"Technical Analysis in Financial Markets"

Assume that at time 0, all agents have equal initial wealth. Thus for all j we have W_j,0=ω₀ and

E(W_j,t|I_t)=ω₀

t-1

∏

i=0

(

1+r^F + (r_t-i^P-r^F) E(y_t-1-i)

)

. (54)

According to (6.54) the expectation is equal for all agents at time t, E(W_j,t|I_t)=ω_t, under the assumption that all agents have the same wealth at time 0. Finally we now have found that E(W_j,t^h|I_t)=q_t^h ω_t ∀ j.
The variance of W_j,t^h conditioned on I_t is equal to:

V(W_j,t^h\|I_t)	=	E((W_j,t^h)²\|I_t)-E²(W_j,t^h\|I_t)
	=	q_t^h E(W_j,t²\|I_t)- (q_t^h)² E²(W_j,t\|I_t)
	=	q_t^h V(W_j,t\|I_t) + q_t^h (1-q_t^h) E²(W_j,t\|I_t).

(55)

The expectation of the squared value of the wealth of agent j at time t conditioned on I_t and his wealth at t-1 is equal to:

E(W_j,t²|I_t, W_j,t-1)=R² W_j,t-1²+2R W_j,t-1² (r_t^P-r^F) E(y_t-1) + W_j,t-1² (r_t^P-r^F)² E(y_t-1²),

and the expectation only conditioned on I_t is equal to:

E(W_j,t²|I_t)=[R² +2R (r_t^P-r^F) E(y_t-1) + (r_t^P-r^F)² E(y_t-1²)] E(W_j,t-1²|I_t-1),

which iterates to:

E(W_j,t²|I_t)=W_j,0²

t-1

∏

i=0

[R² +2R (r_t-i^P-r^F) E(y_t-1-i) + (r_t-i^P-r^F)² E(y_t-1-i²)], (56)

where W_j,0 is the initial wealth of investor j. The square of the expectation of the wealth of agent j at time t is equal to:

E²(W_j,t|I_t)=[R² +2R (r_t^P-r^F) E(y_t-1) + (r_t^P-r^F)² E²(y_t-1)] E²(W_j,t-1|I_t-1),

which iterates to:

E²(W_j,t|I_t)= W_j,0²

t-1

∏

i=0

[R² +2R (r_t-i^P-r^F) E(y_t-1-i) + (r_t-i^P-r^F)² E²(y_t-1-i)]. (57)

Substituting (6.56) and (6.57) in (6.55) gives the variance of the wealth of agent j assigned to belief h at time t conditioned on I_t. If W_j,0=ω₀ for all agents, then the variance is

280