Rare Event Risk and Heterogeneous Beliefs:
The Case of Incomplete Markets
Stephan Dieckmann∗
November 5, 2009
Abstract
This paper provides an equilibrium model subject to heterogeneous beliefs about the
likelihood of rare events. I explore asset pricing implications in an incomplete capital
market and the effects of market completion. Without explicit rare event insurance,
investors insure themselves indirectly through the stock and money markets, the risk
premium is countercyclical, and flight to quality effects arise. Upon market completion,
the risk premium increases as investors increase their exposure to rare event risk. While
market completion leads to a more efficient allocation based on investors’ anticipatory
utilities, its effect on ex-post efficiency is ambiguous.
∗
Dieckmann is affiliated with the Wharton School at University of Pennsylvania, Philadelphia, PA 19104-
6367, e-mail: [email protected]. I am grateful to Suleyman Basak, Nicole Branger, Jaime
Casassus, Jeffrey Coles, David Cummins, Pierre Collin-Dufresne, Michael Gallmeyer, Geoffrey Heal, Burton Hollifield, Paul Malatesta (Managing Editor), Fulvio Ortu, Chester Spatt, Sanjay Srivastava, Suresh
Sundaresan, Gordon Woo, an anonymous referee, and seminar participants at Arizona State University, the
Bank for International Settlements, Bocconi University, Carnegie Mellon University, Columbia University,
Georgia State University, London Business School, the NBER Insurance meeting, the University of Houston
and the University of Rochester for helpful discussions and comments. A special thanks to the referee for
suggesting to investigate the efficiency properties.
Rare Event Risk and Heterogeneous Beliefs:
The Case of Incomplete Markets
Abstract
This paper provides an equilibrium model subject to heterogeneous beliefs about the
likelihood of rare events. I explore asset pricing implications in an incomplete capital
market and the effects of market completion. Without explicit rare event insurance,
investors insure themselves indirectly through the stock and money markets, the risk
premium is countercyclical, and flight to quality effects arise. Upon market completion,
the risk premium increases as investors increase their exposure to rare event risk. While
market completion leads to a more efficient allocation based on investors’ anticipatory
utilities, its effect on ex-post efficiency is ambiguous.
Keywords: Rare Event Risk, Heterogeneous Beliefs, Incomplete Markets, Catastrophe Insurance, Exchange Economy.
JEL Classification: D51, D52, G11, G12, G13.
I.
Introduction
This paper provides an equilibrium model in which an economy is subject to rare event
risk in addition to regular economic risk. A rare event in this context can be understood
as an extreme jump in economic fundamentals caused by a severe natural or man-made
catastrophe. Barro (2006) provides empirical evidence of rare events over the last century
with a per capita GDP impact ranging between -15% and -64%. In contrast to Barro (2006),
I explore the implications of rare events stemming from investor heterogeneity. Specifically,
investors might not agree on the likelihood or impact of a rare event – a natural source of
heterogeneity since the true statistical properties of rare events are unknown or difficult to
estimate. Since individuals observe only very few occurrences during a lifetime, it is likely
that investors act according to their subjective assessment of the distributional properties,
as argued by Hinich (2003).
The goal of the paper is to address research questions in an incomplete capital market. For
example, in line with the ongoing debate about non-availability of rare event insurance,
Cummins (2006) points out the challenge of catastrophic events for the insurance and reinsurance industry, and the possibility of a large impact catastrophe leading to systematic risk
in the stock market and insurance prices. However, the implications of incomplete insurance
markets on other areas of capital markets is largely unexplored. Another example is the
market for contingent claims such as credit derivatives, allowing investors to bet on rare and
adverse states of the economy. Upon introduction, the credit derivatives market displayed
explosive growth, raising the question to which extent such a market is welfare enhancing.
To improve our understanding of this, I also show the effects of market completion and its
corresponding efficiency properties.
I find that investors (economic agents) who anticipate less frequent rare events are willing
to provide insurance to those agents anticipating a higher frequency. Under the absence of
insurance, agents insure themselves indirectly through the stock and money market, and
portfolio allocations mirror a synthetic put option. I find that the risk premium increases
upon market completion, and the increase can be as large as 30%. This is rather counterintuitive as one might expect additional compensation in the case of sub-optimal risk
1
sharing, and it is surprising from the perspective of the incomplete market literature on
limited participation and idiosyncratic shocks. In my model, however, heterogenous agents
will speculate according to their beliefs to such an extent that incomplete markets lead to
a smoother wealth process. I also show how this finding is linked to the trade-off between
rare event risk and regular economic risk, and it is robust to the case of heterogenous beliefs
about the impact of a rare event, instead of the frequency.
Rare events, heterogenous beliefs and an incomplete market also lead to two features frequently documented in financial markets: Countercyclical risk premiums, see for example
Ferson and Harvey (1991), and flight to quality episodes, as documented in Krishnamurthy
(2002) and Longstaff (2004). Intuition for the countercyclical behavior even in normal economic times can be derived from agents’ portfolio holdings and a financial wealth effect.
Agents who are less worried about rare events decide to increase their stock market exposure and borrow in the money market. On the other hand, agents who worry more have less
exposure in the stock market and serve as a lender. As a result, any positive (negative) economic shock improves relative financial wealth of agents who are less (more) worried about
rare events, leading to a lower (higher) degree of anticipated systematic risk.
Flight to quality is commonly understood as a flow of funds from risky to safer investments in
turbulent economic times – the corresponding price effects include sharp declines in riskless
interest rates due to the higher demand for safer assets. The dynamic nature of this model
allows me to study price effects as well as portfolio effects. The price effect is unambiguous. I
find that interest rates decline significantly at a rare event. At a -20% shock in fundamentals,
the effect can be as large as -90 basis points, even without considering learning. With a
revision in beliefs, the effect is as large as -120 basis points. The portfolio effect, on the
other hand, is ambiguous. I observe large portfolio adjustments in the money market, the
open interest can increase but it can also decrease. Diverse effects can occur in the stock
market. For example, the model can generate the interesting prediction in which an agent,
being already short in the stock market, sells even more stock following a market crash.
My incomplete market results also inform the recent debate about providing catastrophe
insurance. Brown et al. (2004) provide empirical evidence on the economic effects of the
2
passage of the Terrorism Risk Insurance Act (TRIA) in 2002. The passage of TRIA had
an adverse effect on stock prices in the 10 industries most likely to be affected by terrorism
insurance. They argue that their findings could be driven by lower estimates of future cash
flows, resulting in lower stock prices and negative abnormal returns around the time the
TRIA was adopted. However, the first passage of TRIA can also be interpreted as the
transition from an incomplete to a complete (or less incomplete) capital market. Cummins
(2006) points out that a considerable amount of insurance offered under TRIA would not
have been available in the absence of the Federal “make available” rule, as also shown by
more recent data on terrorism insurance take-up rates. My model predicts the risk premium
(discount rate) is larger upon completing the market, and ceteris paribus the stock price to
be lower. A broader example is that of option introduction. Sorescu (2000) and Ho and Liu
(1997) re-examine the effect of option introduction on the underlying stock price, and find
significant evidence of a negative abnormal performance post 1981.
But given the asset pricing implications, to which extent is completing the market welfare
improving? First, I illustrate how market completion leads to a more efficient allocation given
the agents’ anticipated utilities. For example, suppose trading in credit derivatives occurs
because of significant dispersion in beliefs about adverse states of the economy - assuming
some investors believe rare events occur once in a century, but others are more pessimistic
and believe they occur up to 10 times in a century. Then my results show that the credit
derivatives market leads to a gain in anticipatory utility up to 1% per annum expressed in
consumption equivalents. Second, I also show the properties of market completion based on
ex-post utilities. As a modeler, I can impose which agents have the correct beliefs and which
agents have incorrect beliefs. Having incorrect beliefs creates an ex-post cost, a disutility
that can be larger with complete compared to incomplete markets due to more risk sharing
opportunities. As a result, complete markets might not be more efficient ex-post, and I
illustrate an example in which the complete market is less efficient regardless which agent
holds the correct belief. However, this is only an exercise a paternal agent could undertake;
the agents within the model would be pleased to complete the market.
The theoretical contribution of this paper is to explicitly solve an equilibrium model in
3
which exogenous risk is assumed to follow a jump-diffusion process. It has the convenient
feature in that it captures the skewed law of motion typical for rare events, and it has
been considered in several areas in financial economics. For example, Naik and Lee (1990)
study the equilibrium prices of options in a continuous-time analogue of Rietz (1988) with
jumps in consumption growth. Longstaff and Piazzesi (2004) study a model in which the
jump in firms’ dividends can differ from the jump in consumption to explain the size of the
equity premium. Liu et al. (2005) add model uncertainty to the framework of Naik and Lee
(1990) and study the option volatility smirk. Barro (2006), in discrete time, revisits Rietz’s
(1988) model and introduces a default probability. All these models generate shadow prices
and no actual demand for zero net-supply securities due to the presence of homogenous
agents. Hence, an equilibrium solution with potentially incomplete markets is equivalent to
the complete market case.
