RISK-AVERSE TRADERS WITH INSIDE INFORMATION
Paolo Vitale
University of Cambridge
This Version: September 1995
Abstract
This chapter generalises Kyle's (1985) model to the case in which the informed
trader is risk-averse. Solution methods are based on Whittle's (1990) analysis of
LEQG dynamic programming problems. Risk-aversion increases the informativeness of prices and the eÆciency of the market; it also reduces its liquidity in the
rst auctions of the model and increases it in the last ones. The volume of the
transactions aects the liquidity and eÆciency of the market and increases these
eects.
I
thank my thesis supervisor William Perraudin and participants at seminars at Cambridge University
and Birkbeck College, London, for very useful discussions and comments. The Paper was prepared as part
of the ESRC/Newton Trust project on Financial Econometrics (ESRC Grant no. R00023 4641). Additional nancial support from the San Paolo Bank, Italy, is gratefully acknowledged. Correspondence should
be addressed to Paolo Vitale, 308 King's College, CB2 1ST, Cambridge UK (Tel. 44-1223-501298, Email:
[email protected]).
1
1
Introduction
In this chapter, we analyse the consequences of risk-aversion on the characteristics of a dealer
market with informed trading, studying its eects on the way private information is diused
in the market. A sequential trading model of a centralised market is developed along the lines
proposed by Kyle (1985). This popular model represents a cornerstone of the market microstructure theory and has been generalised in many directions: Holden and Subrahmanyam
(1992) introduce imperfect competition among a number of insiders; Foster and Viswanathan
(1993a) extend the class of distributions from which fundamental uncertainty derives; Caballe
and Krishnan (1994) consider a multi-security market with several insiders. In this chapter,
we replace the standard assumption of expected prot maximisation, assuming the informed
agent's objective function is given by a constant absolute risk-aversion (CARA) utility function
of the total expected prots. Risk-aversion seems a constant aspect of security markets and
studying its eects on market micro structure is an important subject of research.
Indeed, this area of research has been left relatively unexplored because this apparently
innocuous modication of the basic model produces a dramatic increase in the complexity of
the analysis. We, actually, employ a fairly new and innovative method for solving dynamic
programming problems for the class of Linear Exponential Quadratic Gaussian problems. Because this method has not been applied in nance we carefully present the Denitions and
Lemmas relevant for its application. The solution we obtain retains the simple formulation of
the original model but the implications for the dynamics of the state variables, the liquidity
and eÆciency indicators are quite dierent.
In particular, we prove that the informed trader (insider) reveals her private information to
the market maker at a higher speed and that prices incorporate more information that in the
risk-neutral case. This implies that in a dynamic setting with sequential equilibria risk-aversion
brings about an important increase in the eÆciency of prices and social welfare. We also show
that the dierences between the neutral and averse cases are increasing with the dimension of
the market and that when the insider is risk-averse the volume of trading is larger and decreases
over time.
These conclusions contradict the general intuition that risk-aversion reduces the quantity of
trading and the eÆciency of the market (see Subrahmanyam (1991)). In eect, in a one-period
2
(\static") setting this intuition is conrmed as the risk-averse informed trader places a smaller
market order and retains more of her informational advantage. We investigate in details both
the one-period and the multi-period settings giving account of the dierences and proposing
their interpretations.
The chapter is organised as follows. In the next section we present the model in a one-period
setting and in a multi-period one and characterise the linear equilibrium. We also present the
assumptions, Denitions and Lemmas relevant for the application of Whittle's method, that is
used in the proof of the main Proposition of the chapter. In the following section we describe the
linear equilibrium of the multi-period formulation simulating the dynamics of the parameters
of the model. A conclusion briey summarises the results of our analysis, while an appendix
gives full details of the proof of the main Propositions of the chapter.
2
The Model
In the model, we follow Kyle in assuming that a risky asset is exchanged for a numeraire among
three kinds of agents: a single insider who retains some private information on the liquidation
(fundamental) value of the risky asset; a group of uninformed (liquidity) traders, that trade
purely for liquidity reasons; a dealer (market maker) that sets the price of the asset according
to an eÆciency condition: this will imply that the expected prots of the market maker are
zero, as one would expect in a competitive market, and that the prots of the insider are the
losses of the liquidity traders.
Trading takes place sequentially at dierent auctions until the liquidation value is announced: this hypothesis permits the formulation of a model based on the sequential equilibrium concept of Kreps and Wilson (1982). At any round of trading, we assume that, rst,
customers, the insider and the uninformed traders, choose their market orders and then that
these orders are batched and passed to the market maker, who will x the asset price equal to
the expected liquidation value of the asset given the history of the order ow. We assume, in
fact, that all the information the market maker possesses is contained in the observation of the
total sum of the market orders, i.e. that the market maker cannot observe the identity of the
customers placing the orders. Therefore, any uctuation in the asset price is just a consequence
of the order ow movements.
3
These hypotheses regarding the trade protocol are crucial aspects of the model: relaxing
the assumption of anonymity would allow the market maker to detect quickly the identity of
the informed trader, while assuming that customers x the market orders given the transaction
price would imply an opportunity of innite prots for the informed trader.
The uninformed traders are supposed to transact for liquidity reasons and their marker
orders are assumed to be independent of past and present values of prices and quantities. This
implies that the insider is not able to predict the liquidity trading and does not have perfect
foresight on future prices. Her objective is that of nding the optimal trading strategies that will
permit maximising her expected utility while progressively revealing her private information.
While in Kyle's formulation the informed trader is risk-neutral and therefore maximises the
total expected prots function, we assume her objective function is a CARA utility function
and study the eects of this hypothesis on the implications of the model. In this formulation, we
assume that the random variables are normally distributed in order to nd a linear solution of
the dynamic programming problem as this assumption permits, despite the objective function
is no more quadratic, keeping a linear structure of the equilibrium of the model.
Using the model, we examine the implications of risk-aversion for the characteristics of the
market. In particular, we study its eects on the market liquidity, eÆciency and resiliency.
The depth of the market (a measure of the market liquidity) is given by the inverse of the
sensitiveness of the price to the order ow, its eÆciency is measured by the public and private
information incorporated into the prices and its resiliency by the speed of convergence of prices
to the \true" fundamental values. In substance, we examine questions such as: Is the market
more or less liquid when agents are risk-averse? Are prices relatively more informative than
in the risk-neutral case? Is the volume of trading an important factor for the liquidity and
resiliency of the market? What is the pattern of the volume of transactions during a trading
day? It could be the case that introducing risk-aversion gives some insights in some of the
market aspects not addressed by Kyle.
In the linear equilibrium of Kyle's model the optimal strategy of the informed trader is to
feed the market gradually in order to keep constant the informational content of the order ow;
in this way the insider is able to hide herself optimally and protect her informational advantage.
