Non-Information Trading Risk

R.Carrizosa

A. Dontoh

J. Ronen

April, 24th 2008

Preliminary - Please do not quote

Department of Accounting

Stern School of Business, New York University

New York, NY 10012.

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Abstract

In this paper we present a model that shows volume to be indirectly related to a risk factor

that explains expected returns. Specifically, our analysis suggests that trading that is not

based on information on the fundamental value of the stock (non-information based trading)

contributes to equilibrium returns variance and hence increase risk premia. Within a noisy

rational expectations model we show that NIB trading, to the extent it is undiversifiable,

introduces risk in that it brings about common volatility in returns that is not captured fully

in the markets return variation or the Fama-French factors. Comparative statics show that

the variability of NIB trading is positively associated with expected returns. To the extent

that NIB trading is positively correlated with total volume (scaled by shares outstanding),

our model’s inferences are consistent with findings cited above: positive contemporaneous

association between trading volume and average returns.

JEL Classification: M41 - AccountingKey Words: Noisy Rational expectations equilibrium, Non-information based trading, RiskPremium, Returns Volatility.

1. Introduction

Past studies have identified much regularity in average stock returns. For example, DeBondt

and Thaler (1985) document long-term price reversals where long-term past losers outperform

long-term past winners over the following three to five years. Similarly, Jegadeesh (1990)

and Lehmann (1990) report price reversals at monthly and weekly intervals. In contrast,

Jegadeesh and Titman (1993) find that short-term returns tend to continue; that is, stocks

with higher returns in the previous 12 months tend to have higher future returns., Fama

and French (1996) show that a three-factor model of returns does not explain intermediate-

Horizon price momentum. Lee and Swaminathan (2000) introduced trading volume as a

link between ”momentum” and ”value” strategies. Specifically, they show that firms with

high (low) past turnover ratios exhibit many glamour (value) characteristics, earned lower

(higher) future returns, and have consistently more negative (positive) earnings surprises over

the next eight quarters. They also show that high (low) volume winners (losers) experience

faster reversals.1

Contrary to the liquidity explanation (Amihud and Mendelson, 1986), Lee and Swami-

nathan (2000) find that high (low) volume stocks earned higher (lower) average returns in

each of the five years prior to portfolio formation, that trading volume is only weakly cor-

related with traditional liquidity proxies and that the volume effect is robust to various risk

adjustments. The authors’ explanation attributes the effect of volume to providing informa-

tion about relative under-or overvaluation of stocks: that high-volume stocks are generally

glamour stocks and low-volume stocks are generally value or neglected stocks.

In this paper we present a model that shows volume to be indirectly related to a risk

1Prior research on the effects of volume (e.g. Datar, Naik and Radcliffe (1998) show that low (high) volume firms earnhigher (lower) future returns. Lee and Swaminathan (2000) showed that the volume effect is observable over the next three tofive years and is more pronounced among the extreme winner and loser portfolios; they claim their evidence contradicts thecommon interpretation of trading volume as simply a liquidity proxy.

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factor that explains expected returns. Specifically, our analysis suggests that trading that

is not based on information on the fundamental value of the stock (non-information based

trading, henceforth NIB) captures common variation in returns incremental to the often

used four factors (the excess return on a broad market portfolio (Rm-Rf); the difference

between the return on a portfolio of small stocks and the return on a portfolio of large

stocks (SMB, small minus big); the difference between the return on a portfolio of high-

book-to-market stocks and the return on a portfolio of low-book-to-market stocks (HML,

high minus low); and a momentum factor. Within a noisy rational expectations model we

show that NIB trading, to the extent it is undiversifiable, introduces risk in that it gives rise

to common volatility in returns that is not captured fully in the markets return variation.

Comparative statics show that the variability of NIB trading is positively associated with

expected returns. To the extent that NIB trading is positively correlated with total volume

(scaled by shares outstanding), our model’s inferences are consistent with findings cited

above: positive contemporaneous association between trading volume and average returns.

It is well known in Finance and Economics that in market equilibrium settings where

traders learn from price in forming expectations about the fundamental values of assets, the

presence of liquidity or noise traders (i.e. traders who base their trading on factors other

than information about the fundamental value of the asset) resulting in noisy equilibrium

prices. Such non-information based trading is a fundamental observed feature of securities

markets which Grossman (1995) characterized as follows

“In general, there may be many reasons for trade other than information. After all, the

traditional view of the market is of a location where resources are reallocated. Reasons forthese non- informational trades include cross-sectional changes in wealth, risk-preferences,liquidity needs, unanticipated investment opportunities and all other factors that do notdirectly relate to the payoffs of traded securities.”

For instance, in response to random shocks in their wealth or preferences, traders may

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re-optimize their global portfolios including non-financial assets. The results of such reop-

timizations, when restricted to a single market such as the stock market, may appear as

random perturbations in asset-holdings that are unrelated to information about underlying

market values. A similar notion is embedded in the concept of market created risk succinctly

stated by Krause and Smith (1989, p. 558) as follows:

“. . . however, uncertainty about future prices can also reflect uncertainty about whatwe call the ’state of the market’: the beliefs, preferences and endowments of the otherparticipants in the economy. Even if all investors’ probability beliefs about ultimate payoffs,as well as the knowledge that these beliefs would not change in the future were common

knowledge, uncertainty about future prices would still be present as long as investors hadimperfect information about the state of the market. We refer to this source of uncertaintyas ‘market created risk’ to emphasize that its source is investors themselves, rather than the

stochastic process describing the ultimate cash payouts to securities.”