The models proposed by Dieckmann and Gallmeyer (2005) and Bates (2008) are closer
to this paper in considering investor heterogeneity and actual demand for risky zero-net
supply securities. The former generates demand for rare event insurance by introducing
heterogeneity in risk aversion. The latter imposes a crash-averse utility function which
is convex in the number of jumps realized along the aggregate dividend path, and both
papers do not study an incomplete market. This paper, on the other hand, applies standard
assumptions on preferences and homogeneity in risk aversion. More recently, Gabaix (2009)
proposes a model with time-varying rare event risk and homogenous agents to explain several
market phenomena. Interestingly, heterogeneous beliefs deliver time-varying rare event risk
premia endogenously due to the redistribution of wealth over time.
The agents’ optimization problem is non-standard in two respects. I show how an equilibrium can be attained by introducing a representative agent with a stochastic weight that
can capture heterogenous beliefs plus an incomplete market. Heterogeneity in beliefs has
become a standard setting, see for example Detemple and Murthy (1997), Scheinkman and
Xiong (2003), or Kogan et al. (2006), especially the “agreeing to disagree” framework. Those
studies allow for some form of heterogeneity in beliefs (and potential market frictions) in the
context of regular economic risk, but not in the presence of rare event risk. While the
4
“agreeing to disagree” framework requires that investors do not have rational expectations, I
consider this a natural setting in particular for the case of rare event risk, for which it would
take a long time for different priors to converge.
The more complicated case arises where heterogenous beliefs are coupled with an additional
market friction. I borrow a technique used by Cvitanic and Karatzas (1992) and Basak
and Croitoru (2000) to solve for this case. Under logarithmic preferences, the minimaxobjective allows me to directly identify a unique state price density process such that standard
martingale methods apply. From this perspective, this friction is different than a short sale
constraint in the stock market under heterogeneous beliefs, see for example Gallmeyer and
Hollifield (2008), for which the market price of risk for non-participating agents can typically
be written down directly. In my model, however, agents hold a risky portfolio even in the
case of an incomplete market.
The rest of the paper is structured as follows: The model is developed in Section II, where
I first present the incomplete market solution, and then the effects of market completion.
Section III discusses results in the following order: Subsection III.A contains results about
prices and portfolio allocations, III.B shows the efficiency analysis of market completion, and
III.C illustrates the flight to quality effects. Section IV concludes, and the Appendix shows
an extension of the original model to incorporate uncertainty and heterogeneous beliefs about
the rare event severity, while agreeing on the likelihood of occurrence.
II.
The Economy
This section describes a continuous time generalization of a Lucas (1978) pure exchange
economy. Two agents (i = 1, 2) observe the realization of an aggregate output process with
the following exogenous dynamics:
(1)
de(t)
=µ
be dt + σe dB(t) + κe dN (t, λ).
e(t−)
The economy is subject to two sources of uncertainty. First, regular economic risk enters
through a one-dimensional standard Brownian motion B(t), defined on a probability space
5
¡
¢
ΩB , F B , P B . Second, rare event risk enters through a one-dimensional Poisson process
¡
¢
N (t) with intensity λ, defined on probability space ΩN , F N , P N . I define (Ω, F, P) as the
ª
©
product probability space and the filtration of the combined history as {Ft } = FtB × FtN .
The coefficients µ
be , σe and κe serve as the deterministic growth rate, the instantaneous
volatility and the rare event jump size, respectively. I restrict the jump size to κe ∈ (−1, 0)
in order to induce negative jumps and to ensure that the output process remains strictly
positive. The initial value and the instantaneous volatility are strictly positive, e(0) > 0,
σe > 0, and agents are assumed to agree on µ
be , κe and σe but have incomplete information
about the true rare event frequency λ.
Agents are heterogenous since they have different subjective beliefs about the frequency
of rare events. Specifically, λi (t) = Ei,t [λ] is agent i’s conditional estimate of λ, where Ei [·]
denotes the expectation relative to Pi . The agents’ perceived measures, P1 and P2 , and the
true probability measure are equivalent. Under each agent’s beliefs the aggregate output
process obeys
(2)
de(t)
=µ
be dt + σe dB(t) + κe dN (t, λi (t)).
e(t−)
Both agents see precisely the same path of aggregate output, but belief that the discontinuous
movements they observe are generated by the frequency λi (t). Although agents agree on the
growth rate in normal economic times, they “agree to disagree” on the total expected growth
rate. To see this, one can write equation (2) alternatively as
de(t)
= (µe,i (t) − κe λi (t))dt + σe dB(t) + κe dN (t, λi (t)),
e(t−)
where µe,i (t) serves as agent i’s total expected growth rate of aggregate output, implying
that µ
be = µe,i (t) − κe λi (t). Hence, the difference in expected growth rates can be expressed
in terms of the dispersion in beliefs given by µe,1 (t) − µe,2 (t) = κe (λ1 (t) − λ2 (t)).
A.
Learning about Rare Events
Unless agents are dogmatic, they learn about the true frequency of rare events over time.
For simplicity, I assume the true frequency is either high or low, given by λH or λL . Each
6
agent forms her conditional estimate as
H
H
L
λi (t) = pH
i (t)λ + (1 − pi (t))λ ,
(3)
H
where pH
i (t) is agent i’s prior belief of λ . The information utilized to form the posterior
belief is, whether a rare event in the aggregate output process occurred, or not. Regular
economic risk, on the other hand, does not lead to new information about rare events in
continuous time. Hence, each agent’s probability changes according to
H
H
dpH
i (t) = dpi (t)|dN (t)=1 + dpi (t)|dN (t)=0 .
(4)
Using Bayes’ rule leads to
H
H
dpH
i (t)|dN (t)=1 = pi (t + dt)|dN (t)=1 − pi (t)
H
pH
i (t)λ
= H
− pH
i (t),
L
pi (t)λH + (1 − pH
(t))λ
i
(5)
as well as
(6)
H
H
dpH
i (t)|dN (t)=0 = pi (t + dt)|dN =0 − pi (t)
H
pH
i (t)(1 − λ dt)
= H
− pH
i (t).
L dt)
pi (t)(1 − λH dt) + (1 − pH
(t))(1
−
λ
i
Such an updating process has an intuitive pattern. An agent who assigns unity to one
of the frequencies has a dogmatic belief and never updates. In contrast, an agent will learn
about the true frequency in case of strictly positive support for both states. As long as no
rare event occurs, she slowly revises her belief about the frequency downward. If a rare event
occurs, however, she dramatically revises her belief upwards. Moreover, in a finite horizon
setting agents will never be certain, and with different initial priors the dispersion never
reaches zero, such that a substantial degree of heterogeneity can remain at the end of the
agents’ lifetimes.1
B.
The Capital Market
Both agents trade continuously in a capital market in order to share regular economic risk
and rare event risk, and to finance their optimal consumption profile. This capital market,
1
This way to model learning is similar to work by Collin-Dufresne et al. (2004), who use learning to
capture contagion effects in the perceived likelihood of corporate default.
7
however, might be incomplete.
The capital market consists of a riskless money market and a stock market. The riskless
money market trades in zero-net supply and pays the interest rate r, such that
(7)
dB0 (t) = B0 (t)r(t)dt,
B0 (0) = 1.
The stock market is given by a unit-supply stock, a claim to the aggregate output process.
From the perspective of agent i, it has the posited dynamics given by
(8)
dS(t) + e(t)dt
= µ
bS (t)dt + σS (t)dB(t) + κS (t)dN (t, λi (t)).
S(t−)
The interest rate r, the return µ
bS , the volatility σS as well as the jump size κS are posited
to be {Ft } measurable, and to be determined endogenously in equilibrium.
It is easy to see this capital market is incomplete, since the two sources of uncertainty
need to be shared through one risky security. However, upon market completion the capital
market also offers the ability to trade in a rare event insurance product. From the perspective
of agent i, it has the assumed price process
(9)
dP (t)
= µ
bP (t)dt + κP (t)dN (t, λi (t)),
P (t−)
P (0) > 0.
This security has some convenient characteristics to study the research questions under
consideration. It does not contain any continuous source of uncertainty, and its price process
is affected only by the occurrence of a rare event, which allows me to label it an insurance
product. The buyer of security P gets rewarded amount µ
bP every instant of time, but faces
the risk that its value drops to (1 + κP )P at a rare event. This security is in zero-net supply
and pays no dividend. Therefore, the initial level P (0) as well as the jump size, κP < 0, can
be freely chosen, but µ
bP is determined endogenously.2
A central feature of the model is that investor heterogeneity generates actual demand for
this insurance product after market completion. In other words, market incompleteness stems
from restricted participation in the insurance product; participation in the stock market,
however, is unrestricted.
2
An investment in P is comparable to the purchase of a catastrophe bond or a hybrid security that is
linked to adverse states of the economy. If one were to isolate the insurance derivative from the bond, then
the price for the insurance derivative is simply given by the spread above the riskless rate, µ
bP − r.
8
C.
The Agent’s Optimization Problem
Each agent solves an optimization problem over a finite horizon in a standard expected utility
framework. Both agents are risk averse and have logarithmic preferences. They choose a
non-negative consumption process ci and a portfolio process πi , and are initially endowed
with a fraction of the unit-supply stock, xS,i , such that xS,1 + xS,2 = 1.