In the presence of risk-aversion, this is not necessarily true as the variance of the returns is now
a relevant factor in the insider maximisation problem. In fact, risk-aversion brings about two
4
eects: it reduces (i) the propensity to trade and (ii) the intertemporal substitution between
current and future prots of the informed trader.1 This two eects have important consequences
for the dynamics of the characteristics of the market.
The characteristics of the market are a function of the composition of the order ow: if
the fraction of informed trading is smaller, then the price is less sensitive to the order ow.
While the propensity eect would reduce the informativeness of the order ow, the second
would increase it. In fact, while a smaller propensity to trade implies that the insider trades
less intensively, a smaller intertemporal substitution means that the insider prefers to take
sooner advantage of her private information. The two eects have opposite signs and it is not
quite clear which one should prevail. Anyway, we could argue that in a multi-period model the
second eect should dominate in the rst auctions when there is more uncertainty about the
price movements, while the second should prevail in the last rounds of trading.
Another interesting aspect of the risk-neutral model is that the optimal strategy of the
insider is to keep constant the relative fraction of her trades with respect to that of the liquidity
traders; therefore, the volume of transactions is not a relevant factor in the price movements.
In the risk-averse case this is no longer the case as the volume of trading aects the variability
of the returns and hence the decision of the informed trader. While it is diÆcult to predict the
eect of an increase in the variance of returns, it is clear that the larger the volume of trading
the larger the departure from the dynamics of the risk-neutral case.
2.1
A Single Auction Equilibrium
In this section, we begin by considering a simple one-period formulation of the model with
risk-aversion. This is the simple case studied by Subrahmanyam (1991), which we present for
exposition reasons. It will be, in fact, clear later that in a fully dynamic setting dierent results
hold. We assume the liquidation (fundamental) value of the asset, f , is normally distributed
with mean f and variance 0.2 This random variable is independent of the random quantity
transacted by the liquidity traders, x , that in turn is distributed as a normal of mean 0 and
l
1 In eect, in a dynamic set-up, the coeÆcient of risk-aversion in the CARA utility function is also an index
of the attitude towards intertemporal substitution in consumption, so that the two terms can be considered as
synonymous. We follow the nance literature and refer to risk-aversion.
2 The notation we use follows quite closely that proposed by Kyle.
5
variance 2. Initially, the informed trader chooses her market order, x , on the basis of the
knowledge of f . The market maker subsequently xes the price of the asset conditional on the
information he has obtained from the order ow.
i
l
We indicate with X the insider's net demand function, i.e. the optimal market order given
the information she possesses: as this is given by f , X is a function of the liquidation value,
x = X (f ). On the other hand, the pricing rule of the market maker, S , is a function of the
order ow he observes, s = S (x), where x = x +x . The prots received by the insider,
= (f s)x , are a function of these two strategies, = (X; S ), while the utility she will
maximise is a function of these prots and depends on her risk-aversion, v = V ((X; S ); ).
Given these formulations for the agents' strategies we modify the equilibrium concept proposed
by Kyle to take account of the dierent objective function of the insider.
i
i
l
i
Denition 1 A Nash equilibrium for the dealer market with a risk-averse insider is a pair of
strategies (X,S) such that the following two conditions hold.
(1) The insider maximises her conditional expected prots. That is, for any alternative trading
strategy X 0 , and any f :
0
E [V ((X; S ); ) j f ] E [V ((X ; S ); ) j f ]:
(1)
(2) The market maker sets the price according to a weak-form eÆciency condition:
s = E [f
j x]:
(2)
The equilibrium concept dened can be interpreted in a game theoretic framework if we assume
that competition among market makers leads to zero expected prots condition. In this case
our weak-form eÆciency condition withholds and the equilibrium that we dene corresponds
to a Nash equilibrium in simple strategies. In the analysis below we concentrate on the subset
of linear equilibria.
Denition 2 A linear Nash equilibrium is a pair of strategies (X,S) that satises Denition 1
and such that there exists for which the following holds:
s = f + (x):
6
(3)
It is not diÆcult to characterise the linear equilibrium of this formulation of the model
under the assumption that the insider's utility function is a CARA utility function of the
prots (v = expf g). In fact, the following Proposition proved in the appendix holds.
Proposition 1 If the informed trader's utility function is negative exponential with a positive
coeÆcient of risk-aversion , there exists a unique linear Nash equilibrium. The linear Nash
equilibrium is given by the following equations:
s
xi
= f + (x);
= (f f);
(4)
(5)
where the coeÆcients and satisfy the following equation system:
= 2(1 +11 2 ) ;
2
0
= 2 + 2 :
l
0
l
(6)
(7)
We can now compare the solution of the risk-averse case with that of the original model proposed
by Kyle, comparing the dierent values of the coeÆcients of the linear equilibria in the two
formulations. In the risk-neutral case, an explicit solution can be obtained immediately. In
fact, we have:
( =2 )1 2 ; 1 = 0 ;
= (2 =0 )1 2 ;
= 0
2
2
where 1 = var(f j x). In the risk-averse case, a simple analytic solution for these parameters
is not available. Despite analytical results can be obtained (Subrahmanyam (1991)), we can
simply solve numerically the system of equations (6) and (7), deriving these coeÆcients for
dierent values of the parameters and 2 , and obtain the same conclusions. Results of the
numerical calculations are presented in Table 1 for 0 = 1.
l
=
l
=
l
As it is clear from Table 1, in the presence of risk-aversion, we have a smaller , a smaller and a larger 1 . Now, the insider trades less aggressively and the order ow is less informative
than in the risk-neutral case. Therefore, the market maker's pricing rule is less sensitive to the
order ow, i.e. the market is deeper (, the inverse of the market depth, is smaller), and less
information is incorporated into the asset price. This is reected in the larger value of 1, the
conditional variance of the prediction error, for larger values of . In other words, the eÆciency
of the market, measured by the inverse of 1, is smaller in the risk-averse case.
7
Table 1: Comparison of Market Characteristics
l2 = 1
1
0
1.0000
0.5000
0.5000
0.05 0.1 0.2
0.33
0.5
1
0.9877 0.9759 0.9535 0.9266 0.8949 0.8191
0.4999 0.4998 0.4994 0.4985 0.4969 0.4902
0.5061 0.5121 0.5237 0.5380 0.5552 0.5984
1
0
1.4142
0.3536
0.5000
0.05 0.1 0.2
0.33
0.5
1
1.3899 1.3667 1.3239 1.2741 1.2173 1.0900
0.3535 0.3533 0.3528 0.3516 0.3496 0.3419
0.5087 0.5171 0.5330 0.5520 0.5744 0.6274
l2 = 2
In the risk-neutral case the volume of transactions is not relevant for the informativeness of
the order ow, as 1 is unaected by changes in 2 , the parameter that denes the \dimension"
of the market. In fact, the optimal strategy for the insider is to keep constant the relative size
of her own market order with respect to that of the liquidity traders when 2 changes. This is
no longer the case in the presence of risk-aversion, as the value of 1 depends on the degree of
risk-aversion and the volume of transactions. The relation between 1 and 0 is in fact:
2 2
1 = ++122 20 :
2
This equation conrms that the eÆciency of the market, i.e. the proportion of public and
private information incorporated into the price, is independent of the volume of transactions
only when = 0 and that it is smaller when the insider is risk-averse. Besides, Table 1 shows
that 1 increases with 2 and , indicating that the insider will trade less when the volume of
transactions in the market rises or her risk-aversion is larger. In synthesis, risk-aversion in the
\static" formulation of Kyle's model brings about a reduction of the eÆciency of the market.
