property that volatility grows at an increasing rate in the sense that
t
grows at an
increasing rate over time. In essence, once investors exhibit PEF, investors over-emphasis on
earnings grows inde…nitely. By expanding on the class of equilibria characterized in Lemma
1, however, we can demonstrate the possibility of time varying degrees of PEF. For example,
the degree of PEF could begin at a positive level and then revert at some time to a steady
state level ( = 0).
4.1
Alternative Price Paths
To characterize a equilibrium in which the degree of PEF varies over time we must expand
the set of equilibrium beyond all PEF equilibria characterized in Lemma 1. To do so, we
combine PEF equilibrium pricing paths to create a new equilibrium price path exhibiting
variation in PEF. These alternative price paths are not simple linear equilibria, although
they are linear with the exception of any point in time at which the pricing function changes
from one PEF pricing function to another.
Before proceeding, it is useful to de…ne a bit of notation by letting P ("t ;
0;
) denote a
PEF equilibrium pricing function as described in Lemma 1 that is characterized by the an
initial constant term of
0
and a degree of PEF . With this notation established, conjecture
an equilibrium pricing that is characterized by a shift in the degree of PEF from
17
1
to
2
at
time
> 1:
Pt =
where
P ("t ;
01 ; 1 )
P ("t ;
02 ("
for all t <
,
);
for all t
2)
satis…es
02 ("T )
P (" ;
02 ("
);
2)
= P (" ;
01 ; 1 )
for all realizations "T , which implies that
02 ("T )
=
01
+
v
(
(1 + r)
1
(2J + K
1)
2
The conjectured equilibrium exhibits shifts at time
acterized by a degree of …xation
1,
acterized by degree of …xation
P ("t ;
2,
of the pricing function at time
of the constant term,
time
02 ("
P ("t ;
01 ; 1 ),
02 ("
);
2)
2 ))
+
1
(
1
2 )"
:
from one PEF pricing function charto another PEF pricing function char-
02 ("T ); 2 ),
although a complete parameterization
is not known until " is realized because the initial value
), is a function of " . Furthermore, the pricing function at
yields an identical distribution for time
cause P (" ;
(2J + K
= P (" ;
01 ; 1 )
the pricing function anticipated at time
prices as the initial pricing function be-
for all realizations for " :Because of this feature,
by young investors at time
1 can still be
expressed as a simple linear function of the realization of abnormal earnings at time
as P (" ;
01 ; 1 )).
The time
(i.e.,
+ 1 linear pricing function anticipated by investors at time
, however, is a function of the realization " because the intercept is a function of " .
Nonetheless, the pricing function at time
+ 1 is still linear in "
+1 .
In order to prove that the conjectured equilibrium is indeed an equilibrium, we must show
that supply equals demand at every point in time, consistent with the earlier de…nition of an
equilibrium. It follows directly from Lemma 1 that supply must equal demand for t <
18
1:
Otherwise, P ("t ;
01 ; 1 )
would not be an equilibrium pricing function. In addition, it follows
directly from Lemma 1 that supply must equal demand for all t >
is an equilibrium pricing function. Finally, at t =
1 if P ("t ;
01 ; 1 )
because P (" ;
02 ("
);
);
2)
1 young investor demands as a
function of price are the same as they would be if the pricing function for
characterized by P ("t ;
02 ("
2)
" . As a consequence, the market also clears a time
= P (" ;
01 ; 1 )
continued to be
for any realization
1 at price P ("t ;
01 ; 1 )
so the
conjectured equilibrium is indeed an equilibrium.
Proposition 1 follows naturally from the fact that we have characterized an equilibrium
exhibiting one degree of PEF initially followed by another degree of PEF.
Proposition 1. There exists an equilibrium in which the extent of PEF and the associated
price variability change over time.
Proposition 1 has the important implication that equilibrium price paths exist where price
volatility varies over time. In fact, we can use the same logic employed to derive the equilibrium price path with a single switch from degree of PEF to construct paths with any number
n 2 f1; 2; 3; :::g of changes in the degree of PEF. Furthermore, in the path we constructed,
we assume the degree of PEF changed with probability 1 at time . It is also possible to
construct equilibrium price paths in which the degree of PEF changes with some probability
less than 1 such as when, say, the change is conditioned upon the realization for "T (e.g.,
the change occurs when " is more or less than some value). Hence, our analysis suggests
that the degree of PEF can vary over time. Unfortunately, there are no standard equilibrium
conditions that allow us to predict when changes in the degree of PEF occur nor the direction
of those changes if they do occur.
