Asset Pricing in Multiperiod Securities MarketsAuthor(s): Gary ChamberlainSource: Econometrica, Vol. 56, No. 6 (Nov., 1988), pp. 1283-1300Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/1913098Accessed: 15/02/2010 06:50

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Econometrica, Vol. 56, No. 6 (November, 1988), 1283-1300

ASSET PRICING IN MULTIPERIOD SECURITIES MARKETS

BY GARY CHAMBERLAIN'

The intertemporal models developed in this paper have been stimulated by the capital asset pricing model (CAPM) of Sharpe and Lintner, and the role of factor structure in Ross' arbitrage pricing theory (APT). Suppose that some (one-dimensional) stochastic process S provides a sufficient statistic for aggregate consumption. Then a (heuristic) dynamic programming argument shows that S, market wealth, and the wealth derivative of the value function (for any agent) are all locally perfectly correlated. It follows from Merton's work that there is a linear relationship between the local mean return on a security and the local covariance of that return with the return on the market portfolio. The formal development uses martingale representation and martingale projection to obtain an intertemporal CAPM; the history of a scalar Brownian motion plays the role of a sufficient statistic.

We motivate the assumption of a sufficient statistic for aggregate consumption by considering a countable set of securities whose payoffs have an approximate factor structure, where the factor components are in the information set generated by an N-dimensional Brownian motion and the idiosyncratic components are weakly correlated. The approximate factor structure on the security payoffs implies that the rates of return have (locally) an approximate factor structure. The role of the market portfolio can now be played by a set of N well-diversified portfolios.

KEywoRDs: Intertemporal CAPM, sufficient statistic, factor structure, martingale repre- sentation and projection.

1. INTRODUCTION AND SUMMARY

THIS PAPER BUILDS upon the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965), and the role of factor structure in Ross's (1976, 1977) arbitrage pricing theory. The CAPM formula crystalizes important insights about asset price relationships and has provided guidance for extensive empirical work. Our objective is to obtain a similar pricing formula in an intertemporal context.

In the static CAPM, agents are interested in certain consumption at date 0 and state contingent consumption at date T. The basic result is a linear relationship between the expected change in the price of a security and the covariance of that price change with the change in market wealth.2 Now consider an intertemporal model in which state-contingent consumption and security payoffs still occur only at the terminal date T, but securities can be traded at intermediate dates in response to new information about the security payoffs. In a dynamic program- ming approach, the agent chooses a portfolio at date ti to maximize the conditional expectation of a value function defined over wealth at ti1,. However, this value function depends also upon the information available at ti+1. Ad- ditional state variables must therefore be introduced to summarize changes in the investment opportunity set, as in Merton (1973). In Merton's continuous trading

1 1 am grateful to participants at a number of seminars for discussions and comments. Financial support was provided by the National Science Foundation, the University of Wisconsin Graduate School, an Alfred P. Sloan Research Fellowship, and by the Center for Advanced Study in the Behavioral Sciences.

2 For overviews and critiques of the CAPM, see Ross (1978) and Roll (1977). 1283

1284 GARY CHAMBERLAIN

model, the pricing formula contains covariances of price changes with the change in market wealth and also with the changes in the state variables. Implementing this approach with information on individual securities requires a large number of state variables and does not appear to yield a simple pricing formula.

In this paper, we allow for a rich evolution of information about individual securities and explore assumptions that lead to simplifications in the pricing formula. As in the CAPM, markets are frictionless, with no transactions costs and no restrictions on short sales. All agents share the same probability assess- ments. The evolution of information about the security payoffs is modeled by a diffusion process of high dimension; combined with continuous trading, this allows us to exploit the local linearity of the Ito calculus.

A powerful assumption is that there is a (scalar) stochastic process S that provides a sufficient statistic for aggregate consumption; i.e., the conditional distribution of aggregate consumption at T depends on date t information only through S,. With complete markets and with risk-averse expected utility prefer- ences over consumption claims at T, the optimal claim for any agent is some function of aggregate consumption, as in Breeden and Litzenberger (1978). Then a dynamic programming argument shows that S, market wealth, and the wealth derivative of the value function (for any agent) are all locally perfectly correlated. An intertemporal CAPM follows-a linear relationship between the local mean of the price change on a security and the local covariance of that price change with the change in market wealth:

(1.1) E (dZkt IFt) = t COV (dZkt, dWt IFt), where we are conditioning on the information Ft available at date t.

