holds, which means that the long run growth rate of price is less than the discount rate rf, then the price is equal to the net present value of all future dividends

Pt*=
 
lim
k → ∞
k
∑
j=1
Et(Dt+j)
Rj
.
This price is called the efficient markets hypothesis (EMH) fundamental rational expectations price, or fundamental price for short.

We will focus on the case where for all beliefs h expectations on dividend are equal: Eth(Dt+1)=Et(Dt+1) and where the dividend process is iid with mean D. The fundamental price is then constant and equal to

P*=
D
r
.

6.3.4  A heterogeneous agents model with fundamentalists
versus moving average traders

Utility maximizing belief: the fundamental trader

Fundamentalists expect that prices return to the fundamental value with speed v, that is
Etfund(Pt+1)=P*+v(Pt-1-P*), 0≤ v ≤ 1.
If v=1, then the fundamental traders make naive price expectations and if v=0, then the fundamental traders expect the price to be always equal to the fundamental value. The fraction of individual wealth invested in the risky asset is then equal to
ytfund(Pt)=
1
Pt
Et(Pt+1+Dt+1)-R
a σ2
=
1
Pt
(P*+v(Pt-1-P*)+
D
)-R
a σ2
,
where a>0, σ2>0 and R=1+rf>1.

Non utility maximizing belief: the exponential moving average trader

Moving average traders buy (sell) if the price crosses the moving average from below (above). We use the exponential moving average MAt=µ Pt + (1-µ) MAt-1, where the exponential smoothing constant 0<µ<1. The fraction of individual wealth invested in the risky asset is then equal to
ytma = 2 γ
xt+1
1+xt+12
, where xt+1=
1
λ
Pt-1-MAt-2
MAt-2
, γ>0 and λ>0.

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