Trading signals are generated by the crossing of the price through the moving average. If the price crosses the moving average upwards, i.e. Pt>MAt ∧ Pt-1≤ MAt-1, then a buy signal is generated and at time t+1 a long position in the market is taken. If the price crosses the moving average downwards, i.e. Pt<MAt ∧ Pt-1≥ MAt-1, then a sell signal is generated and at time t+1 a short position in the market is taken. The magnitude of the position held in the market can also be conditioned on the distance between the price and the moving average. If Pt is close to MAt, small positions should be held, because it is uncertain whether the strategy generated correct signals. It also seems reasonable to assume that if Pt is very far away from MAt, then small positions should be held, because the price exploded too fast away from MAt.

To satisfy above conditions the demand of the moving average forecasting rule, as a fraction of individual wealth at time t, ytMA, is defined as a continuous function of past prices and moving averages in the following way:

xt-1=
1
λ (1-µ)
Pt-1-MAt-1
MAt-2
=
1
λ
Pt-1-MAt-2
MAt-2
;
ytMA=f(xt-1)=2 γ
xt-1
1+xt-12
,
    (26)
where λ>0, γ>0. Notice that in contrast to the fraction yth(Pt) in (6.24), the fraction ytMA of wealth invested by moving average traders in the risky asset does not depend upon the (unknown) market equilibrium price Pt, but only upon past price observations and moving averages.

The demand function (6.26) has the following properties (see figure 6.2 for illustration):

  • 0cm ytMA<0 if Pt-1<MAt-1
  • ytMA=0 if Pt-1=MAt-1
  • ytMA>0 if Pt-1>MAt-1
  • limPt-1→ ∞ ytMA=0
  • limPt-1↓ 0 ytMA=-2 γ   λ/1+λ2
  • d ytMA/d Pt-1= 2 γ (1/λ MAt-2) 1-x2/(1+x2)2
  • minimum: (Pt-1, ytMA)=((1-λ)MAt-2, -γ)
  • maximum: (Pt-1, ytMA)=((1+λ)MAt-2, γ)
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