We can motivate the assumption of a sufficient statistic for aggregate consump- tion by considering a large number of securities with a factor structure on their payoffs. We assume that the payoffs can be decomposed into dk = fk + ek, where fk is the factor component and ek is the idiosyncratic component. The idiosyn- cratic components are weakly correlated, so that their risk is negligible in a well-diversified portfolio. The factor structure restriction is that the conditional distribution of the factor components depends on information available at t only via a small number (N) of state variables. A law of large numbers eliminates the idiosyncratic components in (per capita) aggregate supply, as in Connor (1984), so that aggregate consumption is in the factor space, and there is an N-dimen- sional sufficient statistic for it.

If N = 1, we have the intertemporal CAPM in (1.1). If N > 1, then we cannot base risk adjustment simply on the market wealth portfolio. We can, however, use N well-diversified portfolios for this purpose. In addition, we show how the exogenous specification of a factor structure on security payoffs implies a local factor structure for the endogenous security price (or return) processes.

There is a quite different motivation for the sufficient statistic assumption that is based on multivariate normality. Suppose that there is enough multivariate normality connecting the security payoffs with the diffusion information process so that the conditional distribution of the aggregate payoff given date t informa-

ASSET PRICING 1285

tion is normal with a variance that is a deterministic function of t. Then the conditional mean provides a scalar sufficient statistic process S and (1.1) follows. Although multivariate normality has a long history of providing asset pricing restrictions, and provides new insights here, it is quite restrictive; furthermore, unlike the factor model, the argument does not appear to extend to multiple consumption dates.

In terms of testable restrictions, the basic implication of (1.1) is that security price processes (or gains processes when we allow for intermediate dividends) are martingales after a risk adjustment that depends on market wealth. The difficulty is that the risk adjustment involves local covariances. It may be possible to obtain suitable estimates based on frequent sampling of the price process, but we do not pursue these inference issues in any detail in the paper. With N > 1, similar issues arise in obtaining estimates of local covariances involving appropriately chosen well-diversified portfolios.

The paper follows Merton (1971, 1973) and Cox, Ingersoll, and Ross (1985a. b) in adopting a framework with diffusion information and continuous trading.3 In the formal development, however, rather than use dynamic programming, we have found it convenient to use martingale representation theory together with a martingale projection argument. This leads to direct proofs under weaker condi- tions than are needed for dynamic programming. The Markov restriction on the sufficient statistic process is not essential; the whole path up to date t of an N-dimensional Brownian motion can play the role of a sufficient statistic. Martingale representation theory has been used previously by Harrison and Kreps (1979) and Kreps (1982) to provide the foundation for the Black-Scholes (1973) option pricing formula. Duffie and Huang (1985) and Duffie (1986) have shown how these representations relate to the number of continuously traded securities needed to implement an Arrow-Debreu equilibrium.4

Breeden's (1979) "consumption-,B" pricing formula, with the extensions of Grossman and Shiller (1982), addresses many of the limitations of the CAPM. The formula is based on preferences of the form EfoTu(c,, t) dt, where cumulative consumption is assumed to be an absolutely continuous process with a consump- tion rate c following an Ito process. Then the aggregate consumption rate can play the role of market wealth in (1.1). This has the advantage of not requiring a sufficient statistic for aggregate consumption. It is useful, however, to consider alternative specifications for consumption, since the time-additive restriction on preferences becomes severe in a continuous time setting. Furthermore, aggregate consumption data have limitations if one's objective is to work with daily (or more frequent) data on security price fluctuations.

3Equilibrium models in a discrete time setting have been provided by Lucas (1978), Brock (1982), and Prescott and Mehra (1980). Constantinides (1982) shows how complete markets can be used to construct a representative agent of the sort needed for these models. Simplifications can be obtained by using special functional forms for preferences, as in Stapleton and Subrahmanyam (1978) for the case of exponential utility.

4Cox and Huang (1986), Huang (1987), and Pliska (1986) have used martingale representations in portfolio theory.

1286 GARY CHAMBERLAIN

The plan of the paper is as follows. Section 2 sets up the price system and follows Harrison and Kreps (1979) in defining a new probability measure under which the security prices are martingales. Section 3 derives the structure of optimal claims and establishes a mutual fund result. Section 4 derives the pricing formula and provides a dynamic programming interpretation. Section 5 develops the factor model and extends the results to allow for consumption and dividends at intermediate dates.