Also relevant is the observation by Black (1986, p. 533) that:

“volatilities will change over time. The volatility of the value of a firm is affected by thingslike the rate of arrival of information about the firm and the firm’s leverage. All the factorsaffecting the volatility of the firm’s value will change. The volatility of price will changefor all these reasons and for other reasons as well. Anything that changes the amount or

character of noise trading will change the volatility of price.”

1. 1. Definition of NIB Risk

For the purposes of this study, we define the information content of both accounting numbers

and prices as the degree to which these measures (earnings or prices) reflect the fundamental

value of the firm. In other words, the information content of a signal is determined by its

ability to predict future flows. Thus, an increase in NIB trading would cause the predictive

content of prices or accounting numbers to degrade. Our analytical results show that an

increase in NIB trading, such as those by liquidity and noise traders, causes stock prices, but

not accounting information, to become more noisy and hence less informative about future

payoffs.

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Consequently, an increase in NIB trading increases the volatility of stock prices. The

process works as follows: NIB induced noise in prices distorts the theoretical relationships

between prices and fundamental value and causes the equilibrium price variance and return

volatility to increase resulting in higher firm risk. To the extent that the NIB trading noise

is not ideosyncratic and hence cannot be diversified away by forming portfolios, thereby

inducing investors to demand a higher expected return to compensate for the added risk.

2. The Model

To formally derive such price behavior in equilibrium, we require a multi-period economy

with risky assets and many traders who anticipate a future public information disclosure. In

addition, the public disclosure must be accompanied by a round of trade and a new disclosure-

contingent price. We construct such a model using the noisy rational-expectations framework

wherein additional information disclosures in each period moves observed equilibrium prices

closer to the fundamental value of the asset, x. The innovation of our model is the occurrence

of new NIB shocks that affects prices every period. This limits the ability of equilibrium

prices to convey information about fundamental values.

We consider a three-date, two-trading-rounds, noisy rational expectations equilibrium

model of trading and prices with a risky asset and many traders similar to Dontoh et. al

(2003). The risky asset has a normally distributed liquidating value of x units (per share),

with mean 0 for simplicity and variance σ2x. Each trader has access to three sources of infor-

mation about the liquidating value of the risky asset, x. They acquire private information,

yi0 , before markets open for trade. Private signals about asset values are identically and

independently distributed across traders, and given by yi0 = x + εi where εi ∼ N(0, σ2εi),

and are independently distributed from x. For simplicity, we assume that the precision of

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the private signal is identical across investors, which implies that σ2εi

= σ2εj

= σ2ε for all

individuals i and j.

One round of trade then takes place with the equilibrium price P0 at time t0 providing

an additional source of information. The demands for the risky and riskless securities are

chosen to maximize the expected utility of end-of-final-period-wealth with the knowledge

among traders that there will be a further round of trading following anticipated future

public disclosure available at time t1. The public information disclosure at time 1 is denoted

by y1 and is defined as: y1 = x + u1.

The public signal y1 reflects the liquidating value with noise u1 in which u1 ∼ N(0, σ2u).

The risky asset yields a payoff of x at time t = 2, when the final period wealth Wi2 is

consumed. Wi2 therefore consists of the returns on investment in the risky asset in periods

0, 1 and 2. Denoting these investment levels by zi0 and zi1, the realized returns on these

holdings are: zi0(P1 − P0) and zi1(x − P1) respectively. Therefore, ending wealth Wi2 can

be expressed as:

Wi2 = zi0(P1 − P0) + zi1(x− P1).

Traders’ utility functions are negative exponential in end-of-final-period wealth Wi2, with a

constant absolute risk aversion coefficient ρi expressed as:

EU(Wi2) = −E[

exp(−ρi

{zi0(P1 − P0) + zi1(x − P1)

}) ].

Each trader has access to private and public information sources. Private information, yi0 ,

is acquired at time t = 0; public announcement y1 is available at t = 1. In addition to

the private information, yi0 , is acquired at time t = 0; public announcement y1 is available

at t = 1, the equilibrium prices P0 and P1 (at times t = 0, 1 respectively) also provide

information to the traders. Therefore, the information sets available to individual i at times

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t = 0, 1, denoted by Ii0 and Ii1, respectively, are given by:

Ii1 = {yi0 , y1, P0, P1, }; Ii0 = {yi0 , P0, }.

As is standard in rational-expectations models (see, for instance, Admati [1985]), we

assume that all variables are jointly normal and that the equilibrium we seek involves price

functions that are linear in signals and aggregate supply of the risky asset. We shall also

assume, as is common in rational expectations studies, a “large” economy where individual

traders are price-takers; the average of the traders’ private information is the true underlying

asset value x and the average of the trader’s net demands (or supplies) is equal to the per-

capita excess supply (or demand) Zt in period t. 5 The determination of an equilibrium

begins with a specification of beliefs regarding the relationship between information and

prices. The key assumption is that these conjectures are linear and may be written, using

suitable coefficients Aij, as:

P0 = A01 x − A02 Z0,

P1 = A11 x + A12 y + A13 P0 −A14 τ (1)

The solution then proceeds through backward optimization beginning with the determination

of price and trader-demands following the second public announcement. We then solve for

individual demands in earlier periods, treating future demands as random variables. Then

we use market clearing to derive equilibrium prices. Finally, we ensure that the equilib-

rium prices are consistent with the original beliefs regarding the structure of prices (as in

1 above). However, the solution procedure is technically complex and it is difficult to rule

out (multiple) equilibria, some of which are supported by implausible beliefs. For this rea-

son, we restrict attention to equilibria where the price evolution follows the plausible pattern5If there are N traders, in equilibrium, the average per-capita noisy supply (or demand), Zt, satisfies Zt = 1

N

∑n

i=1zit

where zit denotes trader i’s demand in period t. It is standard to extend this approach to a continuum of traders and write

Zt =∫ 1

0zitdi and x =

∫ 1

0yi0di despite some associated technical complications (see Judd [1985]). We note that our general

results will hold with finitely many traders as long as individual traders behave as price-takers.

