Asset pricing with flexible beliefs

A Appendix: Proofs and Lemmata

Proof of Proposition 1

Part a: Investors’ optimization (Assumption 3) and market clearing (Assumption 7) imply: pt=ω1RE(pt+1+dt+1|Ωtϕ)+(1−ω)1RE(pt+1+dt+1|Ωtτ)−ωσϕ2λRπqt, where ω≡πστ2πστ2+(1−π)σϕ2. Notice that the solution of the latter expression with respect to π gives π=σϕ2ωστ2(1−ω)+σϕ2ω. By taking expectations conditional on Ωtτ, using the law of iterated expectations (since Ωtτ is a subset of Ωtϕ) and solving for 1RE(pt+1+dt+1|Ωtτ) we obtain that

(24)pt=1RE(pt+1+dt+1|Ωtτ)−ωσϕ2λRπE(qt|Ωtτ). (24)

By using the initial expression again we have that prices satisfy: pt=1RE(pt+1+dt+1|Ωtϕ)+(1−ω)λRπσϕ2E(qt|Ωtτ)−λRπσϕ2qt. From Assumption 5a and by taking expectations conditional on time we obtain that

(25)E(pt|t)=1RE(pt+1+dt+1|t)+(1−ω)λRπσϕ2E(qt|t)−λRπσϕ2E(qt|t). (25)

By subtracting this from prices we obtain that p¯t=1RE(p¯t+1+d¯t+1|Ωtϕ)+(1−ω)λRπσϕ2E(ζt|Ωtτ)−λRπσϕ2ζt. By using our assumptions on information (Assumption 2) and dividends (Assumption 4) we get that

(26)p¯t=1RE(p¯t+1|Ωtϕ)+1Rρ1d¯t+(1−ω)λRπσϕ2E(ζt|Ωtτ)−λRπσϕ2ζt. (26)

Forward iteration on prices, provided that the discount factor is time invariant and greater than unity (Assumption 1) delivers that p¯t=lims→∞1RsE(p¯t+s|Ωtϕ)+ρ1R[∑s=0∞(1Rρ1)s]d¯t+(1−ω)λRπσϕ2[∑s=0∞1RsE(E(ζt+s|Ωt+sτ)|Ωtϕ)]−λRπσϕ2ζt. Since Ponzi schemes are ruled out by Assumption 5c we have that p¯t=ρ1R−ρ1d¯t−(1−ω)λRπσϕ2×[∑s=0∞ 1RsE[ζt+s−E(ζt+s|Ωt+sτ)|Ωtϕ]]−λRπωσϕ2ζt. At this point we have to calculate the term in brackets. Lemma 2 (presented below) shows that under our assumptions it holds that: E[ζt+s−E(ζt+s|Ωt+sτ)|Ωtϕ]=γsE[ζt−E(ζt|Ωtτ)], where ∣γ∣<1. Consequently, the price law of motion becomes: p¯t=ρ1R−ρ1d¯t−(1−ω)λRπσϕ2RR−γ[ζt−E(ζt|Ωtτ)]−λRπωσϕ2ζt. Lemma 3 (presented below) and (48) facilitate the expression of ζt−E(ζt|Ωtτ) as a function of ξ_t, which stands for the technical trader’s errors for dividends at t after he observes prices at t, and particularly as:

(27)ζt−E(ζt|Ωtτ)=ρ1R−ρ1Rπ[(1−ω)RR−γ+ω]λσϕ2ξt, (27)

where ξt=dt−E(dt|Ωtτ). By substituting ζt−E(ζt|Ωtτ) from (27) in the latter expression we obtain that

(28)p¯t=ρ1R−ρ1d¯t−(1−ω)ρ1R−ρ1RR−ωγξt−λRπωσϕ2ζt, (28)

where the law of motion for ξ_t is described in Lemma 3 as:

