In this section, we first present the model setup for the economy including hetero- geneous beliefs and the relative performance.
3.2.1 Economy
We consider a continuous time, finite horizon [0,T] economy with two assets that are risky and risk-free, respectively. We interpret the risky asset as a stock that has
the following dynamics
dSt
St
=µs,tdt+σs,tdBt (3.1)
where σs,t > 0 and Bt is the standard Brownian motion defined on the filtered
probability space (Ω,F,{Ft},P). Note that the Brownian motion Bt is the only
source of uncertainty in this economy. The driftµs,tand diffusionσs,tare determined
in equilibrium. The stock is in positive net supply and pays the liquidating dividend
DT at timeT. We assumeDt follows a geometric Brownian motion
dDt
Dt
=µDdt+σDdBt (3.2)
whereµD and σD are positive constants. The risk-free asset, interpreted as a bond,
is in zero net supply and has a constant returnr. For simplicity, we assume r= 0.
There are two groups of agents, interpreted as fund managers, in the market who optimally allocate their fund between the risky and the risk-free assets. Each group has infinite number of managers that form a continuum with measure 1. Because there are two groups, with a little abuse of notations, we use subscript ofito denote the manager in group i, i∈(1,2)30. Each manager iinvests a fraction, θi,t, of her
investment wealthWi,t on the stock. Hence, Wi,t follows
dWi,t =θi,tWi,t(µs,tdt+σs,tdBt) (3.3)
We assume that the managers have the same initial endowment, which means that each manager hasWi,0= S20 initial wealth.
3.2.2 Relative Performance and Objective Function
In reality, two types of compensation contracts exist for fund managers: fixed (pro- portional to Asset Under Management), or performance based. Under fixed contract, managers care about fund flows, which according to empirical evidence31, depend
on a fund’s relative performance compared to peers. Under performance based con- tract, a manager’s performance is compared to benchmarks that also include peer performance. Consistent with the compensation contracts in the fund management industry observed by Ma, Tang and Gomez (2012), we assume that the managers receive a timeT bonus or penalty that is related to their relative performance com- paring to their peers. We assume that a manager receives bonus if her own type
30
Thus, managerimeans an individual manager who belongs to groupi.
beats the other type of managers, and suffers penalty if the other type beats her own type of managers.
We define the functional form of the bonus/penalty in the following way: first de- noteWi,T as the worth of manageri’s portfolio at timeT, andRi,T as the aggregate
return of all managers in groupirelative to the aggregate return of all managers in group j, where i= 1, j = 2 or vice versa. Manager i’s relative performance, Ri,T,
is defined as:
Ri,T =
Wi,T/Wi,0 Wj,T/Wj,0
(3.4)
From our previous assumption, W1,0 =W2,0 = S20. Then Ri,T = WWi,Tj,T depends only
on the ratio of their performance because they start with the same initial wealth. Then define the bonus/penalty as:
BPi,T =Wi,T(Rki,T −1) (3.5)
where k > 1. If Wi,T > Wj,T, then Ri,T > 1 and BPi,T > 0, so that manager i
receives a bonus. IfWi,T < Wj,T, then Ri,T <1 and BPi,T <0, so that manageri
receives a penalty. The assumption k > 1 ensures the bonus/penalty is increasing and convex in manager i’s relative performance, which is consistent with empiri- cal results. Following this assumption, manager i’s wealth from investment plus bonus/penalty adds up to Wi,T +BPi,T =Wi,TRki,T, which is the objective in her
optimization problem. Denotefi,T := (Ri,T)k, then a CRRA manageri’s objective
function32 is then:
vi,T =
(Wi,Tfi,T)1−γ
1−γ (3.6)
3.2.3 Heterogeneous Beliefs
Manager i has the probability space Ω,Fi,
Fi t ,Pi
. Following the standard fil- tering theorem, the dividend process under fund manageri’s belief follows
dDt
Dt
=µi,Ddt+σDdBi,t (3.7)
By Girsanov’s theorem, dBi,t = dBt+ηidt is the Brownian motion in manager i’s
probability space , and ηi = µD
−µi,D
σD . For two groups of agents, 1 and 2, equation (3.7) implies
dB2,t=dB1,t+µdt (3.8)
where
µ= µ1,D−µ2,D
σD
(3.9) (3.9) represents the investors’ disagreement on the drift of the dividend process, normalized by its diffusion term. µ >0 implies that the agents in group 1 are more optimistic and vice versa. Given the priors of agents,µis an exogenous parameter.33 Under the subjective measures of groups 1 and 2, the stock has the dynamics
dSt = St[µs,tdt+σs,tdBt]
= St[µi,tdt+σs,tdBi,t], f or i= 1,2 (3.10)
The two groups of agents must agree with the price, so we have the relationship between the perceived means
µ1,t−µ2,t =σs,tµ (3.11)
Because the market is complete, there exists a unique state price density process,
πi, for each manageri
dπi,t πi,t =−κi,tdBi,t (3.12) where κi,t = µi,t σs,t (3.13) is the perceived market price of risk (Sharpe ratio) for group 1 and 2 respectively. We also have κ1 −κ2 = µ which is the measure of the disagreement between the
agents’ perceived market price of risk.