Moreover, it is conrmed that increases in the volatility of the liquidity trading reduce price
eÆciency.
l
l
l
l
l
8
2.2
A Sequential Auction Equilibrium
We now consider the analysis of the model when formulated in a multi-period framework, since
we are interested in assessing if our earlier results continue to hold in a more general case
where a sequence of auctions takes place. Actually, we argue that only when we consider a
fully dynamic version of Kyle's model we can investigate the interesting question of the optimal
speed of information revelation. In fact, when more rounds of trading are allowed, the informed
trader has to decide how to diuse information into the market gradually. She will take account
of the eect of her market order on both the present and future values of the asset price, while
the asset price xed by the market maker, will reect all the history of the order ow.
We assume in this formulation that the interval of time between the moment the liquidation
value is revealed to the insider and that in which the same value is announced to the market
is divided in T rounds of trading, that with indicates with the subscripts f1; : : : ; T g. As in the
one-shot formulation, any round of trading, t, is divided in two stages: in the rst the customers
choose simultaneously their market orders, x and x respectively, while in the second the
market maker sets the asset price s . The market order placed by the liquidity traders follows
a simple random walk and therefore its innovation, , cannot be predicted by the observation
of the past values. We follow Kyle in assuming that the variance of the innovation depends
on the interval between two auctions. In the simplest case the auctions are equally spaced and
therefore / N (0; 2 ), where is the time distance between two trading rounds.
i
t
l
t
t
l
t
l
t
l
t
l
T
T
We now indicate with X and S the trading policy of the insider and the pricing rule of market maker for the auction t. At time t the insider will have observed both the liquidation value,
f , and the past prices, while the market maker the history of past total market orders. Therefore for any t, X and S will be functions of these two sets of values: x = X (s1 ; : : : ; s 1 ; f )
and s = S (x1; : : : ; x ). The trading strategy of the informed trader, and the general pricing
rules for the market maker are dened as follows:
t
t
t
t
t
i
t
t
t
t
t
X = fX1 ; : : : ; XT g;
S = fS1 ; : : : ; ST g:
Given these sets of strategies we can dene the future prots the informed trader can obtain in
the rounds of trading from time t onward:
t =
T
X
(f ss)xis:
s=t
9
where, as in the one-shot formulation, these prots depend on the two agents' strategies, =
(X; S ). We assume that the interest rate is null and that the informed trader's objective
function is an utility function, V , of these prots and with coeÆcient of risk-aversion , such
that the sequential equilibrium proposed by Kyle is now reformulated in the following way:
t
t
Denition 3 A sequential Nash equilibrium for the dealer market with a risk-averse insider is
a pair of strategies (X; S ) such that the following two conditions hold.
(1) The insider maximises her conditional expected prots. That is, for any round of trading t,
0
0
0
for any alternative trading strategy X , with X1 = X1 ; : : : ; Xt 1 = Xt 1 , and for any value of
s1 ; : : : ; st 1 and of f :
0
E [V (t (X; S ); ) j s1 ; : : : ; st 1 ; f ] E [V (t (X ; S ); ) j s1 ; : : : ; st 1 ; f ]:
(8)
(2) The market maker sets the price according to a weak-form eÆciency condition. That is, for
any round of trading t:
st = E [f j x1 ; : : : ; xt ]:
(9)
As in the one-shot case, we concentrate on the linear solution of the Nash equilibrium problem.
We have:
Denition 4 A linear Markov sequential Nash equilibrium is a pair of linear strategies (X; S )
that satises Denition 3 and such that there exist constants 1 ; : : : ; T for which the following
Markov system holds; for any t:
st = st 1 + t xt :
(10)
We can now state Proposition 2, the main result of the chapter, which characterises a linear
Markov equilibrium in the multi-period formulation of the model:
Proposition 2 If the informed trader's utility function is negative exponential3 with coeÆcient
of risk-aversion , there exists a unique linear Markov sequential Nash equilibrium. In this
equilibrium, there are constants t , t , t , and t such that for any t:
x =
i
t
t T (f
st 1 );
(11)
3 This choice is crucial, because in this way the optimal trading rule of the insider will not be function of her
wealth and the linear structure of the equilibrium solution will be preserved.
10
s = x ;
= var(f j x1; : : : ; x ):
t
t
t
t
Given the initial value of 0 the constants t , t , t and
following recursive system; for any t:
t
= 2 (1 1 2)+22 2
= 2 1+ 2 ;
1
=
;
t T
t t
t
t
T
t
t
2
t
subject to T
t
t
t
(15)
l
t t
t
= 0 and the second order condition:
1
(1 ) + 2 2
t
(14)
;
(16)
1
;
) + 2 2 ]
= 2[2 (1
t
T
l
t
l
t 1
t
are the unique solution of the
t
l
t
(12)
(13)
t
t
l
T
l
(17)
T
2t > 0:
(18)
The proof of this Proposition comprises three steps. First, we solve with a backward induction
argument the maximisation problem of the insider given the pricing rule of the market maker.
In the second step, we use the market eÆciency condition to show that the pricing rule of the
market maker satises equation (10) and derive the exact formulation of . In the last step, we
show that there is a unique solution of the recursive system characterising the linear Markov
equilibrium.
t
The most delicate part of the proof is the solution of the dynamic programming problem of
the insider. The informed trader's optimal trading strategy will maximise an utility function
that is no longer quadratic and, therefore, the usual simple solution method based on Riccati's
equations can no longer be used.4 Nor is it possible to use the Certainty Equivalence Principle
in the Bellman equation of the dynamic programming problem.
Generally to solve this sort of dynamic programming problems the adopted strategy is based
on the following backward induction argument: the actual form of the future value function
of the Bellman equation is \guessed" and the associated optimal policy function is obtained;
the consistency of the chosen formulation for the future value function is then checked. In this
4 It is
immediate to derive the solution of the risk-neutral version of the model just applying these recursive
equations.
11
respect, it is possible to rely on some general results that give the functional forms of the future
value functions associated with dierent classes of utility functions (see Hakansson (1970) for
the discrete time model cases and Merton (1971) for the continuous counterparts). However, in
the specic case of constant absolute risk-aversion, we can rely on an alternative method based
on some results suggested by Whittle (1990). As the method is quite general, simple and new
to the nance audience, we rst present it briey and then go through the proof of Proposition
2. Readers not interested in it and the proof of Proposition 2 can skip to section 1.3.
2.3
Risk-Sensitive Optimal Control
We now dene a specic class of optimal control problems, generally referred as Linear Exponential Quadratic Gaussian.