(29)ξt=γξt−1+γρ1εt+λσϕ2(ρ1−γ)(R−ωγ)(R−ρ1)(R−γ)Rπρ12ζt, (29)

where

(30)γ=σετ2−σε2ρ1σετ2<ρ1, (30)

where σετ2 satisfies

(31)0=θ02(σετ2)2+[(1−ρ12)θ12σζ2−θ02σε2]σετ2−θ12σζ2σε2, (31)

where θ0=ρ1R−ρ1 and θ1=−[(1−ω)RR−γ+ω]λRπσϕ2. By lagging (28) one period, multiplying it by γ and subtracting from (28) we get that p¯t−γp¯t−1=ρ1R−ρ1d¯t−ρ1R−ρ1γd¯t−1−(1−ω)ρ1R−ρ1RR−ωγ(ξt−γξt−1)−ωλσϕ2Rπζt+ωλσϕ2Rπγζt−1. From this expression and by using (29) we eliminate ξ_t obtaining that

(32)p¯t=γp¯t−1+ρ1R−ρ1(ρ1−γ)d¯t−1+[R(ρ1−γ)+ωγ(R−ρ1)(R−ρ1)(R−ωγ)]εt−[(ρ1−γ)R+γω(R−ρ1)(R−γ)ρ1]λσϕ2Rπζt+ωγλσϕ2Rπζt−1. (32)

At this point we need to calculate the terms σϕ2 and στ2 and substitute them in the previous expression to deliver the final form of the coefficients (restrictions to the coefficients) of the detrended price law of motion. The innovation in one period ahead payoff conditional on fundamentalists’ information, p¯t+1+d¯t+1−E(p¯t+1+d¯t+1|Ωtϕ), is equal to RR−ρ1R−γR−ωγεt+1−[(1−ω)(ρ1−γ)R(R−γ)ρ1+ω]λRπσϕ2ζt+1. Since ε_t+1 and ζ_t+1 are uncorrelated its variance conditional on fundamentalists’ information is

(33)σϕ2=(RR−ρ1)2(R−γR−ωγ)2σε2+[R−(R−ρ1)(R−ωγ)(R−γ)]2(λσϕ2Rπρ1)2σζ2. (33)

Similarly, conditional on technical traders’ information the payoff innovation, p¯t+1+d¯t+1−E(p¯t+1+d¯t+1|Ωtτ), is equal to [1−(1−ω)γR−ωγ]Rρ1R−ρ1ξt+RR−ωγR−γR−ρ1εt+1−[(1−ω)(ρ1−γ)R(R−γ)ρ1+ω]λσϕ2Rπζt+1. Because ξ_t depends only on current and past ε and ζ [see (29)], ξ_t, ε_t+1 and ζ_t+1 are mutually independent, which implies that E(ξtεt+1|Ωtτ)=E(ξtζt+1|Ωtτ)=0. Since^[11]E(εt+12|Ωtτ)=σε2 and E(ζt+12|Ωtτ)=σζ2 the payoff variance conditional on technical traders information is στ2=σϕ2+[[1−(1−ω)γR−ωγ]Rρ1R−ρ1]2E(ξt2|Ωtτ). From Lemma 3 we get that E(ξt2|Ωtτ)=σετ2−σε2ρ12. By substituting σετ2 with γ into the previous formula and using (30) the payoff variance conditional on technical traders information becomes

(34)στ2=σϕ2+(R−γR−ωγ)2(RR−ρ1)2γρ1σε2(1−γρ1). (34)

By substituting the quadratic term (σϕ2)2 in (33) with its equivalent expression from (31), which – by observing that (30) implies that σετ2=σε21−ρ1γ- becomes:

(35)γ=(1−γρ1)(ρ1−γ)(R−ρ1ρ1)2(λσζσϕ2Rπ)2(R−ωγR−γ)21σε2, (35)

we obtain

(36)σϕ2=[1+[1−(R−ρ1)R(R−ωγ)(R−γ)]2γ(1−γρ1)(ρ1−γ)](RR−ρ1)2(R−γR−ωγ)2σε2. (36)

In order to obtain λσ_ζ we use (35) while the restrictions in the ALM of detrended prices follow from the substitution of the term λσζσϕ2 in (32) with their equivalent expression obtained in (35).