19
4.2
A Path with a Focal Steady State
Proposition 1 suggests the possibility of an equilibrium price path characterized by periods
exhibiting PEF and periods exhibiting the steady-state level of volatility (i.e.,
t
).
=
Among the paths exhibiting the steady state volatility, however, the steady state price path
(
t
=
and
t
= a) might be viewed as focal due to its inherent simplicity and stability.
To study the properties of an equilibrium pricing function in which the steady state path
is “focal,” we consider an equilibrium in which the price path begins on the steady state,
characterized by pricing function P ("t ; ; 0) ; and then enters a period with PEF of degree
at time , characterized by the pricing function P ("t ;
0;
) where
P (" ; ; 0). The pricing path continues to be de…ned by P ("t ;
in which P ("
+m ;
; 0) = P ("
+m ;
0;
0;
0
satis…es P (" ;
0;
)=
) until the …rst date +m
), after which the pricing function is again de…ned
by the steady state pricing function P ("; ; 0).
Assume at time t =
investors deviate from the steady state path and begin to exhibit
PEF. Therefore, we know that at time t =
the prices along the steady-state and the PEF
path must be equal, i.e.,
+ " =
" ,
+
(18)
or, equivalently,
.
" =
(19)
Similarly, in order for the two paths to meet again at some future date t =
+ m, it must
be the case that
"
+m
+m
=
.
(20)
+m
It follows that the cumulative-innovations change in abnormal earnings between
20
and +m,
Pm
i=1
m i
t+i
="
m
+m
Xm
" , must satisfy
m i
m
+m
m
,
(21)
Corollary 3. Assume investors are strictly risk averse and at time t =
the price path
i=1
+i
="
+m
" =
+m
which we show in Corollary 3 is always negative.
changes from steady state to one where investors exhibit PEF, i.e.,
price to converge back to the steady-state price at time t =
earnings at t = +m must equal "
+m
where "
+m
<
m
> 0. For the equilibrium
+ m, where m > 0, abnormal
" . Furthermore, "
+m
is decreasing
in m and approaches negative in…nity as m approaches in…nity.
The …rst statement in Corollary 3 implies that earnings news must be bad for the PEF
and steady-state paths to meet. Hence, once investors exhibit PEF, a return to steady-state
fundamental pricing must be associated with bad news. The second statement implies that,
as the period of PEF increases in duration, the worse the news needs to be to converge back
to a steady state. Given that extreme realizations are less likely, this observation suggests
that the “likelihood”of convergence back to steady-state prices “decreases”as the duration
of PEF increases. Of course, given our assumption that abnormal earnings are distributed
continuously whereas trading occurs in discrete intervals, the probability of precise convergence is always 0. If abnormal earnings were discrete or trading were continuous, however,
the probability of precise convergence would not be 0 and we expect the insight in Corollary
3 would continue to hold.
21
5
Welfare Analysis
While our analysis o¤ers a theoretical rationale for the possibility of periods of PEF, it also
o¤ers a theoretical rationale for a wide variety of potential equilibria. A common approach
for assessing which equilibria are more focal, and, hence, more likely to be played, is to
assume that economic agents gravitate towards equilibria that are preferred. Consequently,
we conduct a welfare analysis certain whether investors prefer one equilibrium over another.
5.1
Ex Ante Investor Welfare
The expected utility at time t for a generation-t investor equals
Xt = qt e + "t +
Given Pt =
t
+
t+1
+
t+1
"t + (1 + r) (w
as well as the solutions for qt ,
t "t
t,
qt2
q t Pt )
and
1
t
(exp[
v
1+
2
1) where
Xt ]
2
t+1
.
(22)
as stated in equations (5),
(7), and (8), respectively, we can rewrite Xt as follows:
v
Xt =
2
1+
+
1+r
t+1
!2
+ (1 + r) w.
(23)
Note that Xt is independent of "t in equilibrium, which implies that investor welfare is
independent of "t in equilibrium. Hence, the expected utility of generation t investors prior
to knowledge of "t is the same as with knowledge of "t . It follows from eqn. (23) that the
investors prefer a more volatile equilibrium path because, within the context of this model,
they receive su¢ cient compensation for the additional uncertainty assumed in equilibrium. It
follows that, if investors could choose the equilibrium to play ex ante, they would be unlikely
to choose the steady-state equilibrium, which minimizes the variation in prices. Formally,
we have Corollary 4A.