2. CONTINUOUS TRADING: MARTINGALES AND THE PRICE SYSTEM

There is a complete probability space (S2, F, P) and a Hilbert space H of F-measurable random variables that are square-integrable, with a mean-square inner product: (x, y) = E(xy). The information structure is given by a filtration: F = { F, 0 < t < T } is an increasing family of sub-u-fields with FT = F; date 0 events are certain in that P(A) = 0 or 1 for A E Fo.5

There is a single consumption good and agents are interested in certain consumption at date 0 and state-contingent consumption at date T. There are K + 1 securities representing claims to the good at T. Trades can take place at any date in [0, T]. The stochastic process Z, = (Zo, ..., ZKt) gives the prices at t of the K + 1 securities; Z is adapted to F (i.e., Z, is Ft-measurable, which we denote by Z, E F). The kth security pays dk E H units of the consumption good at T. The 0th security is a riskless asset with do(w) = 1 for all states w E Q2. We use the 0th security as numeraire so that ZO = 1 (O < t < T), and we set ZkT= dk. Assume that EZ, < 2o.

A trading strategy Ot = { Kt . strKa } is a (K + 1)-dimensional stochastic pro- cess, where 0kt gives the number of shares of the k th security held at date t. There is a set 0 of admissible trading strategies. A claim x E H is said to be marketed at t = 0, which we denote by x E M, if there is a trading strategy 0 E 0 such that ST ZT -kO_kTZkT = X with probability one (almost surely or a.s.). In order to avoid arbitrage opportunities, we assume that if 0S ZT = S ZT a.s., then SO ZO= SOs ZO; then we can define the date 0 price of a claim in M as ,T(x)=0. Zo, where 0 - 0 and OT ZT=x a.s.

We shall assume there is a p E H such that p > 0 a.s. and iT(x) = E(px) for x C M. Then define a new probability measure P* with P*(A) = fAp dP and E*(x) = fxdP*, and assume that Z is a (P*, F)-martingale: E*(ZtlFs)= Z= (O < s < t < T). Harrison and Kreps (1979) provide general conditions on prefer- ences that imply the existence of p, and they show that the martingale measure property for P* directly follows if there are no arbitrage opportunities.

Let M2* denote the space of square-integrable (P*, F)-martingales.6 We shall assume that p is uniformly bounded above and away from 0: there is a 8 E R such that P{0 < 8 < p

ASSET PRICING 1287

We shall assume that all martingales (under P or P*) adapted to F have continuous sample paths a.s.'An example is the filtration generated by a vector of independent Brownian motions.

Consider a simple trading strategy (adapted to F) that only revises the portfolio at a finite set of dates; then the strategy is self-financing if the portfolio value at t is the initial value plus the accumulated capital gains and losses. Since Z is a (P*, F)-martingale, so is the portfolio value process 0- Z. We follow Harrison and Pliska (1981) by taking this as the appropriate definition of "self-financing." Finite variance is convenient, and so we assume that 0- Z E M2* for all 0 E 9.

3. PREFERENCES AND THE REPRESENTATION OF OPTIMAL CLAIMS

3.1. The Structure of Optimal Claims The preferences of the j th agent are given by a utility function vj: R x H -* R (j= 1. J). They are risk averse in the following sense: if x, x E H and

x=x+?e, where E(eI x)=0, then vj(c,x) > vj(c, x) for all c E R, with strict inequality unless e = 0 a.s. This holds, for example, if vj(c,x) = E[u1(c, x)], where uj(c, *) is strictly concave in its second argument (and supposing that the expectation exists).

The j th agent has an endowment consisting of - units of the consumption good and #jk units of the k th security at t = 0. So his budget constraint at t = 0 is determined by aj = qc-5 + 0j ZO, where q E R is the price at t = 0 of the consumption good at t = 0. He solves the following problem:

maxv1(c, x) C, X

subjectto:c R, x M, qc?+ (x)

1288 GARY CHAMBERLAIN

representative agent- with time-additive, strictly concave, expected utility prefer- ences-who supports the equilibrium allocation: p(w) = h[x-(w)] for (almost) all states w, where h: R R is the representative marginal utility and x-= =X.x is aggregate consumption.8 Then, given Lemma 1, optimal claims are functions of aggregate consumption, as in Breeden and Litzenberger (1978). This construction is not needed to obtain our pricing formula, but it does enrich the interpretation of p.

3.2. Martingale Representation and Sufficienit StatistiCs

I want to introduce a condition which, combined with martiingale representa- tion theory, implies that there are sufficient statistics for the distribution of p conditional on Ft.