Using martingale techniques, see for example Karatzas et al. (1987), Cox and Huang
(1989) as well as Bardhan and Chao (1996) for the case of discontinuous processes, the
static version of each agent’s optimization problem can be reduced to
·Z
(10)
maxEi
ci
¸
T
log(ci (t))dt
·Z
¸
T
s.t. Ei
ηi (t)ci (t)dt = xS,i ηi (0)S(0).
0
0
Such a maximization problem is well-defined in the case of a complete market. However, multiplicity of budget constraints in the case of market incompleteness requires more structure
on the optimization problem.
I first introduce an agent-specific state price density process ηi given by
µ
¶
λQ,i (t)
dηi (t)
(11)
= −r(t)dt + (λi (t) − λQ,i (t))dt − θi (t)dB(t) +
− 1 dN (t, λi (t)),
ηi (t−)
λi (t)
with an initial value of ηi (0) = 1. The parameters θi and λQ,i serve as the market price
of regular economic risk associated with the Brownian motion, and the risk neutral rare
event intensity associated with the Poisson process (the market price of rare event risk),
respectively.3
3
The change of measure from the perspective of agent i is given by
µ Z t
¶
Z
1 t
B
2
θi (s)dB(s) −
Zi (t) = exp −
θi (s) ds
2 0
0
for the case of regular economic risk, and given by
µZ
ZiN (t)
= exp
t
(λi (s) − λQ,i (s))ds
0
¶ N (t,λ
Yi (t)) µ
λQ,i (s)
λi (s)
¶
for the case of rare event risk, see for example Protter (1990). Under standard regularity conditions, ZiB (t),
ZiN (t), and ZiB (t)ZiN (t) are martingales under each agent’s belief, and the state price density process in
equation (11) follows directly from the dynamics of ZiB (t)ZiN (t) deflated by the price of the money market
B0 (t). The initial value is ηi (0) = 1 since B0 (0) = 1.
9
The optimal consumption policy can be determined from the inverse of each agent’s marginal
utility and yields ci (t) = (yi ηi (t))−1 , where yi serves as the Lagrangian multiplier from the
constrained optimization. After solving for the value of yi , each agent’s level of financial
wealth simplifies to Wi (t) = ci (t)(T − t). The dynamics of financial wealth, while focusing
only on the terms generated by Brownian and Poisson uncertainty, relate to the inverse of
the agent-specific state price density process given by
µ
¶
dWi (t)
λi (t)
(12)
= (...)dt + θi (t)dB(t) +
− 1 dN (t, λi (t)).
Wi (t−)
λQ,i (t)
As usual, one can compare the process in equation (12) to the process of the financing portfolio to recover optimal portfolio holdings. In the incomplete capital market, the
financing portfolio will follow
(13)
dWi (t)
= (...)dt + πS,i (t)σS (t)dB(t) + πS,i (t)κS (t)dN (t, λi (t)),
Wi (t−)
and after market completion yields
(14)
dWi (t)
= (...)dt + πS,i (t)σS (t)dB(t) + (πS,i (t)κS (t) + πP,i (t)κP (t))dN (t, λi (t)),
Wi (t−)
where the fraction of agent i’s wealth invested in the stock market and the insurance product
is denoted by πS,i and πP,i , respectively.
D.
Stock Price Properties and Portfolio Holdings
The equilibrium solution for the stock price follows directly from each agent’s optimization
problem after imposing good market clearing, c1 (t)(T − t) + c2 (t)(T − t) = e(t)(T − t) = S(t).
Hence, the coefficients of the stock price process are the same as the exogenous parameters
of the output process, a common result due to the constant price dividend ratio under
logarithmic preferences, and valid even in an incomplete market in this case.
Lemma 1 The coefficients of the stock price process in equation (8) are given by
(15)
µ
bS (t) = µ
be , σS (t) = σe , κS (t) = κe .
10
Given the financial wealth processes in equations (12), (14) and (13), the portfolio holdings
in the incomplete market are given by
θi (t)
1
πS,i (t) =
=
σS (t)
κS (t)
(16)
µ
¶
λi (t)
−1 .
λQ,i (t)
After market completion, the stock market holdings adjust to
(17)
πS,i (t) =
θi (t)
,
σS (t)
and the portfolio holdings in the rare event insurance product are given by
µ
¶
1
λi (t)
κS (t)θi (t)
(18)
πP,i (t) =
−1 −
.
κP (t) λQ,i (t)
κP (t)σS (t)
E.
Equilibrium in the Incomplete Market
Definition 1 Given both agents’ preferences and endowments, a Walrasian equilibrium in
the incomplete market is a collection of allocations (c1 , π1 ) and (c2 , π2 ) , and a price system
for the capital market (θ1 , θ2 , λQ,1 , λQ,2 , r), such that (c1 , π1 ) and (c2 , π2 ) are optimal solutions
to the agents’ optimization problems. All markets clear at t ∈ [0, T ]:
c1 (t) + c2 (t) = e(t),
W1 (t) + W2 (t) = S(t),
πS,1 (t)W1 (t) + πS,2 (t)W2 (t) = S(t).
The crucial feature of the incomplete market economy is the indeterminacy of a unique
solution for the market price of regular economic risk and the risk neutral rare event intensity
solely from the stock price process. Intuitively, the stock market excess return will have a
two-factor representation given by
(19)
µ
bS (t) − r(t) = σS (t)θi (t) − κS (t)λQ,i (t),
such that solving for θi and λQ,i solely from equation (19) would lead to an infinite set of
solutions.
11
To solve for the equilibrium, I first appeal to a technique used by Cvitanić and Karatzas
(1992) and Basak and Croitoru (2000).4 I assume there exists a solution to each agent’s
minimax problem which can be formulated as
(20)
½
·Z
min maxEi
θi ,λQ,i
ci
T
¸
log ci (t)dt
·Z
T
s.t. Ei
0
¸
¾
ηi (t)ci (t)dt ≤ xS,i ηi (0)S(0) ,
0
subject to the constraint of the excess return of the unconstrained stock market in equation
(19). The maximization problem can be solved as in a complete market case, given any pair
of θi , λQ,i , and Proposition 1 summarizes the solution of the minimization problem.
Proposition 1 Given the capital market coefficients and υ(t) = κS (r(t) − µ
bS (t)), each
agent’s minimax problem is solved by
¶
µ
q
−1
2
2
2
2
2
(21)
θi (t) =
υ(t) + σS (t) − 4κS (t)λi (t)σS (t) + (−υ(t) + σS (t)) ,
2κS (t)σS (t)
(22)
1
λQ,i (t) = 2
2κS (t)
µ
¶
q
2
2
2
2
2
υ(t) − σS (t) + 4κS (t)λi (t)σS (t) + (−υ(t) + σS (t)) .
This solution satisfies the restriction in equation (16), implying that πS,i generates optimal
exposure with respect to regular economic risk as well as rare event risk. The values for θi
and λQ,i are precisely those such that zero demand in the insurance product is a solution for
agent i’s portfolio holding process.
Second, it is convenient to appeal to a representative agent (RA) construction using a
state-dependent weight, given by φ. An RA utility function in which the first agent’s weight
is normalized to unity can be formulated as
U (e(t), φ(t)) = max [log(c1 (t)) + φ(t) log(c2 (t))] , s.t. c1 (t) + c2 (t) = e(t).
4
Basak and Croitoru (2000) solve a consumption/ investment problem under heterogeneity in beliefs, in
which one agent faces a potential portfolio constraint in the stock market. Cvitanić and Karatzas (1992)
solve a model with different interest rates for borrowing and lending. The solution coincides with a complete
market problem under fictitious price dynamics, in which the portfolio constraints are not binding and
standard martingale methods as in Cox and Huang (1989) or Karatzas et al. (1987) apply. The technique
has been extended to more general price processes. For an example with logarithmic utility see Goll and
Kallsen (2003).
12
Optimality and consumption good clearing imply that the RA’s marginal utility equates to
agent 1’s state price density, such that
(23)
η1 (t) =
Uc (e(t), φ(t))
φ(0)
, and η2 (t) =
η1 (t),
Uc (e(0), φ(0))
φ(t)
and the weight is given by
(24)
φ(t) =
y1 η1 (t)
W2 (t)
=
.
y2 η2 (t)
W1 (t)
The approach to formulate an RA with state-dependent weight was introduced by Cuoco
and He (1994), more recent examples can be found in Basak and Cuoco (1998) or Buraschi
and Jilstov (2006). In an economy without disagreement both agents face the same state
price density process and φ reduces to a constant. In a complete market heterogeneous
belief setting, the weight is stochastic and equals the Radon-Nikodym derivative to change
between agents’ perceived probability measures. In this incomplete market case, it captures
not only the disagreement between agents’ beliefs, but also the additional friction stemming
from nonparticipation in the insurance product.
As usual, the optimal consumption policies resulting from this are given by
(25)
c1 (t) =
e(t)
e(t)φ(t)
, and c2 (t) =
.
1 + φ(t)
1 + φ(t)
However, this construction also leads to restrictions for the remaining equilibrium characteristics among both agents as summarized in Proposition 2.
Proposition 2 In equilibrium, the market prices of regular economic risk in equation (21),
and the risk neutral rare event intensities in equation (22) satisfy the conditions,
(26)
σe −
φ(t−)
1
θ1 (t) =
θ2 (t),
1 + φ(t−)
1 + φ(t−)
as well as
(27)
(κe + 1) −
φ(t−) λ2 (t)
1
λ1 (t)
=
,
1 + φ(t−) λQ,1 (t)
1 + φ(t−) λQ,2 (t)
at the equilibrium interest rate
(28)
r(t) = µ
be − σe θi (t) + κe λi,Q (t).