22
Corollary 4A. Generation-t investors always prefer an equilibrium with where investors
exhibit PEF over the steady-state equilibrium prior to observing "t+1 .
5.2
Ex Post Investor Welfare
As should be clear, the realized utility of a generation t investor does not yield the same
preferences over equilibrium paths. In particular, the realized utility of a generation t investor
is
1
(exp[
1), where
Wt ]
Wt = qt e + "t+1 +
t+1
+
t+1 "t+1
+ (1 + r)(w
(24)
qt Pt ).
Note that the di¤erence between Xt as de…ned in eqn. (22) and Wt as de…ned above is that
qt2
"t+1 replaces "t and
Given Pt =
t
+
t "t
v
2
1+
2
t+1
drops out because the investor faces no risk.
along with the solutions for qt ,
t,
and
t
as stated in equations
(5), (7), and (8), respectively, we can rewrite Wt as
Wt = ("t+1
"t ) 1 +
t+1
+ v 1+
2
t+1
+ (1 + r) w.
It follows that there is a unique value for "t+1 , ^", such that generation t investors prefer
an equilibrium where investors exhibit PEF if "t+1 > ^", and the steady-state equilibrium
otherwise. Noting that
t+1
t
is a function of
0
(i.e., it is increasing in
0 ),
^" could be
either positive or negative.
Corollary 4B. After observing "t+1 , generation-t investors prefer the steady-state equilibrium over the an equilibrium where investors exhibit PEF if and only if "t+1 is su¢ ciently
small.
Corollary 4B implies that if generation-t investors could in‡uence the equilibrium being
played when they depart the market, they would wield their in‡uence in favor of the steady
23
state when times are relatively “bad,” and favor an equilibrium where investors exhibit
greater PEF otherwise. This obviously happens because a shareholder who observes a very
negative earnings surprise would like investors interested in buying shares to react as little
as possible to the earnings news.
6
Spillover E¤ects
To this point we have employed a single risky-asset framework to illustrate the possibility
that when investors exhibit PEF, prices manifest more volatility tied to fundamentals than
appears to be warranted by the fundamentals. A common interpretation of the single riskyasset case is that it represents the market portfolio and PEF pertains to economy-wide
earnings disclosures (as opposed to, say, a single …rm’s disclosure). While the single riskyasset framework provides a parsimonious vehicle for illustrating the main points of this study,
a drawback is that it does not provide insights into how PEF involving one risky asset, or
class of risky assets, might spill over and a¤ect the pricing behavior of other assets. Here
we attempt to provide some insights about spillover by extending the model to include two
risky assets: this is su¢ cient to provide insights that involve n risky assets.
Formally, consider our economy with two risky assets, 1 and 2, in addition to the risk-free
asset. Similar to the single risky-asset setting, the earnings for asset i 2 f1; 2g for period t
are
eit = ei + "it , where
"it =
i "it 1
+
We assume the innovation to abnormal earnings,
24
it .
it ,
(25)
(26)
is normally distributed with mean 0
and variance vi for all i 2 f1; 2g and t, the covariance between
it
is independent of
j
1t
and
2t
is c for all t, and
for all i; j 2 f1; 2g, and t 6= . As the following analysis shows, the
covariance of cash ‡ow innovations, c, is an important determinant of the spillover e¤ect.
Finally, we assume that there are
shares of risky asset 1 per-capita and 1
shares of risky
asset 2 per-capita such that, in aggregate, the total number of risky asset shares per-capita is
1. This extension of the model re‡ects at least two spillover scenarios: one where each asset
can be thought of as re‡ecting a large component of the market portfolio, and one where
one asset is an in…nitely small component of a large market and the other is the residual
of the market portfolio. The former setting, which is captured by
assuming a strictly
interior value, characterizes the relation between two industries. The latter setting, which is
captured by
approaching 0 or 1, characterizes the relation between a single risky asset as
an arbitrarily small component of a large market portfolio and the market portfolio itself.
We again restrict attention to the set of linear equilibria where the end-of-period price
for …rm i is a linear function of its abnormal earnings:
Pit =
it
+
it "it .