Suppose that B = (B1... BN) is a vector of independent Brownian motions under P; we shall follow the convention that a Brownian motion has initial value equal to zero. Let FtB denote the a-field generated by { Bs, 0 < s < t }, let FB= FT, and let FB denote the corresponding filtration.9 Let M2 denote the space of square-integrable (P, F)-martingales.

CONDITION (R N): There is an N-dimensional vector B of independent Brownian motions in M2 such that p E FB.

Let H2 denote the set of predictable processes 4 with EJfT4S- ds < oc."') The martingale representation theorem of Kunita and Watanabe (1967) implies that if x E H,

N

E(xI )E(x)+ t fa,s dB,ls E (x) + fa dB,

for some a E 1712 (i.e., each component of a is in II). If x e H(p) and if Condition (RrN) holds, then x = E(xIFB) and, since B is a (P, F)-martingale,

8Such constructions have been used by Dybvig and Ross (1982), Dybvig and Ingersoll (1982), Constantinides (1982), and Huang (1987). An argument exploiting the Riesz representation for prices proceeds as follows: Suppose that preferences of the jth agent are given by r, (e, x-) = (( c) + E[u,(x)], where u,: R - R is increasing and strictly concave wxith derivative u

ASSET PRICING 1289

E(x IFt) = E(xIFtB). Hence the path of B up to t is a sufficient statistic for the distribution of p conditional on F,

Suppose that F is generated by an L-dimensional Brownian motion, with L > N. Then one can construct counterexamples to show that given y E H, there need not be an N-dimensional Brownian motion in M2 such that y E FB. Hence Condition (RN) must be regarded as restrictive.

There is a special case in which (RN) is satisfied with N = 1. Suppose that F= FU, where U is an L-dimensional vector of independent P-Brownian mo- tions, and suppose that there is some measurable function g: R -* R with a measurable inverse such that g(p) and U form a Gaussian system; i.e., the distribution of (g(p), Ut(l),..., Ut(,)) is multivariate normal for any finite set of dates t(i) E [0, T]. Then we have the following representation for g(p):

L

g(p)=E[g(p)] + E ans dUns' n=1

where an: [0, T] -* R is a deterministic function.1' Define y, = (L= la2) 1/2 and, to avoid some complications, suppose that Y> 0 (0 t T). Then Bt

=n _1Jf a,s dUnS is a scalar Brownian motion in M2 and

g(p) = E [ g(p)] + J Ys dBs 12

Since y is deterministic, it follows that g(p) E FB and so p E FB. Note that, given a representative agent, this argument applies if some invertible function of aggregate consumption and U form a Gaussian system.

We shall return to the representation problem in Section 5; where we develop a large economy, factor model and show how Condition (RN) relates to Connor's (1984) condition that aggregate supply be well-diversified.

Now we shall show how Condition (RN) combines with Lemma 1 to restrict the values of optimal portfolios. First we need a representation for H(,p) using the martingale measure P*; the proof follows Duffie and Huang (1985, Proposi- tion 6.3) and is in the Appendix.

LEMMA 2: Suppose that Condition (RN) holds. There is a y = (y, ..., YN) e HI2 such that (i) Y, Bt + Joys ds is an AN-dimensional vector of independent Brownian motions in M2*; (ii) if x E H(p), then x = E*(x) + foTas dYs for some a EH 12.

LEMMA 3: Suppose that Condition (RN) holds and that H(p) c M. Let xJ* be the claim chosen by the jth agent, and let ej* E 9 be the corresponding trading

11 This follows from the L-dimensional version of Lipster and Shiryayev (1977, Theorem 5.6; 1978, p. 1 3). 12 See Lipster and Shiryayev (1977, Lemma 6.9).

1290 GARY CHAMBERLAIN

strategy: 6T ZT= xJ* a.s. Then there is an a1j E Hl2 such that (3.1) -*. zt=*.Zf+ ta dYs (0

ASSET PRICING 1291

subspace. It becomes restrictive if Y is as in Lemma 2, for then (i) the mutual fund result implies that W, - WV ( S(Y); (ii) the residual Vk is also a martingale under the original measure P; and (iii) Vk is uncorrelated under P with the stable subspace generated by B. These are the key points in the following result.

THEOREM 1: Suppose that Condition (RN) holds and that H( p) c M. Then

(4.2) , = WO+fsdBs+f t- y ds, o o

(4.3) Zkt=ZkOf+ IksdBsf 4Iks.Ysds+ Vkt (k=1,.,K;O-t -T),

where a, y, and 1k are in H2, and Vk is a P-martingale in M2 that is uncorrelated with the stable subspace generated by B.