Part b: We conjecture that the trend of prices is of the same functional form as the trend in dividends and supply: E(pt|t)=θ0+θ1t+θ2eθ3t, for some θ₀, …, θ₃ that we will now determine. By substituting this conjecture into (25) together with the deterministic trend of dividends and quantities supplied (Assumption 4b and Assumption 6, respectively) we obtain that θ0+θ1t+θ2eθ3t=1R(θ0+θ1)+1Rθ1t+1Rθ2eθ3eθ3t+1R(δ0+δ1−ωπλσϕ2μ0)+1R(δ1−ωπλσϕ2μ1)t−1Rωπλσϕ2μ2eδt, which should hold for any t, implying that the respective expressions that multiply time trends should be zero, which gives rise to a system of five equations in five unknowns: θ0=1R(θ0+θ1)+1R(δ0+δ1−λωπσϕ2μ0),θ1=1Rθ1+1R(δ1−λωπσϕ2μ1),θ2=1Rθ2eθ3−1Rλωπσϕ2μ2,θ₃=δ. By solving this system for θ₀, …, θ₃ we obtain the expressions for the coefficients of the conjectured deterministic price trend as: θ0=1(R−1)2(δ1−λωπσϕ2μ1)+1R−1(δ0+δ1−λωπσϕ2μ0),θ1=1R−1(δ1−λωπσϕ2μ1),θ2=δ2eδ−λωπσϕ2μ2R−eδ,θ₃=δ. To complete the proof of part (b) of Proposition 1 we have to show also that this form is unique, whose proof follows.

The general form of the trend in prices consistent with Assumptions 1–7 is given in (25) as: E(pt|t)=1RE(pt+1|t)+1RE(dt+1|t)−λRπωσϕ2E(qt|t). By repeatedly substituting for prices and using the law of iterated expectations we obtain that E(pt|t)=lims→∞1RsE(pt+s|t)+∑s=11RsE(dt+s|t)−λRπωσϕ2∑s=01RsE(qt+s|t). From Assumption 4b and Assumption 6 we have that E(pt|t)=lims→∞1RsE(pt+s|t)+∑s=11Rs[δ0+δ1(t+s)+δ2eδ(t+s)]−λRπωσϕ2∑s=01Rs[μ0+μ1(t+s)+μ2eδ(t+s)]. Because δ<r_f (Assumption 4b) and R>1 (Assumption 1), the infinite sums will take the form: ∑s=11Rs[δ0+δ1(t+s)+δ2eδ(t+s)]−λRπωσϕ2∑s=01Rs[μ0+μ1(t+s)+μ2eδ(t+s)]=c0+c1t+c2eδt, where c₀, …, c₂ are constants. The expression for the trend in prices can be rewritten as: E(pt|t)=lims→∞1RsE(pt+s|t)+c0+c1t+c2eδt. If the trend in prices grows at a rate larger than R, then lims→∞1RsE(pt+s|t)=∞. If the trend in prices grows at a rate equal to R then lims→∞1RsE(pt+s|t) is equal to a constant. If the trend in prices grows at a rate lower than R then lims→∞1RsE(pt+s|t) is equal to zero. The trend in prices therefore exists only if lims→∞1RsE(pt+s|t)=κ, where κ is a constant and the form of the trend is the same with the conjectured one, a fact that delivers uniqueness.

Proof of Proposition 3

Part a: With no technical traders in the market (π=1) the market clearing equation (7) becomes: pt=1RE(pt+1+dt+1|Ωtϕ)−σϕ2λRqt. From this expression, the assumption of subjective rationality (Assumption d of the Proposition) and (18) we get that pt=1REtϕ(pt+1|Ωtϕ)+1Rϱt+1Rρ1dt+1Rρ2(t+1)+1Rρ3eρ4(t+1)−(σϕϕ)2λRqt, where (σϕϕ)2 stands for the payoff variance conditional on the rational subjective beliefs of fundamentalists. Under rationality the actual payoff variance is given by σϕ2 in (33) with ω=π=1 and since it does not depend on perceptions on ρ₀ it also gives (σϕϕ)2. By using (i) that subjective beliefs are thought to be constant and preclude Ponzi schemes (Assumptions b and c of the Proposition), (ii) the form of the process of dividends (Assumption 4) and quantities supplied (Assumption 6) and (iii) that the risk free rate is time invariant and positive (Assumption 1) we can apply forward iteration on the price equation and exploit the law of iterated expectations, Etϕ[Et+hϕ(pt+h+1|Ωt+hϕ)|Ωtϕ]=Etϕ(pt+h+1|Ωtϕ), in order to obtain that