13
The interest rate has two terms that stem from the precautionary savings motive, one for
each source of uncertainty. Although the equilibrium characteristics cannot explicitly be
written in terms of exogenous primitives, the implementation reduces to a standard fixed
point problem.
F.
Market Completion
Definition 2 A Walrasian equilibrium after market completion is a collection of allocations
(c1 , π1 ) and (c2 , π2 ) , and a price system for the capital market (θ, λQ , r), such that (c1 , π1 )
and (c2 , π2 ) are optimal solutions to the agents’ optimization problems. All markets clear at
t ∈ [0, T ]:
c1 (t) + c2 (t) = e(t),
W1 (t) + W2 (t) = S(t),
πS,1 (t)W1 (t) + πS,2 (t)W2 (t) = S(t),
πP,1 (t)W1 (t) + πP,2 (t)W2 (t) = 0.
Upon market completion, agents can now trade in the rare event insurance product P .
Given the agent-specific density process, the premium of this security is given by
(29)
µ
bP (t) − r(t) = −κP (t)λQ,i (t),
such that an agent buying security P acts as the seller of insurance and gets rewarded the
excess premium −κP λQ,i per instant of time. However, agreement on the price path of P in
normal times requires agreement on the risk neutral rare event intensity, λQ,1 = λQ,2 = λQ .
This in turn requires agreement on the market price of normal economic risk, θ1 = θ2 = θ,
simply from equation (19). Hence, the market prices of risk are not agent-specific anymore;
Proposition 3 summarizes the equilibrium characterization.
Proposition 3 In the complete market, the market price of normal economic risk, the risk
neutral rare event intensity, and the interest rate are given by
(30)
θ(t) = σe ,
14
(31)
λQ (t) =
(32)
1
λ1 (t)
φ(t−) λ2 (t)
+
,
1 + φ(t−) κe + 1 1 + φ(t−) κe + 1
r(t) = µ
be − σe θ(t) + κe λQ (t).
The market price of normal economic risk reduces to well-known solution under logarithmic
preferences. The risk neutral rare event intensity, however, is a wealth-weighted average of
each individual’s risk neutral rare event intensity. Both economies have equivalent limiting
cases summarized in the following Corollary, they are nested within the results obtained by
Naik and Lee (1990).
Corollary 1 As the weight in the RA formulation approaches the boundaries, i.e. φ(t) → 0
or φ(t) → ∞, or for the case in which there is no disagreement among agents, λ1 (t) =
λ2 (t), both economies converge to a homogeneous agent economy populated by an agent with
logarithmic utility. The equilibrium characteristics converge to θ = σe , λQ (t) =
r(t) = µ
be − σe2 −
κ2e λi (t)
.
κe +1
λi (t)
,
κe +1
and
There is no demand for zero-net supply securities in the capital
market: all wealth is invested in the stock market.
G.
Stochastic Weighting Process
The stochastic weight in the RA construction is the endogenous state variable of this economy, it relates to the ratio of financial wealth of agent 2 relative to agent 1. It fully describes
the dynamic nature of the equilibrium, and our understanding of the model can be improved
by knowing its dynamic process, summarized in Lemma 2.
Lemma 2 The dynamic process of the weight φ(t) in the complete market is given by
·
¸
dφ(t)
λ2 (t)
(33)
= [λ1 (t) − λ2 (t)] dt +
− 1 dN (t, λi (t)),
φ(t−)
λ1 (t)
and in the incomplete market by
(34)
·
¸
λQ,2 (t)λ1 (t) − λQ,1 (t)λ2 (t)
dφ(t)
=
dt
φ(t−)
λQ,2 (t)
¸
·
λQ,1 (t)λ2 (t)
− 1 dN (t, λi (t)).
+ [θ2 (t) − θ1 (t)] dB(t) +
λ1 (t)λQ,2 (t)
15
The dynamic process captures the disagreement between agents in the complete market, and
the wealth ratio is stochastic due to rare event risk. However, Brownian motion risk enters
the process in the incomplete market, such that the wealth ratio randomly fluctuates even
in normal economic times.
The initial value is exogenously given by φ(0) = W2 (0)/W1 (0), and existence of the
equilibrium follows form the positivity of φ(t) at all points in time. Furthermore, φ(t) is
also a martingale under the first agent’s measure P1 in either economy (assuming standard
regularity conditions), which is a crucial property for consistency between the stock price
i
hR
T
value given by S(t)ηi (t) = Ei,t t ηi (s)e(s)ds , and the value that follows directly from
consumption good clearing.5
III.
Results and Analysis
Without loss of generality, I assume that agent 1 anticipates a higher frequency of rare events
than agent 2, and present the results within the following set of parameters: I consider rare
events with an impact of κe = −.20, occurring at a frequency between λ1 (t) = λH = .10
and λ2 (t) = λL = .01, i.e. every 10 or 100 years on average. Barro (2006) estimates an
ex-post probability of 1.5% - 2% per annum, based on rare events during the last century.
While his estimate is closer to the lower bound of λL , I assume that worries about a rare
event occurrence can be as high a λH . Barro (2006) also estimates a mean impact size of
-29%, hence the value of κe considered here is slightly more conservative. Sections III.A
and III.B show results for which agents are assumed to be dogmatic. In Section III.C,
however, I present the capital market adjustments in rare event times, with and without
learning. Furthermore, I assume that regular economic risk has an annualized volatility of
σe = .04, with a growth rate in normal economic times of µ
be = .04, which is approximately
the magnitude of output data in developed countries.
hR
i
T
Recall that the expectation E1,t t η1 (s)e(s)ds can be written (using agent 1’s first-order conhR
i
T
1
dition) as E1,t t c1 (s)y
e(s)ds
, which (using agent 1’s optimal consumption share) simplifies to
1
hR
i
hR
i
T
T 1+φ(s)
E1,t t 1+φ(s)
e(s)ds
=
E
ds
. Hence, the process of φ(t) needs to be martingale under the
1,t
e(s)y1
y1
t
5
first agent’s belief such that the stock price value equals S(t) = e(t)(T − t).
16
A.
Portfolio Allocations and Risk Premia
1.
Risk Sharing through Stock and Money Markets
I first investigate the equilibrium allocations in the incomplete capital market, to see how
agents act if they must share all risk through the stock market and money market. From
Proposition 2 one can see that market incompleteness drives a wedge between the market
prices of risk, which will reflect itself in agents’ portfolio allocations, as shown in Figure 1.
Agent 1 does not invest all her wealth in the stock market and acts as a lender in the money
market. For stronger degrees of heterogeneity this can even lead to a short position in the
stock. For example, suppose both agents are equally present in terms of financial wealth,
then agent 1 sells up to 1.5 times her financial wealth through the stock market. Consistent
with market clearing, agent 2 is willing to accept the opposite exposure in the stock market,
and is a borrower in the money market for the purpose of leverage.
Upon market completion, both agents invest their entire financial wealth in the stock market,
and evaluate their rare event risk exposure separately from normal economic risk. Agent 1
anticipates to be undercompensated through the stock market, has too much rare event risk
exposure and therefore serves as the buyer of insurance. Agent 2 anticipates to be overcompensated through the stock market, and is willing to provide insurance against market
crashes, as summarized in Figure 2.
Intuitively, the difference in allocations between the incomplete and complete capital market
mirrors a synthetic put option, an indirect way of implementing portfolio insurance. The
agent who has reduced stock holdings is the put option buyer; the agent who obtains leverage
and finances through the money market acts as the seller.
2.
Equilibrium Risk Premium
A direct consequence of the incomplete market portfolio allocations is the continuous redistribution of wealth, generating a countercyclical risk premium. Formally, the covariation of
risk premium and aggregate output in normal economic times is given by
17
(35)
d(b
µS (t) − r(t))C
de(t)C
∂(b
µS (t) − r(t))
=
φ(t)(θ2 (t) − θ1 (t))σe .
e(t−)
∂φ(t)
The difference in market prices of regular economic risk, θ2 − θ1 , is strictly positive such
that a positive shock in output improves the financial wealth of agent 2 relative to agent 1.
Since agent 2 anticipates a lower rare event frequency than agent 1, her belief matters more
in relative terms leading to an increasing interest rate and a decreasing risk premium. In
turn, the partial derivative
∂(b
µS −r)
∂φ
is strictly negative, such that the entire covariation term
is strictly negative as well. Furthermore, the stronger the dispersion in beliefs, the more
negative the degree of covariation.
The channel through which countercyclicality occurs is comparable to Basak and Cuoco
(1998) who analyze limited participation in the stock market. In their model, a positive
shock to the economy improves the wealth ratio of the participating agents relative to the
non-participating. This lowers the overall degree of limited participation, and thereby the
required risk premium. While there are different theories delivering countercyclical risk
premia in general, to my knowledge, none has shown that the incomplete market considered
in this paper can contribute to this phenomenon.