PROOF: We shall apply Lemma 4 with Y as in Lemma 2. (i) Summing (3.1) over the J agents gives

(4.4) Wt = WO- tas dYs

with a = >J=a1 i H.2 Combining Yt = Bt + fotys ds with (4.4) and (4.1) gives (4.2) and (4.3).

(ii) Note that p'- is dP/dP*. Since p' E H(p), Lemma 2 implies that

Lt-E*(p-llF) =E*(p-) + frs dYYs

for some 1 E H.2- Vk in (4.1) is a P-martingale if LVk is a P*-martingale (Durrett (1984, p. 83)); this follows since L

-Lo ( S(Y) and Vk is uncorrelated with S(Y) under P*.

(iii) Note that KBn, Vk) = [Yn - fyns ds, Vk] = [Yn, Vk] = yn, Vk)* = 0 (n = 1,., N), where the equality of the sharp brackets and the square brackets follows since all of the martingales are continuous, and we have used the facts that (a) [ , ] is invariant under an equivalent change of measure, and (b) [f Yn ds, Vk] = 0 since fYns ds is a continuous process with finite variation; hence K ft. dns, Vk) = frPs d

1292 GARY CHAMBERLAIN

a local CAPM:

(4.5) Et(dZkt) = Ot Coyt (dZk,, dWt), where ,=-y, a7 1.15

When N = 1, the pricing formula implies that Zkt - fjI3ks,a i' dW, is a P- martingale. This is empirically testable (as T -* oo) if a suitable estimate for Bakt a- can be found. Note that (4.1) and (4.4) imply that

ASSET PRICING 1293

Since XJ' and J' are P-martingales, (4.6) Jt'Et(dX1) + Cov1(dX> d17,') = 0.

Let x denote the claim to aggregate consumption and suppose, corresponding to the case N = 1, that there is a one-dimensional sufficient statistic S for its conditional distribution: given any g: R -- R, there is an m: R2 R such that Etg(x) = m(St, t) (so that Etg(x) = E[g(x)ISJ]). Since u'(x*) is a function of aggregate consumption (by the representative agent argument), we have Jt = Et[u'(x*)] = ml(St, t). Since the price of x is W, (the value of the market portfolio), the sufficiency property implies that

M2(S1, t) _ Wt = Et [x5u ( x*) /Jt' = ( --m3(St, t ),

where mi: R2 R(l = 1, 2, 3). Hence dm3

dWt= A dSt + X1 dt, as / 3ml

dJt ! dSt+X 2dt, and so

Covt (dXt, dJt') = Tt Covt (dXt, dW), where Tt = (dm1/3S)/(Am3dS). Then applying (4.6) with Zkt, dk in place of Xt, x gives

E, (dZko) = t, Covt (dZk,, dWt), where ot= -t/J7.

Under our condition that p EFB, the whole history of B up to date t plays the role of a sufficient statistic. This is weaker than the Markov restriction that the current value of some process S, is sufficient. In addition, our use of martingale representation and projection has provided a direct proof under weaker regularity conditions than are needed for dynamic programming.

If N > 1, then we cannot base risk-adjustment simply on the market wealth process W. The next section shows how N well-diversified portfolios can be used for this purpose.

5. LARGE ECONOMY FACTOR MODELS

In addition to developing a factor model, I want to indicate how the results extend to more general consumption spaces.

Suppose that there is a fixed set of dates 0 < T(1) < ... < (L) = T at which consumption and dividend payments may occur. (We allow consumption at t = 0 but not dividend payments.) Define Hi = { y E H: y e F(i) } and HT = H1 X ... XHL. There is a countable set of securities. A share of the kth security

1294 GARY CHAMBERLAIN

pays dki e Hi units of the consumption good at T(i); the securities are sold ex-dividend, so that ZkT = 0 (k = 0,1,...). The 0th security pays no dividends until the terminal date, when it pays one unit of the consumption good: d01 = 0 (i = 1,..., L - 1) and doL = 1. We let the 0th security be numeraire, so that Zot = I ( -

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mate factor structure. In order to link this to the flow of information, we shall consider the martingales E(ekil FJ), and we need the following definition of weakly correlated martingales (let X denote Lebesgue measure on [0, T]):

DEFINITION 1: A set of martingales { Q1, Q2, ... } in M2 is weakly correlated if: (i) KQj, Qk) is an absolutely continuous process with d(Qj, Qk)tldt ajkt a.e. - P x X.17 (ii) Let the n x n matrix Zrnt have ajkt for its (j, k) element; then the maximal eigenvalue of ?rt is uniformly bounded by t E R for n = 1,2,.... and a.e. - P x X.