(27)
We rule out equilibria where …rm i’s price is a function of j’s abnormal earnings, which
would essentially be a sunspot equilibrium for …rm i. In particular, in the presence of …rm
i’s abnormal earnings, …rm j’s abnormal earnings have no incremental information content
for i’s future cash ‡ows. In any linear equilibrium of the form in (27), the demands by a
new investor in period t are:
q1t
=
q2t
=
e1 +
1 "1t
+
1t+1
+
1t+1 1 "1t
c 1+
v1 1 +
e2 +
2 "2t
+
2t+1
+
2t+1 2 "2t
1t+1
2
1t+1
c 1+
v2 1 +
25
1t+1
2
:
2t+1
1+
2t+1
q2t
(1 + r) P1t
1+
2t+1
q1t
(1 + r) P2t
and
.
(28)
(29)
Market clearing requires that q1t =
and q2t = 1
, which implies that following conditions
have to hold in equilibrium
1t
=
2t
=
1
2
1 + 1t+1
,
(1 + r)
1 + 2t+1
,
(1 + r)
e1 +
1t
(31)
v1
1t+1
=
1+
2
1t+1
+ c (1
) 1+
1t+1
1+
1t+1
1+
2t+1
(1 + r)
e2 +
2t
(30)
v2 (1
2t+1
=
) 1+
2
2t+1
+c
1+
2t+1
(1 + r)
Eqns. (30) and (31) are identical in structure to the equilibrium condition for
t
, and (32)
.
(33)
in the single
risky-asset setting, eqn. (7). This implies that we can apply results about PEF-coe¢ cient
behavior derived in the single-asset setting to either …rm’s pricing function in the dual-asset
setting. Furthermore, the fact that the two coe¢ cients on abnormal earnings do not interact
with one another implies that each risky asset coe¢ cient exhibits di¤ering degrees of PEF.
Finally, with any set of coe¢ cients for abnormal earnings, we can characterize the intercepts,
(32) and (33), in the same manner as in the single risky-asset setting, with the caveat that
two, as opposed to one, abnormal-earnings-asset coe¢ cients determine the value each period.
Hence, we can derive dual-asset variations of Lemmas 1, 2, and 3.
Given these dual-asset variations, we turn to discussing the spillover e¤ects of a change
in the degree of PEF in the pricing behavior of one asset to the pricing behavior of the
other asset. Due to the fact that the coe¢ cients on abnormal earnings do not interact, an
increase in the degree of PEF for one asset, say asset 1, increases asset 1’s price response to
its abnormal earnings,
earnings,
2t ,
1t ,
but has no e¤ect on asset 2’s price response to asset 2’s abnormal
regardless of the degree of covariation in the innovations to abnormal earnings.
26
Hence, changes in the price volatility for asset 1 that are not due to changes in the volatility
of fundamentals need not be associated with changes in the price volatility for asset 2.
Inspection of eqns. (32) and (33), however, reveals a relation between the price level of
asset 2 and the degree of PEF for asset 1. In particular, increasing the degree of PEF for
asset 1 in‡uences the price level of the asset through the intercept in the pricing function,
2t .
The e¤ect on the intercept depends critically on the covariance of the earnings/dividend
‡ows. In particular, for any linear equilibrium and any two degrees of PEF,
equilibrium
=
2t
2t
2
2
=
ei
r
and
2,
the
satis…es
+ (1 + r)t (
+ c
where
1
2)
20
+ v2 (1
(1 + r)t+1 1
(1 + r
) t (1
1+r
1+r
2 vi (1
)+c
r
t
)
2
(2Jt + Kt 2 )
(1 + r)t+1 (1 + r)t
( 1 + 2) +
2
2t
)
1+r
is the steady-state intercept and
2t
1 2
20
!
; (34)
is the initial
intercept.
If the covariance between the earnings/dividends of the two risky assets is negative,
increases in the degree of PEF for asset 1 cause the expected price level for asset 2 to be
lower each period than would otherwise be the case. Furthermore, the degree to which the
expected price level declines increases over time. The reasoning behind this price-level e¤ect
stems directly from the fact that asset 2 serves as a hedge of asset 1 due to the assumed
negative covariance between the two earnings ‡ows. In particular, as asset 1 becomes more
sensitive to its earnings due to a higher degree of PEF, the variance of asset 1’s payo¤s
increases. Because the covariance between the two earnings is negative, investments in asset
2 serve as an increasingly important hedge against the increasingly volatile asset 1, which
causes asset 2’s price levels and associated expected price changes to decline. If, on the other
27
hand, the covariance between the earnings of the two risky assets are positive, the opposite
e¤ect of asset 1’s PEF arises; the expected price level increases over time to compensate
investors for the greater degree of undiversi…able risk caused by asset 1’s PEF.
