Chapter I: History-Dependent Risk Aversion and the Reinforcement Effect
1.5 Behavioral Foundations of the History-Dependent Model
Moreover, when returns are very high, agents happen to be very risk-averse (Point 1 to Point 2 in Figure 1.5), so they underappreciate high returns.
Another empirical fact about the stock market is the predictability of asset returns, which refers to predictable dynamics of asset returns in the following sense:
when asset prices are high, subsequent long-horizon asset returns tend to be low.
To check the predictability, the following simple regression is usually used:
log(Rt+1) =a+blog pt
zt
+t+1,
whereRt+1= pt+1p+tzt+1 is the asset return. The predictability of asset returns means that empiricallybis negative, since a high pztt must be followed by a lowRt+1. From the simulation, I find that ˆa = 4.9 and ˆb = −0.7 (with standard error 0.06). This regression suggests that a 10% increase in the price-dividend ratio implies a 7%
decrease in the next period return.
Second, I state an axiom for simple expected utility. Simple expected utility is captured by an axiom on called strong independence, which states that if a lottery X today is equivalent to an outcome z tomorrow and a lotteryY today is equivalent to an outcomet tomorrow; then a mixtureαX +(1−α)Y is equivalent to a compound lottery(α,(0;z),1−α,(0;t)). Formally,
Axiom 2 (Strong Independence) For anyX,Y ∈∆(R+),z,t ∈R+, andα ∈ (0,1), if(X; 0) (0;z) and(Y; 0) (0;t),then(αX+(1−α)Y; 0) (α,(0;z),1−α,(0;t)).
Strong independence is slightly stronger than the independence axiom of ex- pected utility theory. I now can state the first characterization result.
Theorem 2 (Discounted Expected Utility) A continuous preference relationon L is represented by a history-dependent model {V0, β,V(x,X)} and satisfies time consistency; that is, V0(z) = V(x,X)(z) for any z ∈ R+, the axiom for history independence, and strong independence if and only if there exists a continuous functionu :R+ →R+such that for anyL= (pi,(xi;Zi))in=1,L0= (p0k,(x0k;Z0k))km=1 ∈ L,
L L0iff
n
X
i=1
pi u(xi)+ βEu(Zi) ≥
m
X
k=1
p0k u(x0k)+ βEu(Zk0).
Theorem 2 formally shows that the history-dependent model (1.3) is a result of dropping two properties of DEU, history-independence and simple expected utility.
Now I turn to the second approach: imposing three axioms on . The first axiom is calledregularity, a collection of four standard postulates.
Axiom 3 (Regularity) A preference relation on L satisfies the following four conditions.
1. The preference relation is complete, transitive, and continuous.
2. (Deterministic Monotonicity) For anyz,z0∈R+ withz > z0andX ∈∆(R+), (z; 0) (z0; 0)and pi,(xi;z),(X−i; 0) pi,(xi;z0),(X−i; 0).
3. (Discounted Utility) There exist a utility functionu:R+ →R+and a discount factor β ∈ (0,1)such that for any(x;z),(x0;z0) ∈R2+.
(x;z) (x0;z0)iffu(x)+ βu(z) ≥u(x0)+ βu(z0), (1.10) 4. (Expected Utility Aggregator) There existsU2 : R+ → R such that for any
Z = (rk,zk)mk=1,Z0= (r0k,z0k)mk00=1 ∈∆(R+),
(rk,(0;zk))km=1 (r0k,(0;z0k))km=10 iffEU2(Z) ≥ EU2(Z0). (1.11)
The first part of regularity collects completeness, transitivity, and continuity. I also assume a very weak form of monotonicity calleddeterministic monotonicity. The second half of regularity imposes the other two properties of DEU, discounted utility and expected utility aggregator. Specifically, the third part states that the agent uses discounted utility theory when she aggregates utilities of today and tomorrow;
i.e., the utility of(x;z)isu(x)+ βu(z)(discounted utility). The fourth part states that compound lotteries are evaluated by expected utility theory; i.e, the utility of a compound lottery (rk,(0;zk))mk=1isPm
k=1rkU2(zk)(expected utility aggregator).
The next two axioms are novel axioms. The second axiom (Axiom 4) is calledseparability, which consists of two properties of separability. Separability is essential for studying history-dependent risk aversion because it allows me to define a risk preferences for each history independently of other histories. Separability also implies time consistency.
Axiom 4 (Separability) A preference relation on L satisfies the following two properties.
