Let us model more closely price expectation dynamics with two investor groups:
chartists and fundamentalists, where chartists use previous price data while funda- mentalists rely on fundamental values.
The steady state of the dynamical system is characterized by the discounted dividends rule, and the stability of it will depend on the relative proportions of investors with different types of price expectation. To keep things simple, we assume that given his price expectations every trader maximizes a mean-variance utility for one period ahead. The model we develop is similar to the famous Brock and Homes [BH97] model.
As before, we assume that the economy follows a discrete time tree model with a finite number of states in each period. Since the maximization problems span two- periods only, we switch back to the notation of Chap.4. The following then defines the maximization problem for any traderiD 1; : : : ;Iin any period of timetwith sD1; : : : ;Sstates of the world17:
The utility of agentiis of the mean-variance type18: Ui.ci1; : : : ;ciS/D.ci1; : : : ;ciS/ i
22.ci1; : : : ;ciS/:
Note that his consumption in states is given bycis D i;cRisiwi;f and recall the budget constraint19 PK
kD0Oi;k D 1. Since we did not state the time dependence explicitly, let us point out that the consumption in statesis given by the consumption rate of that period applied to the wealth in state s, which is obtained from the previous period financial wealth20multiplied by the gross return obtained in state
17To simplify matters we first suppress all time and uncertainty dependence in this generic one step ahead optimization problem.
18Ris;kis the return agentiexpects to get from assetkif statesoccurs.
19Recall that in Chap.4kD0was the risk-free asset.
20I.e. the wealth not consumed, but spent on financial assets.
5.6 Dynamics of Price Expectations 241
s:RisOi D PK
kD0Ri;k.s/Oi;k. Hence, for any portfolioO we get a utility from that portfolio21:
Ui.Oi;0; : : : ;Oi;K/Dwi;f
.i;cRiOi/ iwi;f
2 2.i;cRiOi/
:
The solution is as before22 OiD
COV.Ri/1.Ri/Rf ii;cwi;f ; which we can write in economic terms as
iD 1
i
COV.Ai/1
.Ai/Rfq :
Recall that in the multi-period model payoffs are given by dividends and resale prices, i.e.
Ai.!t/DDi.!t/Cqi.!t/:
We assume that conditionally on!tagents agree on the dividends, and we assume point expectations,23 i.e.qit is independent of.!t/. Then, the demand in periodt given the expectations for the next period is
OitD 1
iit;cwit;f
.qt/ .COV.DtC1//1
.DtC1/CqitC1Rfqt
:
Normalizing the supply of each asset to 1, assuming short-run equilibrium and defining
N1D XI
iD1
. i/1
21To be mathematically correct one should introduce a different symbol for a utility function if it depends on different variables than before. However, adding more notation will be confusing to many readers.
22Again, the notation is used:.Ri/is a vector inRkbutRf is a scalar. A more correct notation would be.RQi/Rf1, whereRQidenotes the matrix of risk assets and12Rkis a vector with 1 in each entry.
23The first assumption is not restrictive, since in case two agents were to disagree on the dividends in one of the states, one might introduce more states and let the agents disagree over the occurrence of the states. The second assumption is strong. The only excuse we have is that it is sufficient to generate interesting dynamics – and that it is convenient in the Brock-Homes-Model.
gives XI iD1
it;cwit;f DqtD.qt/ .COV.DtC1//1 .DtC1/ N C
XI iD1
qitC1
i Rfqt
N
! : Multiplying both sides by.q/k1andCOV.DtC1/and definingDM D PK
kD1Dk, for any assetkgives
qkt D .DktC1/ NtCOV.DktC1;DMtC1/CPI
iD1ıiqitC;k1 Rf
;
where
ıiD N
i:
Hence, the price of any asset k in periodt is given by the discounted expected dividends minus the risk of those dividends relative to the market dividends plus the average expected price for the next period.
Before we analyze the dynamics of the model we first characterize its steady state stationary solution. To this end we assume that the trading strategiesOi, the consumption ratesi;c, the expected dividends.D/and the covarianceCOV.D/
are all stationary. Then, the price equation reduces to:
qNkD .Dk/ NCOV.Dk;DM/ rf
;
which is equal to the discounted expected dividends of the constant payoff .Dk/ N COV.Dk;DM/
discounted atRf D1Crf. RecallRf D1Crf and substitute N D .DM/rfqM
2.DM/ ; from summing the above formula overk. Then we obtain
qNk
Dk rf
Dˇk
qNM
DM rf
;
where
ˇkD COV.Dk;DM/ 2.DM/ :
5.6 Dynamics of Price Expectations 243 Hence, we have derived a Security Market Line formula similar to that of the static CAPM, but in terms of first principles: dividends and the risk-free rate!
Now we model some structure on the price expectations more explicitly. There are two types of traders: chartistsi 2 Cand fundamentalistsi2 F. Chartists only use price data as an input for forming their price expectations while fundamentalists compare current prices to the long term steady state prices. Chartists form the price expectations
qitC;k1 Dqkt Cai;k.qkt qkt1/
withai;k > 0being a momentum chartist andai;k < 0being a reversal chartist.
Fundamentalists form the price expectations
qitC;k1 DqktCbi;k.qNkt qkt/
withbi;k> 0being value investors andbi;k< 0being growth investors.
Note that the price dynamics developed above is an inhomogeneous first order difference equation. Such a dynamical system converges to its steady stateqNiff the absolute value of the coefficient in front of the price variablePI
iD1ıiqitC;k1is smaller than one.24Thus we can ignore those forms that do not depend on prices. Inserting the expectation functions we get:
Rfqkt DX
i2C
ıi
qktCai;k.qkt qkt1/ CX
i2F
ıi
qkt Cbi;k.qNkt qkt/ :
Rearranging while ignoring constant terms we get:
X
i2C
ıit.1Cai;k/CX
i2F
ıti.1bi;k/Rf
!
qkt D X
i2C
ıtiai;k
! qkt1
or qkt D
P
i2Cıtiai;k P
i2Cıitai;kP
i2Fıitbi;krf
qkt1:
24For a proof, we show that the iterationXnC1 WD aXnCbalways converges ifjaj< 1. To see this, we compute the fixed point of the iteration asX Db=.1a/, i.e. ifXn Db=.1a/then XnC1 DXn. Then we consider the squared difference ofXn to the fixed point and show with a small computation that it is decreasing. From this we can deduce thatXnindeed converges. We can apply this auxiliary result to the dynamic system above.
Thus, in the stability analysis of the dynamical system we need to consider four cases in which we would get stability of the steady state:
Case 1
(numerator and denominator positive)
This happens, e.g. with strong momentum and weak value. Consequently stability occurs iff
X
i2F
ıibi;kCRf < 0;
which is unlikely sinceRf > 1.
Case 2
(numerator positive and denominator negative)
This happens, e.g. with medium momentum and strong growth. Consequently stability occurs iff
2X
i2C
ıiai;k<X
i2F
ıibi;kCrf;
which cannot be since in this case X
i2C
ıtiai;k> 0 and X
i2F
ıtibi;k< 0:
Case 3
(numerator negative and denominator positive)
This happens, e.g. with reversal and strong growth. Consequently stability occurs iff
X
i2F
ıibi;kCrf < 2X
i2C
ıiai;k;
which is well possible.
Case 4
(numerator and denominator negative)
This happens, e.g. with reversal and value or weak growth. Stability occurs iff 2X
i2C
ıiai;k<X
i2F
ıibi;kCrf;
which is possible if the reversal is not too strong relative to value.