Let I = { k(j), j = 1, 2, ... } be a subsequence of the nonnegative integers that indexes the securities in positive net supply.

DEFINITION 2: Let B = (B1,..., BN) be an N-dimensional vector of indepen- dent Brownian motions in M2. The dividend process for the securities in positive net supply has an approximate N-factor structure generated by B if: (i) dk1 = fki + eki (kE1, i= 1,..., L) wherefk = (fkl.**, fkL)cE HT and fk E FB. (ii) Define UkiEM2 by Ukjt=E(eki1Ft); then KBn, Uki) = O. (iii) Choose any vector k = (k1, ... IL) of bounded predictable processes and define Qk e M2 by Qkt = Zfr 1 fJpi, dUkis; then { Qk, k E I } is weakly correlated.

This definition combines a dimension restriction on the factor space with the notion that the predictions of the idiosyncratic components should be weakly correlated with each other and uncorrelated with the factor space.

DEFINITION 3: (a) A sequence of trading strategies { an } E a is well-diversified if: (i) the components of 9fn = (6wn,..., ,nn) are bounded processes; (ii) EJoTQn - Q,n ds 0 as n -x o; (iii) kn = ? (O < t < T) if k 0 I (i.e., on employs only the securities in positive net supply).

(b) A claim x E M is well-diversified if there is a well-diversified sequence of trading strategies { 6 } E such that 917 generates x n E M and xn -> x in HT as n X.18

The following result is proved in the Appendix.

THEOREM 2: Suppose that B is an N-dimensional vector of independent Brownian motions in M2 with p E FR. If the dividend process for the securities in positive net

17We use a.e. - P x X to denote "almost everywhere with respect to the product measure P x X." lx The role of well-diversified claims in a static factor model is developed in Chamberlain (1983),

where it is shown that exact arbitrage pricing corresponds to the presence of a well-diversified claim on the mean-variance frontier.

1296 GARY CHAMBERLAIN

supply has an approximate N-factor structure generated by B, and if X is the gains process for a well-diversified claim x E M, then

(5.1) X = faxsdBs+ fa|xs ysds, 0 0

(5.2) Gkt f'Iks dBs+ j'I3ks Ys ds + Vkt (k = 1, 2,.. .; 0 < t < T),

where ax, y, and P3k are in H2, and Vk is a P-martingale in M2 that is uncorrelated with the stable subspace generated by B. In addition, the martingales { Vk, k E I) are weakly correlated.

Let Xm, be the gains process for the well-diversified claim xm. Then the differential form of (5.1) is

dXmt =amt dBt +mt , Yt dt.

Let At be the N x N matrix with amt as its mth column. If At is nonsingular, then

N Z-7,dG*,= , cmtdXmt + Z,t dVkt

m=1

where f3kt = Zkt A7t- l1k So the local rate of return on the k th security is a linear combination of N well-diversified returns plus a martingale increment. If there is a 8 E R such that P{O < 8 < Zkt, k E I, 0 < t < T ) = 1, then the martingales { fZks dVk k E I) are weakly correlated; the differential counterpart is that the conditional covariance matrix of Zj,' dVkt (k E I) has uniformly bounded eigen- values. Then the local rates of return for the securities in positive net supply have an approximate factor structure conditional on current information. The pricing restriction is that

N E(Zkt 'dGk,tFt) E fmtCov(Zt1 dGkt,dXmtIFt)X

m=1

where pt =At-y,. In proving the theorem, we need to take eki and decompose its gains process

into a factor component and an idiosyncratic component. There is a factor component since p E EFB implies qi E EFB. A related problem is treated in Reisman (1985). We show that

E*(qiekilF,) = E*(qieki) + Ukis Qis + J Qis dUkis, where Q,t = E*(qilF,) is the date t price of a sure claim to a unit of the consumption good at T(i), and Ukj,= E(ekiJFt). Then it follows that Vk,=

5/ 1 fotQis dUkis. As with Theorem 1, an interpretation of the p E FB assumption is that the B

process provides a sufficient statistic for the conditional distribution of the

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aggregate consumption stream. As before, a (heuristic) dynamic programming argument can be used to obtain local linearity between conditional mean returns and conditional covariances with dB. But now we have a way to link the B process to observables even when N > 1. In addition, we can relate the restriction that p E FB to the restriction that aggregate supply is well-diversified. This is our final topic.