Denition 5 (LEQG) An optimal control problem is said to be Linear Exponential Quadratic
Gaussian if the expected value of the following criterion function
C
exp ;
(19)
2
where > 0, is maximised on T periods with respect to the control variables ut (t = 1; : : : ; T ),
under the following conditions:
(i) In any period t, ut can take any value in some nite-dimensional vector space.
(ii) C is a cost function that can be expressed in the following form:
C = Q(UT ; );
where UT = (u1 ; u2 ; : : : ; uT ) and is a vector-valued noise vector.
(iii) Q is quadratic in all the arguments and positive denite in UT .
(iv) is normally distributed with mean zero and covariance matrix V independent of the policy.
(v) The observable variables, yt , are reducible to linear functions of .
Under the conditions of Denition 5, let the total stress function be
S = C 1 D;
where D = 0 V 1 , and let the optimal policy function at time t be
u = U (W );
t
t
12
t
where W = fU 1 ; Y g, U = (u1 ; u2 ; : : : ; u ) and Y = (y1 ; y2; : : : ; y ). Then, one may obtain
the following Lemma, that denes a modied Risk-Sensitive Certainty Equivalence Principle
(RSCEP) for the class of LEQG problems.
t
t
t
t
t
t
t
Lemma 1 (RSCEP) The optimal value of the vector ut is determined by simultaneously minimising S with respect to ut ; ut+1 ; : : : ; uT and maximising it with respect to the unobservable values yt+1 ; yt+2 ; : : : ; yT +1 . In other words, we obtain an optimal current decision by minimising
with respect to all decision currently unmade (future values of the control variables) and maximising with respect to all the quantity currently unobservable (future values of the observable
yt ).
The proofs of this and the following Lemma are given in Whittle (1990).
This Lemma extends the Certainty Equivalence Principle of the Linear Quadratic Gaussian
problems: the normally distributed unobservable variables are no longer replaced by their
expected values, but by those that maximise the total stress in order to compensate for riskaversion. The Risk-Sensitive Certainty Equivalence Principle is particularly useful when we
consider a Markov LEQG problem.
Denition 6 A LEQG problem is Markovian if it satises the following conditions:
(i) the vector of state variables, zt , is governed by the following plant equation:
zt = At 1 zt 1 + Bt 1 ut 1 + t ;
(ii) the vector of observable variables is given by:
yt = Ct 1 zt 1 + t ;
0
with t = (t ; t ) / N (0; );
(iii) the cost and discrepancy components of the total stress are decomposable in the following
way:
C
=
D =
T
X
ct + CT +1 ;
t=1
T
+1
X
D1 + dt ;
t=2
where ct is a quadratic function in the control and the state variables (ut ; zt ), dt is a quadratic
form in t and D1 is a quadratic form in the vector z1 and its estimate z^1 .
13
Therefore, the total stress function of a Markov LEQG problem is now given by:
S=
T
T
+1
X
X
ct + CT +1 1 [D1 +
dt ]:
t=1
t=2
In order to optimise S with respect to the unobservable values and the control variables, we
can split this expression for the total stress into two components, which we can assign to past
and future at time t. We dene the extremised past stress and future stress at time t as follows.
Denition 7 At time t the extremised past stress and future stress, Pt (zt ; Wt ) and Ft (zt ), are
given by the following expressions:
t 1
t
hX
i
X
Pt (zt ; Wt ) = maxz1 ;:::;zt 1
ch 1 (D1 + dh ) ;
h=2
8 h=1
9
T
T
+1 i=
<
hX
X
Ft (zt ) = minut ;:::;uT :maxzt+1 ;:::;zT +1 ;yt+1 ;:::;yT +1
ch + CT +1 1
dh ; :
h=t
h=t+1
In accordance to this Denition, the extremised past stress and future stress are obtained by
minimising with respect to the future values of the control and maximising with respect to the
past and future values of the state variables. Only the current values of the state variables, z ,
appear in both the expressions and are left undetermined. By holding z free at any time t, it is
possible to separate the problem of the stress optimisation between past and future. In eect,
the following Lemma, which suggests a new method to nd the optimal control at any time t
and denes a Separation Principle (SP) for the LEQG problem, holds.
t
t
Lemma 2 [SP] The evaluation of the extremised past stress and future stress can be decoupled
if these evaluations are made conditional on the current values of the state variables, zt . These
partially extremised stress functions, Pt (zt ; Wt ) and Ft (zt ), relate to estimation and control
respectively. In fact, the evaluation of Pt summarises the eect of past observations, while the
evaluation of Ft implies the calculation of the optimal control ut (zt ), which would be optimal
if zt were known. The calculations of Pt (zt ; Wt ) and Ft (zt ) are then recoupled, maximising
Pt (zt ; Wt ) +Ft (zt ) with respect to zt ; this yields the minimal stress estimate z^t . The optimal
value of the control vector at time t is then given by ut (^zt ).
An important property of the extremised past and future stress is that they respect the
following recursions:
P (z ; W ) = max t 1 [c 1 1 d + P 1 (z 1 ; W 1 )];
t
t
t
z
t
t
14
t
t
t
Ft (zt )
n
= min t max t+1
u
z
o
1 dt+1 + Ft+1 (zt+1 )] ;
t+1 [c
;y
t
with boundary conditions:
P1 (z1 ; W1 )
FT +1 (zT +1 )
=
=
1 D1 ;
CT +1 :
This property is particularly useful in the evaluation of estremised past and future stress
and the implementation of Lemma 2. In particular, as a corollary of this Lemma, we note
that when all state variables are observable (y = z ), we only need to evaluate the extremised
future stress and therefore the calculation of the optimal control path is much easier. Besides,
considering the plant equation for the state vector, we can substitute +1 for z +1 so that the
extremised future stress recursion is now as follows:
n
o
F (z ) = min t max t+1 [c 1 d +1 + F +1 (z +1 )] :
t
t
t
t
t
u
t
t
t
t
t
In conclusion we obtain a recursion similar to the Bellman equation for the value function of
dynamic programming: given the extremised future stress at time t + 1, we obtain the optimal
control at time t by solving the extremised future stress recursion. We rst maximise with
respect to +1 and then minimise the value in curly brackets with respect to u . In the next
section, we show that the maximisation problem of the insider can be formulated in the above
way and thus prove Proposition 2.
t
2.4
t
Proof of Proposition 1.2
STEP 1.