(37)pt=f(t)+RR−ρ11R−1ϱt+ρ1R−ρ1dt−(σϕϕ)2λRζt, (37)

where f(t) is a deterministic function of time, where f(t)=R(R−1)2ρ2+(RR−ρ1)×1(R−1)2ρ1ρ2−λ(σϕϕ)21R−1[μ0+μ11(R−1)]+ρ2R−1t+(RR−ρ1)1R(R−1)ρ1ρ2t−λ(σϕϕ)21R−1μ1t−λ(σϕϕ)2μ2R−eμ3eδt.

Tedious calculations show that when ρ_t=ρ₀, (rational expectations price law of motion) the price trend can be expressed as:

(38)E(pt|t)=f(t)+RR−ρ11R−1ρ0+ρ1R−ρ1E(dt|t). (38)

By subtracting (38) from (37) we obtain the deviations, p¯t, of the beliefs updating price law of motion (37) from the trend of the rational expectations price law of motion, where p¯t=RR−ρ11R−1(ϱt−ϱ0)+ρ1R−ρ1d¯t−(σϕϕ)2λRζt.

Let us now denote by σu2 the variance of the last term (−(σϕϕ)2λRζt), which obviously has zero mean, i.e., σu2=E(((σϕϕ)2λRζt)2)=(σϕϕ)4(λσζ)2R2. Since (σϕϕ)2 is given by (33) with ω=π=1, by solving with respect to λσ_ζ we obtain that λσζ=Rσu(RR−ρ1)2σε2+σu2. By substituting now σ_u with ρ1(R−ρ1)γσε(1−γρ1)(ρ1−γ) we obtain

(39)λσζ=Rρ1(R−ρ1)γ(1−γρ1)(ρ1−γ)[R2(1−γρ1)(ρ1−γ)+ρ12γ]σε, (39)

Notice that this substitution is consistent with the restriction on the coefficient α_4,t of the price law of motion of the Proposition 3(a) and completes the proof of part a since it holds provided γ∈(0, ρ₁) and (39) holds.

Part b: The statement regarding perceived detrended prices follows from the assumption of subjective rationality (Assumption d of the Proposition) and the fact that under rational expectations the PLM for detrended prices does not depend on perceptions about ρ₀. The statement regarding the perceived trend follows from the assumption of subjective rationality and the fact that ρ₀ appears in (38) and therefore needs to be replaced in that equation by ρ_t.

