apoorva javadekar's - comments on lewellen shanken

Learning, Asset Pricing Testsand Market Efficiency

Apoorva JavadekarPresented for EC794

Boston University

May 2, 2012

() Learning & Asset Prices May 2, 2012 1 / 22

Motivation

Large literature on stock returns predictability

D/P ratio (Campbell, Cochrane)Consumption - Wealth Ratio (Latteu, Ludvigson)Aggregate Market Liquidity (Adrian-Shin)

Main Concern- Predictability in the data 6= predictability by investors

Why? Parameter uncertainty: Agents learn about nature ofuncertainty over time and prices react to learning

Learning ⇒ future returns are a function of past prices and dividends⇒ Predictability observed in the data

But, this predictability can not be exploited by investor: He is unsureif he overestimates or underestimates the parameters


Motivating Example

suppose dividends are i.i.d over time with mean δ and variance σ2

known variance but unknown mean

After each realized dividend dt , investor updates his belief aboutmean δ.

Price of a stock is an increasing function of belief about δ

With i.i.d nature, best guess δ̃t = dt at t

If dt > δ, then price > fundamental value.

But by i.i.d nature, average dividends in the future are lower than δ⇒ negative price change in future accompanied by lower returns

Result: Higher prices predict lower returns in future and it isobserved in the data. But this predictability can not be exploitedby investor. (He is simply not sure if observed dividend is aboveor below true mean)


Few Early Remarks

Predictability arises solely from learning (Not from persistence inthe dividend process)

Predictability in data - But it can’t be exploited by investor

Predictable equilibrium returns even when investors are rational

Prices are functions of beliefs at any point in time. Beliefs areinfluenced by true DGP in long run and hence prices converge tofundamental value in LR (only if parameters are time invariant)

Conclusion: Parameter Uncertainty creates wedge betweendistribution of returns in data and distribution perceived by investors


Model - Set - up and Pricing

Riskless asset paying net return of r every period

N risky assets with i.i.d joint dividend process given by

dt v MVN[δ,Σ] (1)

Investors: Risk Neutral (we want to focus on effects of learning on toasset pricing.)

General pricing result: price is equal to expected discounted valueof future dividends. (only mean matters, not the variance)

Perfect Information: Investor knows both δ and Σ

pt = p =δ

r(2)

Price is constant

⇒ No predictability in returns (dividends are i.i.d and price isconstant)


Pricing - Parameter Uncertainty

Known Σ and Unknown δ; Investor has a diffuse (Normal) prior over δ

Posterior Distribution: after observing d1, d2,...,dt , subjectivedistribution of δ is given by

δ vs MVN

[dt ,

1

tΣ

](3)

Predictive distribution of dt+1 at time t

dt+1 vs MVN

[dt ,

t + 1

tΣ

](4)

Pricing: applying same logic

pt =1

rdt (5)


Price properties - Basic

pt =1

rdt (6)

Result 1: Equilibrium price is a function of beliefs

Result 2: Price change if average of dividend realization is differentthan current average of dividends.

Current ’best’ belief about mean level of dividend is average ofdividends observed till the date (given i.i.d nature)

Changes in price:

pt+1 − pt =1

r(dt+1 − dt) (7)


Predictability - Price Changes

Result 3: Prices are martingale under subjective distribution

E st (pt+1 − pt) =

1

rE st (dt+1 − dt) =

1

r(dt − dt) = 0 (8)

Result 4: Past dividends predict future prices under true distribution

Expected price changes under true distribution: Depends uponexpectation of dt+1 under true distribution

Et(pt+1 − pt) =1

rEt(dt+1 − dt)

=1

rEt

(δ + tdtt + 1

− dt

)=

1

r(t + 1)(δ − dt)


Predictability - Returns

Result 5: Past dividends predict future returns in the data.

Et(Rt+1) = Et(dt+1 + pt+1 − pt)

= δ +1

r(t + 1)(δ − dt)

Higher past dividends above mean predict lower future returns belowmean

Result 6: Forecast Errors are correlated with past cash flows

Define URt+1 = Rt+1 - E st (Rt+1)

Et(URt+1) =

(1 +

1

r(t + 1)

)(δ − dt) (9)

Market Efficiency ⇒ Forecast errors not predictable. But we haverationality + predictable forecast errors


Autocorrelation and Variance - Returns

Result 7: Returns are negatively autocorrelated

cov(Rt ,Rt+1) = − 1

r(t)(t + 1)σ2 (10)

Why? Higher returns today imply more likely negative surprisetomorrow which pulls down the price

⇒ Prices will be mean reverting in observed data ex - post

Volatility:

vart(pt+1) =

(1

r(t + 1)

)2

σ2 (11)


Learning - Third Factor Explaining Time Variation

Prices are backward looking

Price volatility arises as a result of learning solely: Not because ofchanges in discount rates or changes in mean growth rate of dividends

Wrong Conclusion from data: Looking back at the data,econometrician perceives changes in prices as driven by changes inexpected returns. (Econometric tests will tell that there was nochange in mean rate of dividends at any point in time)