Well-Diversified Supply: Connor (1984) introduced the condition that per capita supply be well-diversified, and he used it to obtain an exact pricing formula in a static factor model. I want to sketch how his analysis can be applied to our model.

Consider a sequence of economies in which the n th economy has n fn identical agents of type j, where X>'= on - 1 and 4nj converges in R to oj as n x-* . Only the first n + 1 securities are available in the nth economy. Simplify notation by excluding consumption at t = 0 and by assuming that the net supply of each security is one share. Then per capita supply of the single perishable good at t = T(i) is k = 7dki/n.

Suppose that n=odk1/n converges in H to d1 as n > co. Assume, without loss of generality, that E(eki) = 0. If the dividend process has an approximate N-factor structure generated by B, then Yk=Oek1/n converges in H to 0, and so d E F4B. This restriction on d. is our counterpart to Connor's condition that per capita supply is well-diversified.

The preferences of type j agents are given by the utility function vj(x)= ?2r 1E[uji(xi)I, where uj1: R -- R is increasing and strictly concave with deriva- tive u>. Let xj E HT be the solution to max v.(x) subject to x E M, E(p - x) < a e R, and assume that Y_J=10jxJ*= di a.s. If M = HT (complete markets) then {xj, = 1,...J } is a Pareto-efficient allocation for the limit economy; i.e., there is no allocation {x,} E H T that (i) satisfies the resource constraint for the limit economy: EJ=10jxji < d1 a.s. (i = 1,..., L); and (ii) dominates {Xj* x : vj(x1) > v1(x1*) (1= 1,..., J).

Consider the allocation {Xj}, where x = E(xjFT(i)). This satisfies the re- .~~~~~~~~

J source constraint since

J J Z . =E EfZ4)jx*I FB(i) = E(dilIi)) =Ti

Furthermore, vj(x)d> vj(x*) unless Xj =:x- a.s. Hence Pareto-efficiency of {X

1298 GARY CHAMBERLAIN

APPENDIX

PROOF OF LEMMA 2: Let Q, = E( p I F,j). The Kunita-Watanabe result implies that

Q =1? + tf dB,

where f E I12 and 4, E F5B. Define y, = - Q 14,; then Girsanov's theorem (Lipster and Shiryayev (1977, Theorem 6.4)) implies that

Y,=- B,+ fysds

is a vector of independent Brownian motions in M2*. Then any x cE H that is measurable with respect to FB can be represented as a stochastic integral over Y (Lipster and Shiryayev (1977, Theorem 5.20) or Kallianpur (1980, Theorem 8.3.1)). Q. E. D.

PROOF OF THEOREM 2: As in Lemma 2 (but with dP* = pl, dP), let Y, = B, + f'Y, ds be a vector of independent Brownian motions in M2*. (The lower limit of integration is always 0 and will be omitted.)

(i) To obtain (5.2), follow the proof of Theorem 1: use Lemma 4 to project Gk onto S(Y); the residual P*-martingale Vk is also a P-martingale since dP/dP* - p_ 1 and E*(pf 115) - E*(p- 1) E S(Y).

(ii) We shall show that

Vk = E3 f'Qi, dUki,= f' Q dUks,

where Uk,,, = E(eki I FI) and Qi, = E*(qi I F,). Then it will follow that { Vk-, k I IJ is weakly correlated, since Q is a bounded process.

Note that Gk, =E*(q fk + q ek IF) - ZkO for k E 1. Since q, fk E F', Lemma 2 implies that E*(q Ifk I F) - E*(q Ifk) is in S(Y). Hence the residual from the projection of E*(q * ek I FS) on S(Y) must be V,,.

Uki is in M2* and is uncorrelated with S(Y). (The argument follows the proof of Theorem 1 except that now we are going from P to P*: pl, =dP*/dP, E(pu IFtS) - E(pL) G S(B), and Uki is uncorrelated with S( B) in M2 due to the approximate factor structure.) Since q c FB, Lemma 2 implies that Qit - Qio E S(Y) and so Uki and Qi are uncorrelated in Al2*: E*(qiekilF) = E*(QiTUkiT F) = Qi,Uki,. Now apply the integration by parts form of Ito's formula (Dellacherie and Meyer (1 982, p. 327)):