After market completion, however, each agent’s consumption profile is perfectly correlated
with the stock market, and no redistribution of wealth occurs due to output shocks in
normal economic times. Hence, countercyclicality is not present. But how does market
completion affect the general level of the risk premium? The main result is an increasing
risk premium due to market completion. Figure 3 shows the equilibrium risk premium in the
incomplete market, as well as the percentage increase after introducing rare event insurance.
For example, suppose agents are equally present in terms of wealth, then the risk premium
can be up to 20% higher in the complete market, and even 30% higher if agent 2 represents
a larger fraction. Such an increase is large considering the moderate degree of risk aversion
in the case of logarithmic preferences.
To understand this result, it is analytically convenient to show how the risk premium changes
once a portfolio constraint on P becomes marginally binding. There can be two reasons for
18
lower demand or supply for the insurance product. From the perspective of the buyer,
demand will be reduced if either insurance is more expensive, or if less rare event exposure
is present in the stock market.6 From the perspective of the seller, supply will be reduced
if either insurance is less expensive, or if more rare event exposure is present in the stock
market. The equilibrium restrictions in equations (26) and (27) allow me to study how
agents are willing to exchange risk exposures without violating market clearing, i.e.
(36)
dθ1 (t) = −φ(t)dθ2 (t),
dλQ,1 (t) = −φ(t)
λ2 (t)
dλQ,2 (t).
λ1 (t)
Not surprisingly, both agents are willing to trade-off regular economic risk at rate φ, their
ratio of financial wealth. This, however, is not true for the trade-off of rare event risk, as
agents are willing to exchange rare event risk at rate φ λλ21 . A graphical representation of this
relation can be found in Figure 4. Furthermore, both agents need to agree on the change of
the total risk premium in the unconstrained stock market given by
(37)
σe dθ1 (t) − κe dλQ,1 (t) = σe dθ2 (t) − κe dλQ,2 (t).
This allows me to derive an explicit relation for the trade-off between normal economic risk
and rare event risk given by
(38)
dθ1 (t) =
κe
σe
³
λ1 (t)
λ2 (t)
´
+ φ(t)
1 + φ(t)
dλQ,1 (t).
With this I can rewrite the change in the risk premium as
µ
¶
λ1 (t)
κe
(39)
σe dθ1 (t) − κe dλQ,1 (t) =
− 1 dλQ,1 (t).
1 + φ(t) λ2 (t)
The right hand side is strictly negative under the assumption of a strictly negative jump
size κe . From the perspective of the buyer of insurance, the decrease in the market price of
regular economic risk, θ1 , overcompensates for the increase in the market price of rare event
risk, λQ,1 . Alternatively, from the perspective of the seller, the decrease in the market price
6
Formally,
−1 λi (t)
dπP,i (t)
|λ (t)=λQ (t) =
> 0,
dλQ,i (t) Q,i
κe λQ (t)2
dπP,i (t)
−1
|θ (t)=θ(t) =
< 0.
dθi (t) i
σe
The partial derivatives of the portfolio demand functions are evaluated at the complete market values.
19
of rare event risk overcompensates for the increase in the market price of regular economic
risk. Hence, the total risk premium decreases once the constraint is binding.7
In general, three outcomes can occur while moving from an incomplete to a complete
market equilibrium – the risk premium can increase, it can decrease, or it is unaffected. Of
course all three have been obtained in equilibrium models, so I will position my finding with
respect to some previously studied frictions. For example, an increasing equity premium has
been observed with an endowment structure based on pure diffusive risk by Detemple and
Murthy (1997) and Gallmeyer and Hollifield (2008) in the case of heterogenous beliefs with
short sale constraints in the stock market. In the current paper, however, agents are limited
in their ability to trade a zero-net supply security and all agents hold a risky portfolio even
in incomplete markets.
My result is in contrast to findings by Basak and Cuoco (1998), who find a decrease in the
equity premium after relaxing the participation constraint. The same can occur in models
with suboptimal risk sharing in the presence of idiosyncratic income shocks. For example,
Constantinides and Duffie (1996) and Mankiw (1986) have shown that risk premiums are
higher if uninsurable idiosyncratic shocks become more volatile during recessional states of
the economy. My model does not consider idiosyncratic shocks, nor does it assume limited
participation in the stock market.
As mentioned earlier, the increased risk premium is consistent with the empirical evidence
on the passage of TRIA, or more generally the effect of option introduction on the underlying
stock price. None of these studies, however, examines whether the discount rate changes
upon market completion. I therefore replicate part of the regression analysis in Brown et al.
(2004), focusing on the five industry portfolios most affected by terrorism insurance. In
7
The derivation above does not assume κe < 0. Suppose, ceteris paribus, that the rare event in fundamen-
tals is not a catastrophe but a positive jump in output, κe > 0. As the portfolio constraint on the insurance
contract is binding, it will again be optimal for agent 1 to increase λQ,1 , and for agent 2 to decrease λQ,2 ,
compared to the complete market case. Then equation (39) shows that the total risk premium increases
in the case of an incomplete market. In other words, relaxing the participation constraint will lead to a
decreasing risk premium in the case of rare events with a positive impact, another prediction that arises
from this analysis.
20
addition to the standard market model and the indicator variable capturing the passage of
TRIA, I also allow for an interaction term among both. I find that the covariation of the
industry and market return does appear to he higher around the passage of TRIA.8
B.
Risk Exposure and Efficiency Properties
1.
Agent’s Exposure to Rare Event Risk
A result of the theoretical analysis is that the complete market risk neutral intensity is
bounded by the risk neutral intensities in the incomplete market, which are bounded by the
values of a single-agent economy. The following ordering occurs:
λ2 (t)
λ1 (t)
< λQ,2 (t) < λQ (t) < λQ,1 (t) <
.
κe + 1
κe + 1
(40)
This set of inequalities has a meaningful counterpart in the interpretation of each agent’s
optimal consumption (financial wealth) process as given in equation (12). Each agent faces
rare event risk exposure of the magnitude
µ
(41)
λi
λQ,i
− 1, such that
¶ µ
¶
µ
¶ µ
¶
λ2 (t)
λ2 (t)
λ1 (t)
λ1 (t)
−1 <
− 1 < κe <
−1 <
−1 .
λQ (t)
λQ,2 (t)
λQ,1 (t)
λQ (t)
The graphical representation of this relation is shown in Figure 5. After completing the
market, agents will bet on their beliefs to a large extent by taking a position in the capital
market. Since agent 1 is anticipating a higher frequency than agent 2, she is willing to accept
rare event risk exposure less negative than the impact size in aggregate output. Conversely,
agent 2 is willing to accept rare event risk exposure more negative. In the incomplete market,
however, agents have limited risk sharing possibilities: Agent 1 (agent 2) is still willing to
accept rare event risk exposure less negative (more negative) than present in output, but
not as much as compared to after market completion.
For example, suppose agent 1 represents 20% of the economy, agent 2 the remaining 80%.
Agent 1 buys such a large amount of insurance after introduction, such that her consumption
would increase by up to 200% at the occurrence of a rare event. Without explicit insurance,
8
Detailed test results are available upon request.
21
she would still benefit, but only by 70%. As a result, the wealth transfer among agents at
the occurrence of a rare event is larger after market completion compared to before.
Given heterogenous beliefs, I have identified several channels through which market completion will affect each agent’s optimal consumption profile, through the risk premium, through
changes in the covariation with regular economic shocks, and through larger wealth transfers
at a market crash. But what is their effect from a welfare perspective?
2.
Efficiency Properties of Market Completion
I first illustrate the effect of market completion based on the agents’ subjective beliefs. The
consumption multiplier that equates the indirect utility of the complete and incomplete
market economy is given by
¸
·Z t+1
·Z
complete
(42)
Et,i
log(ci
(t))dt = Et,i
t
t+1
t
¸
log(ki cincomplete
(t))dt
i
,
where the optimal consumption policies are those determined in Section II. Since consumption is proportional to wealth, ki corresponds to the wealth enhancement given to agent i
per year by completing the market. For this analysis, it is assumed that beliefs are constant
over the one year period. Please note, expectations are formed with respect to each agent’s
own information set, hence I refer to this as the effect on agents’ anticipatory utilities. The
results are shown in Figure 6: For example, agent 2, being the seller of insurance, perceives a
utility gain up to 2% p.a. Next, I compute the total effect as the wealth-weighted average of
individual effects given by (k1 −1)/(1+φ)+(k2 −1)φ/(1+φ), which is essentially the effect of
market completion as perceived by the representative agent. The largest effect occurs when
agent 1 controls 25% of financial wealth in the economy, and agent 2 controls the remaining
fraction. In that case, the total effect on anticipatory utility is close to 1% p.a. The total
effect is strictly positive, which is not surprising given that the complete market solution is
the most efficient allocation based on agents’ beliefs.
An alternative view of efficiency arises from an ex-post perspective. Not all agents’ anticipatory utility can realize itself ex-post, as some agents will bear a disutility from having
22
incorrect beliefs. Following Brunermeier and Parker (2005), one possibility to measure this
ex-post cost is to compute indirect utilities with respect to the true probability. This should
be interpreted with care, as it requires a paternal agent to know the true probability to
condition on in equation (42). The results are reported in Figure 6, showing that the total
effect of market completion might not improve efficiency ex-post.