Hence, Learning is a completely new factor explaining pricemovements (different from Campbell - Shiller decomposition)


Simulated Path of Prices

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.5

0

0.5

1

1.5

2

2.5

3Price Through Time

Time

Pric

es a

nd R

etur

ns

Student Version of MATLAB


Asset Pricing: Cross Sectional Properties

Consider a multi asset model with risk averse agent (CARA Utilityfunction)

Given his subjective expectations about returns, he chooses a M-Vefficient portfolio (Under normality of prices and returns)

Conditional Variance of Returns

vart(Rt+1) =

[1 +

1

r(t + 1)

]2Σ (12)

All the Variances and covariances are scaled up because of parameteruncertainty (No asymmetry)

⇒ β are same as in no uncertainty case


Cross Sectional Properties II

Parameter Uncertainty does not imply MV inefficiency:Investors still hold MV efficient portfolios

CAPM holds under subjective distributions

E st (Rt+1) = rpt + β [E s

t (RM,t+1 − rpM,t)] (13)

Expected ex-post deviations from CAPM under subjective beliefs iszero

But, it is not zero under true distribution: Lagged prices anddividends explain cross sectional deviation from CAPM

Why? Simply because δ is unknown


Results Till Now and Direction Ahead

Result: Past prices and dividends predict future returns under truedistribution

Assumptions: i.i.d returns, constant parameters (δ, σ), diffuse priors

Extensions:Time varying parameters for dividends: Learning is never completeInformative Priors: Agents have some information about changes inparametersGeneral Dividend processes

Goal: How does Asset prices behave in Long run?


Impact of Informative Priors

Suppose investor has prior about δ: mean of δ∗ and variance of σ2/h.

Higher the h, higher the accuracy of prior

Predictive Distribution: Average of prior and realizations of dividends

dt+1 vs N

[h

t + hδ∗ +

t

t + hdt ,

t + h + 1

t + hσ2]

(14)

Variance converges to true variance in the LR (as previously)

Impact of prior diminishes over time, but never disappears

Pricing: Under risk neutrality

pt =h

t+hδ∗ + t

t+hdt

r(15)

⇒ Prior acts exactly like observing dividends for h periods with meanof δ∗

All the results carry over


Time Varying Parameters: Set Up

Why? Otherwise uncertainty diminishes over the period ⇒ no LReffects

How?Variance of dividends is constant through timeMean: True mean is redrawn from N(δ∗, σ

s ) after every K periodsSequence of short ”regimes”

What Agent Learns in the LR? True Distribution of mean ofdividend ⇒ In the LR, prior at the start of each regime is that δ vN(δ∗, σ

s )

More complicated Assumptions ⇒ even true distribution may not beknown in the LR

Reduced Form Version: Assume that prior is that δ v N(δ∗,σh ): hcould be different than s


Time Varying Parameters: Implications

Predictive Distribution: The predictive conditional mean for nextperiod dividend after t periods in the regime

mt =h

t + hδ∗ +

t

t + hdt (16)

where dt is the average dividends in the regime

Pricing of stock

pt =1

(1 + r)K−t

(δ∗

r

)+

K−t+1∑i=1

mt

(1 + r)i(17)

Price= PV(dividends in this regime expected at mt ) + PV(perpetualstream of δ∗)


Time Varying Parameters: Challenges

Econometrics: Difficult to tell whether conditional or unconditionalcovariance is influential for empirical test

Consider a model with only one regime

Covariance of prices and future returns is conditional on current (theonly) short term mean

For multiple regime: Econometrician implicitly conditions on severalvalues of mean dividend drawn from the distribution

A better ideas: Perform simulation


Results: Simulation

Rt+1 = α + β (Dt/Pt) + εt (18)

Table : Slope Coefficients: Returns on Lagged DY

Regimes s h=16 h=25 h=49

K =38 16 0.74 0.36 -0.3725 0.95 0.70 0.0949 1.28 1.04 0.68

K =19 16 0.42 -0.31 -2.0425 1.01 0.50 -0.9049 1.64 1.30 0.44


Results: Simulation

Increase in K ⇒ increase in slope coefficients: Longer the regime,better the learning

As h increase (for a given s), accuracy of prior increase andpredictability increase

As s increase (for a given h), dispersion of the draws of meanreduces and predictability improves

h > s ⇒ economy more volatile than expected ⇒ slow learning

Future returns are decreasing function of current s/h ratio


Comments

Elegant idea

But difficult to take to data

Ideally, we would like to isolate the impact of learning on price variation(Tractability of Shiller’s approximation)Paper provides no guidance on this front.

Heterogeneous Priors:Will prices reflect ”best” (more accurate) available prior? (Recall fromMicro: Prices could be revealing information)In that case, how much influence learning of ”Naive” agent can haveon prices? Probably not much ⇒ Quantitatively may not be important


apoorva javadekar's - comments on lewellen shanken

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