QitUkit = QiOUkiO + f Qis dUk s + f Uki, dQi,. Since QiUki and JQi5 dUkis are in M2*, we have fLUk , dQi, C M2*. Since Qi, = Qio + fto, dY} for some f C112, we can use a localization argument to show that Uki4 C H2,'9 and then -IUki, dQ,= f'Uki,os dYs follows from associativity of the stochastic integral (Dellacherie and Meyer (1982, p. 340)). Hence Yi I f Ukis dQis is in S(Y). Since Y -IfQisdUks J Qs dUAs is uncorrelated with S(Y), it follows that fQ5 dUks is the residual from the projection of E*(q - ek I F) on S(Y), and so

Q, dUks = Vk. (iii) Proof of (5.1): Let {f 9 } and { x, } be the sequences specified in Definition 3. Then

n

qx = E*(q X x,) + f Ton dGks k=l

19 Drop the k and i subscripts to simplify writing. Since U is locally bounded (Dellacherie and Meyer (1982, no. VII.32)), there is a sequence of stopping times { Ti } and constants n,, E R such that ,Tn T and I Ul is bounded above by D, on [0, Tj. By Jensen's inequality, E*( TU dQ )2 >

E T(EJ~J dQ F)2 )2 f 2n TU 2d E . s, dS).

ASSET PRICING 1299

From (i), GA = Ak + VA, where A, E S( Y); since 0' is bounded, it follows that

X, = E*(q. x, j)-E*(q x,) = C,,, + Z f'9 /dVA., where C(, E S(Y). Since { Vk, k E I } is weakly correlated in M2,

,, ~~~~2 E( f fT s dVk EA fJ)5 l,A d (AJ) VAk

< tEfT TP * ." ds -O

as n - oc. Hence X7, - C -O in M2* (i.e., I X,, - CI 1 1 =- L E*( X,T - GT 2 ]1/2 0). Since X',X in M2* and since S(Y) is closed in M2*, C,, e S(Y) implies XE- S(Y). Then (5.1) directly follows.

Q. E.D.

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Article Contentsp. 1283p. 1284p. 1285p. 1286p. 1287p. 1288p. 1289p. 1290p. 1291p. 1292p. 1293p. 1294p. 1295p. 1296p. 1297p. 1298p. 1299p. 1300

Issue Table of ContentsEconometrica, Vol. 56, No. 6 (Nov., 1988), pp. iii-v+1247-1493+1504Volume Information [pp. iii - v]Front MatterFull Extraction of the Surplus in Bayesian and Dominant Strategy Auctions [pp. 1247 - 1257]The Structure of Nash Equilibrium in Repeated Games with Finite Automata [pp. 1259 - 1281]Asset Pricing in Multiperiod Securities Markets [pp. 1283 - 1300]Household Economies of Scale in Consumption: Theory and Evidence [pp. 1301 - 1314]Nonparametric Analysis of Technical and Allocative Efficiencies in Production [pp. 1315 - 1332]Trends versus Random Walks in Time Series Analysis [pp. 1333 - 1354]Testing for Structural Change in Dynamic Models [pp. 1355 - 1369]Estimating Vector Autoregressions with Panel Data [pp. 1371 - 1395]Asymptotic Normality, When Regressors Have a Unit Root [pp. 1397 - 1417]Chi-Square Diagnostic Tests for Econometric Models: Theory [pp. 1419 - 1453]Notes and CommentsEquity and Efficiency in Public Sector Pricing: A Case for Stochastic Rationing [pp. 1455 - 1465]The Optimal Depletion and Exploration of a Nonrenewable Resource [pp. 1467 - 1471]

Nomination of Fellows, 1988 [p. 1473]European Meeting of the Econometric Society, Munich, Federal Republic of Germany, 1989: Announcement and Call for Papers [p. 1473]North American Winter Meeting of the Econometric Society: Announcement and Call for Papers [pp. 1473 - 1475]9th Latin American Meeting of the Econometric Society--1989, Santiago, Chile: Announcement and Call for Papers [pp. 1475 - 1477]1989 Far Eastern Meeting of the Econometric Society: Announcement and Call for Papers [p. 1478]1989 North American Summer Meeting of the Econometric Society: Call for Papers [pp. 1478 - 1479]Australasian Meeting of the Econometric Society: Announcement and Call for Papers [pp. 1479 - 1480]Accepted Manuscripts [p. 1480]News Notes [pp. 1481 - 1482]Program for the 1988 North American Summer Meetings of the Econometric Society [pp. 1483 - 1491]ErratumDissolving a Partnership Efficiently [p. 1493]

Submission of Manuscripts to Econometrica [p. 1504]Back Matter