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of evolving towards the true fundamental value as summarized in our first assumption below.

Assumption 1 (Belief Evolution)

The coefficients in Equation 1, Aij, are all positive.

The key point is that the condition, At1 ≥ 0 ensures that the information in the market

clearing price Pt concerning x, on average, is positively correlated with x. For example,

E[x|y, P0, P1] − E[x|y, P0] = A11(x − E[x|y, y2, P0, P1])

is positively (or zero) correlated with x as long as A11 ≥ 0. The requirement that the other

coefficients, Atj, j ≥ 1 be positive states that equilibrium prices behave “normally” in the

sense that they have a positive relation with information signals and a negative relation with

the excess supply (see also the solution in Section 4.).

2.1 Structure of the NIB Supply

The linear rational expectations structure requires that the aggregate per-capita supply, Zt,

be distributed as jointly normal variables and independent of all other information variables.

Given the joint normality, we can always write aggregate supply of the risky asset at time t,

in general, as a function of a persistent noise component Z0, and period specific transitory

component τt. Let Z0 = C0/λ and let

Zt = γt,t−1Zt−1 + γt,tτt

= γt,0C0 + γt,1τ + γt,2τ2 + · · ·+ γt,kτk + · · · + γt,t−1τt−1 + γt,t (2)

for 0 ≤ k ≤ t, where for an appropriate choice of λ such that 0 ≤ λ ≤ 1 (see lemma 1 below).

The main results in the paper can be derived with very minor restrictions on the general

linear structure of NIB trading outlined in the following assumptions as stated below. As

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formally introduced below in Assumptions 2 and 3, we impose certain restrictions on the

γt,k coefficients in Equation (2) that ensure that the noise process has desirable economic

properties.

2.2 Assumption 2 (Decay of NIB Supply Shocks)

Effects on NIB of period specific transitory supply shocks , τ , as well as of the persistent

component, Z0, decay over time such that for t ≥ 1 and k ≤ t, we have 0 ≤ γt,k < γt,k+1 ≤ 1

Assumption 2 ensures that the effects of previous transitory NIB supply shocks as well as

the persistent component Z0 diminish with the passage of time. This condition guarantees

that temporal transitory supply shocks, τt continue to affect the NIB supply in subsequent

periods but with diminished effect.

The supply of the risky asset, Zt, should be interpreted as the number of shares made

available for trade by NIB traders; i.e., it represents the number of shares that NIB traders

choose not to hold in a given period, and hence should be absorbed by the informed, non-

NIB traders. The specification in Assumption 2 accords the temporal evolution of Zt a

considerable degree of generality. For example, γt,k = 0 when k > 0 implies that the number

of shares NIB traders make available for trade is constant across periods. On the other

hand, γt,k = 1 implies that the change in the number of shares NIB traders make available

for trade is independently determined each period; and finally, γt,k ∈ (0, 1) implies that

the fraction γt,k of shares made available for trade (and hence not held) by NIB traders in

period k continues to be made available for trade in period t, in addition to a random change.

2.3 Variance of the NIB Supply

We consider a setting where the (common knowledge) uncertainty introduced by NIB

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noise is exogenous and has the following distributional properties:

Z0 ∼ N(0, σ2Z0

); τt ∼ N(0, σ2τ ) for all t

and where τt is independent of Z0 as well as τt−k, for every k ≤ t. This is the sole information

that traders possess about Zt at all times.

Assumption 3 - Constant Variance

The aggregate uncertain supply is such that

σ2Zt

= σ2Zt−k

for all t ≥ 1 and any 1 ≤ k ≤ t.

Assumption 3 formalizes the notions that (1) the uncertainty inherent in Zt remains con-

stant over time. The following lemma establishes existence of parameter sets that satisfy the

conditions of Assumption 2 and Assumption 3.

Lemma 1“The conditions of Assumption 2 and Assumption 3 are satisfied by the following set of

parameters:

γt,k = bt−k√

(1 − b2) and λ = 1√(1−b2)

, for every t ≥ 1 and any 0 ≤ k ≤ t, such that

Zt = γt,0 λZ0 +t−1∑k=0

bt−k√

(1 − b2)τk +√

(1 − b2)τt, where 0 ≤ b ≤ 1, (3)

whenever the NIB supply follows an AR1 (first order autoregressive) stochastic process of the

form: Zt = bZt−1 + γt,tτt .

Proof: See Appendix A

The conditions identified in the lemma are quite general. For example, b = 1 yields

Zt = Z0 for every t. That is to say, the NIB supply shock is comprised of the initial

persistent shock, non-decaying with the passage of time. The NIB supply shock will be

perfectly correlated across time as in the equilibrium analyzed in Kim and Verrecchia (1991).

At the other extreme, when b = 0, we get Zt being i.i.d and the NIB supply shock is

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comprised of only period specific transitory components as in the equilibria analyzed in Kim

and Verrecchia 1997.