We can show that in our model the insider faces an optimal control problem which
belongs to the Markov LEQG class with observable state variables. These are given by the
vector of the fundamental value, the asset holding of the insider and the asset price before an
auction is called, z = (f; x 1; s 1 )0 , while the control variable is given by her market order,
u = x . Assuming that equation (10) holds, the plant equation governing the state variables
is given by:
f = f;
x 1 = x 2 + x 1 ;
s 1 = s 2 + 1 x 1 + 1 1 :
(20)
t
t
i
t
t
i
t
i
t
i
t
t
t
i
t
t
15
i
t
t
l
t
Now, the insider maximises the expectation of the following objective function exp[ 1],
which corresponds to the form (19) with
C=
where c = 2(s 1 + x
t
t
t
T
X
ct ;
t=1
f )xit + 2t xit lt , s0 = f and CT +1 = 0.
i
t
The cost function is quadratic in z and in x . It is not necessarily positive denite in
x (see Denition 5), but it will possess a minimum provided that a second order condition
is met and Lemma 1 will still apply. Moreover, the problem is Markovian: Lemma 2 can then
be applied and, as the informed trader has perfect knowledge of the current state variables, we
can concentrate on the solution of the extremised future stress recursion. Given that +1 = and that 1d +1 = l21T 2 , the recursion is given by:
i
t
t
i
t
t
t
Ft (zt ) = minxit
t
l
t
(
max t [2(s 1 + x
t
f )x
i
t
t
i
t
1
2
+ 2 x i l
t t
t
T
2
l
t
+ F +1 (z +1 )]
t
t
)
:
(21)
As in this class of problems the extremised future stress function is quadratic in the variables
of z we assume that F +1 (z +1 ) = 2 (s f )2 with 0; then, maximising the right hand
side of (21) we obtain:
t
t
t
t
lt = t ft (1
where:
t
t
2 )x 2 (s
i
t
t t
t =
l
t
t
t
1 f )g;
l2 T
1 + 2t 2t l2 T :
Note that we have maximum as the second order condition holds: in fact, the second derivative
of the argument in brackets in equation (21) is (2 2 +1=(2 )). Plugging our expression
for in the right hand side of (21) we can show that minimising the expression in curly brackets
in (21) with respect to x ,5 under the second order condition (18), gives the optimal market
order of the insider at time t, x = (f s 1 ), where:6
1 2 =
2 (1 ) + 22 :
t
t
l
T
l
t
i
t
i
t
t
T
t
T
t
t
t
t t
t
t
l
T
5 For the proof of all these steps see the appendix.
6 The fact that t T does not depend on the insider's wealth depends on the particular choice of the utility
function: the CARA utility function we are using permits maintaining a linear equilibrium of the model even in
the presence of risk-aversion.
16
Condition (18) guarantees that the expected prots of the insider are bounded and that we
actually have a maximum for the expected utility function. Intuitively, condition (18) rules out
situations in which the informed trader destabilises the price in the rst auctions with large
unprotable market orders and then gains huge benets in the following periods. In fact, when
is large and is small it is suÆcient a small market order and it is not very \expensive" in
terms of lost utility to destabilise the price in the t-th auction. In this case the destabilisation
can take place. Instead, if is large with respect to it is not convenient for the insider
to move the asset price away from its liquidation value, because the cost-opportunity of doing
so, measured by , is too large. The second order condition simply places an upper limit to
the admissible value of , that is a decreasing function of , ruling out these destabilising
schemes.
For = 0, we end up with the same result as in Kyle,7 while for > 0 we have a smaller ,
given and . The presence of risk-aversion gives the expected result: the informed traders
are less keen on trading given the same relation between s 1 and the liquidation value, f , i.e.
given the market's depth. Plugging the optimal values of x in the right hand side of (21) we
can prove that F (z ) = 2 1 (s 1 f )2, where:
1
1=
2[2 (1 ) + 2 2 ] ;
with boundary condition = 0 as C +1 = 0. Again, we end up with the same result as in
Kyle for = 0. Moreover, for 1 = 0 and = 1, we obtain the same result as the single
auction case.
t
t
t
t
t
t
t
t
t
t
t
t
i
t
t
t
t
t
t
T
t
t
t
t
l
T
T
T
STEP 2.
The proof that the pricing strategy of the market maker satises Denition 4 is
based on two arguments. First, the competition among market makers forces the dealer to
set the price equal to the expected value of the asset given the information contained in the
order ow, under the hypothesis that the other market makers can observe it. Second, a simple
application of the Kalman lter (see Harvey (1989)), under the assumption that equation (11)
holds, shows that s = x , where is given by equation (16),8 and that the variance of
the error is given by equation (15): therefore the solutions of the two problems of the insider
and the dealer are mutually consistent.
7 In fact, for ! 0 the extremised future stress recursion tends to the Bellman equation of the risk-neutral
t
t
t
t
case.
8 It is the choice of a normal distribution for the liquidation value f and the innovation in the asset holding of
the liquidity traders that permits having a value of t independent of the state vector zt and keeping the linear
Markov form of the equilibrium.
17
STEP 3.
We now show that the recursive system governing the parameters of the linear Markov
equilibrium has a unique solution which satises the boundary conditions and the second order
condition (18). From equations (16) and (14) we can obtain the following cubic equation in :
2 4
2
(22)
2(1 )(1 2 ) = 1 + 3 :
Given positive values of and , this cubic equation possesses two negative roots and one
positive if 2 2 is negative; it possesses two positive roots and one negative if instead
2 2 is positive. Negative values of are not admissible as the second order condition
(18) is not satised. Moreover, in the case in which we have two positive roots the larger
does not satisfy (18), so that given and there is only one admissible . It then follows
that given there is only one set of parameters which solves the recursive system backwards.
Finally, since 0 is uniquely determined from 1 in equation (15), we have a one-to-one relation
between and 0, and the proof is completed.
t
t
t
t
t
l
t
T
t
T
l
t
l
t
t
t
T
l
T
t
t
t
t
T
T
3
Results
Proposition 2 shows that risk-aversion makes the insider care about the variance of her prots.
The uncertainty the insider faces results from the randomness of liquidity traders orders. Given
the information set of the insider at time t (not including the current asset price), the expected
value of the asset, s , is E [s j I ] = s 1 + x and consequently the conditional variance
of this value is var[s j I ] = 2 2 . This conditional variance and the insider's risk-aversion
now enter in the specication of the coeÆcient , which denes the optimal trading rule of
the insider, and hence aect the equilibrium of the model.
t
t
t
t
t
t
l
t
i
t
t
T
t
T
Ceteris paribus, risk-aversion reduces the informativeness of the order ow and the liquidity
of the market, since, for given values of and , a risk-neutral insider will submit a larger
market order than a risk-averse trader. Anyway, this does not imply that a risk-averse insider will trade less, because her trading will depend on the dynamics of the entire set of the
coeÆcients of the linear Markov equilibrium.
t
t
To obtain general conclusions, we have to investigate the dynamics of the coeÆcients of the
linear Markov solution for dierent choices of the parameters of the model 0, 2 and . In
particular, and may be regarded as indicators of market depth (liquidity), eÆciency and
l
t
t
18
resiliency: as already indicated, the depth of the market is the minimum size of the market
order required to change a price in a given amount, its eÆciency measures the private and
public information incorporated into the price and its resiliency is the speed of convergence of
the price of the asset to its fundamental value. In the dealer market the depth and eÆciency
corresponds to the inverse of and respectively, while the resiliency is measured by the
speed of convergence of to zero.
t
t
t
Combining equations (14), (15) and (16) we obtain
= (1
t
t t T )t 1 :
This implies that the variance of the asset price is monotonically decreasing in t. As one would
expect, information is gradually incorporated into the price of the asset as it is disclosed through
time by the order ow. Therefore, the market turns out to be resilient. Even if trading by the
informed trader is small relatively to that of the liquidity traders, it is still suÆcient to drive
the asset price to its fundamental value.
