"Technical Analysis in Financial Markets"

where the � sign depends on the sign of the expected excess profit on the risky asset by belief h. The capital allocation line (CAL) of agent j with belief h is derived by substituting (6.4) in the conditional expectations equation (6.2) which yields

E_j,t^h(W_j,t+1)=RW_j,t + S_t^h V_j,t^h(W_j,t+1) if E_t^h(P_t+1+D_t+1-RP_t)≥ 0; (5)

E_j,t^h(W_j,t+1)=RW_j,t - S_t^hV_j,t^h(W_j,t+1) if E_t^h(P_t+1+D_t+1-RP_t)<0; (6)

S_t^h=

E_t^h(P_t+1+D_t+1-RP_t)

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

E_t^h(P_t+1+D_t+1-RP_t)/P_t

V_t^h(P_t+1+D_t+1)/P_t²

E_t^h(r_t+1^P-r^f)

V_t^h(r_t+1^P)

Here r_t+1^P is the return of the risky asset at time t+1 and |S_t^h| is the reward to variability ratio, or stated differently, the extra expected return to be gained per extra point of expected risk to be taken. The CAL shows the relation between the expected wealth and the expected dispersion of the wealth by agent j. The CAL is always an increasing function of V_j,t^h(W_j,t+1), which means that the more risk the agent expects to take, the more he expects to earn.

BH assume that each agent has constant absolute risk aversion (CARA) and that the utility of the asset allocation decision of agent j who invests according to belief h at time t is given by

U_j,t^h=E_j,t^h(W_j,t+1)-

a_j

V_j,t^h(W_j,t+1), (7)

where a_j is the risk aversion parameter of agent j. Every agent chooses an asset allocation that maximizes his utility, that is

Max_{z_j,t^h} E_j,t^h(W_j,t+1)-

a_j

V_j,t^h(W_j,t+1) under the CAL (6.5) or (6.6). (8)

This maximization yields the optimal choice of the number of stocks to be bought or sold short

z_j,t^h(P_t)=

E_t^h(P_t+1+D_t+1-RP_t)

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

, (9)

where z_j,t^h(P_t) ∈ IR is the demand for shares as a continuous monotonically decreasing function of P_t. If z_j,t^h>0, then a long position in the market is taken and if z_j,t^h<0, then a short position in the market is taken. If it is assumed that all agents have the same risk aversion parameter a_j=a, then all agents with the same belief buy or sell short the same number of shares irrespective of their wealth. If j ∈ belief h, then z_j,t=z_j,t^h=z_t^h, where z_t^h is the number of shares recommended to be bought or sold short by belief h at time t.

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