3. Noisy Rational Expectations Equilibrium

This section develops the relationship between prices and demands arising from utility max-

imizations under a rational expectations structure. 6

3. 1. Equilibrium Prices in Period t = 1

Standard results show that there exists a noisy rational-expectations equilibrium at time 1.

The equilibrium price P1 and associated optimal demand configuration {zi1} are given by:

zi1 =1

ρiV1

[E(x|Ii1) − P1

]

where V1 is the variance of E(x|Ii1). This conditional expectation is linear in the information

variables. The conditional expectation can be evaluated directly (see Appendix B), and

aggregating across traders and applying market clearing (that is, setting average demand

equal to the per-capita noisy supply), yields a solution:

P1 = A11 x + A12 y + A13 P0 − A15 τ (4)

that is consistent with the conjectures stated earlier.

3. 2. Equilibrium prices in Period t = 0

At t = 0, both the demands at time 1 have to be treated as random variables. The associated

equilibrium price is of the same form as the conjectured relation:

P0 = A01 x − A02 Z0 (5)

Details of the solution derivation are discussed in Appendix C.

6These results are general and hold for any equilibrium that satisfies Assumption 1.

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4. Equilibrium Solution

It is possible to derive a closed-form equilibrium solution with prices explicitly determined in

terms of the parameters. Under the conditions established in Assumptions 1-3, we show in

Appendices B and C that there exists a Noisy Rational Expectations Equilibrium demands

zit and prices Pt where:

zi1 =1

ρi[s(yi0 − bq0 − (1 − b)q1)]

zi0 =1

ρi[s(yi0 − q0)]

and where

P1 = V1 [sx + suy1 + sq0q0 + sq1q1] − ρV1Z1. (6)

P0 = Kx + (T + α11)Z0. (7)

Where the T, K and α11 are all positive and their closed-form expressions are provided in

Appendix C. The remaining parameters are expressed as follows:

sx = 1σ2

x; s = 1

σ2ε; su = 1

σ2u1

= 1σ2

u2

; sZ0 = sZ = 1σ2

Z0

= sτt = sτ , q0 = x− ρsZ0;

q1 = x − ρs

(√1+b1−b

)τ.

ρ is the average level of risk-tolerance (see Appendix A), b is the autocorrelation parameter

in NIB trading and s0 and s1 are the associated “precisions” and expressible as: s0 =

s2sZ

ρ2 ; s1 = s2(1−b)sτ

ρ2(1+b)and

V1 =1

sx + s + s0 + s1 + su; V0 =

1

sx + s + s0

respectively denote the posterior variances of x at time t=1 and t=0. Derivation of the

equilibrium solutions are provided in Appendices B & C.

Apart from providing comparability with earlier studies, this equilibrium has the addi-

tional desirable property of being “continuous” at b = 0. That is, setting b = 0 in the

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solutions in (1) yields the standard equilibrium with constant noise Z0 and no transitory

shocks τt. Under the equilibrium price structure described in (1), Assumption 1 may be

verified directly.

5. NIB Trading Noise and Expected Returns

In the following discussion, we investigate the the effect of the NIB trading noise on expected

returns. Gross asset returns at time 1 can be decomposed into two components: expected

and unexpected returns as follows:

(P1 − P0)

P0=

(E(P1|Ii0) − P0))

P0+

(P1 − E(P1|Ii0))

P0(8)

where(E(P1|Ii0 )−P0)

P0is the expected time 1 return predicted at time 1 and

(P1−E(P1|Ii0 ))

P0is

the unexpected return. We wish to examine the effect that NIB trading has on expected

returns. To do so, we rearrange the preceding expression to get

(E(P1|Ii0) − P0)

P0

=(P1 − P0)

P0

− (P1 − E(P1|Ii0))

P0

(9)

In other words, expected return is simply the total or gross return less the portion that

was not anticipated i.e., the abnormal return. We show in the following proposition that the

above expression (9) increases in NIB trading holding gross returns constant.

Proposition 1For a given total return, the equilibrium expected return is increasing in the magnitude of

NIB trading.

Proof: Appendix D

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The result follows from the fact that the equilibrium price variance increases in NIB

trading. Hence, high NIB trading activity results in higher firm risk. As a further investi-

gation of the effects of NIB trading noise on firm risk, we conducted a numerical simulation

of the relationship between the equilibrium expected return(E(P1|Ii0 )−P0)

P0and the variance

of the NIB trading noise. The purpose of the simulation is to investigate if the volatility

(variance) of NIB trading contributes to the equilibrium returns variance and hence the risk

premium associated with the firm. To perform the simulation, we set all other parameters

and variables to unity with the exception of b that was set to its midpoint value of 0.5.

[INSERT FIGURE 1 ABOUT HERE]

The simulation demonstrates that firm risk, measured by the variance of equilibrium

returns, is increasing in the NIB trading variance. If NIB trading contributes to systematic

price volatility, investors would demand higher expected to compensate for the increased

risk. While our model features a single risk asset, it is plausible to believe that at least part

of this NIB risk is non diversifiable. For example, cross sectional changes in wealth may cause

resource reallocation across economy sectors with some resources invested or withdrawn from

the stocks of different industries or sectors. These capital movements may be manifested

in common variability across firms within a sector or industry but not necessarily in its

variability with the stock market as a whole. Hence, such common variability may not be

captured by the usual measures of systematic risk (β). This is however an empirical issue

which we investigate in the following sections.