Note, moreover, that the value of , which indicates how aggressively the insider will trade,
will determine, together with , how much of her private information will be disclosed in the
t-th round of trading. indicates that the depth of the market is an inverse function of the
private information still retained by the insider and of her trading intensity. In other words,
more informed trading (a larger ) or more fundamentl uncertainty (a larger ) leads the
market maker to reduce the market liquidity (a larger ). We now solve numerically the set
of dierence equations characterising the equilibrium values of the dealer market.
t
t
t
T
t
t
By continuity with the \static" formulation, we expect to have conclusions similar to those
of section 1.2.1 when few periods of trading are possible. Conversely, we expect to have new
results when a longer horizon is considered. In order to investigate this possibility the model
is simulated for the following parameter values:9 0 = 1; 2 = 1 and 2; = 0, 0:1 and 1;
T = 200. The resulting dynamics of the intensity of trading, , and the eÆciency and
l
t
9 The
T
algorithm we have implemented to nd the solution of the non-linear dierence equations for the parameters of the equilibrium is simple. Given t and t , there is a unique positive value of t satisfying the
second order condition (18). This value is given by the appropriate root of equation (22). It is then immediate
to obtain t T and, through backward iteration, t 1 and t 1 . Since we have a nal value for T , we can
dene a numerical function of T , G, that gives the initial variance of the liquidation value: 0 = G(T ). Given
that G(T ) is increasing in T it is easy to nd the root of the numerical equation 0 = G(T ) that gives the
unique value of T consistent with the boundary value 0 .
0
19
liquidity coeÆcients of the market, and for 2 = 1 and dierent values of are given in
Figures 1, 2 and 3. For = 0, i.e. in the standard Kyle's model, information is disclosed at a
constant speed and the depth of the market is substantially constant, since the derivative of with respect time is constant and so is . This is because the insider nds optimal to trade
with constant intensity, since this permits to hiding better her market orders.
t
t
l
t
t
In the risk-averse case, instead, the informed trader places larger market orders in the rst
auctions, as is larger than for = 0. This is because the intertemporal substitution
between present and future prots is reduced by risk-aversion and, therefore, the insider prefers
exploiting her information advantage earlier. As a consequence, she trades more aggressively,
the order ow is more informative and the market maker learns at a higher speed the liquidation
value of the asset. This implies that the conditional variance of the prediction error, , declines
more rapidly and that the price is more sensitive to the order ow, since the market maker will
adjust it more in response to a given total market order. In other words, the market is more
eÆcient ( is smaller) and less liquid ( is larger).
t
T
t
t
t
As the market maker progressively learns the fundamental value of the asset, the variability
in the price and the uncertainty about the future prots falls over time and consequently the
propensity eect brought about by risk-aversion starts dominating that of the intertemporal
substitution between present and future prots. Despite the fact that is increasing in t
and larger in the presence of risk-aversion, as time elapses the informational content of the order
ow decreases ( declines through time) and hence the decline in the value of is smaller. In
the end, in the risk-averse case the information gain from the order ow becomes smaller than
that of the risk-neutral one and the depth of the market is larger ( is now smaller for > 0).
Anyway, despite the reduction in the information gain, the informativeness of prices is always
larger in the risk-averse case as is always smaller for larger than 0.
t
t
T
t
t
t
This decline in in Figure 3 is consistent with the results of Table 1, where we found that
in the single auction equilibrium the dealer market is deeper with a risk-averse insider. In the
one-shot case, the intertemporal substitution between present and future prots eect is absent
and risk-aversion just reduces the propensity to trade. By continuity with the \static" case,
when T is small risk-aversion should still reduce the resiliency and eÆciency of the market, so
that the conclusions proposed so far cannot be true for all the possible values of the parameters
of the model. In eect, in the left panel of Figure 4, in which T = 2, we can see that the
conditional variance of the liquidation value is smaller for = 0. Anyway, as the right panel
t
20
clearly indicates, already for T = 5 the situation is reversed. We can thus claim that the
analysis carried out when 200 rounds of trading are possible indicates the general eects of
risk-aversion when a fully dynamic formulation of the model is studied.
From the market eÆciency condition of equation (9), it is clear that the asset price follows
a random walk, and in the limit a Brownian motion. But, while in the risk-neutral case the
conditional variance of the price is substantially constant, as is constant most of the time,
in the risk-averse case, it is decreasing over time. For the same reason we have a decreasing
pattern of the size of the total order ow, x . This positive correlation between price or returns
volatility and trading volume is consistent with other models of the market micro-structure
theory, such as those of Admati and Peiderer (1988) and Holden and Subrahmanyam (1992).
Assuming that private information is accumulated every trading day before the market opens,
the model predicts patterns in returns volatility and transactions size that have been reported
in several empirical papers. Among these various papers, Foster and Viswanathan (1993b) and
Gerety and Mulherin (1994), studying the NYSE market, nd evidence that trading volume
is higher at the beginning of the day and cannot reject the hypothesis that there is intraday
variation in the returns volatility.
t
t
From Figures 2 and 3, it is clear that the dierence in the dynamics of the coeÆcients and between the risk-neutral and the risk-averse case are larger as increases. The larger
the measure of risk-aversion, the stronger the propensity and the intertemporal substitution
eects. This is true even when we increase 2 , the variance of the liquidity trading, that is a
measure of the uncertainty of the insider. In the case of risk-neutrality, instead, a variation
in the trading volume does not aect the informativeness of prices, but increases the market
depth proportionally: for a doubling in is halved; a larger trading volume by the liquidity
traders allows the insider to protect better her informational advantage and as a consequence
the market maker will defend himself reducing the market liquidity. In the risk-averse case,
the protection given by larger liquidity market orders is not completely exploited because it is
associated to a larger uncertainty about future prots. From Figure 5, it is clear that when we
increase the uncertainty about the future prices the intertemporal substitution between current
and future prots is reduced and the eÆciency and resiliency of the market are increased. The
propensity to trade is reduced as well and therefore the moment from which the information
gain is smaller in the risk-averse case is anticipated (compare Figure 6 with Figure 3). Thus,
an increase in 2 augments the eect of risk-aversion on the market characteristics and the
t
t
l
l
t
l
21
patterns of trading. This result is conrmed by Foster and Viswanathan (1993b), that nd
stronger evidence of the reported patterns of the volume and of the volatility for the most
actively traded stocks.