How is this possible? With or without complete markets, one agent is incorrect regarding
her belief about the frequency. After introducing an insurance market, we can expect a
utility gain for one agent. But this gain might not be sufficient to compensate the other
agent for the disutility of having the incorrect belief, a disutility that can be larger in the
complete versus the incomplete market. For example, suppose the buyer assumes the correct
belief (λ = .10), then the seller receives insufficient compensation for providing insurance
compared to the subjective beliefs impounded in the price per contract. Of course the seller
also has the incorrect belief in the incomplete market, but limited risk sharing possibilities
protect the seller from the consequences. While the magnitude of this ex-post effect crucially
depends on the assumption about which agent is correct, a notable result emerges that is
belief independent: There exists a region in which the effect of market completion on total
ex-post utility is negative regardless of whether agent 1 or agent 2 holds the correct belief.
If xS,1 < .25, then regardless of whether .01 or .10 is the true lambda, the utility loss of the
agent having the incorrect belief will be larger than the utility gain from its counterpart.
This illustration points toward an interesting feature of heterogenous belief models, and
the path from an incomplete to a complete capital market. It shows that such a model can
contain a welfare effect similar to the “Hirshleifer effect” in incomplete market models subject
to informational differences, see for example Marı́n and Rahi (2000), where less information
and an incomplete set of securities might be preferable. In my example, however, such a
feature arises in an “agreeing to disagree” setting.
C.
Dynamic Behavior in Rare Event Times
A common understanding of market participants is that flight to quality effects are portfolio adjustments of investors moving from riskier to safer investments, resulting in a sharp
23
decrease of riskless interest rates. But such a partial equilibrium argument leaves us uninformed about the investor engaging in the opposite transaction. As risky securities are in
fixed supply, who are the buyers of stocks in flight to quality times? If the demand for safer
investments is increasing, then who engages as a lender? Having an equilibrium model, I
show what flight to quality effects emerge at the occurrence of a rare event.
1.
Price Effects
I first show how the riskless interest rate adjusts, see Figure 7. Without learning, the interest
rate is not affected in case of homogeneity. However, the interest rate obeys a negative jump
in case of heterogeneity, even without a revision in beliefs. The reason is a wealth effect:
Assuming λ1 > λ2 , then the jump size in the weighting process is strictly negative. As agent
1 gains financial wealth relative to agent 2, the overall degree of anticipated rare event risk
increases, causing the interest rate to fall. For the chosen set of parameters, the drop in
interest rates can be as large as -90 basis points.
Learning amplifies this drop in interest rates. Suppose both agents allow for a small possibility that the other agent is correct in her belief – agent 1 assigns a 99% probability that λH is
correct, but also assigns a 1% chance that λL is the true frequency, and agent 2 assumes the
opposite. As as result, agents will update their beliefs over time, with a substantial upward
revision at a rare event; the average upward revision over an agent’s lifetime (80 years) is
approximately +.01. Taking this into account, the drop in interest rates can be as large as
-120 basis points. Furthermore, it is not surprising that this flight to quality effect can be
even stronger in the complete market, knowing that wealth transfers are larger after market
completion. For example, if agent 1 controls 20% of the economy, agent 2 the remaining
80%, then the total effect of wealth transfer plus learning will be as large as -140 basis points.
These flight to quality effects are comparable to Chang and Sundaresan (2005), who
model an equilibrium setting with default. The default mechanism affects the interest rate
and risk premium, such that the borrower’s shadow riskless rate decreases in the flight to
quality region. Vayanos (2004) proposes an equilibrium model based on transaction costs
and stochastic volatility. In his model, the riskless rate is exogenous and agents require a
24
higher risk premium in volatile times due to increased risk aversion. Both models, however,
assume homogeneous agents, and risk aversion in my model is not affected at a rare event.
2.
Portfolio Effects
Figure 8 displays the change in stock holdings of agent 1 given by
(43)
π1,S (t)W1 (t) π1,S (t−)W1 (t−)
π1,S (t)
π1,S (t−)
−
=
−
,
S(t)
S(t−)
1 + φ(t) 1 + φ(t−)
showing that the magnitude of portfolio rebalancing can be larger in the incomplete market,
and the direction is ambiguous. For example, agent 1, being already short in the stock
market, will sell even more stock following a market crash. This occurs when agent 2
controls most of the wealth in the economy. Multiple effects are combined here: First, a
wealth transfer occurs from agent 2 to agent 1, as agent 1 is indirectly insured. In addition,
agent 1 is willing to take a short position for high degrees of heterogeneity and low levels of
wealth. If the latter overcompensates the former, then a market crash can generate a trade
in which agent 1 is the seller of shares. This can not happen after market completion, as
agent 1 will always increase her stock holdings at a rare event.
I also find that flight to quality does not need to be associated with a larger amount of
borrowing and lending. When agent 1 controls most of the wealth in the economy, x(S, 1) >
.5, then a rare event effectively shrinks the degree of heterogeneity, leading to smaller open
interest in the money market. After market completion, all effects are absorbed in the open
interest of the insurance product, and a similar pattern occurs: The change in open interest
given by
(44)
π1,P (t)
π1,P (t−)
π1,P (t)W1 (t) π1,P (t−)W1 (t−)
−
=
−
,
S(t)
S(t−)
1 + φ(t) 1 + φ(t−)
can increase or decrease at a rare event. First, agent 1 gains exposure towards rare event risk
through trade in the stock market, and would like to purchase additional insurance. Second,
the premium for insurance will increase as a higher weight is put on agent 1’s beliefs. Both
effects point in opposite demand directions and might offset each other. For low levels of
agent 1’s wealth, x(S, 1) < .1, the open interest will increase although insurance has become
more expensive. Otherwise, flight to quality is associated with a smaller amount of open
25
interest. The revision in beliefs plays a minor role in these portfolio effects, it does matter
of course, but the major source is wealth transfers.
IV.
Conclusion
My analysis highlights several outcomes of rare event risk and heterogeneous beliefs in the
context of an incomplete capital market: Investors will try to insure themselves indirectly
through the stock market and the money market, and risk premia behave countercyclically
with respect to normal economic shocks. Upon market completion, risk premia increase
along with investors’ exposure to rare event risk. Although market completion leads to
a more efficient allocation based on investors’ anticipatory utilities, complete markets can
facilitate wealth transfers to such a large extent, that investors with incorrect beliefs are
better protected in an incomplete market.
The findings are delivered in a two-agent expected utility framework under logarithmic
preferences. Relaxing this assumption to more general utility functions is a potential fruitful
path to extend this study. While this is a straightforward task for a complete market
setting under heterogenous beliefs, it is a challenge in the case of a dynamic equilibrium
with incomplete markets of this type.
26
Appendix
A.
Proofs
Proof of Proposition 1. The first-order condition of equation (10) as well as the interior
problem of equation (20) yield ci (t) =
(20), the objective function reduces to
1
.
yi ηi (t)
·Z
¸
T
min Ei
θi ,λQ,i
After substituting the solution into equation
− log ηi (t)dt .
0
Due to the properties of the logarithmic utility function, I can simplify to
·
µ
¶
¸
θi2 (t)
λQ,i (t)
λi (t) ,
min r(t) +
− (λi (t) − λQ,i (t)) − log
θi ,λQ,i
2
λi (t)
subject to the constraint in equation (19). Although this problem is quadratic in the respective choice variables, one solution can be ruled out due to negative values for risk neutral
rare event intensities, and the unique minimum is given in equations (21) and (22).
Proof of Proposition 2.
With agent-specific values of θi and λQ,i as in equation (11),
the dynamic process of the RA’s stochastic weight is given by
£
¤
dφ(t)
= (λ1 (t) − λQ,1 (t)) − (λ2 (t) − λQ,2 (t)) + θ22 (t) − θ2 (t)θ1 (t) dt
φ(t−)
·
¸
λQ,1 (t)λ2 (t)
+ [θ2 (t) − θ1 (t)] dB(t) +
− 1 dN (t, λi (t)).
λ1 (t)λQ,2 (t)
Positivity of the stochastic weight follows from the fact that physical as well as risk neutral
rare event intensities remain strictly positive, and hence the jump size can not exceed 100%. Optimality and consumption good clearing imply that the RA consumes the aggregate
dividend, and the RA’s marginal utility equates to agent 1’s state price density. Hence,
i
h
dη1 (t) = d 1+φ(t)
e(t)
=
¤
φ(t) £
(λ1 (t) − λQ,1 (t)) − (λ2 (t) − λQ,2 (t)) + θ22 (t) − θ2 (t)θ1 (t) dt
e(t)
φ(t)
1 + φ(t)
+
[θ2 (t) − θ1 (t)] dB(t) −
[b
µe dt + σe dB(t)]
e(t)
e(t)
φ(t)
1 + φ(t) 2
−
σe [θ2 (t) − θ1 (t)] dt +
σe dt
e(t)
e(t)
h
i


λQ,1 (t)λ2 (t)
φ(t−)
1
+
1 + φ(t−)  1+φ(t−) 1+φ(t−) λ1 (t)λQ,2 (t)
− 1 dN (t, λ1 (t)).
e(t−)
(κe + 1)
27
After summarizing terms, comparing the non-stochastic term of the above with equation
(11) leads to the identification of the interest rate, the comparison of the dB(t) terms leads
to the restriction in equation (26), and the comparison of the dN (t, λ1 ) terms leads to the
restriction in equation (27). Equation (26) can also be derived directly using market clearing
conditions and Lemma 1, since πS,1 (t) + πS,2 (t)φ(t) = 1 + φ(t).