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6. Empirical Calibration

In the following empirical analysis, we investigate whether NIB trading does contribute to

overall firm risk and whether this risk is indeed priced. Specifically, we investigate whether

the magnitude of NIB trading can explain observed abnormal returns after controlling for

other risk factors. To do so, we estimate abnormal returns using the Fama-French four

factor model. Briefly, the the Fama-French model expands the classical CAPM expected

returns model to include three additional risk premium factors: the difference in returns

between small and large firms (SMB); the difference in returns between high and low market

to book firms (HML) and a momentum factor UMD to proxy for the tendency for positive

serial correlation in stock returns in addition to the CAPM excess return factor (Rm-Rf) as

follows:

r − Rf = α + β1(Rm −Rf) + β2xSMB + β3xHML + β4UMD (10)

where r is the portfolio’s return rate, Rf is the risk-free return rate, and Rm is the market

return. Together, β1(Rm−Rf)+β2xSMB+β3xHML+β4UMD represent the risk premium

associated with the portfolio excess return (r−Rf) and α is the ”unexplained” or abnormal

return. The ”four factor” beta is similar to the classical beta with three additional factors.

SMB and HML stand for ”small [cap] minus big” and ”high [book/price] minus low”; they

measure the historic excess returns of small caps and ”value” stocks over the market as a

whole.

6. 1. Calculation of the NIB metric - TURN .

Using analyst earnings (actual and forecast) data from I/B/E/S, and trading volume from

CRSP, we obtain firm level measures of NIB for each year from 1995 to 2006. The model

used to estimate NIB relates trading volume to variables designed to capture firm specific

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information. The firm-year measure of NIB is defined as the component not explained

by the information variables (α1 for annual NIB). To facilitate estimation, we limit the

sample to firms with a minimum of 5 analyst revisions for the upcoming fiscal year end,

at least 3 analysts issuing earnings forecasts in the year, and a minimum of 125 days of

trading volume. The independent variables include earnings surprises (with 2 leads/lags),

and separate forecast revision variables for each analyst that issues a revision in the year

(with 1 lead/lag). The number of independent variables varies across firm years depending

on the number of analysts covering each firm. Using daily observations, we estimate the

following for each firm-year:

TURNt = α1 + α2SURPt−2 + α3SURPt−1 + α4SURPt + α5SURPt+1 + α6SURPt+2

+∑K−1

i=0 [α3i+7REVit−1 + α3i+8REVit + α3i+9REVit+1] + εt

where SURPt = |EarnActualt−EarnMeanFcastt |Pt−1

, TURNt = SharesT radedt

SharesOustandingt,

REVit = |FCASTit−FCASTit−1|Pt−1

,

K=number of analysts issuing forecasts for the firm

6. 2. Discussion of Results.

Table 1 shows that the correlation matrix between the NIB trading variable (TURN) ,

unexpected and raw returns, and the Fama-French return factors. With the exception of the

momentum factor, the NIB trading variable is significantly correlated with the Fama-French

factors. The significance of the correlation with α, the abnormal return unexplained by

the Fama-French Four Factor model, suggests that NIB trading is a potential risk premium

factor that can be used to improve calculation of expected returns. Figure 2 shows that NIB

trading decreases in firm sized measured by total assets. This implies that NIB trading is

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greater in smaller, possibly more risky firms. Figure 3a and Figure 3b seem to support this

interpretation by showing NIB to be increasing in portfolios with higher earnings volatility

(Figure 3a) and high growth firms, i.e., firms with high market to book ratio (figure 3b).

Figures 2, 3a and 3b suggest that NIB trading is indeed related to systematic firm risk

factors.

To explore this, we first investigate the relation between NIB trading and (raw) returns.

The results are presented in Figure 4. It shows that total returns are generally increasing in

the level of NIB trading. To rule out the possibility that NIB trading might just be proxying

for the Fama-French factors, we run the Fama-French Four Factor model in expression (10)

and regressed NIB trading on the abnormal return component not explained by their model.

The results are presented in Figure 5. It shows that NIB trading is increasing in the Fama-

French abnormal return component and hence it represents risk premium factor not captured

by the Fama-French model.

7. Conclusion

The paper has presented a model that shows NIB trading volume to be indirectly related to

a risk factor that explains expected returns. Both the analyticl and the empirical calibration

suggest that NIB trading captures common variation in returns incremental to the often

used Fama-French four factor expected returns model. Within a noisy rational expectations

model, we show that NIB trading, to the extent it is undiversifiable, introduces risk in

that it gives rise to common volatility in returns that is not captured fully in the markets

return variation. Comparative statics show that the variability of NIB trading is positively

associated with expected returns. To the extent that NIB trading is positively correlated

with total volume (scaled by shares outstanding), our model’s inferences are consistent with

18

findings cited above: positive contemporaneous association between trading volume and

average returns.

19

APPENDIX

A Proof of Lemma 1

Proof:

The first part of the proof is straightforward. For a given period t, it is easy to see that

since b ≤ 1, the loading on the transitory component τ , γt,k = bt−k√

(1 − b2), decreases with

k for a given t, since t ≥ 1 and 0 ≤ k ≤ t, . This reduces the weight assigned to earlier

realizations of τ . Similarly, the loading on the persistent component Z0 is such that γt,0 =

bt√

(1 − b2) < γt−k,0 = bt−k√

(1 − b2) and hence decreases as k decreases thereby satisfying

the conditions in Assumption 2. Next, we need to show that σ2z0

= σ2Zt

for all t ≥ 1. Observe

that γt,0 = γt,k=0 = bt√

(1 − b2). Hence, the coefficient on Z0 is simply λ(bt√

(1 − b2))

= bt

since λ = 1√(1−b2)

. Consequently, for any k ≥ 0, σ2Zt−k

= b2(t−k)(1− b2)σ2τt−k

. Making use of

the assumption that σ2z0

= σ2τt

for all t, we get

var(Zt) = σ2z0

(b2t + b2(t−1)(1 − b2) + b2(t−2)(1 − b2) + · · · + b2(t−(t−1))(1 − b2) + (1 − b2)

)= σ2

z0

((b2t − b2(t−1)+2) + (b2(t−1) − b2(t−2)+2) + · · · + (b2(t−(t−1)) − b2) + 1

)

= σ2z0

(1 +

t∑k=1

(b2(t−(k−1)) − b2(t−k)+2)))

= σ2z0

(A1)

for every Zt t ≥ 1 and for any 0 ≥ b ≤ 1.