Conclusion
We have developed a model of an auction market in which a risk-averse informed trader maximises a CARA utility function over a number of trading rounds exploiting her informational
advantage. At the same time, the market maker sets the price according to a weak-form efciency condition. Hence, the insider exercises her informational monopoly while her private
information is progressively incorporated into the asset price. Thus, this setting allows for informational asymmetries while still accomplishing market eÆciency. The main conclusions of
our analysis are the following.
1. An important aspect of Kyle's framework is that the liquidity and eÆciency characteristics
of the market are endogenously determined in the model, by the combined actions of the
insider and the dealer. In our extension of Kyle's model, we allow the eÆciency and
liquidity of the market to be aected by the level of risk-aversion. The eect is similar
to that of imperfect competition among a number of insiders studied by Holden and
Subrahmanyam (1992): the dealer market is substantially more eÆcient because private
information is diused in the market at a higher speed. While in their setting this result
comes from imperfect competition, and perhaps is more obvious, in our extension it is a
consequence of risk-aversion.
2. The present generalisation of Kyle's model may explain important regularities observed in
security markets, concerning the conditional variance of prices and the volume of trading,
and imply that risk-aversion has signicant eects on the distribution of income among
market participants. In particular, risk-aversion introduces a new role for uncertainty: in
fact, the volume of liquidity trading, that is a measure of the uncertainty of the insider,
aects the market eÆciency as the larger the volume of liquidity trading, the higher the
resiliency of the market. This can be interpreted as an indication that agents facing
liquidity constrains had better trade in larger markets, because they will face smaller
losses.
22
3. The analysis of the risk-aversion within a micro-structural perspective indicates that it
does not necessarily reduce the eÆciency of security markets as one would expect. In fact,
when risk-aversion is combined with strategic behaviour, it increases the informativeness
of price. Moreover, if risk-averse agents are concerned with the uncertainty of their future
returns, any institutional intervention that increases such uncertainty would have positive
eects on the eÆciency of the market.
23
Appendix
Proof of Proposition 1.
Let us assume that the pricing function of the market maker is given by equation (4), then the informed
trader's optimal policy will maximise the following function:
E f exp[ (f f + x)xi ] j f g:
It is easy to verify that in this case the optimal market order for the informed trader is given by equation
(5), where the coeÆcient of trading intensity is given by equation(6). Hence, applying the projection
theorem, we have that:
E [f j x] = f + x;
where is given by equation (7). As this expectation corresponds to the asset price, the two strategies
for the market maker and the insider are mutually consistent. The equation system (6) and (7) possesses
two dierent pairs of real solutions. In fact, substituting for in equation (7) we obtain the following
quartic equation in :
(23)
4l22 (1 + 21 l2)2 = (1 + l2 )0 :
The left hand side of (23) is a quartic function with positive rst and second derivatives, while the right
hand side is straight line with positive intercept: thus the equation admits two roots, one positive and
one negative. Anyway, the negative one is not admissible. As the left hand side of the equation takes
only non-negative values, the negative root will respect the condition 1 + l2 > 0 and will violate the
second order condition of the maximisation problem of the informed trader, that is (1 + 21 l2 ) > 0.
Instead, for > 0 the second order condition is satised. This completes the proof.
Proof of Proposition 2: detailed calculations.
We now present the detailed calculations of the optimal trading strategy of the insider outlined in Step
1 of the proof of Proposition 2. In particular, we need to solve the extremised future stress recursion
given in equation (21) to nd the solution of LEQG problem of the insider.
1
i
i
i l
l2
Ft (zt ) = minxit maxlt [2(st 1 + t xt f )xt + 2t xt t
+ Ft+1 (zt+1 )] :
(24)
l2 T t
Let us suppose that Ft+1 (zt+1) = 2t(st f )2, we can write:
n
o
Ft (zt )
l
i
i
=
min
max
H
(
;
x
;
z
)
;
(25)
l
t
t
xt
t
t
t
2
where
1 l2
2
Ht (lt ; xit ; zt ) = [(st 1 + t xit f )xit + t xit lt
2l2T t t(st f ) ]:
Therefore lt solves:
maxlt Ht (lt; xit ; zt) = maxlt [t ltxit 212T lt2 t(st 1 + t xit + t lt f )2]:
l
24
Taking the rst derivative of Ht (lt; xit ; zt) with respect to lt we have:
@Ht (lt ; xit ; zt)
1 l 2tt(st 1 + t xi + t l
=
t xit
t
t
2
l
l T t
@t
Therefore we have:
lt = t ft (1 2t t )xit 2t t (st 1 f )g;
where
l2 T
t =
1 + 2t2t l2 T :