Proof of Proposition 2

The law of motion of prices under Proposition 1 can be expressed by the system of equations (28 and 29). By taking expectations conditional on the information sets of technical traders and fundamentalists and utilizing that ξt=d¯t−E(d¯t|Ωtτ) (Lemma 3) we can express the expectations of fundamentalists and technical traders, respectively as: E(pt+dt|Ωt−1ϕ)=ft+RR−ρ1ρ1d¯t−1−(1−ω)ρ1R−ρ1RR−ωγγξt−1 and E(pt+dt|Ωt−1τ)=ft+RR−ρ1ρ1(d¯t−1−ξt−1). By replacing these expressions in the pricing equation (7) with time varying proportion of fundamentalists in the market, π_t, we get the price law of motion under evolution as: pt=1Rft+1+ρ1R−ρ1d¯t−{ωt(1−ω)γR−ωγ+(1−ωt)}ρ1R−ρ1ξt−ωtλσϕ2Rπtqt, where ft+1=RE(pt|t)+ωλσϕ2πE(qt|t) and E(p_t∣t) stands for the price trend under asymmetric information. Therefore, the deviation of the price law of motion under evolution from this trend is p¯t=ρ1R−ρ1d¯t−(ωt(1−ω)γR−ωγ+(1−ωt))ρ1R−ρ1ξt+ωλσϕ2RπE(qt|t)−ωtλσϕ2Rπtqt. This expression does not involve any lagged values of p¯t and depends partly on the law of motion of ξ_t, which remains time invariant. It therefore allows comparison of these prices with the prices from asymmetric information, p¯tA, given in (28). To show this, we solve (28) for the term ρ1R−ρ1d¯t and substitute it back in the previous equation, which yields that p¯t=p¯tA+(R−γ)ρ1(R−ρ1)(R−ωγ)(ωt−ω)ξt+(ωπ−ωtπt)λσϕ2Rqt, which delivers after some elementary transformations (noticing that ωπ−ωtπt=ωπgt and ωt−ω=σϕ2στ2−σϕ2ωπgt, where g_t is as described in (16) of Proposition 2 our result.

Lemma 1In the environment of Proposition 1, assuming that each type of trader has equal wealth Wtϕ=Wtτ at time t, the certainty equivalent difference in the value of fundamentalists’ relative to technical traders information, ctinfo is given by: ctinfo=ctϕ−ctτ=λσϕ22(xtϕ−xtτ)2>0, where xtϕ−xtτ=(στ2−σϕ2)qt+(R−γ)Rρ1λ(R−ωγ)(R−ρ1)ξtπστ2+(1−π)σϕ2 and ξt≡d¯t−E(d¯t|Ωtτ).

Proof. From Assumption 3 the conditional certainty equivalent of the trader i, cti, is cti=E(Wt+1i|Ωti)−λ2var(Wt+1i|Ωti), for i={τ, ϕ}. Using the law of motion of wealth, i.e., Wt+1i=(Wti−xtipt)R+xti(pt+1+dt+1), the certainty equivalent is equal to (Wti−xtipt)R+xti(Ept+1+dt+1|Ωtϕ)−λ2(xti)2σϕ2, for i={τ, ϕ}. By assuming equal wealths (i.e., Wtϕ=Wtτ) the difference in the certainty equivalent between rational traders takes the form: ctϕ−ctτ=(xtϕ−xtτ)[E(pt+1+dt+1|Ωtϕ)−Rpt]+λ2[(xtτ)2−(xtϕ)2]σϕ2. From the formula for the demand of fundamentalists, xtϕ, we have that E(pt+1+dt+1−Rpt|Ωtϕ)=λσϕ2xtϕ, which we substitute in the previous expression and obtain that ctϕ−ctτ=λσϕ22(xtϕ−xtτ)2. In order to find the difference between the demand of rational traders we use the market clearing equation, i.e., πxtϕ+(1−π)xtτ=qt, which can be transformed to xtϕ−xtτ=1π(qt−xtτ), so the certainty equivalent differential can be written as:

(40)ctϕ−ctτ=λσϕ22π2(qt−xtτ)2. (40)

Because xtτ=1λστ2E(pt+1+dt+1−Rpt|Ωtτ), we have to calculate E(pt+1+dt+1−Rpt|Ωtτ). This is given by (24, Section A) as: E(pt+1+dt+1−Rpt|Ωtτ)=ωσϕ2λπE(qt|Ωtτ). Because E(qt|Ωtτ)=qt−[ζt−E(ζt|Ωtτ)], the demand of technical traders, xtτ, is equal to ωσϕ2πστ2qt−ωσϕ2πστ2[ζt−E(ζt|Ωtτ)]. By utilizing now (27, Section A) we obtain that xtτ=ωσϕ2πστ2qt−ωστ2(R−γ)Rρ1λ(R−ωγ)(R−ρ1)ξt. By substituting this in (40) we obtain our final result.■

Lemma 2In the environment of Proposition 1 it holds that: E[ζt+s−E(ζt+s|Ωt+sτ)|Ωtϕ]=γsE[ζt−E(ζt|Ωtτ)], |γ|<1.

