"Technical Analysis in Financial Markets"

We assume that the agents have a constant relative risk aversion so that the utility of the capital allocation decision by agent j with belief h is given by

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

a_j

V_j,t+1^h(r_j,t^c), (21)

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

Max_{y_j,t^h} E_j,t^h(r_j,t+1^c)-

a_j

V_j,t^h(r_j,t+1^c) under CAL (6.19) or (6.20). (22)

The first order condition of (6.22) is

d U_j,t^h

d y_j,t^h

=E_t^h(r_t+1^P-r^F) - a_j y_j,t^h V_t^h(r_t+1^P)=0.

This implies that the optimal fraction of individual wealth invested in the risky asset by agent j with belief h as function of the price P_t is equal to

y_j,t^h(P_t)=

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

a_j V_t^h(r_t+1^P)

. (23)

Since the second order condition

d² U_j,t^h

d (y_j,t^h)²

=-a_j V_t^h(r_t+1^P)<0

is satisfied, utility is maximized. If y_j,t^h>0, then a long position in the risky asset is held. If y_j,t^h<0, then a short position in the risky asset is held. If we assume that all agents have the same risk aversion parameter a_j=a, then all agents with the same belief h invest the same fraction of their individual wealth in the risky asset. If agent j has belief h, then y_j,t=y_j,t^h=y_t^h, where y_t^h is the optimal fraction of wealth invested in the risky asset at time t recommended by belief h. Under the assumption that a_j=a it is also true that U_j,t^h=U_t^h for all j. Further we assume that the conditional variance V_t^h(r_t+1^P)=σ² is constant through time and equal for all beliefs. y_t^h(P_t) can now be rewritten as:

y_t^h(P_t)=

P_t

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

a σ²

(24)

which is a convex monotonically decreasing function of P_t. Note that we can bring P_t outside the expectations formula, because the price is not a random variable, but an equilibrium price set by the market auctioneer. Stated differently, the fraction of wealth invested in the risky asset at time t depends on the price set by the market at time t and the forecast or belief about the price at time t+1, based on all available information until time t but not including P_t. Figure 6.1 illustrates the demand function of the mean-variance utility maximizing belief.

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