1. (Separability between Parallel Histories) For any (pi,(xi;Zi))in=1 ∈ L and Y,Y0∈∆(R+),
pi,(xi;Y),(X−i; 0) pi,(xi;Y0),(X−i; 0)
iff pi,(xi;Y),(pk,(xk;Zk))k,i pi,(xi;Y0),(pk,(xk;Zk))k,i. 2. (Weak Separability between Today and Tomorrow) For any (pi,(xi;zi))in=1 ∈
∆(R+×R+)and µ,z ∈R+,
i) (X; 0) (µ; 0)if and only if(X;z) (µ;z)and
ii) (pi,(0;zi))in=1 (0;µ)if and only if(pi,(xi;zi))in=1 (X;µ).
Suppose the agent receives either of two simple lotteriesY andY0after a history (xi,X). Separability between parallel histories requires that a comparison between the two simple lotteriesY andY0 cannot be affected by what she would receive in histories other than(xi,X). This axiom is essentially a dynamic version of an axiom calledreplacement separability, introduced in Machina (1989). Weak separability between today and tomorrow essentially requires that the utility of a deterministic outcome z is history-independent (recall time consistency); i.e., the utility of z is not affected by a lottery X and a deterministic outcome µ. Specifically, i) states that if µis the certainty equivalent of a simple lottery X, then µis still the certainty equivalent ofXeven if the agent will receive a deterministic outcomezin the second period. Moreover, ii) states that ifµis the certainty equivalent of a compound lottery (pi,(0;zi))in=1, then (0;µ) is still the certainty equivalent of (pi,(0;zi))in=1 even if the agent receives a lotteryX in the first period.
The third axiom (Axiom 5) is called additivity.21 Suppose a risky option that gives z with probability p tomorrow is equally preferred to a safe option µtoday.
Similarly, suppose a risky option that gives z0with probability 1−p tomorrow is equally preferred to a safe optionλtomorrow. Additivity requires that a combination of the two risky options is equally preferred to a combination of the two safe options;
that is, receivingzwith probabilitypandz0with probability 1−ptomorrow is equally preferred to receiving µtoday andλtomorrow.
Axiom 5 (Additivity) For any p,(0;z),1− p,(0;z0) ∈ ∆(R+×R+) and λ, µ ∈ R+, and for anyi,j,
if p,(0;z),1−p,(0; 0) ∼ (µ; 0) and p,(0; 0),1−p,(0;z0) ∼ (0;λ), then p,(0;z),1− p,(0;z0) ∼ (µ;λ).
Finally, I can state the second characterization theorem.
Theorem 3 (History-Dependent Model) A preference relation on L satisfies regularity, separability, and additivity if and only if there are strictly increasing
21Additivity is not essential to Theorem 1. I can relax it and Theorem 1 can be modified accordingly.
continuous functionsV0:∆(R+)→ R+andV(x,X):∆(R+)→ R+such that for any L = (pi,(xi;Zi))in=1,L0= (p0k,(x0k;Z0k))km=1 ∈L,
L L0iffV0(X)+ β
n
X
i=1
piV(xi,X)(Zi) ≥V0(X0)+ β
m
X
k=1
p0kV(x0
k,X0)(Zk0), (1.12) andV0(z)=V(x,X)(z)for each z ∈R+.
I also have a strong uniqueness result.
Proposition 1 (Uniqueness) Take any preference relation on L that satisfies regularity 1-2. If it is represented by triplets (V0, β,{V(x,X)}) and (V0
0, β0,{V(x,0 X)}) that satisfy time consistency,V0(0) =V0
0(0), andV0(1) =V0
0(1), then the two triplets are identical.
I conclude this section by illustrating that Theorem 1 can be stated in terms of axioms oninstead of using the history-dependent model{V(x,X)}. Since dynamic monotonicity is already defined on the primitive , and the history-dependent model (1.3) is characterized by Theorem 3, it is sufficient to define the RE in terms of conditions on.22 It turns out, the RE is equivalent to the following condition.
Definition 5 (Reinforcement Effect) For any simple lotteryX ∈∆(R+)andxi,xj ∈ supp(X)withxi > xj, for any Z ∈∆(R+)and µ∈R+,
if pj,(xj;Z),(X−j; 0) pj,(xj;µ),(X−j; 0), then pi,(xi;Z),(X−i; 0) pi,(xi;µ),(X−i; 0).
1.6 Specifying Risk Preferences: Expected Utility, Disappointment Aversion,