Proof. From Assumption 5d we have that prices (as well as deterministically detrended prices) and noise in the supply are jointly normal distributed. This is formally written as: [ζtP¯]~MVN([00], [σζ2cζpc′ζpΣP¯]), where P¯=[p¯t p¯t−1 p¯t−2 ..]′ stands for all current and past detrended prices, ΣP¯=E(P¯P¯′) stands for their variance-covariance matrix (time invariant) and c_ζp stands for the row vector of the covariances of ζ_t with p¯t−s, s>0. In particular, because ζ_t is independent of past prices, cζp has a single nonzero element, i.e.: cζp=[cov(ζt, p¯t) 0 0 0 .]. From the properties of the multivariate normal distribution, the distribution of ζ_t conditional on P¯ is normal and has a mean E(ζt|P¯) that is given by: E(ζt|P¯)=E(ζt)+cζpΣp−1P¯=cζpΣp−1P¯=∑s=0∞csp¯t−s where cs=cov(ζt, p¯t)(ΣP¯−1)1,s and (ΣP¯−1)1,s denotes the element of the inverse variance covariance matrix (ΣP¯−1) in row 1 and column s. Since P¯ coincides with the information set of technical traders, we can write that E(ζt|Ωtτ)=∑s=0∞csp¯t−s. Substituting the last expression in equation (26, in the proof of Proposition 1, Section A) we obtain:

(41)p¯t=1RE(p¯t+1|Ωtϕ)+1Rρ1d¯t+(1−ω)λRπσϕ2∑s=0∞csp¯t−s−λRπσϕ2ζt (41)

and by forward iteration we obtain that E(p¯t+1|Ωtϕ)=(∏j=1T−1κj)E(pt+T|Ωtϕ)+∑h=0T−2(ρ1T−h(∏j=h+1T−1κj))d¯t+(1−ω)λπσϕ2∑s=0∞(∑h=0T(cT−h(∏j=h+1T−1κj)))p¯t−s where the κ’s are constants whose general formula is given (recursively) by: κs=aR×(1 −κs−1ab1 −κs−1κs−2ab2 −κs−1κs−2κs−3ab3−..)−1,a=11−(1−ω)λRπσϕ2c0,bi=(1−ω)λRπσϕ2ci and c_i has been defined previously. Provided that limT→∞(∏j=1T−1κj)=0, the limit of the previous expression becomes: E(p¯t+1|Ωtϕ)=φ0d¯t+(1−ω)λπσϕ2∑s=0∞φ1sp¯t−s, where φ0=limT→∞∑h=0T−2(ρ1T−h(∏j=h+1T−1κj))<∞ and φ1s=limT→∞∑h=0T(cT−h(∏j=h+1T−1κj))<∞. Substituting E(p¯t+1|Ωtϕ) in (41) and rearranging terms we obtain that p¯t−(1−ω)×λRπσϕ2×∑s=0∞(cs+φ1s)×p¯t−s=1R[φ0+ρ1]d¯t−λRπσϕ2ζt. Since p¯t−(1−ω)×λRπσϕ2×∑s=0∞(cs+φ1s)p¯t−s∈Ωtτ, the previous expression can be equivalently written as yt=θ0d¯t+θ1ζt, where θ0=1R[φ0+ρ1] and θ1=−λRπσϕ2, so the conditions of the Lemma 3 (presented below) are met and the proof holds from Lemma 3d.■

Lemma 3Consider the state space representation of a system that consists of the state equation: d¯t+1=ρ1d¯t+εt+1 and the observation equation: yt=θ0d¯t+θ1ζt, where ρ1∈(0,1]. The state equation describes the law of motion for the unobserved state variable d¯ (Assumption 4a) over time (extended to include the case ρ1=1) and the observation equation defines the relationship between the observed variable y and the state variable. Define the prediction error ξt as follows:

(42)ξt=d¯t−E(d¯t|Ωtτ), (42)