Note that in discussing spillover, we provided no context. That is, we do not distinguish
between a setting where both assets represent a large component of the market (e.g., each
represents a large sector of the economy) versus a setting where one asset represents a single
equity security and the other represents the residual of a large, diversi…ed, equity market
portfolio. The previous discussion explains well the former case. The latter case, however,
requires that we set asset 1’s size to one of the extremes,
case where
= 0 or
= 1. Consider …rst the
= 1 so asset 1 represents the equity market portfolio and asset 2 represents
a single equity security. Consistent with the traditional capital asset pricing model, asset
2’s price is not determined by its own variance. Instead, its price is determined by how
its earnings covary with the market earnings, asset 1. Hence, consistent with our generic
discussion above, any PEF with respect to asset 1 a¤ects asset 2 prices through the covariance
of the two asset’s earnings. Consider next the case where
= 0, so asset 1 now represents
the single equity security and asset 2 represents the equity market portfolio. In this case, if
investors exhibit PEF in response to asset 1 there is no e¤ect on asset 2 because asset 1 is
an arbitrarily small element of the broader market. Note as well that asset 1’s price level
(i.e.,
1t )
is also not in‡uenced by the degree of PEF with respect to asset 1. This result
arises because asset 1 is simply too small for its own price variance to matter.
To summarize, the main insights generated by considering two risky assets are formalized
in Observation 2.
Observation 2. Allow for the possibility of two risky assets. One asset’s degree of PEF
28
has no direct e¤ect on the price volatility of the other asset:
@
@
it
j
= 0, where i 6= j. One
asset’s degree of PEF, however, a¤ects the price level and expected change in price of the
other asset if and only if the asset that exhibits PEF is large and the earnings for each asset
have a non-zero covariance:
@
@
it
j
>0
@
@
it
j
< 0 and
@ 2 it
@ j @t
>0
@ 2 it
@ j @t
< 0 , where i 6= j, if
asset j is measurable and c > 0 (c < 0):
7
Conclusion
Using an overlapping-generations modeling framework, we consider the implications that
arise from investors projecting that investors in subsequent periods will place too much emphasis on earnings information. We describe this phenomenon as one where current investors
exhibit “projected earnings …xation”(PEF). Our analysis shows that when current investors
exhibit PEF, current investors themselves place too much emphasis on current earnings.
As a consequence, equilibria can arise where the response to earnings disclosures becomes
increasingly more exaggerated than the variability of fundamentals would suggest. In addition, our analysis demonstrates that the reaction to earnings disclosures of stochastically
stable fundamentals can vary over time, with, for example, episodes of PEF where prices
appear highly variable and inconsistent with standard valuation frameworks, and periods
when prices are more stable and consistent with valuation frameworks.
While our modeling framework does not provide predictions regarding the particular
equilibrium price path investor trade takes, it does yield a number of empirical insights.
First, our model suggests that volatility can be time varying even if fundamentals manifest
constant volatility. Second, our model suggests that the associations between news events,
such as earnings releases, and equity prices can be time varying, even if the relation between
29
the news events and economic fundamentals is stable. Third, our model predicts returns
will be higher over periods of higher volatility, which is consistent with empirical evidence.
Finally, our model predicts that price levels are expected, on average, to be higher in periods
of high volatility and will converge back towards fundamental steady-state prices when news
is bad.
As an extension, we consider a setting with two risky assets in order to assess how PEF
in the market for one asset can spill over to a¤ect the pricing of another asset. We show that
increasing PEF with respect to one asset need not have any impact on the price volatility
of the other asset, irrespective of the correlation between the assets’fundamental earnings
‡ows. Increasing PEF with respect to one asset, however, does in‡uence the price level
and expected price change of the other asset if the former asset is large and the correlation
between the assets’fundamental ‡ows are non zero.
30
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Proofs
Lemma 1. The proof follows directly from the fact that the coe¢ cient speci…cations
satisfy the equilibrium conditions (7) and (8) for any t.