B Deriving the Equilibrium

We begin by outlining the technical result that underlies all our computations. Let Θn

represent coordinates in �n, A a positive definite symmetric n × n matrix, Qn an arbitrary

n-vector and C some constant. Then the integral:

∫�n

exp{−1

2(Θ′

nAΘn + 2Q′nΘn + 2C)

}dΘn

may be evaluated as: √(2π)n|A| exp

(1

2Q′

nA−1Qn − C)

(A2)

20

This evaluation is based on factoring the positive definite symmetric matrix A as ΦΦ′ and

setting Un = Φ′Θn + Φ−1Qn (Anderson 1971).

The derivation of the equilibrium involves working backwards starting with period 2 and

ending at period 0. We therefore begin by applying this technique to the decision problem

faced by a typical trader i at the second period.

B.1. Derivation of time 2 risky-asset demands

The general method outlined above is needed mainly for calculating time 1 demands. How-

ever, we begin by applying it at time 2, where more direct procedures are available, so we

can illustrate the technique in a univariate setting. The final wealth of trader i (where we

have taken the initial endowment to be 0), Wi2, is given by:

Wi2 = zi0(P1 − P0) + zi1(x − P1)

= zi0(P1 − P0) + zi1(E(x|Ii1) − P1) + zi1(x −E(x|Ii1))

and trader i’s decision problem at time 1 is to maximize E[−exp(ρiWi2)|Ii1] by suitably

choosing zi1.

E( − exp {−ρiWi2 } |Ii1)

=−1

|Ω1| 12 2π12

∫ ∞

−∞exp

{−ρi [Wi1 + zi1(E[x|Ii1] − P1) + zi1Θi1] −

1

2

[Θ′

i1Ω−1

1 Θi1

] }dΘ

where: Θi1 = x−E(x|Ii1), Ω1 = V1 = var(x|Ii1), Wi1 = zi0(P1 − P0) + zi1(P1 − P1). Setting

A1 = Ω−11 , Qi1 = ρizi1 and Ci1 = Wi1 + zi1(E[x|Ii1] − P1) and applying (A2),

E[−exp(Wi2)|Ii1] = −exp(

1

2ρ2

i z2i1V1 −Ci1

). (A3)

Collecting the terms that involve zi1, the problem reduces to:

maxzi1

1

2ρ2

i V1z2i1− ρizi1(E[x|Ii1] − P1).

This yields the solution in Equation A4:

zi1 = (ρiV1)−1(E(x|Ii1) − P1). (A4)

In order to compute E(x|Ii1), define:

q0 ≡ 1A01

(P0) = x − B0Z0

q1 ≡ 1

A11 −A14B0

(P1 − A12 y − A13 P0 − A14

B0q0) = x − B1τ1

(A5)

21

where B0 = A02A01

, B1 = A14B0A11B0−A14

. The qi can be used to replace Pi in the information

sets, i.e., Ii0 = {yi0, q0}, Ii1 = {yi0, y1, q0, q1}. Using this specification, the joint distribution

of x with time 1 information variables is multivariate normal with variance-covariance matrix:

∑=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x yi0 =x+εi y1=x+u1 q0=x−B0Z q1=x−B1τ1

x| σ2x σ2

x σ2x σ2

x σ2x

yi0 | σ2x σ2

x + σ2ε σ2

x σ2x σ2

x

y1| σ2x σ2

x σ2x + σ2

u σ2x σ2

x

q0| σ2x σ2

x σ2x σ2

x + B20σ

2z σ2

x

q1| σ2x σ2

x σ2x σ2

x σ2x + B2

1σ2τ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Let Σ12 ≡ Σ′21 = (σ2

x, σ2x, σ

2x, σ

2x) and: Σ22 be the 4 × 4 matrix obtained from

∑above

by eliminating the first row and first column. Then the coefficients of the conditioning

information variables are determined by Σ12Σ−122 while the posterior variance of x at time 2,

V1 satisfies var[x − E(x|Ii2)] = σ2x − Σ12 Σ−1

22 Σ21. Denoting the respective precisions by:

sx =1

σ2x

; s =1

σ2εi

; su =1

σ2u

; sq0 =1

B20σ

2Z0

; sq1 =1

B21σ

2τ

,

the coefficients in the conditional expectation may be evaluated as:

E[x|Ii1] = V1 [syi0 + suy1 + sq0q0 + sq1q1] (A6)

with the posterior variance of x, V1, given by:

V1 =1

su + sx + s + sq0 + sq1

. (A7)

Averaging (A4) over i and using market clearing now leads to:

P1 = V1 [sx + suy1 + sq0q0 + sq1q1] − ρV1Z1. (A8)