The second order derivative is:
@ 2 Ht
= (2t2t + 1=l2T ):
@ 2 l
f ):
t
And therefore for t > 0 we are sure we have a maximum. Plugging lt in (25) we end up with:
Ft (zt )
i
2 = minxit Mt(xt ; zt);
where
Mt(xit ; zt)
(26)
h
= (st 1 + t xit f )xit + t 2t (1 2tt )xit2 2tt 2t (st 1 f )xit +
1 2ft(1 2tt )xi 2tt(st 1 f )g2 +
t
2l2T t
i
t f(1 2tt 2t )(st 1 f ) + t [1 + t t (1 2t t )]xit g2 :
Taking the rst derivative of Mt with respect to xit we have:
@Mt
@ xit
= (st
1 f ) + 2t xit
2tt2t (st
1 f ) + 2t 2t (1
2tt)xit +
1 2t(1 2tt )[t (1 2tt )xi 2tt (st 1 f )] +
t
2
l T t
2tt[1 + tt (1 2tt )]f(1 2tt2t )(st 1 f ) + t [1 + tt (1
2tt )]xit g =
f1 2t t 2t + 22 t t2 2t (1 2t t ) 2t t [1 + t t (1 2t t )](1 2t 2t t )g(st 1 f )
T
l
+ f2t 21T t22t (1 2tt )2 + 2t2t (1 2tt ) 2t2t [1 + t t(1 2tt )]2 gxit:
l
This corresponds to:
2
@Mt
=
(1
2
t t )f1 2tt 2t + 2 t t2 2t 2t t 2t (1 2t 2t t )g(st 1 f ) +
i
l T
@ xt
+ f2t(1 tt ) + 2t2t (1 2tt )2[1 t ( 212 + t2t )]gxit = 0: (27)
l
25
T
In fact:
1 2tt2t + 22T tt22t (1 2tt ) 2tt [1 + tt (1 2tt )](1 2t2t t) =
l
2
1 2ttt + 22T tt22t (1 2tt ) 2tt (1 2t2t t) +
l
2tt2t (1 2tt)(1 2t2t t ) =
(1 2tt ) 2tt2t (1 2tt ) + 22T tt22t (1 2tt) +
l
2tt2t (1 2tt )(1 2t2t t);
while:
2t
1 t2 2t (1
T
1 t2 2t (1
T
2t
2tt )2 + 2t2t (1 2tt ) 2t2t [1 + 2tt (1 2tt) + t22t (1 2tt)2 ]
l2
= 2t
2t
2tt )2 + 2t2t (1 2tt ) 2t2t [1 + tt (1 2tt )]2 =
l2
1 t2 2t (1
T
1 t2 2t (1
T
l2
2tt)2 2tt24t (1 2tt)2 2t2t + 2t2t (1 2tt)2 =
t t ) + 2t 2t (1
2t(1
Now:
2
In fact, given t , we have:
2
l2 T
while:
since
t t2 2t
2tt )2 + 2t2t (1 2tt ) 2t2t +
4tt3t (1 2tt) 2tt24t (1 2tt)2 =
2tt)2 2tt24t (1 2tt)2 2t2t + 2(t2t 2tt 3t )(1 2tt) =
l2
2t
1 t2 2t (1
T
l2
l2 T
t t2 2t
2tt )2[1
t ( 212 T
l
+ t2t )]:
2tt2t (1 2t2t t) = 0:
(1 2t2t t) = 1 + 2t122 T
t
l
2tt2t
2
t 2t 2 l4 T
2t2t l2 T = 0;
=
1 + 2t2t l2 T
l2 (1 + 2t2t l2 T )2 (1 + 2t 2t l2 T )2
1
t (
1
2 1
2l2 T + t t ) = 2 ;
t (1 + 2t 2t l2 T )
2l2T
26
= 12 :
Therefore, equation (27) reduces to:
@Mt
= (1 2tt )(1 2tt2t )(st 1 f ) + f2t(1 tt ) + t2t (1 2tt)2 gxit
@ xit
2t(1 tt ) + 2t l2T xi = 0:
t t
(
s
f
)
+
(28)
= 1 + 12t22
t
1
t
2
1 + 2t2t l2T
t
l T
In fact:
2 2
2
2
2 2
2t(1 t t) + t2t (1 2tt )2 = 2t(1 t t)(1 + 2tt1+l 2T )2+2 l T t (1 + 4t t 4tt)
t t
T
l
2
t (1 t t ) + 2t l2 T
=
1 + 2t2t l2 T :
Therefore, considering that t > 0 we can simplify (28) to:
@Mt
= (1 2tt)(st 1 f ) + [2t(1 t t) + l2 T 2t ]xit = 0:
@ xit
Therefore
xit = tT (f st 1);
(29)
where
1 2tt
t T =
2t(1 tt ) + l2T 2t :
The second order derivative of (26) is then:
@ 2 Mt 2t (1 t t ) + l2 T 2t
= 1 + 2t2 2T :
@ 2 xit
t
l
Hence, for t > 0 we have a minimum if
2t(1 tt ) + l2 T 2t > 0:
Consider now Ft(zt); this is obtained plugging (29) in (26). We have:
Ft (zt )
2 2
2
2 2 2 2
2
2 = t T t (f st 1 ) tT (f st 1) + t T tt (1 2t2t)(f st 1)2 +
+ 2ttT t t(f st 1) +
1 2ftT t (1 2tt ) + 2tt g2(f st 1)2 +
2l2T t
t ft [1 + t t (1 2tt )]t T (1 2t t 2t )g2 (f st 1 )2 :
(30)
This implies that:
Ft (zt ) = 2t 1(f st 1 )2 :
(31)
Therefore, the extremised future stress specication is consistent. We now nd the recursive equation
for t 1 and the conditions to have a positive value for this coeÆcient. We have:
t2 2T [t + t 2t (1 2t t )] t T (1 2t t 2t ) =
= (t t22T
t T )(1
2tt 2t ) + t2 2T t2t =
27
2tt)(1 + t l2T ) + 2t (1 2tt )2l2 T =
(1 + 2t2t l2 T )[2t (1 tt ) + l2T 2t ]2
2tt) 2t3t (1 2tt)l2 T :
= (1 +2t(1 2 2
t t ) + l2 T 2t ]2
t t
l T )[2t (1
=
In fact:
While:
t (1
(32)
1 2tt2t = 1 + 2 12 2
t t
T
l
2t (1 2t t )2 l2 T
t2 2T t 2t =
(1 + 2t2t l2 T )[2t (1 tt ) + l2T 2t ]2
(1 2tt )[t (1 2tt ) 2t + 2t2t 2t l2T ]
t T (t T t 1) =
[2t(1 t t) + l2 T 2t ]2
2
= [2t(1(1 2tt))(1+ +2tl2 ]T2 ) :
t
t t
T t
l
t T t (1
In fact:
2 2
2tt ) + 2tt = 2(1t (1 +2t)+t l 2T )2 :
t
t t
l
T
t
(33)
2
2
2
2tt) + 2tt = t f(1 2t2t ) (1+ 2t[2t)(1+ 2tt)+2 l T t )]g ;
t
t t
T t
l
that is equal to the right hand side of (33). Finally, we have that:
t T t (1
t [1 + t t (1
In fact:
t [1 + t t (1
That is equal to:
2tt)]t T (1 2tt2t ) = 2t(1
t
:
t t ) + l2 T 2t
(34)
2tt )]t T (1 2tt2t ) = (tT t 1)(1 2tt2t ) + tT t 2t :
1 + t2t l2 2T =
(1 + 2t2t l2T )
t (1 + t l2 T )(1 2t t ) 2t + 2t 2t l2 T 2t
(1 + 2t2t l2T )[2t(1 tt ) + l2 T 2t ] :
t T t
This corresponds to the right hand side of (34). Plugging (32), (33) and (34) in (30) we have that t 1
is equal to:
2t(1 2tt ) 4t3t (1 2tt )l2T 2t (1 + 2t2t l2T )l2 T 2t2t (1 + 2t2t l2 T )
2(1 + 2t2t l2 T )[2t (1 tt ) + l2T 2t ]2
2tt ) 2t l2T [1 + 2t2t l2 T + 4tt(1 tt )] :
= 2t(1 2(1
+ 2t2t l2T )[2t(1 t t) + l2 T 2t ]2
28
This is equal to:
Thus, if t is larger than zero, t
1
2[2t(1 tt ) + l2T 2t ] :
1 is larger than zero if t (1 t t ) + 21 l2 T 2t > 0.
29
(35)
Figure 1: Dynamics of for = 0, 0:1 and 1; 2 = 1 and T = 200
t
T
l
30
Figure 2: Dynamics of for = 0, 0:1 and 1; 2 = 1 and T = 200
t
l
31
Figure 3: Dynamics of for = 0, 0:1 and 1; 2 = 1 and T = 200
t
l
32
Figure 4: Dynamics of for = 0 and 1 when T = 2 and T = 5; 2 = 1
t
l
33
Figure 5: Dynamics of for 2 = 1 and 2; = 1 and T = 200
t
l
34
Figure 6: Dynamics of for = 0, 0:1, and 1; 2 = 2 and T = 200
t
l
35