"Technical Analysis in Financial Markets"

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.

We 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 portfolio return of agent j at time t who invests according to strategy h is then equal to

E_j^h(r_j,t+1^c|I_t)=E_j,t^h(r_j,t+1^c)=r^F+y_j,t^h E_t^h(r_t+1^P-r^F), (16)

where E_t^h(r_t+1^P-r^F) is the forecast belief h makes about the excess return of the risky asset at time t+1 conditioned on I_t. If the conditional expected excess return of belief h is positive, then the fraction invested in the risky asset is positive (y_j,t^h ≥ 0) and if the conditional expected excess return is strictly negative, then the fraction invested in the risky asset is strictly negative (y_j,t^h<0). Hence agent j with belief h can choose to buy shares or to sell shares short. Agent j does not only forecast his portfolio return but also the dispersion of the portfolio return which is equal to

V_j^h(r_j,t+1^c|I_t)=V_j,t^h(r_j,t+1^c)=(y_j,t^h)² V_t^h(r_t+1^P), (17)

where V_t^h(r_t+1^P) is the forecast of belief h about the dispersion of the excess return of the risky asset. Solving (6.17) for y_j,t^h yields

y_j,t^h=�

V_j,t^h(r_t+1^c)

V_t^h(r_t+1^P)

if V_t^h(r_t+1^P)>0, (18)

where the � sign depends on the conditional expected excess return of the risky asset. The capital allocation line (CAL) can be derived by substituting (6.18) in the conditional expectations equation (6.16), that is

E_j,t^h(r_j,t+1^c)=r^F + S_t^h V_j,t^h(r_j,t+1^c) if E_t^h(r_t+1^P-r^F)≥ 0; (19)

E_j,t^h(r_j,t+1^c)=r^F - S_t^h V_j,t^h(r_j,t+1^c) if E_t^h(r_t+1^P-r^F)<0; (20)

S_t^h=

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

V_t^h(r_t+1^P)

Here |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 return and the expected dispersion of the return. The CAL is always an increasing function of V_j,t^h(r_j,t+1^c). This implies that the more risk the agent expects to take, the more he expects to earn.

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