Corollary 1. The variance in price at time t equals
1+r t
growth in variation between t 1 and t equals
2
tv
=
+
2
1+r t 1
v=
1+r t
2
v, so the
1+r 2(t 1)
1+r
which is increasing in r and decreasing in .
Corollary 2. At time 0 it must be the case that the price in the steady state equals the
price in the presence of PEF or:
0
+ v (2J0 + K0 ) + ( + ) "0 =
(35)
+ "0
or, given that J0 = 0 and K0 = 0,
(
0
(36)
) + "0 = 0.
The time-0 expected time-m > 0 price when investors exhibit PEF less that for the steady
state equals
+ (1 + r)m (
0
) + v (2Jm + Km ) +
+
1+r
m
m
"0
( +
m
"0 ) , (37)
which is proportional to
(
0
)+
v
(2Jm + Km ) + "0 .
(1 + r)m
33
(38)
2
v,
Relying on eqn. (36), eqn. (38) can be shown to be proportional to:
v
(2Jm + Km ) ,
(1 + r)m
which is strictly positive if
(39)
> 0 because Jm > 0 and Km > 0, and 0 if
= 0.
Proposition 1 Conjecture an equilibrium pricing function such that, for all t 2 f0; 1; 2; ::: g,
the price is P ("t ;
+ (1 + r)t (
1t ;
1t ;
01 )
)+ v
01
1
=
1t "t ,
+
1t
where
(2Jt + Kt 1 ) ;and
01
1+r t
2,
0,
2
is set so that P (" ;
02
=
01
+
1
(
6=
;
1
2 )"
1
lieve that at time
degree of PEF,
1
2
1,
;
01 )
+
v
(1+r)
+ (1 + r)t (
=
2t
=
+
1
2J (
" =
1
2)
1
+
1+r t
1,
0,
1
1t
=
is the initial value for the constant, and,
for all t 2 f ; + 1; + 2; :::g, the price is P ("t ;
+
=
1t
2t ;
2t ;
+K (
=
)+ v
02
2
02 )
+
2
1
2
2t
2
+
2t "t
1t
(2Jt + Kt 2 ), and
" = P (" ;
2
2)
where
2
;
;
2
02 )
=
02
or
. Hence, all investors be-
the pricing function changes so that it is characterized by a di¤erent
as opposed to
2
1,
and that the constant term adjusts at time
so that the
time
price is identical to the price that would be realized under the initial pricing function
P ("t ;
1t ;
1t ;
10 ).
In order for this path to constitute an equilibrium, markets have to clear
at any point in time. For t <
Lemma 1. At t =
1 the market clears at price P ("t ;
1
;
clears a time
P ("t ;
1t ;
1t ;
1t ;
01 )
following
1 young investor demands as a function of price are the same as they
would be if the pricing function for
because P (" ;
1t ;
1
;
01 )
= P (" ;
1 at price P (" ;
01 )
continued to be characterized by P (" ;
2
1
;
;
2
1
;
;
02 )
1
;
1
;
10 )
for any realization " . Hence, the market
01 ) :Finally,
for all t
, the market clears at
following Lemma 1.
Corollary 3. Similar to eqn. (35), following equation has to hold for the two price paths
34
to meet at time t:
t
(1 + r) (
) + v (2Jt + Kt ) +
0
t
1+r
(40)
"t = 0.
Additionally, in order for the steady-state price path to meet the one where investors exhibit
PEF at date t + m, it must be the case that
(1 + r)t+m (
0
v
)+
(2Jt+m + Kt+m ) +
(1 + r)m
t
(1 + r) (
0
Using eqn. (40) and the fact that "t+m =
1+r
1+r
) + v (2Jt+m + Kt+m ) +
t
Xm
i=1
i
t+i
=
Pm
m i
i=1
v
t+i
+
t+m
1+r
"t+m = 0 or
t
"t+m
m
m
2Jt+m + Kt+m
(1 + r)m
= 0.
(41)
"t , we can re-express (41) as
(2Jt + Kt ) .
(42)
The proof to Corollary 2 shows that the right hand (42) is negative, which completes the
proof.
Observation 2. The proof follows directly from inspect of eqn. (34).
35
600
High Volatility
Steady State
High Volatility, Risk Neutral
Steady State, Risk Neutral
500
400
300
200
100
0
0
10
20
30
-100
-200
Figure 1: Steady State and High Volatility
40
50