It is also worth deriving an expression for zi1. Comparing Equations (A6) and (A8) and

using Equation (A4), it follows that

zi1 =ρ

ρis(yi0 − x) +

ρ

ρiZ1 =

ρ

ρi(sεi + Z1). (A9)

22

B.2. Initial-Period Price and Demands

Trader i′s decision problem at time 0 concerns the holding zi0 which will maximize expected

terminal wealth based on the information available at time 1. Using iterated expectations,

E( − exp {−Wi2 } | Ii0) = E(E( − exp {−Wi2 } | Ii1) Ii0). Using the expression in (A3) and

substituting for the optimal zi1,

E( − exp {−ρiWi2 } |Ii0)

=−1

|Ωi1|12 2π

∫ ∞

−∞

∫ ∞

−∞exp

{−ρi

[zi0(P1 − P0) +

1

2ρiV1z

2i1

]− 1

2

[Θ′Ω−1

i1Θ] }

dΘ

where Θi = {P1 −E(P1|Ii0), zi1 −E(zi1|Ii0)} are random variables orthogonal to Ii0. Define

P 1i ≡ E(P1|Ii0) and zi1 ≡ E(zi1|Ii0) and let Ωi1 denotes their joint variance-covariance

matrix, that is,

Ωi1 =

⎛⎜⎝ var(P1 − P 1i|Ii0) cov{zi1 − zi1 , P1 − P 1i|Ii0}

cov{zi1 − zi1 , P1 − P 1i|Ii0} var(zi1 − zi1|Ii0)

⎞⎟⎠ denoted by

⎛⎜⎝σ11 σ12

σ12 σ22

⎞⎟⎠ .

In order to solve the maximization problem, first note that:

ρizi0(P1 −P0)+1

2ρ2

i V1z2i1

= ρizi0(P1 −P 1i)+ zi0(P 1i −P0)+1

2ρ2

i V1[(zi1 − zi1)+ zi1]2. (A10)

Let:

Ai1 = Ω−1i1

+

⎡⎢⎢⎢⎣

0 0

0 ρ2i V1

⎤⎥⎥⎥⎦ A−1

i1=

⎡⎢⎢⎢⎣

α11 α12

α12 α22

⎤⎥⎥⎥⎦ (A11)

zi1 = E(zi1 |Ii0); P 1i = E[P1| Ii0] (A12)

Qi1 = {ρizi0 , ρ2i V1zi1} (A13)

Ci1 = ρizi0(P 1i − P0) +1

2ρ2

i V1zi12 + ρiWi0, (A14)

where Wi0 = Bi + zi0(P1 − P0)

Individual i’s first period demand optimization problem can be expressed as:

maxzi0

−1

|Ωi1|12 2π

∫Θi

exp{−1

2(Θ′

iAi1Θi + 2Q′i1Θi + 2Ci1)

}dΘi. (A15)

23

Because Ω−1i1

is positive definite while the matrix

⎡⎢⎢⎢⎣

0 0

0 ρ2i V1

⎤⎥⎥⎥⎦ is positive (semi) definite, their

sum Ai1 is automatically positive definite. Therefore, from (A2), the maximization problem

may be transformed to:

maxzi0

−|Ai1| 12|Ωi1 |

12

exp(1

2Q′

i1A−1

i1Qi1 − Ci1

).

After substituting back for A−1i1

,Qi1, and Ci1, and gathering terms in zi1, the problem reduces

to:

maxzi0

−exp{−ρi

[zi0(P 1i − P1) − ρ2

i V1α12zi1zi1 −1

2ρiα11z

2i1

]}.

Maximizing and solving for zi0 yields: (P 1i− P0)−ρ2i V1α12zi1 −ρiα11zi0 = 0 or equivalently

zi0 =1

ρiα11

((P 1i − P0) − ρ2

i V1α12zi1

)(A16)

This equation may be rewritten as:

zi0 − czi1 =P 1i − P0

ρiα11where c = −ρiV1α12

α11.

C An Equilibrium solution

In this section we outline how to calculate a complete multi-period equilibrium that yields

the prices in Equations (6) and (7) of the paper. The procedure involves assuming the

conjectures in expression (1) and then verifying that the resulting prices are the same as

the conjectures. To compute the closed-form equilibrium solution, we to derive explicit

expressions for the coefficients αij. Denoting determinants by | · | and using standard matrix

inversion formulae, Equation (A11) yields:

α11 =σ11 + ρ2

i V2|Ωi1 ||Ωi1||Ai1|

α12 =σ12

|Ωi1 ||Ai1|α21 =

σ12

|Ωi1 ||Ai1|α22 =

σ22

|Ωi1 ||Ai1|(A17)

The denominators in the expressions above are positive because the matrices Ωi1 and Ai1

are both positive definite.

24

|Ωi1 ||Ai1| = 1 + ρ2V1var(sx − Z1|Ii0) ρiV1σ12 = ρV1cov{sx − Z1, P1|Ii0}and σ12 = cov{zi1 − E(zi1|Iio), P1 − P 1i|Ii0} and σ22 = var(zi1 − E(zi1|Iio). Note that

although the above depend on ρi, the quantities are identical across traders as a result of

the fact that the equilibrium price is independent of individual preferences and information.

Thus, c is the same for every trader.