Proof of Proposition 3.
Agreement on all price paths in the complete market requires
agreement on the risk neutral rare event intensity, λQ,1 = λQ,2 = λQ , and on the market
price of diffusive risk, θ1 = θ2 = θ at all times. Therefore, the state price density process in
equation (11) reduces to
dηi (t)
= −r(t)dt + (λi (t) − λQ (t))dt − θ(t)dB(t) +
ηi (t−)
µ
¶
λQ (t)
− 1 dN (t, λi (t)).
λi (t)
Applying Ito’s Lemma to the stochastic weight in equation (24) yields
·
¸
dφ(t)
λ2 (t)
= [λ1 (t) − λ2 (t)] dt +
− 1 dN (t, λi (t)).
φ(t−)
λ1 (t)
As before, positivity of the stochastic weight follows from the fact that λ1 (t) and λ2 (t) are
strictly positive, hence the jump size can not exceed -100%. In this case, the RA’s marginal
utility has the dynamic process given by
i
h
1+φ(t)
d e(t)
£
¤
φ(t−)
i =
h
[λ1 (t) − λ2 (t)] dt − µ
be − σe2 dt − σe dB(t)
1+φ(t−)
1 + φ(t−)
e(t−)
h
i


φ(t−)
λ2 (t)
1
+
1+φ(t−)
1+φ(t−) λ1 (t)
+
− 1 dN (t, λ1 ).
(κe + 1)
The solution for the risk neutral rare event intensity follows from the comparison of the
dN (t, λ1 ) terms, the solution for the market price of diffusive risk requires the comparison
of the dB(t) terms, and the interest rate can be determined through the comparison of the
non-stochastic terms.
28
B.
Uncertainty about Rare Event Impact
This Appendix shows an extension of the original setup. I assume that agents agree on the
frequency of rare events, but instead face potential uncertainty about the impact of a rare
event, which can be of high or low magnitude. Furthermore, I allow for heterogenous beliefs
about the likelihood of a high versus low impact. Suppose the aggregate output process is
given by
de(t)
=µ
be dt + σe dB(t) + κe (t)dN (t, λ),
e(t−)
where the rare event impact size κe (t) can take on either high or low negative values, κH
e or
κLe with probability p and 1 − p. Each agent has a dogmatic belief about the distribution of
L
κe (t), (κH
e , κe ) ∼ (pi , 1 − pi ). A convenient way to solve this problem is to decompose the
randomness of the jump size into two independent Poisson processes with a constant jump
L
size, and define pi λ = λH
i and (1 − pi )λ = λi . From the perspective of agent i the aggregate
output process obeys
de(t)
L
=µ
be dt + σe dB(t) + κH
e dN (pi λ, t) + κe dN (t, (1 − pi )λ), (i = 1, 2).
e(t−)
I assume no common jumps among the two Poisson processes. As before, a riskless money
market trades in zero-net supply and pays interest rate r(t). The unit-supply stock has the
posited dynamics from agent i’s perspective given by
dS(t) + e(t)dt
H
L
L
=µ
bS (t)dt + σS (t)dB(t) + κH
S (t)dN (t, λi ) + κS (t)dN (t, λi ).
S(t−)
After market completion, agents have the opportunity to trade in two insurance products,
one for each rare event size, given by
dP H (t)
H
= µ
bP H (t)dt + κH
P (t)dN (t, λi ),
P H (t−)
dP L (t)
= µ
bP L (t)dt + κLP (t)dN (t, λLi ),
L
P (t−)
P H (0) > 0,
P L (0) > 0.
The agent-specific state price density in equation (11) also extends to two Poisson processes
and is now given by
dηi (t)
H
L
L
= −r(t)dt + (λH
i − λQ (t))dt + (λi − λQ (t))dt − θ(t)dB(t)
ηi (t−)
Ã
!
Ã
!
L
λH
(t)
λ
(t)
Q,i
Q,i
+
− 1 dN (t, λH
− 1 dN (t, λLi ),
i )+
L
λH
λ
i
i
29
L
H
L
where λH
Q and λQ are the risk neutral rare event frequencies corresponding to λi and λi ,
respectively. The fraction of agent i’s wealth invested in the stock and the two insurance
products is denoted by πS,i , πP H ,i , and πP L ,i , respectively. The coefficients of the stock price
H
L
L
process are given by µ
bS (t) = µ
be , σS (t) = σe , κH
S (t) = κe , κS (t) = κe .
1.
Equilibrium in the Incomplete Market
In this extension, I assume non-availability of insurance for rare events with a high impact, but availability of insurance for rare events with a low impact. For convenience, I
directly write down the corresponding equilibrium restrictions derived from market clearing
conditions in the following Proposition.
Proposition 4 The incomplete market equilibrium corresponds to the set of six values (θ1 (t),
H
L
L
θ2 (t), λH
Q,1 (t), λQ,2 (t), λQ,1 (t), λQ,2 (t)) which solves the set of six equations given by
φ(t−)
1
θ1 (t) =
θ2 (t),
1 + φ(t−)
1 + φ(t−)
!
!
Ã
Ã
θ1 (t)
1
θ2 (t)
1
λH
λH
1
2
= H
−1 ,
= H
−1 ,
σe
κe
σe
κe
λH
λH
Q,1 (t)
Q,2 (t)
!
!
Ã
Ã
1
φ(t−)
λL1
λL2
−1 +
− 1 = 1 + φ(t−),
κLe λLQ,1 (t)
κLe
λLQ,2 (t)
σe −
H
L L
H H
L L
L L
L L
σe θ1 (t) − κH
e λQ,1 (t) − κe λQ,1 (t) = σe θ2 (t) − κe λQ,2 (t) − κe λQ,2 (t), κe λQ,1 (t) = κe λQ,2 (t),
L
subject to the constraints λH
Q,i (t) > 0, λQ,i (t) > 0. The equilibrium interest rate is given by
H
L L
r(t) = µ
be − σe θi (t) + κH
e λQ,i (t) + κe λQ,i (t).
As in the original setup, agents indirectly insure each other through the stock and money
markets. The portfolio allocations are such that the agent who assigns a higher probability
to κH
e reduces the exposure in the stock market. The result of a smaller risk premium in the
incomplete market compared to a complete market is robust to the extension considered here.
Suppose p1 > p2 , then from the perspective of the potential buyer of insurance against κH
e ,
the decrease in the market price of regular economic risk, θ1 , overcompensates the increase
30
the market price of rare event risk, λH
Q,1 . Alternatively, from the perspective of the potential
seller, the decrease in λH
Q,2 overcompensates the increase in θ2 .
The finding of countercyclical variation of the risk premium is also valid in this extension.
As before, regular economic risk enters the stochastic weighting process. Assuming p1 > p2 ,
then the difference in market prices of regular economic risk, θ2 − θ1 , is strictly positive as
agent 2 holds a leveraged position in the stock market, leading to the same countercyclical
channel as in the original setup.
2.
Market Completion
Proposition 5 After market completion, the market price of regular economic risk, the risk
neutral rare event intensities, and the interest rate are given by
θ(t) = σe , λH
Q (t) =
H
λH
λL1 + φ(t−)λL2
1 + φ(t−)λ2
L
,
λ
(t)
=
,
Q
(1 + φ(t−))(κH
(1 + φ(t−))(κLe + 1)
e + 1)
H
L L
r(t) = µ
be − σe θ(t) + κH
e λQ (t) + κe λQ (t).
The market price of regular economic risk equals the solution in the original setup. The
risk neutral rare event intensities correspond to wealth-weighted averages of each agent’s
beliefs about each impact size. Both agents invest all their financial wealth in the stock
market after market completion. The allocations in the insurance market are as follows:
The agent who assigns a higher probability to the high impact size is the buyer of insurance
L
against κH
e , and the seller of insurance against κe . Conversely, the agent who assigns a lower
probability to the high impact size is the seller of insurance against κH
e , and the buyer of
insurance against κLe . Under homogeneity, p1 = p2 , we do not observe actual holdings in the
insurance market.
This solution can be mapped back into the problem based on one Poisson process and a
random impact size. In that case, the equilibrium characteristics do not consist of two risk
neutral rare event intensities, but one risk neutral intensity, λQ , corresponding to λ, and a
31
risk neutral probability for the distribution of impact sizes, q, as given in Lemma 3. The
market price of regular economic risk is not affected by this mapping.
Lemma 3 After market completion, the risk neutral rare event intensity and the risk neutral
probability for the distribution of impact sizes are given by
¡
¢
L
H
λ( κH
e + 1 (1 + φ(t−)) + (p1 + φ(t−)p2 )(κe − κe )
λQ (t) =
,
L
(κH
e + 1) (κe + 1) (1 + φ(t−))
q(t) =
(p1 + φ(t−)p2 )(κLe + 1)
.
