"Technical Analysis in Financial Markets"

where R=(1+r^F) is the risk free gross return, r^F is the risk free net return assumed to be constant, P_t is the equilibrium price of the risky asset at time t and D_t is the dividend paid at time t. The term (P_t+1+D_t+1-R P_t) is equal to the excess profit of one long position in the risky asset.

BH make the following assumptions regarding the trading process. All agents are price takers. That is, an agent cannot influence the market's equilibrium price by his individual investment decision. The demand for the risky asset z_j,t is a continuous monotonically decreasing function of the price P_t at time t. Further, the model follows a Walrasian equilibrium price scenario. Before the setting of the equilibrium price at time t, each agent j chooses a trading strategy h and makes an optimal investment decision z_j,t^h in the time interval (t-1, t). Expectations about future prices and dividends are made on the basis of the information set of past equilibrium prices and dividends {P_t-i, D_t-i: i ≥ 1 } (note that P_t and D_t are not included). Through the market mechanism an equilibrium price is set so that the market clears. Dividends D_t paid at time t can immediately be reinvested at time t.

Demand

BH define the information set I_t={P_t-i, D_t-i: i ≥ 1 } ∪ {P_t, D_t}, where {P_t-i: i ≥ 1 } are past equilibrium prices, {D_t-i: i ≥ 1 } are past dividends, {D_t} is current dividend, but where {P_t} is not yet necessarily the equilibrium price. The conditional expected wealth of agent j at time t who invests according to strategy h is then equal to

E_j^h(W_j,t+1|I_t)=E_j,t^h(W_j,t+1)=RW_j,t + z_j,t^h E_t^h(P_t+1+D_t+1-RP_t), (2)

where E_t^h(P_t+1+D_t+1-RP_t) is the forecast strategy h makes about the excess profit of the risky asset at time t+1 conditioned on I_t. The agent also makes a forecast about the dispersion of his expected wealth conditioned on I_t

V_j^h(W_j,t+1|I_t)=V_j,t^h(W_j,t+1)=(z_j,t^h)² V_t^h(P_t+1+D_t+1), (3)

where V_t^h(P_t+1+D_t+1) is the forecast of belief h about the dispersion of expected price plus dividend. It is assumed that if the conditional expected excess profit of belief h is positive, then the agent holds a long position in the market (z_j,t^h ≥ 0) and if the conditional expected excess profit is strictly negative, then the agent holds a short position in the market (z_j,t^h<0), so that conditional expected wealth is always equal or larger than RW_j,t.

Solving the conditional variance equation (6.3) for z_j,t^h yields

z_j,t^h=�

V_j,t^h(W_j,t+1)

V_t^h(P_t+1+D_t+1)

, (4)

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