From above, we get

α11 =σ11+ρ2

i V1|Ωi1 |1+ρ2

i V1σ22

α12 = σ12

1+ρ2i V1σ22

. (A18)

To obtain closed-form expressions of the above, we will need the definitions,

q0 = x − ρ

sZ0; q1 = x − ρ

s

⎛⎝√

1 + b

1 − b

⎞⎠ τ

and the closed-form expressions for B0and B1 which we derive as follows. Summing the

first period equilibrium demands over i, we get Z0 = sρ(x − q0) = s

ρB0Z0 which implies that

sρB0 = 1 and hence

B0 =ρ

s

Similarly, the sum of the first period equilibrium demands yields

Z1 = sρ(x− q1) = s

ρ(x − bq0 − (1 − b)q1) which simplifies to Z1 = s

ρ(bB0Z0 +(1−b)B1τ ) after

substituting for q0 and q1 from expression A5. From Lemma 1, we know that generally,

Zt = btZ0 +∑t−1

k=1 bt−k√

(1 − b2)τk +√

(1 − b2)τt, where 0 ≤ b ≤ 1,. Hence, for t = 1, 2, 3, we

have Z1 = bZ0 +√

(1 − b2)τ. Equating the one expressions for Z1, we get

s

ρ(bB0Z0 + (1 − b)B1τ = bZ0 − s

ρ(bB0Z0 +

√(1 − b2)τ

Substituting for B0 = ρs, and solving for B1 yields

B1 =ρ

s(1 − b)τ

(−s

ρbρ

sZ0 + bZ0 +

√(1 − b2)τ

)=

√(1 − b2)

sρ(1 − b)

=ρ

s

√(1 + b)√(1 − b)

We are now in a position to evaluate σ11 = var(P1−P 1i|Ii0), σ22 = var(zi1−E(zi1|Ii0)), and

σ12 = cov{zi1 − E(zi1|Ii0), P1 − P 1i|Ii0} explicitly. To calculate the variance-covariance

25

matrix associated with the variables in A16, we first record the posterior variances of x

given information set Iit :

V1 =1

su + sx + s + sq0 + sq1

and let V0 =1

sx + s + sq0

represent the posterior variance of x given the information set Ii0 at time 0. Given the fact

that

1

V1− 1

V0= su+sq1 and hence V0−V1 = V1V0(su+sq1) or equivalently V0 = V1+V1V0(su+sq1)

allows us to calculate

σ11 = var(P1 − P 1i|Ii0)

= V 21

(s2

u

(V0 +

1

su

)+ (sq1 + s(1 − b))

2

(V0 +

1

sq1

)+ 2V0su (sq1 + s(1 − b))

)

= V1su(V1V0(sq1 + su) + V1) + V1suV1V0sq1 + 2s(1 − b)V1(V1V0(sq1 + su) + V1)

+V 21

(sq1 + s2

q1V0 +

1

sq1

s2(1 − b)2 + s2(1 − b)2V0

)

= (V0 − V1) + 2V0V1s(1 − b) + V 21 (1 − b)2s2

(V0 +

1

sq1

),

σ22 = var(zi1 − E(zi1|Iio)) =1

ρ2i

s2(1 − b)2

(V0 +

1

sq1

)

σ12 = cov{zi1 − E(zi1|Iio), P1 − P 1i|Ii0}

=1

ρis(1 − b)

[V1V0su + V1((sq1 + s(1 − b)(V0 +

1

sq1

)

]

=1

ρis(1 − b)

(V0 + V1s(1 − b)(V0 +

1

sq1

)

)

Making appropriate substitutions and simplifying yield

α11 =σ11+ρ2

i V1|Ωi1 |1+ρ2

i V1σ11=

(V0 − V1) + 2V0V1s(1 − b) + V1V0s2(1 − b)2 1

sq1

V1s2(1 − b)2(V0 + 1sq1

)

α12 = σ12

1+ρiV1σ22=

sρi

s(1 − b)(V0 + V1s(1 − b)(V0 + 1

sq1))

1 + V1s2(1 − b)2(V0 + 1sq1

). (A19)

26

Let

k1 = V1 ((s + sq0)(sq1 + s + su)) , k2 = sρ(1 − b)(1 − (s + sq0))

t1 = V1B0(s + sq0 + sq1 + su) + bρ, t2 = sρ(bBo + (1 − b)sq0B0)

K = k1 + ρk2α12 and T = t1 + ρt2α12

Given that the information set {y1, P0, P1} is informationally equivalent to {y, q0, q1}, we

obtain (using Equation A16 and integrating over i)

P0 = Kx− (T + α11)Z0 (A20)

verifying the solution for P0 in expression 5. Substituting back the solution for P0 into

expression (A16) results in:

zi0 =s

ρi(yi0 − q0) (A21)

D Proof of Proposition 1

We need to show that for a fixed raw return, (P1−P0)P0

, ddτ

(E(P1|Ii0 )−P0

P0) => 0. We begin with

an orthogonal representation of expression 6 in the paper as follows:

P1 = V1 (J1x + J2u1 + J3Z0 + J4τ ) (A22)

J1 = su + sq0 + sq1 + s

J2 = su

J3 = −(sq0 + bs)B0

J4 = −(sq1 + (1 − b)s)B1 (A23)

and where B0 = ρs; B1 = ρ

s

(√1+b1−b

).

Recall that(E(P1|Ii0) − P0)

P0=

(P1 − P0)

P0− (P1 − E(P1|Ii0))

P0.

Since neither P0 nor E(P1|Ii0) depend on τ , we can differentiateE(P1|Ii0)−P0

P0with respect

to τ directly, holding (P1−P0)P0

fixed, to get

d

dτ(E(P1|Ii0) − P0

P0

) = − d

dτP1 = −V1J4 > 0. (A24)

since V1 > 0 and J4 < 0

27

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