L
((1 − p1 ) + φ(t−)(1 − p2 ))(κH
e + 1) + (p1 + φ(t−)p2 )(κe + 1)
The interest rate can be written equivalently as r(t) = µ
be − σe θ(t) + κQ (t)λQ (t), where
κQ (t) denotes the risk neutral mean of the distribution of impact sizes given by κQ (t) =
L
q(t)κH
e + (1 − q(t))κe . The two-point distribution (q(t), 1 − q(t)) constitutes a valid probability
measure.
C.
Figures
32
Agent 1 Stock Market
Agent 2 Stock Market
1
4.5
4
0
3.5
pi (S,2)
pi (S,1)
−1
−2
3
2.5
−3
2
−4
−5
0
1.5
0.1
0.2
0.3
0.4
0.5 0.6
x(S,1)
0.7
0.8
0.9
1
0
1
0.1
0.2
Agent 1 Money Market
0.3
0.4
0.5 0.6
x(S,2)
0.7
0.8
0.9
1
0.8
0.9
1
Agent 2 Money Market
6
0
−0.5
5
−1
pi (B,2)
pi (B,1)
4
3
−1.5
−2
2
−2.5
1
0
0
−3
0.1
0.2
0.3
0.4
0.5 0.6
x(S,1)
0.7
0.8
0.9
1
−3.5
0
0.1
0.2
0.3
0.4
0.5 0.6
x(S,2)
Figure 1: Equilibrium Allocation in Incomplete Market Economy.
0.7
The graphs display
the fraction of each agent’s wealth invested in the stock market and the money market. The x-axis
corresponds to each agent’s initial endowment, xS,i ∈ (0, 1). Parameter values of the economy are:
µ
be = .04, σe = .04, κe = −.20, λ2 = .01. The lines correspond to three levels of dispersion in beliefs,
dotted λ1 =.04, dashed λ1 =.07, solid λ1 =.10, respectively.
33
Agent 2 Stock Market
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
pi (S,2)
pi (S,1)
Agent 1 Stock Market
2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.1
0.2
0.3
0.4
0.5 0.6
x(S,1)
0.7
0.8
0.9
0
0
1
0.1
0.2
4
−5
3.5
−10
3
−15
2.5
−20
1.5
−30
1
−35
0.5
0.1
0.2
0.3
0.4
0.5 0.6
x(S,1)
0.7
0.8
0.9
0
0
1
0.1
0.2
35
−0.5
30
−1
25
−1.5
20
−2.5
10
−3
5
−3.5
0.2
0.3
0.4
0.5 0.6
x(S,1)
0.7
0.8
0.9
1
0.3
0.4
0.5 0.6
x(S,2)
0.7
0.8
0.9
1
0.8
0.9
1
−2
15
0.1
0.7
Agent 2 Money Market
0
pi (B,2)
pi (B,1)
Agent 1 Money Market
40
0
0
0.5 0.6
x(S,2)
2
−25
−40
0
0.4
Agent 2 Insurance Market
0
pi (P,2)
pi (P,1)
Agent 1 Insurance Market
0.3
0.8
0.9
1
−4
0
0.1
0.2
0.3
0.4
0.5 0.6
x(S,2)
0.7
Figure 2: Equilibrium Allocation after Market Completion. The graphs display the fraction of each agent’s wealth invested in the stock market, insurance market and money market after
market completion. The x-axis corresponds to each agent’s initial endowment, xS,i ∈ (0, 1). Parameter values of the economy are: µ
be = .04, σe = .04, κe = −.20, λ2 = .01. The lines correspond
to three levels of dispersion in beliefs, dotted λ1 =.04, dashed λ1 =.07, solid λ1 =.10, respectively.
34
Risk Premium (incomplete market)
3
Risk Premium in %
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
x(S,1)
0.7
0.8
0.9
1
0.8
0.9
1
Effect of Market Completion
1.35
Ratio of Risk Premium
1.3
1.25
1.2
1.15
1.1
1.05
1
0
0.1
0.2
0.3
0.4
0.5
0.6
x(S,1)
0.7
Figure 3: Equilibrium Risk Premia. The upper graph shows the equilibrium risk premium
in the incomplete market, µ
bS - r, and the lower graph the effect of market completion. The
effect of market completion is shown as the ratio of post/pre completion risk premium. The
x-axis corresponds to the initial endowment of agent 1. Parameter values of the economy are:
µ
be = .04, σe = .04, κe = −.20, λ2 = .01. The lines correspond to three levels of dispersion in beliefs,
dotted λ1 =.04, dashed λ1 =.07, solid λ1 =.10, respectively.
35
Rare Event Risk
0.14
0.12
lambda (Q,2)
0.1
0.08
0.06
0.04
0.02
0.06
0.065
0.07
0.075
0.08
lambda (Q,1)
0.085
0.09
Normal Economic Risk
0.15
theta (2)
0.1
0.05
0
−0.05
−0.1
−0.15
−0.1
−0.05
0
0.05
theta (1)
0.1
0.15
0.2
Figure 4: Market Prices of Rare Event Risk and Normal Economic Risk. The graphs
show the path of the equilibrium characteristics due to market completion, the incomplete market
economy corresponds to the square, the complete market economy to the circle. The upper graph
shows the market prices of normal economic risk, θ1 and θ2 , the lower graph the risk neutral rare
event intensities, λQ,1 and λQ,2 . Parameter values of the economy are: µ
be = .04, σe = .04, κe =
−.20, λ1 = .1, λ2 = .01. Both agents have equal financial wealth, i.e. φ = 1.
36
Rare Event Risk
7
Agent
Agent
Agent
Agent
lambda(i)/lambda(Q,i)−1
6
5
1
2
1
2
(incomplete)
(incomplete)
(complete)
(complete)
4
3
2
1
0
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
x(S,1)
0.7
0.8
0.9
1
Normal Economic Risk
0.2
0.15
0.1
theta(i)
0.05
0
−0.05
−0.1
−0.15
−0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
x(S,1)
0.7
0.8
0.9
1
Figure 5: Equilibrium Risk Exposure. The graphs show each agent’s equilibrium risk exposure
to rare event risk and normal economic risk, respectively, in the incomplete market and after market
completion. The x-axis corresponds to the initial endowment of agent 1. Parameter values of the
economy are: µ
be = .04, σe = .04, κe = −.20, λ1 = .10, λ2 = .01.
37
Effect on Anticipatory Utilities
1.1
Agent 1
Agent 2
1.08
Multiplier k
1.06
1.04
1.02
1
0.98
0
0.1
0.2
0.3
0.4
0.5
0.6
x(S,1)
0.7
0.8
0.9
1
Total Effect
1
Total Effect in %
0
−1
−2
−3
Ex−ante
Ex−post, lambda = .01
Ex−post, lambda = .10
−4
−5
0
0.1
0.2
0.3
0.4
0.5
0.6
x(S,1)
0.7
0.8
0.9
1
Figure 6: Efficiency Analysis of Market Completion. The top graph shows the consumption
multiplier ki based on each agent’s anticipatory utility. The bottom graph (solid line) shows
the total effect of market completion on ex-ante utility. The bottom graph (dotted lines) show
the total effect on ex-post utility incorporating that one of the agents has incorrect beliefs. The
x-axis corresponds to the initial endowment of agent 1. Parameter values of the economy are:
µ
be = .04, σe = .04, κe = −.20, λ1 = .10, λ2 = .01.
38
Change in Interest Rate − Incomplete Market
Change in Basis Points
0
−50
−100
−150
0
0.2
0.4
0.6
0.8
1.0
x(S,1)
Change in Interest Rate − Complete Market
Change in Basis Points
0
−50
−100
−150
0
0.2
0.4
0.6
0.8
1.0
x(S,1)
Figure 7: Flight to Quality - Price Effects. The graphs show the adjustment of the equilibrium
interest rate in basis points at the occurrence of a rare event, r(t) − r(t−), in the incomplete market
and after market completion, respectively. The x-axis corresponds to the initial endowment of
agent 1. Parameter values of the economy are: µ
be = .04, σe = .04, κe = −.20, λ1 = .1, λ2 = .01.
The solid line corresponds to dogmatic beliefs, the dotted line corresponds to an upward revision
of beliefs of .01 for both agents at the rare event.
39
Stock Market (Incomplete Market)
Stock Market (Complete Market)
1.5
Change in Share Ownership
Change in Share Ownership
1.5
1
0.5
0
−0.5
0
0.2
0.4
0.6
0.8
1
0.5
0
−0.5
0
1.0
0.2
0.4
x(S,1)
0.8
1.0
Insurance Market (Complete Market)
2
1.5
1.5
1
1
Change in Open Interest
Change in Open Interest
Money Market (Incomplete Market)
2
0.5
0
−0.5
−1
−1.5
−2
0
0.6
x(S,1)
0.5
0
−0.5
−1
−1.5
0.2
0.4
0.6
0.8
1.0
x(S,1)
−2
0
0.2
0.4
0.6
0.8
1.0
x(S,1)
Figure 8: Flight to Quality - Portfolio Effects. The graphs show the portfolio adjustments at
the occurrence of a rare event, in the incomplete market and after market completion, respectively.
The x-axis corresponds to the initial endowment of agent 1. Parameter values of the economy are:
µ
be = .04, σe = .04, κe = −.20, λ1 = .1, λ2 = .01. The solid line corresponds to dogmatic beliefs, the
dotted line corresponds to an upward revision of beliefs of .01 for both agents at the rare event.
40
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